Marko Jakšic Inventory Models with Uncertain Supply Univer sity of Ljubljana FACULTY OF ECONOMICS Publishing Marko Jakšic Inventory Models with Uncertain Supply Publisher: Faculty of Economics Ljubljana, Publishing Office For the publisher: Metka Tekavcic, dean Editorial board: Mojca Marc (president), Mateja Bodlaj, Nadja Dobnik, Marko Košak, Tanja Mihalic, Aleš Popovic, Tjaša Redek Reviewers: Peter Trkman Matjaž Roblek Ljubljana, 2018 The book is available online: http://www.ef.uni-lj.si/zaloznistvo/raziskovalne_publikacije Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici v Ljubljani COBISS.SI-ID=295514368 ISBN 978-961-240-338-6 (pdf) All rights reserved. No part of this publication may be reproduced or transmitted in any form by any means, electronic, mechanical or otherwise, including (but not limited to) photocopy, recordings or any information or retrieval system, without the express written permission of the author or copyright holder. Contents 1 Introduction 1 1.1 Outline of the monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Underlying stochastic capacitated inventory model . . . . . . . . . . . . . . . 5 1.2.1 Underlying model description . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Stochastic capacitated inventory models under study . . . . . . . . . . . . . 9 1.4 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Preliminaries 13 2.1 Discrete-time model formulation . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Cost parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Optimizing cost in a single period problem . . . . . . . . . . . . . . . . . . . 17 2.4 Optimizing cost in a .nite-horizon problem . . . . . . . . . . . . . . . . . . . 19 2.5 Finite horizon capacitated inventory problem . . . . . . . . . . . . . . . . . . 24 3 Inventory management with advance capacity information 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Analysis of the optimal policy . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Value of ACI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.1 E.ect of demand and capacity mismatch . . . . . . . . . . . . . . . . 38 3.4.2 E.ect of volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.3 E.ect of utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 “Zero-full” supply capacity availability . . . . . . . . . . . . . . . . . . . . . 47 3.6 Relation between ACI and ADI . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Inventory management with advance supply information 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Analysis of the optimal policy . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 Insights from the myopic policy . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Value of ASI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 Dual sourcing: trading o. stochastic capacity limitations and long lead times 76 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Characterization of the near-optimal myopic policy . . . . . . . . . . . . . . 82 5.4 Value of dual sourcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.1 Optimal costs of dual sourcing . . . . . . . . . . . . . . . . . . . . . . 87 5.4.2 Relative utilization of the two supply sources . . . . . . . . . . . . . 88 5.4.3 Bene.ts of dual sourcing . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 Value of advance capacity information . . . . . . . . . . . . . . . . . . . . . 92 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6 Conclusions 96 Appendices 100 A Chapter 3 Proofs 101 B Chapter 4 Proofs 107 C Chapter 5 Proofs 112 Bibliography 116 List of Figures 1.1 Extent of information sharing in supply chains. . . . . . . . . . . . . . . . . 3 1.2 Schemes of the stochastic capacitated inventory model. . . . . . . . . . . . . 6 1.3 Schemes of the models under study. . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Supply chain (a) without and (b) with ACI sharing. . . . . . . . . . . . . . . 29 3.2 Advance capacity information. . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Illustration of the optimal ordering policy. . . . . . . . . . . . . . . . . . . . 35 3.4 Expected demand and capacity pattern, and optimal base stock level ˆy(n=0) (a) Exp. 1 (b) Exp. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Expected demand and capacity pattern, and optimal base stock level ˆy(n=0) (a) Exp. 3 (b) Exp. 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Expected demand and capacity pattern, and optimal base stock level ˆy(n=0) (a) Exp. 5 (b) Exp. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.7 The relative and the absolute value of ACI for CVD = 0.45 and CVQ = 0.45 . 46 3.8 The relative and the absolute value of ACI for CVD = 0 and CVQ = 0.45 . . 46 3.9 Example of end-customer demand and retailers supply pattern. . . . . . . . . 48 3.10 Relative value of ACI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.11 Absolute change in the value of ACI . . . . . . . . . . . . . . . . . . . . . . 54 4.1 The time perspective of sharing ASI and ACI. . . . . . . . . . . . . . . . . . 61 4.2 Advance supply information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 (a) Ct(yt, zzt, zt)as a function of yt and zt, and (b) Ct(yt, zzt, zt)as a function of zt for a particular yt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 The relative and the absolute value of ACI for CVD = 0. . . . . . . . . . . . 74 4.5 The relative and the absolute value of ACI for CVD = 0.65. . . . . . . . . . . 74 5.1 Sketch of the supply chain under study. . . . . . . . . . . . . . . . . . . . . . 78 5.2 The optimal inventory positions after ordering and the optimal orders. . . . 83 5.3 The scheme for the myopic inventory policy. . . . . . . . . . . . . . . . . . . 84 5.4 The relative di.erence in costs between the optimal and the myopic policy. . 86 5.5 Optimal system costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.6 The share of inventories replenished from a faster capacitated supply source. 89 5.7 The share of inventories replenished from the faster capacitated supply source. 90 5.8 System costs under ACI for di.erent unreliable supply channel utilizations. . 93 List of Tables 3.1 Summary of the notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Optimal ˆy(n=0) and myopic ˆyM base stock level, optimal system cost, and the value of ASI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Optimal base-stock level ˆy(n=0) , optimal system cost, and the value of ACI . 42 3.4 Optimal system cost and the value of ACI under varying ACI horizon n . . . 45 3.5 Value of ACI for b = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Value of ACI for b = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Value of ACI for b = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1 Summary of the notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Optimal base-stock levels ˆyt(zt-2, zt-1) (L = 3, m = 2, E[D] = 5, CVD = 0, E[Q] = 6, CVQ = 0.33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 The di.erence in the base-stock levels, ˆyt(zt-2 + 1, zt-1)- yˆt(zt-2, zt-1 + 1), and vice versa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 The optimal and myopic base-stock levels, (L = 3, m = 2, E[D]= 5, CVD = 0.5, E[Q]= 6, CVQ = 0.33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Summary of the notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 The optimal and the myopic inventory positions and orders. . . . . . . . . . 86 5.3 The optimal costs and the relative value of dual sourcing. . . . . . . . . . . . 91 Chapter 1 Introduction Inventory and hence inventory management, plays a central role in the operational behavior of a production system or a supply chain. Due to complexities of modern production pro­cesses and the extent of supply chains, inventory appears in di.erent forms at each stage of the supply chain. The fact is that on average 34% of the current assets and 90% of the working capital of a typical company in the United States are invested in inventories (Simchi-Levi et al., 2007). Every company, a party in a supply chain, needs to control its inventory levels by applying some sort of inventory control mechanism. An appropriate selection of this mechanism may have a signi.cant impact on the customer service level and company’s inventory cost, as well as supply chain systemwide cost. Research on inventory management is focused on providing decision making tools, which improve the performance of inventory systems. Unfortunately, devising a good inventory control mechanism is di.cult, since modern companies operate in a highly complex business environment, which exhibits complexities both within the company’s own processes and due to interactions between the various levels in the supply chain as a whole. Multiple models have been developed with a desire to capture the complex interactions between supply chain parties, dealing with an e.cient way of managing information and material .ows through the supply chain. However, while a lot of the research tries to describe these complexities on one side, it fails to capture the dynamics and uncertainties, which are an integral part of the modern business environment, on the other. It is demand uncertainties that attracted the attention .rst. One of the main objective of the companies is to provide good service to its customers. But changes in the customer’s demand may cause di.culties in guaranteeing timely and reliable service, particularly if these changes are uncertain. Production companies or the retailers that provide the product to the end-customers often face seasonal and stochastic demand. A very partial listing that is due to Metters (1998) includes weather related industries (e.g. pharmaceutical products, lawnmowers, canned foods), back-to-school industries (e.g. pencils, clothing), and holiday related industries (e.g. toys, wrapping paper). These companies have to cope with di.cul­ties in managing their inventories in comparison with ones that produce to satisfy stable stationary demand. The consequence of the seasonality is that the production capacities that are likely to be quite stable, are not aligned well with the demand (Fair, 1989; Krane and Braun, 1991). Typically in high demand season the capacity is not su.cient, and the companies have to resort to building up inventory in the low demand season in expectation of high demand later. In such a case it is not su.cient to devise a inventory policy that only seeks to satisfy short-term demand, but the company has to look into future and make the appropriate capacity allocation decisions between the products, and additional decisions on when to start with the inventory build-up to prepare for the high demand season. By having to anticipate far into the future, companies are now also facing higher uncertainties that make it even harder to devise good planning mechanisms. Although capacity is frequently assumed stable, this does not need to hold in general. The aggregate capacity decisions are long-term strategic decisions, and they lack the .exibility in changing the capacity levels quickly, which would enable a company to align its capacities to the changing demand. Looking at production capacity in short-term reveals that the capacity allocated to produce a certain product may vary a lot through time, primarily due to multiple products sharing the same capacity. This may give some additional .exibility in terms of allocating the capacity to the product that needs the most in a certain period. However, for the company to be successful in determining the appropriate product mix in each period, it would need to use a highly evolved production/inventory control system. As it turns out, this is not the case in practice. Bush and Cooper (1988) and Metters (1997) report that .rms typically do not use any formal analytic approach and often carry excess inventory. Other sources of uncertainty in capacity levels are related to changes in workforce (e.g. holiday leaves), machine breakdown and repair, preventive maintenance etc. Variable uncertain capacity is a thing that also can be observed in the supply of products to the end-customers. The retailers may face supply restrictions imposed by their suppliers. This can be due to supplier delivering its products to multiple retailers, or due to overall product unavailability at certain times. Above we have pointed out the di.culties related to variable demand and supply capac­ity that the companies are facing in managing their inventories. Also, when one looks in prescriptive texts for production scheduling, it has been acknowledged in the past that the simple rules used under stable demands and unlimited capacity perform poorly, but ac­ceptable practical rules are not given. However, this is changing in the last two decades. Companies have been extensively adopting and using new planning concepts, like Material Requirements Planning (MRP) and Manufacturing Resource Planning (MRP-II). This was supplemented by introduction of new business practices (e.g. electronic data interchange or EDI, point-of-sales or POS, internet sales) and the ways the companies started to cooperate (e.g. collaborative planning forecasting and replenishment or CPFR, vendor managed inven­tory or VMI) (Lee and Padmanabhan, 1997; McCullen and Towill, 2002). Bollapragada and Rao (2006) point out that particularly through the implementation of Enterprise Resource Planning (ERP), which provides timely access to accurate data, a fresh look at decision support and advanced production planning systems is warranted. This view is not directed only towards accepting more complex planning algorithms, but even more importantly at providing the relevant information that would reduce the uncertainties and stimulate the collaboration between the supply chain parties. The need to share the relevant information through the supply chains has also been widely recognized by the practitioners. Already more then a decade ago, the computer industry has been extensively sharing the informa­tion both on the demand side, as well as on the supply side, as reported by Austin et al. (1998) (Figure 1.1). The largest extent of information sharing is due to sharing capacity information, however due to the sensitivity of information and the rudimentary state of the information technology at the time the capacity information was not shared electronically. /C5CCh/5 5{/Chh CÇÇ{CC/// tÇÇhÇC/hÇh t{Ch/ tÇÇ/Ç/hÇh t{Ch/ th{ /Ç//Ç/{Ç// CÇÇ{CC/// {{{{ ÇÇÇÇÇÇÇ 0% 10% 20% 30% 40% 50% 60% % CÇ/5Chh{/ Figure 1.1 Extent of information sharing in supply chains. Apart from sharing information, there are other possible ways to mitigate the undesirable ef­fects of potential supply shortages, mainly related to production .exibility in the production setting, and multiple sourcing options and capacity reservations in the supply chain setting. By taking advantage of these options companies can alleviate the e.ect of supply shortages that are due to capacity constraints. The dynamic capacity investment or disinvestment problem has been investigated extensively, where we point out the works of Rocklin et al. (1984); Angelus and Porteus (2002); Gans and Zhou (2001) in establishing the optimal poli­cies for managing capacity in a joint capacity and inventory management problem. These works were later extended by inclusion of other means of capacity .exibility, where Pinker and Larson (2003) and Tan and Alp (2005) discuss the option of hiring contingent labor upon the need to raise the capacity. This work is later complemented in Mincsovics et al. (2006), where they assume the lead time for capacity acquisitions. Ryan (2003) gives review of the literature on dynamic capacity expansions with lead times. Yang et al. (2005) introduce a model of a production/inventory system with uncertain capacity levels and the option of subcontracting. This extends the works concentrating on a single supplier setting to a dual or multiple supplier environment. Whittemore and Saunders (1977); Chiang and Gutierrez (1996, 1998); Tagaras and Vlachos (2001) all study a periodic review inventory system in which there are two modes of resupply, namely a regular mode and an emergency mode. Or­ders placed through the emergency channel have a shorter supply lead time but are subject to higher ordering costs compared to orders placed through the regular channel. In Vlachos and Tagaras (2001) they also impose capacity constraints. Minner (2003) presents a good overview of the multiple-supplier inventory models. Finally we point out the stream of mod­eling assuming the option of capacity reservations or supply quantity .exibility contracting. The capacity reservation is the company’s ability to form an agreement with the supplier stating the extent of supply capacity that is to be reserved in advance and the associated costs (Tsay, 1999; Tsay and Lovejoy, 1999; Bassok and Anupindi, 1997; Jin and Wu, 2001; Cachon, 2004; Erkoc and Wu, 2005; Serel, 2007). We feel that studying these models, complements the .ndings of our analysis in a sense that companies should explore all the available options to improve its inventory control. This can be either through improving the collaboration and coordination with their current suppliers or by establishing new alternative ways that give them additional .exibility in managing their inventories. We have demonstrated above that companies are having troubles managing their inventories if they are working in a capacitated supply chain, where it is likely that the available supply may not cover the company’s full needs in certain periods. Thus, our work is directed towards exploring two conceptual ways of improving the performance of inventory system facing uncertain supply capacity availability: • Supply capacity information sharing : sharing the information of the current or future supply capacity availability in a supply chain to improve the performance of the inventory management. • Alternative supply options : taking advantage of the alternative supply capabilities to mitigate the negative e.ect of supply shortages of the primary uncertain supply channel. The remainder of this chapter is organized as follows. In Section 1.2, we describe the un­derlying stochastic capacitated inventory model, and proceed by giving the main model assumptions and parameters, and provide a short literature review of the research track leading to the formulation of the model. Then we give a short description of the four pro­posed inventory models that incorporate either capacity information sharing or alternative supply options in Section 1.3. We continue by formulating the relevant research objectives in Section 1.4, and give the outline of the rest of the monograph in Section 1.1. 1.1 Outline of the monograph The remainder of this monograph is organized as follows. The monograph can be conceptu­ally split into two parts, each consisting of two chapters. Chapters 3 and 4 constitute .rst part, where we study the role of sharing information on supply capacity availability on in­ventory management policies. In the second part we deal with an alternative replenishment option to improve the supply reliability: the dual sourcing setting in Chapter 5. More speci.cally, in Chapter 3 we introduce the model with ACI on future supply capacity availability in detail and focus on establishing the optimal inventory policy and its properties. To quantify the value of ACI, we resort to numerical study, where we present the results for a broad selection of di.erent scenarios, which enable us to establish the important managerial insights. We study the similarities between our ACI model and ADI models. For this purpose we formulate a special version of a capacitated ADI model. Parts of the contents of this chapter appeared in Jak.si.c and Rusjan (2009) and Jak.si.c et al. (2011). In Chapter 4 we study a positive lead time variant of underlying stochastic capacitated inventory model, where advance information is available on the realization of the pipeline orders, denoted as ASI. In an addition to characterizing the optimal policy and its properties, we also study the behavior of the state-dependent myopic policy. In the numerical analysis we estimate the bene.ts obtained through sharing and integrating ASI into the inventory policy. In Chapter 5 we study a dual sourcing inventory setting in which a slower reliable supplier is used to improve the reliability of sourcing through a faster stochastic capacitated supply supplier. Study of the optimal policy is complemented by the development of the near-optimal myopic policy, which proves to serve as a nearly perfect approximation for the optimal policy. Finally, we summarize the main results of the research in Chapter 6. 1.2 Underlying stochastic capacitated inventory model In this section, we describe the stochastic capacitated inventory model that was .rst intro­duced by Ciarallo et al. (1994). The model forms the base setting for the extensions proposed in this monograph, which are motivated in the Introduction and introduced in Section 1.3. We elaborate on the major model assumptions and introduce the relevant system parameters, and follow with a short literature review. 1.2.1 Underlying model description We present the underlying stochastic capacitated inventory model in Figure 4.1. Below we list the major model assumptions and parameters: Figure 1.2 Schemes of the stochastic capacitated inventory model. Single stage, single product: We focus on a single stage in a supply chain by considering an individual company or one stocking point, suggesting that the inventory is kept and reviewed at one location. This company can be either a retailer o.ering the products directly to the end-customers, or a manufacturing company producing the products on the production line. In both cases, we assume that demand from parties lower in a supply chain, and the supply availability of .nished products or components needed in a production process, are exogenous to the company. This means, that we can not in.uence the demand and supply capacity through our actions, e.g. ordering decisions. We assume inventory control of a single item or product, which means that the product is treated independently of other products. Discrete time, periodic review, .nite horizon: We assume discrete time inventory control, and study a .nite horizon inventory problem. We assume periodic review, where inventory is reviewed in regular .xed time intervals, and more importantly, the order decision is given at these prespeci.ed points as well. Without the loss of generality, we set the length of the review period R to 1. Lead time: The supply or replenishment lead time L imposes a time lag between the moment when order is placed with the supplier or manufacturing unit, and the time the products are actually received or produced. L is assumed to be nonnegative and constant, meaning that the delivery time is known with certainty. The entire order is delivered at the same time, where the order quantity may be restricted by the available supply capacity. The only restriction we impose is, that if L is longer than a review period, L needs to be a multiple of R. Demand and supply capacity distribution, demand backorders and lost supply capacity: The demand and supply capacity are assumed to be nonnegative random variables with known probability distributions. We restrict ourselves to the case where both the demand and supply capacity distributions are independent over time periods, but may be non-stationary over time. Demand backordering case is assumed, meaning that any leftover inventory at the end of the period is retained and can be used to satisfy demand in the following period. In the base case, we assume that the unused supply capacity at the supplier is lost to the company. Costs: We consider linear inventory holding cost and backorder costs, where per unit costs are assumed the same in all periods. We do not consider any .xed ordering costs. The objective of the inventory control policy is to minimize the discounted expected cost accu­mulating over a .nite number of future periods. The costs incurred in future periods are discounted to their net present value. 1.2.2 Literature review We proceed with the short literature review of the inventory models that share the basic modeling assumptions with the inventory models proposed in this monograph. We .rst brie.y discuss the models with constant capacity and then focus on research track modeling the stochastic capacity. In the context of our research, we are interested models that do not only recognize that the supply chain’s demand side is facing uncertain market conditions, but also look at the risks of limited or even uncertain supply conditions. The researchers revisited the early stochastic demand models and extended them to incorporate the uncertainty on the supply side. Federgruen and Zipkin (1986a) .rst address the .xed capacity constraint for stationary inventory problems and prove the optimality of the modi.ed base-stock policy. They have considered an in.nite horizon case both under average and discounted cost criterion (Fed­ergruen and Zipkin, 1986b). Tayur (1993) extends this work by developing an algorithm for the computation of the optimal policy parameters and cost. For a speci.c setting of stochastic seasonal demand and .xed production capacity Kapuscinski and Tayur (1998) and Aviv and Federgruen (1997) also show the optimality of a modi.ed base-stock policy. Anticipation of future demand, due to its periodic nature, causes a corresponding increase or decrease in the base-stock level. Later, Ciarallo et al. (1994) were the .rst to capture the uncertainty in supply capacity, by analyzing limited stochastic production capacity model. Particularly the inclusion of limited stochastic capacity is of interest to us, as it is closely related to the work presented in this monograph and we refer to it throughout the monograph. In their view, the random capacity assumption is appropriate for systems where there is uncertainty about which resources within the process will be available, resulting in limits on the ability to produce in any period. As such, stochastic capacity is a very general way of representing a variety of internal uncertainties. As we have argued before, this notion can be generalized even further by including the aspect of varying availability of supply, due to uncertainties in the supply capacity of the supplier. They start their analysis by studying a single period problem, where they show that variable capacity does not a.ect the order policy. The myopic policy of newsvendor type is optimal. Meaning that the decision maker has no incentive to try to produce more than it is dictated by the demand and the costs, and simply has to hope that the capacity is su.cient to produce to the optimal amount. However, in a multiple period situation, one can respond to the possible capacity unavailability by building up inventories in advance. Due to capacity uncertainty the cost function is somewhat complicated, having a quasi-convex form. The available capacity is likely to limit the actual production and as such e.ectively prevents high inventory accumulation. The consequence of capacity shortage is that the cost function levels out for high order sizes, making it nonconvex. In the .nite horizon stationary case they show that the optimal policy remains to be a base-stock policy, where the optimal base-stock level is increased to account for the uncertainty in capacity. More precisely, the additional inventory, above the level needed to cope with uncertain demand, is there to cover the possible capacity shortfalls in future periods. They extend this work by introducing a notion of extended myopic policies, where they show that these policies are optimal if the decision maker considers appropriately de.ned review periods. For the same setting, Iida (2002) obtains upper and lower bounds of the optimal base-stock levels, and shows that for an in.nite horizon problem the upper and the lower bounds of the optimal base-stock levels for the .nite horizon counterparts converge as the planning horizons considered get longer. This allows him to establish minimum planning horizons over which the solution of the .nite horizon problem is close enough to the in.nite horizon case. Thus, it is possible to obtain a policy su.ciently close to the optimal one by solving .nite horizon problems in a rolling horizon manner. Khang and Fujiwara (2000) present su.cient conditions for the myopic policies to be optimal under stochastic supply, however, the demand is assumed deterministic. G¨ull¨u et al. (1997) model supply uncertainty in a di.erent way, by introducing a notion of partial availability, meaning if an order is placed above the available supply capacity there is a positive probability that only the capacity restricted amount will be delivered. Again, it is shown that the optimal policy is a base-stock policy, but they were also able to develop a simple newsvendor like formula to compute the optimal base-stock levels under a more restrictive assumption of two-point stationary supply availability (supply is either fully available or completely fails). To summarize, the models capturing the e.ects of limited capacity all show that carrying extra inventory is needed in comparison with the case of an uncapacitated supply chain. Such strategy would guarantee the optimal level of performance. This suggests that implementing the inventory policies proposed by the classical uncapacitated inventory theory in practical situations leads to a signi.cant deviation from the optimal performance. 1.3 Stochastic capacitated inventory models under study In this section we provide a description of the inventory models studied in this monograph. Proposed inventory models can be characterized as a variants of the stochastic capacitated inventory model introduced in Section 1.2. We represent the proposed inventory models in Figure 1.3. Our focus is on a single party in a supply chain, denoted as a company. The role of a company is to serve the stochastic customer demand. This is done through replenishment of products by placing orders with the supplier. However, the supply availability may be limited due to stochastic capacity availability at the supplier. In the introduction we have formulated two conceptual approaches to tackle the uncertainty in supply capacity availability: supply capacity information sharing and alternative supply options. In line with the .rst approach, companies should exploit the potential of sharing the rele­vant information about the supply conditions to reduce the supply uncertainty. Here, our contribution lies in developing models that would incorporate this information in the inven­tory policy that would capture these potential bene.ts. However, the company that wants to establish means of necessary information sharing with its supplier, has to .nd a way to communicate these bene.ts. We propose two di.erent ways in which supply capacity infor­mation is communicated to the company by its supplier. The di.erence is due to whether the supply capacity information is revealed before the order is placed, or later, during the time the order is being processed at the supplier. In both cases the information is revealed before the order is replenished. Therefore we denote this information as advance information. The capacity availability may vary substantially depending on the current inventory or/and production status at the supplier, and the future production plans. It is reasonable to assume that the supplier has an insight into the current inventory position, status of accepted orders, and the capacity availability for a number of future periods. This allows the supplier to identify empty production capacity slots, quote reliable lead times to new customers’ orders, and thus improve the service to the customers. We denote the .rst way of sharing advance information as advance capacity information (ACI), as the supplier reveals the information about the future capacity availability to the company. The ordering process at the company can be optimize to avoid the supply shortages by accumulating inventories upfront in the periods with adequate capacity. The second way to share advance information, is that supplier’s provides the information on the current order status after the decision maker in the company has already placed the order with the supplier. We denote this way of information sharing as sharing emphadvance supply information (ASI). The second approach is targeted at increasing supply availability through exercising alter­native supply options. We are looking for a way to e.ectively decrease the utilization of the Figure 1.3 Schemes of the models under study. 10 primary stochastic capacitated supply source, and thus improving its supply availability. We proposse the option targeted at expanding the supply base through an alternative supplier in addition to a regular supplier with uncertain supply capacity availability. The alternative supplier is modeled as a fully reliable supplier, but its replenishment lead time is longer. We denote this option as dual sourcing option. Therefore, we seek to develop a dual sourcing inventory policy that would successfully split the order between the two supply sources, taking advantage of fast replenishment of the regular supply channel, and at the same time decreasing the possibility of the supply shortage by exercising a reliable replenishment option available at the slower supplier. To summarize, we propose the following inventory models, where each of the models is analyzed in the succeeding chapters of this monograph: • ACI model (Chapter 3): inventory model with advance capacity information on future supply capacity availability, limiting the orders placed in the near-future periods. • ASI model (Chapter 4): inventory model with advance supply information on supply capacity available for replenishment of the orders already placed, but are currently still in the pipeline. • Dual sourcing model (Chapter 5): inventory model where an alternative reliable yet slower supplier is used to cope with the supply capacity uncertainty of the primary supplier. 1.4 Research objectives The main goal of this monograph is to develop quantitative inventory models that capture the stochastic nature of the demand and supply process, and possible ways of either reducing the uncertainty of supply or taking advantage of alternative supply options, in an integrated manner. The motivation behind this is twofold. Firstly, we would like to enrich the exist­ing capacitated stochastic inventory research literature by developing the new models, and characterizing the resulting optimal or near-optimal inventory policies. Secondly, taking a practitioners’ point of view, we aim to show the potential reduction of inventory costs in comparison to the underlying stochastic capacitated inventory system presented in 1.2, and provide the relevant managerial insights and decision policies that would allow the decision maker to achieve these bene.ts. In accordance with the above, we formulate two groups of research questions that are relevant to all chapters of the monograph. The .rst group of research questions deals with the modeling perspective and the structural analysis of the inventory policies: Q1. How can we incorporate supply capacity information and alternative supply options into the underlying stochastic capacitated inventory model? Q2. What is the structure of the optimal ordering policy and can we derive its properties? Addressing Q1 is relevant due to the fact that the models presented have received none or very limited attention in the inventory control literature. When exploring the inventory problems, the important objective is to characterize the control mechanism that guarantees the optimal behavior of the inventory system. We wish to answer Q2 by explicitly describing the control mechanism behind the optimal inventory control policy. Here, it is helpful if one can show that the optimal policy has a particular structure. Problems similar to the ones studied in this monograph have been known to have a structure of the base-stock policy, thus we wish to con.rm if this also holds in our case. As the complexity of models under study is larger then the complexity of the underlying stochastic capacitated model, similar is expected for the optimal policy structure. The additional complexity is captured by a more comprehensive system’s state description, which will a.ect the optimal policy parameters. The motivation behind developing new inventory control policies lies in the potential bene.ts that can be achieved through their application. By answering the next group of questions, we wish to evaluate these bene.ts and provide the relevant managerial insights, which will serve as guidelines for better ordering decisions: Q3. What is the value of supply capacity information and alternative supply options in terms of inventory cost reduction? Q4. Which are the determining factors of the magnitude of the expected cost bene.ts, or more speci.cally, what is the in.uence of the relevant system parameters? By answering Q3 we want to show that the proposed inventory policies lead to improved performance of the stochastic capacitated inventory system. However, the magnitude of the observed bene.ts may vary a lot depending on the particular system setting. Further elaboration of the in.uence of the system’s parameters on the inventory cost reduction is needed to characterize these settings, which we address with Q4. From the methodological perspective, the analysis of the structural properties of the proposed inventory models, is complemented by the numerical analysis to evaluate the extent and the in.uence of the system parameters on the bene.ts of information sharing and using alternative supply options. Chapter 2 Preliminaries In this chapter, we give some preliminaries on the concepts we will be exploring in further depth later in this monograph. The main objective of this chapter is twofold, we: (1)present a general way in which stochastic inventory models are developed and analyzed, and (2) give some of the fundamental results stemming from the analysis of both the uncapacitated and the capacitated inventory model. Throughout the chapter we remain within the model context presented in Chapter 1, that is single stage, single product, periodic review inventory model. We .rst present a discrete-time model formulation of a general stochastic inventory model. The systems dynamics are described, identi.cation of the control and state variables, and determination of the functional relationships, which describe the evolution of the state vari­ables. We proceed by closer inspection of the relevant cost structure. A short description of the cost parameters is given, and based on those single period inventory costs are assessed. We then give a short review of a single period inventory problem in which we already capture the importance of .nding an optimum balance between the relevant costs, namely inventory holding and backorder costs. The focus of this chapter is on derivation of a multi-period discounted expected optimal cost function by means of a dynamic programming technique. We .rst give a short review of the fundamental results of the uncapacitated inventory model. Showing the idea of the state space reduction, the optimality of the base stock policy and the near-optimal behavior of the myopic policy. This part relies strongly on Zipkin (2000) and Porteus (2002)1, to whom we also refer in some of the proofs. We conclude this section by giving a review of the optimal policy characterization in the case of capacitated inventory problem, where we show the monotonicity results describing the dependency between the optimal base stock level and the capacity limit size. 2.1 Discrete-time model formulation Throughout the monograph we deal with the notion of discrete time inventory control, where all important events occur at prespeci.ed time points. We are concentrating on the stream of research assuming periodic review, which means that the inventory position is reviewed in 1Also, we advise the reader more interested into fundamentals of stochastic inventory control to review these two works. every time period t, where t = 0,1,. . . , T . The planning horizon T may in general be either .nite or in.nite, where we limit ourselves to the .nite horizon case, but also touch upon the known results from the in.nite horizon case upon need. After review, the ordering decision zt is made to raise the inventory position, which is then later used to satisfy the demand dt. The demand is assumed to be a nonnegative random variable, where we resort to the case where dt are independent over t. Due to the multi-period setting, it holds that any leftover inventory at the end of one period is retained and can be used to satisfy demand in the following period. This fact distinguishes the model from a simple single period newsvendor model. The consequence of the inherently stochastic nature of the demand process is that we ran out of stock in the situation where demand exceeds the available inventory on hand. In our model we assume backorders rather than lost sales, which means that any supply shortage carries over to the next period and remains to be satis.ed. In general, we assume that there is a certain positive supply lead time L needed for the order to be delivered. To make e.ective ordering decisions we must have insight into the available inventory. That is inventory already on hand or net inventory xˆt, and the pipeline inventory, which constitutes of orders already given but not yet received. Both together constitute an important state variable, inventory position before ordering xt. Due to the supply lead time, each of the orders remains in the pipeline for L periods. Therefore we can express the inventory position before ordering xt as the sum of net inventory and pipeline inventory. t-1 xt = xˆt + zs. (2.1) s=t-L Note that in the special case of zero lead time, L = 0, there is no pipeline inventory as there are no outstanding orders. (2.1) is then simpli.ed to xt = xˆt, which means that the inventory position before ordering corresponds to the inventory on hand. At the beginning of period t, the starting inventory position xt is reviewed and correspond­ingly the ordering decision zt is made to increase the inventory position to a desired level, by adding the order quantity to xt, yt = xt +zt. The order placed L periods ago is received, and used together with available net inventory to cover the period’s demand dt. Moving from period t to t + 1 the system dynamics is described by the following equation: xt+1 = xt + zt -dt = yt -dt. (2.2) The inventory position after ordering yt is set at the level that is assumed su.cient to cover the lead time demand. That is demand, which is realized in the time interval (t, t +L), and can be expressed as: t+L Dt L = Ds. s=t Based on yt and a certain realization of lead time demand Dt L, we can determine the end­of-period net inventory in period t + L, as the di.erence between the two: xˆt+L+1 = yt -Dt L . (2.3) This corresponds to period t + L being the .rst future period we can a.ect by our ordering decision at time t, since the actual delivery of order zt will occur L periods later in period t + L. End-of-period net inventory gives us the extent of excess inventory or backorders at the end of period t + L. Note above that the end-of-period net inventory in period t + L is by de.nition equal to starting net inventory ˆxt+L+1. The potential excess inventory or backorders are already indicators of the appropriateness of ordering decisions. In an ideal situation we want to avoid either. While in a deterministic setting we could decide on future orders all at once due to the perfect information about future, this is not possible in a stochastic environment. It makes no sense to specify zt early, since through time we obtain relevant demand information and the system evolves accordingly. It is better to react when additional information becomes available. Our next goal is therefore to determine the best ordering policy, given the economics of a situation. Rather than working with physical performance measures (e.g. inventory, backorder levels), we proceed by evaluating those by introducing the relevant costs associated with inventory, backorders, etc. 2.2 Cost parameters In the majority of the backorder inventory models the following cost parameters are as­sumed2: h : inventory holding cost per unit per period b : backorder cost per unit per period c : unit variable order/production cost k : .xed order/production cost The inventory holding cost h is charged per each unit of end-of-period inventory on hand after the current period’s demand is satis.ed. The inventory holding cost represents all the costs associated with the storage of the inventory until it is sold or used. Here, not only 2For a general review of the role and type of inventories, the inventory costs etc., look at any of Nahmias (1993); Chopra and Meindl (2001); Cachon (2004). a mere .nancial holding cost (opportunity cost of capital) is important, but many other reasons that could make the inventory holding cost substantially higher: obsolescence of inventory, inventory might physically perish, inventory requires handling, storage space and other overhead cost (insurance, security, etc). On the contrary, when there is too little inventory to cover the demand, backorder or penalty cost b is charged per each unit of backorder. This backorder cost may incorporate the lost opportunity value of the delayed revenue, the additional cost required to place an expedited order, and a more vague concept of lost goodwill, which is intended to account for any resulting reduction of future demands. When ordering, we also incur variable and .xed ordering cost. When ordering or producing a product, a certain cost is incurred per unit of product. This variable cost can be denoted as the purchase price or a production cost. The .xed ordering cost is incurred if an order is placed, regardless of its size. The cost can be due to incremental time of the buyer placing the extra order, .xed part of the transportation or delivery cost, receiving costs like administration work related to updating the inventory records, and other. In a production setting, the equivalent of the order quantity is the batch size or a production run. The .xed cost associated with the production of a batch is due to setting up the equipment. It can include direct costs, the opportunity cost of time it takes to carry out the setup, and the implicit cost of initiating a production run because of learning and ine.ciencies at the beginning of the run. The above cost parameters need not to be stationary, their values can change through time, ht, bt, etc. Although we make no explicit assumption on cost stationarity throughout the monograph and keep the derivation of the inventory models general in this manner, we con­sider the cost parameters to be stationary in the numerical studies. Therefore, we suppress the subscript t in writing the cost parameters, from now on. Note that, for the purpose of deriving the ordering policies, it is safe to ignore sales revenues and rather concentrate on minimizing the expected cost of running the inventory system (Silver and Peterson, 1985). This is possible since over a long run the total demand is set and also satis.ed. The ordering decisions might a.ect the number of products sold in a certain period (and thus revenue) and change the sales pattern. But these e.ects can be captured through the cost itself. For instance, a delayed revenue is the consequence of incurring backorders, and the e.ect of this lateness can be captured through backorder cost itself. Based on the proposed cost structure, we can develop a single period cost function, which allows us to determine the cost we incur in each period. The total cost associated with ordering at time t is kd(zt) + czt, where we use d(zt)to denote a Heaviside function (1 when zt > 0, 0 otherwise). Meaning that the .xed part of ordering cost is only incurred in the case of positive order size. If no order is placed, the ordering cost is not incurred. In establishing inventory holding and backorder cost in period t, we again proceed with assumption of constant, positive lead time L. Thus, these costs are actually charged at the end of period t + L when the order given at time t arrives and is eventually used to satisfy demand in a corresponding period. Based on (2.3), we write a single period holding and backorder cost function: Cˆt+L(ˆxt+L+1)= h(yt -Dt L)+ + b(Dt L -yt)+ . (2.4) We are now ready to evaluate the costs of running the inventory system over multiple time periods. Our focus lies primarily in establishing an optimal or at least near-optimal decision rule, which, when utilized will guarantee low costs. From here on we will only concentrate on inventory problems in which .xed ordering cost is not substantial (k = 0), and can therefore be neglected. In a periodic review setting this actually means that there is no incentive not to order, except if the current inventory position is already su.ciently high. In case of k > 0, the decision to order a small order might not be optimal due to .xed ordering costs associated with placing the order. 2.3 Optimizing cost in a single period problem Before we proceed with optimizing a multi-period inventory problem, it is worth to look at a simpler variant of stochastic inventory problems -a single period problem. The problem applies in the analysis of items that perish quickly, thus, they cannot be stored in stock for future periods. Practical situations can be observed in setting the initial sizes for high fashion items, ordering policies for food products that become obsolete quickly, or determining run sizes for items with short useful lifetimes, such as newspapers (Silver et al., 1998). Because of this last application, the single-period stochastic inventory model has come to be known as the newsvendor model. The importance of a single period problem analysis for our work is twofold: • the newsvendor model is the simplest problem solved by a critical number, or a single base stock level, and • the relationship of a single period problem analysis to the analysis of the myopic behavior of the multi-period optimal policy. When the decision horizon T = 1 the decision maker is facing a single order decision at time 1. Note, that the notion of a single period corresponds to starting in period 1 and ending in the beginning of period 2. During this period a random demand occurs and the order arrives (when talking about a single period problem it is rational to assume a zero lead time, L = 0, without loss of generality). The system variables and the equations describing the system dynamics as given in (2.2), simplify to x1 = xˆ1, y1 = xˆ1 +z1, where at the end of the period the relevant costs are incurred based on the net inventory ˆx2: xˆ2 = y1 -d1 = xˆ1 + z1 -d1. (2.5) To chose the order quantity, the decision maker aims to balance the order cost, inventory holding and backorder cost, plus some terminal cost. This terminal cost is called the salvage value. The salvage cost equals the order (purchase) cost of leftover inventory, which at the end of period 1 amounts to -c2xˆ2. This case arises when we can obtain reimbursement of the ordering cost for each leftover unit. This corresponds to selling the leftover stock at price equivalent to ordering cost. To determine the optimal policy, that is the optimal order size, we write the optimization problem. We simplify the notation further by suppressing the subscripts in x1 = xˆ1 = x, y1 = y, d1 = d, and by considering stationary costs also c = c1 = c2. The inventory holding and backorder costs are assessed based on the simpli.ed form of (2.4), where after taking the ˆ expectation over possible realizations of random demand D, we have C(y)= EDC(y-D). The minimum expected cost function fs, as a function of starting inventory x and inventory after order y, is expressed in the following form: fs(x, y)= min {c(y-x) + C(y)-.c(y-E(D])}. (2.6) y=x The .rst term is the ordering cost, the second term represents the holding/backorder cost, and the last the salvage value. The .xed order cost charge is irrelevant here, since we have only one ordering opportunity that we have to make use of anyway. We solve (2.6) to .nd the optimal y, by .rst expressing the optimum cost fs(y)= fs(x, y)+ cx as a function of single decision variable y: fs(y)= min {cy(1 -.) + C(y)-.cE[D]}. (2.7) y=0 Lemma 2.1 Let yˆs be the smal lest minimizer of function fs . The fol lowing holds for all t: 1. The optimal ordering policy is a base stock policy with the optimal base stock level yˆs . 2. Under the optimal policy, the inventory position after ordering y is given by x, yˆs = x, y =(2.8) yˆs , x < yˆs . The ordering policy instructs to order only if the initial inventory x is below the target inventory level ˆys . Corresponding order size z is equivalent to the di.erence ˆys - x. If we start o. with a high inventory level above ˆys, the order should not be placed. This is precisely a characterization of a base stock policy with a base stock level ˆys . To compute ˆys, we must solve (fs) ' = 0. Taking the .rst derivative of C(y) gives C ' (y)= (h+b)F(y)-b, where F represents a cumulative distribution function of demand. We have (fs) ' (y)= c(1 -.) + (h+ b)F(y)-b, and ˆys is .nally given by b-c(1 -.) F(ˆy s)= . (2.9) h+ b Some additional assumption have to be made for (2.9). First, b > (1 -.)c ensures that it is not optimal to not order anything and merely incur backorder cost. We also assume that h+ (1 -.)c > 0, which holds in a realistic case of h > 0. This also guarantees that (fs) ' is not strictly decreasing, which would lead to the optimal base stock level of ˆys = 8. 2.4 Optimizing cost in a .nite-horizon problem In this section we present some results of the classical stochastic inventory theory based on Karlin and Scarf (1958); Karlin (1960a); Veinott (1963) that are highly relevant for our work. The idea is to derive the ordering policy, which would enable the decision maker to make optimal ordering decisions in a multi-period non-stationary stochastic demand environment. We continue from the .nite horizon inventory model description given in Sections 2.1 and 2.2. At the end we give some insights into capacitated multi-period inventory problem, based on Federgruen and Zipkin (1986a). Due to constant nonzero lead time the decision maker should protect the system against lead t+L time demand, Dt L = Ds, which is demand realized in the time interval (t, t + L). Since s=t the current order zt a.ects the net inventory at time t + L, and no later order does so, it makes sense to reassign the corresponding inventory holding and backorder cost to period t. The expected cost charged to period t is based on the net inventory at the end of the period t+L and we can write it in the following form by using the de.nition of a single period cost function Cˆt+L(yt)given in (2.4): Ct(yt)= .LEDL Cˆt+L(yt -Dt L). (2.10) t The reassigning of cost is done through discounting with a discount factor . over a relevant lead time period. When thinking of cost that are accrued in the .nite horizon setting (from t onward to the end of horizon T), we see that the cost in time interval [t,t + L)cannot be in.uenced. However, there are also cost that are incurredafter T upto periodT+L, through (2.10) these are a consequence of ordering decisions made in the last L periods before T. Thus, we seek to minimize the total discounted expected cost, which are the consequence of ordering decisions made in [t,T). We proceed bydevelopingthe dynamic programming formulation, which is a term describing the optimization of the performance of a multi-period stochastic system. The goal is to develop a recursive formulation of the minimum cost function as a function of the system variables. When thinking about the adequate representation of the state space at time t, we should include all the relevant information currently available. To assess the inventory cost through (2.10), we need to keep track of net inventory ˆxt and the information about outstanding orders. These are pipeline orders already given in past L periods, but are not yet delivered. We de.ne the vector of outstanding orders, z t = (zt-L,zt-L+1,. . . ,zt-2,zt-1), given in periods t-L,. . . ,t-1. The state space gets updated when we move from t to t+1 in the following way: xˆt+1 = xˆt + zt-L -dt, (2.11) z t+1 = (zt-L+1,zt-L+2,. . . ,zt-1,zt). (2.12) Observe that the order zt is given based on the available information described by the state space variables (ˆxt,z t), then z t is updated with zt and zt-L component is purged out due to order arrival. Before writing the optimum cost formulation, we de.ne the cost function representing all holding and backorder cost in time interval [t + l-1,t + l): C˜t l(yt)= .lEDl Cˆt+s(yt -Dtl), (2.13) t t+l where Dt l = Ds. The full holding and backorder cost in time interval [t,t + L) can be s=t expressed as a sum of (2.13)for l = 0,. . . ,L: L l+t-L-1 ˜C˜l Ct(ˆxt,z t)= (ˆ). t xt + zsl=0 s=t-L To account for the cost in the remaining periods, we write the minimal discounted expected cost function, optimizing the total cost over a .nite planning horizon T from time t onward and starting in the initial state (ˆxt,z t), as: f˜ t(ˆxt,zt,z t)= min{czt+C˜t 0(ˆxt)+.EDt f˜ t+1(ˆxt+zt-L-Dt,zt+1,z t+1)}, if t = T + L, (2.14) zt=0 where the remaining costs from T onward are given by: f˜ T+1(ˆxT+1,zT+1,z T+1)= C˜T+1(ˆxT+1,z T+1). (2.15) At time T + 1, there are no further orders to be placed, but costs continue to accrue until time T + L. Alternatively, we can set f˜ T+L(·)= 0 and apply (2.14) for all t < T + L+ 1, which again leads to (2.15). This is possible since placing orders beyond T is not rational zt>T = 0, doing so would raise the ordering cost due to c = 0. Although we do not consider ordering cost explicitly later in formulations of the proposed models, it is also reasonable to assume this when operating with holding and backorder cost only. By solving (2.14), the optimal order quantities zt can be computed. However, the presented dynamic program is a complex one due to the nontrivial state representation. It can be shown that an equivalent simpler version exists with a single state variable xt, as de.ned in (2.1). The following Theorem is due to Zipkin (2000); Theorem 9.6.1.: Theorem 2.1 The following holds for all t: 1. The relationship between the original dynamic program given by (2.14) and the new formulation ft(xt)is expressed as: f˜ t(ˆxt,zt,z t)= C˜t(ˆxt,z t)+ ft(xt), if 1 = t = T. 2. The optimal order zt in (2.14)is the same one that achieves the minimum in ft(xt). The simpli.ed dynamic programming formulation has the following form: ft(xt)= min{c(yt -xt)+ Ct(yt)+ .EDt ft+1(yt -Dt)}, if 1 = t = T, (2.16) yt=xt where fT+1(·) = 0. ft(xt) represents the optimal cost under a revised accounting scheme, excluding the holding and backorder costs before time t + L. Although the problem complexityis now signi.cantly reduced, it still poses a great di.culty when trying to obtain the optimum order quantities. In general, xt and yt are continuous variables, and so an in.nite number of expectations and in.nite number of minimizations have to be taken in (2.16) for each t. To solve functional equations of this kind typically requires numerical approximation techniques. In our numerical calculations later in this monograph we discretize and truncate the state space. This means that through an appro­priate selection xt andyt values are restrictedto a .nite set ofvalues. This e.ectivelyreduces the number of possible system states. In a similar fashion the restrictions are imposed on possible demand dt and supply capacity realizations zt + . Obviously, care has to be taken in the way we restrict the state space, not to omit the possible directions in which the system can evolve. The solution, the best yt for every xt for each t, written out in a form of table, would be hard to understand and tedious to implement. Similarly as in single period problem (Lemma 2.1), the optimal policy has a characteristic structure. Again, it .ts into the scope of base stock policies. This is proven through the following Theorem, which is due to Porteus (2002); Lemma 4.3 and Theorem 4.2.: Theorem 2.2 The fol lowing holds for all t: 1. The cost-to-go function gt, de.ned as gt(yt)= c(yt) + Ct(yt) + .EDt ft+1(yt -Dt), is a convex function of yt. 2. Optimal ordering policy is a base stock policy. The optimal base stock level yˆt is the smal lest value yt minimizing gt(yt). 3. ft(xt)is a convex function of xt. The structure of the optimal policy is intuitive in the sense that one only orders if the starting inventory position xt does not exceed the target inventory level ˆyt. Otherwise, it is not rational to place the order. However, knowing the optimal policy structure is not su.cient to determine the optimal base stock levels. We still must perform the recursive calculations using numerical techniques. It is worth to compare the optimum cost formulation (2.16) with its single period equivalent given in (2.6). Consider fs(x, y) now as a function of x only, we see that ft(xt) plays the same role as fs(x)in a single period problem. In fact we can show that the smallest value of y that minimizes somewhat transformed Ct +(yt), based on Ct(yt) as de.ned in (2.10) f+ (transformation is due to setting t (xt)= cxt + ft(xt)), exactly matches the solution of a single period problem ˆys from (2.9). Thus, the myopic base stock level ˆyt M = yˆs . The only change in (2.9) is due to the lead time L, which we have assumed in a multi-period setting. This means that a single period cumulative demand distribution F is replaced by Ft,DL , representing the cumulative distribution of lead time demand D[t, t + L). Interestingly, the righthand side of (2.9) is not a.ected, as it does not depend on L: M b-c(1 -.) Ft,DL (ˆyt )= . (2.17) h+ b The corresponding base stock policy minimizes only the current period cost while ignoring the future, so it is called myopic policy. The myopic policy is obviously much simpler that the optimal one, since it only optimizes costs in period-per-period manner, ignoring the future consequences. However, both are closely related, which is bene.cial in certain settings, where a myopic policy exhibits near-optimal behavior. This means that the decision maker would not be worse o. using the myopic policy when deciding on order quantities, rather than the optimal one. The following Theorem due to Zipkin (2000); Theorem 9.4.2, gives some of the policies interrelations: Theorem 2.3 The fol lowing holds for all t: 1. yˆt = yˆt M . 2. If yˆt M = yˆt+1, then yˆt = yˆt M . 3. yˆt = minu=t yˆu M . The .rst part tells us that the optimal base stock level is always at or below the myopic level, never above (Veinott, 1963). This means that taking the future into account can only reduce the base stock level. Part 2 states that both levels are equal, unless, ˆyt M > yˆt+1. The only reason we might choose ˆyt < yˆt M is to reduce the chance that xt+1 > yˆt+1 in the next period, and this concern is only relevant when ˆyt+1 is small. By part 3, ˆyt is bounded below by the smallest subsequent myopic level. These results together imply that, if the ˆyt M are nondecreasing in t, then ˆyt = yˆt M for all t. In such a setting the myopic policy is optimal, which was .rst shown by Karlin (1960b). Although the analysis of an in.nite horizon problem is not within the scope of this mono­graph, it is appropriate to mention the following result, which further enhances the impor­tance of studying the myopic behavior. The stationary base stock policy with a base stock level ˆyt is optimal, where ˆyt = yˆt M for all t. This means the myopic policy is optimal. How­ever, in an in.nite horizon setting the assumption that not only cost parameters, but also the demands are stationary in time, is required. Due to this ˆyt remains constant in time, yˆt = yˆ, and the policy can be described simply as a demand replacement rule. Meaning that each new order just replaces the prior period’s demand. Finally, we give a comment on the cost transformation proposed by Veinott (1963, 1965), which allows us to exclude the presence of variable ordering cost in the functional equa­tions presented above. Through the cost transformation we establish a new optimum cost formulation ft *(xt), derived from ft(xt) de.ned in (2.16), by making the substitution ft *(xt) = cxt + ft(xt). The variable ordering cost term c(yt -xt) term vanishes. The cost transformation is based on the assumption that the decision maker is judged no di.erently whether he obtains inventory from a previous period, or by ordering it. We use the same principle in establishing the dynamic programming formulations in the case of the models presented in this monograph. 2.5 Finite horizon capacitated inventory problem We now shortly review the capacitated version of the multi-period stochastic inventory prob­lem just described. This review is primarily based on Federgruen and Zipkin (1986a,b) where they assume a .xed capacity level. We denote the capacity level as Q, where (Q> 0). The capacity limit e.ectively restricts the order quantity. We have shown that for the uncapac­itated case, the base stock policy is optimal. We know that the base stock policy instructs to order up to the base stock level in each period. However, if capacity is restricted, this might not be possible in the case the starting inventory position xt is low. Thus, we expect a somewhat di.erent structure of the optimal policy. It also remains to be shown, if this new policy has the properties of a base stock policy at all. We rewrite the dynamic programming formulation for the uncapacitated inventory prob­lem given in 2.16, modifying it to account for the capacity limit. The minimal discounted expected cost function ft(xt, Q)is now a function of Qalso: ft(xt, Q)= min {c(yt -xt) + Ct(yt) + .EDt ft+1(yt -Dt, Q)}, if 1 = t = T , (2.18) xt=yt=xt+Q where fT+1(·)= 0. Observe the di.erence in yt range over which minimization is done. It can be again shown that the cost functions are convex and we are faced with a simple minimization problem with a single optimal solution, therefore the optimal policy is a mod­i.ed base stock policy. The term modi.ed is due to the fact that it is not always possible to attain the target base stock level, which distinguishes it from the base stock policy in the uncapacitated setting. Denoting the optimal base stock level ˆyt(Q), inventory position after ordering is given in the following form: . . xt, yˆt(Q)= x, . yt(xt, Q)= yˆt(Q), yˆt(Q)-Q= x < yˆt(Q), (2.19) . . xt + Q, x < yˆt(Q)-Q. Now we look at the dependency of ˆyt(Q) on Q. Intuitively, we would expect that, if Q is low, we would compensate by increasing the base stock level. The following Theorem, which is due to Porteus (2002); Lemma 8.5 and Theorem 8.4, shows that this is the case: Theorem 2.4 The fol lowing holds for all t: 1. The optimal ordering policy is a base stock policy with the base stock level yˆt(Q), which is a decreasing function of Q. 2. ft(xt, Q)is a convex function on {(xt, Q)|x}. The proof in Porteus (2002) relies also on showing that ft is also supermodular, a function property introduced by Topkis (1978)3 . The optimal policy is expressing a monotone be­havior, where the optimal base stock level is a decreasing function of the state -capacity Q. 3For more information on supermodularity/submodularity properties see Topkis (1998). Chapter 3 Inventory management with advance capacity information The importance of sharing information within modern supply chains has been established by both practitioners and researchers. Accurate and timely information helps .rms e.ectively reduce the uncertainties of a volatile and uncertain business environment. The uncertainty of future supply can be reduced if a company is able to obtain advance capacity information (ACI) about future supply/production capacity availability from its supplier. In this chap­ter, we address a periodic-review inventory system under stochastic demand and stochastic limited supply, for which ACI is available. We show that the optimal ordering policy is a state-dependent base-stock policy characterized by a base-stock level that is a function of ACI. We establish a link with inventory models that use advance demand information (ADI) by developing a capacitated inventory system with ADI, and we show that equivalence can only be set under a very speci.c and restrictive assumption, implying that ADI insights will not necessarily hold in the ACI environment. Our numerical results reveal several man­agerial insights. In particular, we show that ACI is most bene.cial when there is su.cient .exibility to react to anticipated demand and supply capacity mismatches. Further, most of the bene.ts can be achieved with only limited future visibility. We also show that the system parameters a.ecting the value of ACI interact in a complex way and therefore need to be considered in an integrated manner. 3.1 Introduction Particularly in the last two decades, companies working in the global business environment have realized the critical importance of e.ectively managing the .ow of materials across the supply chain. Industry experts estimate not only that total supply chain costs represent the majority of operating expenses of most organizations but also that, in some industries, these costs approach 75% of the total operating budget (Monczka et al., 2009). Inventory and hence inventory management play a central role in the operational behavior of a production system or supply chain. The fact is that the average cost of managing and holding inventory in the United States is 30% to 35% of its value, inventory represents about a third the current assets and up to 90% of the working capital of a typical company in the United States is invested in inventories (Jacobs et al., 2008). Due to the complexities of modern production processes and the extent of global supply chains, inventory appears in di.erent forms at each level of the supply chain. Asupply chain member needs to control its inventory levels by applying some sort of inventory control mechanism. The appropriate selection of this mechanism may signi.cantly impact on the customer service level and the members inventory cost, as well as supply chain system-wide costs. Due to the focus on providing a quality service to the customer, it is no surprise that demand uncertainties attracted the initial attention. However, through time companies have realized that the e.ective management of supply is equally important. A look at the supply chains production and supply capacities allocated to produce or deliver a certain product reveals that these are generally far from stable over time. On the contrary, supply capacity may be highly variable for several reasons, like frequent changes in the product mix, particularly in a setting where multiple products share the same capacity, changes in the workforce (e.g. holiday leave), a machine breakdown and repair, preventive maintenance etc. To compensate for these uncertainties an extra inventory needs to be kept. However, there is another, perhaps even more appealing way to tackle uncertainties in sup­ply. Foreknowledge of future supply availability can help managers anticipate possible future supply shortages, while also allowing them to react in a timely manner by either building up stock to prevent future stockouts or reducing stock in the case of favorable supply conditions. Thus, system costs can be reduced by carrying less safety stock while still achieving the same level of performance. These bene.ts should encourage the parties in a supply chain to for­malize their cooperation to enable the requisite information exchange by either implementing necessary information sharing concepts like Electronic Data Interchange (EDI) and Enter­prise Resource Planning (ERP) or by using formal supply contracts. We could argue that extra information is always bene.cial, but further thought has to be put into investigating in which situations the bene.ts of information exchange are substantial and when it is only marginally useful. Many companies nowadays have invested millions to improve inventory management through modern planning systems like ERP systems, which allow them to use one software package with a number of integrated modules, rather than multiple, con.icting systems with di.erent operating platforms and data formats. While Bush and Cooper (1988) and Metters (1997) a decade ago attribute problems in determining the appropriate inven­tory levels to the fact that companies typically do not use any formal analytic approach, the problem persists as companies do not realize that ERP system is only a management tool; a framework that must be e.ectively applied and integrated to achieve success. In this chapter, we study the bene.ts of obtaining advance capacity information (ACI) about future uncertain supply capacity. These bene.ts are assessed based on a comparison between the case where a manufacturer is able to obtain ACI from its supplier, and a base case without information. Our focus is on a manufacturer which raises its inventory posi­tion by placing orders with its supplier. The supplier could be a contract manufacturer to which the manufacturer has outsourced part of its production. As the contract manufac­turer has limited capacity, it pre-allocates its capacity to manufacturers and noti.es each manufacturer about the allocated capacity slot in advance. This practice is common in the semiconductor industry, where semiconductor foundries routinely share their capacity status with their buyers. Lee and Whang (2000) recognize that capacity information helps com­panies cope with volatile demand and can contribute substantially to mitigating potential shortage gaming behavior, thereby countering a potential source of the bullwhip e.ect to which suppliers are particularly prone. In the near future, the supplier can be certain of the capacity share that it can allocate to a particular customer. Similarly, short-term production plans tend to be .xed and uncertainty regarding the size of the available workforce is lower in the near future. Since the supplier has an insight into future capacity availability, it can communicate ACI to its customer (Figure 3.1). As a customer, the manufacturer now faces changing, albeit known, supply capacity availability in near future periods. By anticipat­ing the possible upcoming capacity shortages it can behave proactively by in.ating current orders and taking advantage of currently available surplus capacity. Unused capacity that was allocated to a particular manufacturer in a certain period is assumed to be lost for that manufacturer, rather than backlogged, as the supplier wishes to dampen the volatility of the manufacturer’s demand. Thus, he chooses to o.er its customers advance information rather than capacity .exibility to help them cope with the periods of low supply capacity availabil­ity. The supplier can use the remaining capacity to cover demand from other markets, less important customers etc. The focus of this chapter is on establishing the optimal inventory policy that would allow the manufacturer to improve its inventory control by utilizing the available ACI. In addition, we seek to identify the settings in which ACI is most valuable. We proceed with a brief review of the relevant literature. Although uncapacitated problems form a foundation in the stochastic inventory control research .eld, we are interested in inventory models that simultaneously tackle the capacity that may limit the order size or the amount of products that can be produced. These models not only recognize that the supply chain’s demand side is facing uncertain market conditions, but also look at the risks of limited or even uncertain supply conditions. Researchers are revisiting the early stochastic demand models and extending them to incorporate uncertainty on the supply side. A base-stock policy characterizes the optimal policy for several di.erent capacitated problems. The sense of a base-stock policy di.ers in a resource constrained case from that in an uncapacitated case. In an uncapacitated case, the base-stock level has a clear interpretation -it is the inventory position to order/produce up to. Yet, in the capacitated case it merely represents a target that may or may not be achieved. If the capacity limit in a certain period is known, there is no use in ordering/producing above that level and we are thus talking about a modi.ed base-stock policy. Federgruen and Zipkin (1986a,b) .rst address the .xed capacity constraint for a stationary inventory problem and prove the optimality of the modi.ed base-stock policy. This result is extended by Kapuscinski and Tayur (1998) for a non-stationary system assuming periodic demand, where it is also shown that a modi.ed base-stock policy is optimal. Later, a line of research extends the focus to capture uncertainty in capacity by analyzing models with limited stochastic production capacity (Ciarallo et al., 1994; G¨ull¨u et al., 1997; Khang and Fujiwara, 2000; Iida, 2002). Here we point out the relevance of Ciarallo et al. (1994) to our work. For a .nite horizon stationary inventory model they show that the optimal policy remains a base-stock policy, where the optimal base-stock level is increased to account for possible, albeit uncertain, capacity shortfalls in future periods. In the analysis of a single period problem, they show that stochastic capacity does not a.ect the order policy. The myopic policy of the newsvendor type is optimal, meaning that the decision maker is not better o. by asking for a quantity higher than that of the uncapacitated case. Our model builds on these models by assuming both demand and capacity are non-stationary and stochastic, while additionally modeling that some of the future supply capacity is revealed through ACI. We propose that by reducing supply capacity uncertainty due to ACI we can re.ne the inventory policy and improve its performance. To our knowledge, the relationship between ACI and optimal policy parameters has not yet been explored in the literature1 . The complexity of a capacitated stochastic non-stationary inventory formulation presents a challenge in terms of obtaining an analytical solution for the parameters of the optimal 1We would like to point out the new developments in this .eld since the work presented in this chapter has been published in Jak.si.c et al. (2011). We direct the reader to the papers by Altu.g and Muharremo.glu (2011), C¸ inar and G¨ull¨u (2012) and Atasoy et al. (2012) policy, mainly optimal base-stock levels. Researchers have resorted to developing appli­cable heuristics (Bollapragada and Morton, 1999; Metters, 1997, 1998). In a capacitated non-stationary stochastic setting, Metters (1997) captures both the e.ect of deterministic anticipation (anticipating mismatches in demand and capacity, and reacting by building up the inventory), as well as the e.ect of uncertainty in demand. We note that there is a lack of literature concerning an approximate analysis of inventory systems that also assume uncertain capacity. We propose that an e.ective way of circumventing the uncertainty of supply capacity is to use ACI. Advance information can also be obtained on the demand side and is dealt with by the well-established advance demand information (ADI) research stream where classical stochastic demand models are extended to incorporate ADI (G¨ull¨Ozer, u, 1996; Gallego and ¨ 2001; Wijngaard, 2004; Tan et al., 2007). It is usually assumed that future uncertainty can be reduced due to some customers placing their orders in advance of their needs. This forms a stream of early demand that does not have to be satis.ed immediately. Since this demand is revealed beforehand through ADI we can use ADI to make better ordering decisions. For ¨ a capacitated inventory model with ADI, Ozer and Wei (2004) show that the base-stock policy is optimal, where the optimal base-stock level is an increasing function of the future demand levels revealed through ADI. In terms of modeling, our approach resembles the ADI modeling approach so our work also focuses on presenting possible similarities and relevant distinctions between the two. Our contributions in this study can be summarized as follows: (1) We develop a periodic review inventory model with stochastic demand and limited stochastic supply capacity which enables a decision maker to improve the performance of the inventory control system through the use of ACI. (2) We demonstrate the structural properties of the optimal policy by showing the optimality of a modi.ed base-stock policy with an ACI-dependent base-stock level. Our focus is on showing the dependency of the optimal base-stock level on ACI, which we elaborate on by establishing the monotonicity properties of the base-stock level in relation to ACI. (3) Our computational results provide useful managerial insights into the conditions in which ACI becomes most bene.cial. In particular, we show that the biggest savings can be achieved when one is facing a high level of uncertainty in future supply and can e.ectively reduce this uncertainty by using ACI. However, we also emphasize that the bene.ts depend greatly on the successfulness of the anticipatory inventory build up, which may be limited by the size of the available capacity. In addition, we demonstrate how the value of ACI changes with respect to the length of the ACI horizon, cost parameters and demand uncertainty. (4) We develop a corresponding capacitated ADI inventory model and comment on its characteristics relative to the proposed ACI model. We show that only with the restrictive assumption of constant supply capacity that is always su.cient to cover the early orders recorded through ADI are the two models equivalent in their optimal base-stock level and optimal cost. The remainder of the chapter is organized as follows. We present a model incorporating ACI and its dynamic cost formulation in Section 3.2. The optimal policy and its ACI-related properties are discussed in Section 3.3. In Section 3.4 we present the results of a numerical study and point out additional managerial insights. We study the setting with Bernoulli distributed supply capacity in Section 3.5. In Section 3.6 we look at the di.erences between the ADI modeling and the analysis of ACI model presented in this chapter. Finally, we summarize our .ndings and suggest directions for future research in Section 3.7. 3.2 Model formulation In this section, we introduce the notation and our model. The model under consideration assumes periodic-review, non-stationary stochastic demand, limited non-stationary stochas­tic supply with a zero supply lead time, .nite planning horizon inventory control system. The assumption of zero lead time is not a restrictive assumption as the model can be easily generalized to the positive supply lead time case. The supply capacity is assumed to be exogenous to the manufacturer and unused capacity in a certain period is assumed to be lost for this manufacturer, rather than backlogged. The manager is able to obtain ACI on the available supply capacity for orders to be placed in the future and use it to make better ordering decisions. We introduce a parameter n, which represents the length of the ACI horizon, that is, how far in advance the available supply capacity information is revealed. We assume ACI qt+n is revealed at the start of period t for the supply capacity that limits the order zt+n, that will be placed in period t+n. The model assumes perfect ACI, meaning that in period t the exact supply capacities for future n periods are known. The supply capacities in more distant periods, from t + n + 1 towards the end of the planning horizon, remain uncertain (Figure 3.2). This means that, when placing the order zt in period t, we know the capacity limit qt and ordering above this limit is irrational. Assuming that unmet demand is fully backlogged, the goal is to .nd an optimal policy that minimizes the relevant costs, that is inventory holding costs and backorder costs. We assume an inventory system with zero .xed costs. The model presented is general because no assumptions are made with regard to the nature of the demand and supply process. Both are assumed to be stochastic and with known independent distributions in each time period. The major notation is summarized in Table 3.1 and some is introduced when needed. Table 3.1 Summary of the notation T : number of periods in the .nite planning horizon n : advance capacity information, n = 0 h : inventory holding cost per unit per period b : backorder cost per unit per period a : discount factor (0 = a = 1) xt : inventory position in period t before ordering yt : inventory position in period t after ordering ˆxt : starting net inventory in period t zt : order size in period t Dt : random variable denoting the demand in period t dt : actual demand in period t gt : probability density function of demand in period t Gt : cumulative distribution function of demand in period t Qt : random variable denoting the available supply capacity at time t qt : actual available supply capacity at time t, for which ACI is revealed at time t -n rt : probability density function of supply capacity in period t Rt : cumulative distribution function of supply capacity in period t We assume the following sequence of events. (1) At the start of period t, the decision maker reviews the current inventory position xt and ACI on the supply capacity limit qt+n, for order zt+n that is to be given in period t + n, is revealed. (2) Ordering decision zt is made based on the available supply capacity qt, where zt = qt and correspondingly the inventory position is raised to yt = xt +zt. Unused supply capacity is lost. (3) The order placed at the start of period t is received. (4) At the end of the period previously backordered demand and demand dt are observed and satis.ed from on-hand inventory; unsatis.ed demand is backordered. Inventory holding and backorder costs are incurred based on the end-of-period net inventory. To determine the optimal cost, we not only need to keep track of xt but also the supply capacity available for the current order qt and the supply capacities available for future orders, which constitute ACI. At the start of period t, when the available supply capacity qt+n is already revealed for period t + n, the vector of ACI consists of available supply capacities potentially limiting the size of orders in future n periods: (qt+1, qt+2,. . . , qT), if T -n = t = T , z(3.1) qt = (qt+1, qt+2,. . . , qt+n-1, qt+n), if 1 = t = T -n -1. Information on the current supply capacity qt obviously a.ects the cost, but we chose not to include it in the ACI vector since only ACI for future orders a.ects the structure and parameters of the optimal policy. All together, the state space is represented by an n + 2 -dimensional vector (xt, qt, zqt), where xt and qzt are updated at the start of period t + 1 in the following manner: xt+1 = xt + zt -dt, (3.2) (qt+2, qt+3,. . . , qT), if T -n -1 = t = T -1, z= (3.3) qt+1 (qt+2, qt+3,. . . , qt+n+1), if 1 = t = T -n -2. Note also that the probability distributions of both demand and supply capacity a.ect the optimal cost and optimal policy parameters. The minimal discounted expected cost function, optimizing the cost over .nite planning horizon T from time t onward and starting in the initial state (xt, qt, zqt)can be written as: ft(xt, qt, zqt)= min {Ct(yt)+ xt=yt=xt+qt (3.4) aEDt ft+1(yt -Dt, qt+1, zqt+1), if T -n = t = T , + aEDt,Qt+n+1 ft+1(yt -Dt, qt+1, zqt+1), if 1 = t = T -n -1, JJ8 where Ct(yt)= h 0 yt (yt - dt)gt(dt)ddt + b yt (dt - yt)gt(dt)ddt is the regular loss function, and the ending condition is de.ned as fT+1(·)= 0. 3.3 Analysis of the optimal policy In this section, we .rst characterize the optimal policy, as a solution of the dynamic pro­gramming formulation given in (3.4). We prove the optimality of a state-dependent modi.ed base-stock policy and provide some properties of the optimal policy. See the Appendix A for proofs of the following theorems. Let Jt denote the cost-to-go function of period t de.ned as Ct(yt) + aEDt ft+1(yt -Dt, qt+1, zqt+1), if T -n = t = T , Jt(yt, zqt)= (3.5) Ct(yt) + aEDt,Qt+n+1 ft+1(yt -Dt, qt+1, zqt+1), if 1 = t = T -n -1, and we rewrite the minimal expected cost function ft as ft(xt, qt, zqt)= min Jt(yt, zqt), for 1 = t = T. xt=yt=xt+qt We .rst show the essential convexity results that allow us to establish the optimal policy. Note that the single-period cost function Ct(yt)is convex in yt for any t since it is the usual newsvendor cost function (Porteus, 2002). Lemma 3.1 For any arbitrary value of information horizon n and value of the ACI vector z q, the fol lowing holds for all t: 1. Jt(yt, zqt)is convex in y, 2. ft(xt, qt, zqt)is convex in x. Based on the convexity results, minimizing Jt is a convex optimization problem for any arbitrary ACI horizon parameter n. Theorem 3.1 Let yˆt(zqt)be the smal lest minimizer of the function Jt(yt, zqt). For any zqt, the fol lowing holds for all t: 1. The optimal ordering policy under ACI is a state-dependent modi.ed base-stock policy with the optimal base-stock level yˆt(zqt). 2. Under the optimal policy, the inventory position after ordering yt(xt, qt, zqt)is given by . . xt, yˆt(qzt)= xt, . yt(xt, qt, zqt)= yˆt(zqt), yˆt(qzt)-qt = xt < yˆt(zqt), . . xt + qt, xt < yˆt(zqt)-qt. This modi.ed base-stock policy is characterized by a state-dependent optimal base-stock level ˆyt(zqt), which determines the optimal level of the inventory position after ordering. The optimal base-stock level depends on the future supply availability, that is supply capacities qt+1, qt+2,. . . , qt+n, given by ACI vector. Optimal base-stock level ˆyt(qzt)can be characterized as an unconstrained base-stock level as it might not be reached due to the limited supply capacity. Observe also that ˆyt(zqt) is independent of the available supply capacity qt in period t, which follows directly from Theorem 3.1 and the de.nition of Jt in (3.5). The optimal policy can thus be interpreted in the following way. In the case that the inventory position at the beginning of the period exceeds the optimal base-stock level, the decision maker should not place an order. However, if the inventory position is lower he should raise the inventory position up to the base-stock level if there is enough supply capacity available; if not, he should take advantage of the full supply capacity available for the current order. Our model assumes that also in the case of n = 0 the realization of supply capacity qt, available for the current order, is known at the time the order is placed, while only future supply capacity availability remains uncertain. However, when ACI is not available (No-ACI) the decision maker encounters uncertain supply capacity for the order he is currently placing, as has been modeled by Ciarallo et al. (1994). Due to the zero lead time the updating of the inventory position happens before the current period demand needs to be satis.ed. Therefore, it is intuitively clear that by knowing only the current period’s supply capacity qt one cannot come up with a better ordering decision, thus the optimal base-stock levels and performances should be the same for both models. Theorem 3.2 The ACI model given by (3.4), in the case when n = 0, and the No-ACI model given by (A.4), are equivalent with respect to the optimal base-stock level and the optimal discounted expected cost. In the following theorem we proceed with a characterization of the behavior of the base-stock level in relation to the actual capacity levels revealed through ACI. Intuitively, we expect that when we are facing a possible shortage in supply capacity in future periods we tend to increase the base-stock level. With this we stimulate the inventory build-up to avoid possible backorders which would be the probable consequence of a capacity shortage. Along the same lines of thought, the base-stock level is decreasing when higher supply availability is revealed by ACI. We con.rm these intuitive results in Part 3 of Theorem 3.3 and illustrate the optimal ordering policy in Figure 3.3. From this point we elect to suppress the subscript t in state variables for clarity reasons. We de.ne the .rst derivative of functions ft(x, ·) and Jt(x, ·) with respect to x as ft ' (x, ·) and Jt ' (x, ·). Observe that for any two ACI vectors qz1 and zq2 in period t, zq2 = zq1 holds if and only if each element of zq1 is greater than or equal to the corresponding element of zq2. In Parts 1 and 2 of Theorem 3.3, we show that the rate of the change in costs is higher at higher capacity levels. Observe that Jt and ft are also convex nonincreasing functions in zq and therefore the costs are decreasing with higher future supply capacity availability. The rate of the cost decrease, and thereby the sensitivity of the optimal cost to an increase in capacity, is higher for low capacity and diminishes for higher capacities. Part 3 of Theorem 3.3 suggests that when zq2 = qz1 the decision maker has to raise the base-stock level ˆyt(zq2) above the one that was optimal in the initial setting, ˆyt(zq1). Theorem 3.3 For any q2 = q1 and zq2 = zq1, the fol lowing holds for all t: 1. Jt ' (y, zq2)= Jt ' (y, zq1)for all y, 2. ft ' (x, q2, zq2)= ft ' (x, q1, zq1)for all x, 3. yˆt(qz2)= yˆt(qz1). We proceed by o.ering some additional insights into the monotonicity characteristics of the optimal policy. We continue to focus on how changes in ACI a.ect the optimal base-stock level. In the .rst case, we want to assess whether the base-stock level is a.ected more by a change in supply capacity availability in one of the imminent periods, or whether a change in the available capacity in distant periods is more signi.cant. Let us de.ne unit vector ei with dimensions equal to the dimensionality of the ACI vector (n-dimensional), where its ith component is 1. With vector ei we can target a particular component of the ACI vector. We want to establish how the optimal base-stock level is a.ected by taking away . units of supply capacity in period i from now, in comparison with doing the same thing but one period further in the future. The results of a numerical study suggest that taking away a unit of supply capacity given by ACI in period i a.ects the optimal base-stock level more than taking away a unit of supply capacity which is available in later periods, i + 1 and beyond. We provide the following simple example for the above by setting n = 2 and i = 1. If we expect the capacity shortage in period t+ 1 (system 1), we will increase ˆyt(qt+1 -., qt+2)correspondingly to avoid future backorders (assuming b > h). However, if the shortage is expected to occur one period later in period t+ 2 (system 2), we will only increase ˆyt(qt+1, qt+2 -.) if we anticipate that we will not be able to account for the shortage by increasing the optimal base-stock level in period t+ 1. Therefore there is no reason for system 2 to start o. with more inventory than system 1. Careful observation also reveals that the same holds for a less interesting setting of h > b, even more, both systems will always have the same optimal base-stock level in this case. This suggests that the closer the capacity restriction is to the current period the more we need to take it into account when setting the appropriate base-stock level. We formalize this result in Conjecture 3.1. Conjecture 3.1 The fol lowing holds for all t: yˆt(qz-.ei)= yˆt(qz-.ei+1)for i=t+1,. . . ,t+n-1. Furthermore, we study the sensitivity of the optimal base-stock level to a change in the capacity limit. Assuming the result of Conjecture 3.1 holds, we show in Conjecture 3.2 that the change in the base-stock level should be lower than the change in the capacity limit in absolute terms (see Appendix A for the proof). In other words, each unit decrease in available supply capacity revealed by ACI leads to a lower or at most an equal increase in the optimal base-stock level. Conjecture 3.2 The fol lowing holds for all t: yˆt(qz-.ei)-yˆt(qz)= . for i=t+1,. . . ,t+n. A dampening e.ect is present and, together with the result of Conjecture 3.1, this implies that the exact ACI for a distant future period becomes less relevant to a current ordering decision. This is an important result, suggesting there is a diminishing bene.t in overex­tending the ACI horizon n. This is desirable both in terms of having reliable ACI in a practical setting as well as reducing the complexity of determining the optimal parameters by operating with small values of n to avoid the curse of dimensionality. 3.4 Value of ACI In this section we present the results of a numerical analysis which was carried out to quantify the value of ACI and to gain insights into how the value of ACI changes as some of the system parameters change. Numerical calculations were done by solving the dynamic programming formulation given in (3.4). We .rst introduce the value of ACI and then proceed by analyzing di.erent settings in three subsections: We construct a set of experiments with di.erent demand and capacity patterns. This enables us to describe the e.ect that a period-to-period mismatch between demand and capacity has on the inventory cost in Section 3.4.1. Based on a particular demand and capacity pattern we proceed with a more detailed analysis of the in.uence of the cost structure and the uncertainty of period-to-period demand and capacity in Section 3.4.2. Through an extensive simulation of random demand and capacity patterns we explore the e.ect of utilization more closely and generalize the results in Section 3.4.3. To determine the value of ACI, the performance comparison between our ACI model and the case of a capacitated stochastic supply model with no ACI is of interest to us. It follows from the result of Theorem 3.2 that it is su.cient to compare the n > 0 setting with the n = 0 setting to determine the full value of ACI. We de.ne the relative value of ACI for n > 0, %VACI , as the relative di.erence between the optimal expected cost of managing the system where n = 0, and the system where we have an insight into future supply availability n > 0: (n=0) (n>0) %VACI (n > 0) = ft -ft . (3.6) (n=0) ft We also de.ne the marginal change in the value of ACI, .VACI . With this we measure the extra bene.t gained by extending the length of the ACI horizon by one time period, from n to n + 1: (n) (n+1) .VACI (n + 1) = ft -ft 3.4.1 E.ect of demand and capacity mismatch We proceed with constructing a set of six experiments (experiments number 1-6) with di.er­ent demand and capacity patterns. The remaining parameters are set at the .xed value of: T = 8, n = 1, . = 0.99, h = 1, b = 20, Normal demand and capacity both with a coe.cient of variation (CV ) of 0.1. We give a graphical illustration of demand and supply capacity patterns, by plotting the expected demand and expected supply capacity for each of the periods in Figures 3.4-3.6. We give the optimal base stock levels ˆy(n=0) for n = 0 setting. For comparison also the myopic base stock levels ˆyM are given. The myopic base stock level is de.ned as: M b Ft,DL (ˆyt )= . (3.7) h+ b The myopic base stock levels indicate how high should the inventory position be to cover the additional risks due to demand uncertainty, and depending on the backorder and holding cost ratio. However, ˆyM completely neglects the e.ect of limited supply capacity. The optimal inventory costs are presented in Table 3.3, where also the value of ASI, %VASI , and the absolute change in the value of ASI, .VASI , is given. Let us .rst observe the di.erences in inventory cost between the proposed settings. In general the inventory cost are higher if the supply capacity is highly utilized (experiment number 2, 3 and 4). However, although the average utilization2 in all three cases is 100%, there are still considerable cost di.erences. These are mainly due to period-to-period mismatch between demand and supply capacity pattern. In experiment number 3 presented in Figure 3.5 (a), 2De.ned as the sum of average demands over the sum of average supply capacities over the whole planning horizon. which represents a kind of of the ”worst case” scenario, we are .rst faced with multiple periods of high demand and inadequate capacity. Therefore the extent of the backorders accumulated in these .rst periods is high, thus inventory cost are high. We see that the optimal policy instructs that we raise the base stock levels in the beginning periods. By doing this, we aim to use as much of the available capacity as possible. However, the probability to achieve these target base stock levels is minor, thus we cannot avoid the backorders. If we reverse this setting in experiment number 4 presented in Figure 3.5 (b), we can use the early excess capacity to build up the necessary inventory to cope with the subsequent capacity shortage. This in turn greatly reduces the cost. We gradually increase the base stock levels as we approach the over-utilized periods. After the peak demand periods, towards the end of the planning horizon, the base stock level drops to the myopic optimal level in the last period3 . Observe also that the optimal base stock level in the .rst period is 12, which is above the myopic base stock level of 6. We know that in an uncapacitated setting ˆy lies at or below ˆyM . We obviously see that this is not the case in the capacitated system. That is why we see the optimal base stock levels increasing as we move towards capacitated periods. In the .rst four periods the available surplus in capacity (in cumulative terms) is not enough to build up enough inventory to cope with high demand in last periods. A high starting base stock level is a sign that the pre-build phase should commence earlier, and if so, additional savings could be achieved. Exp. 1 (b) Exp. 2. We can further con.rm the inventory build up insight by inspecting the experiment number 1 in Figure 3.4 (a), where we are faced with two demand peaks in periods 4 and 7 (Figure 3.4a). To avoid the probable backorders in the two critical and the following periods, the rational thing to do is to pre-build the inventory. Looking at the myopic base stock level and 3In period T, at the end of the planning horizon, the myopic and the optimal solution converge. Since this is the last period, we only need to optimize over a single period, which means that the myopic optimization is optimal. Figure 3.6 Expected demand and capacity pattern, and optimal base stock level ˆy(n=0) (a) Exp. 5 (b) Exp. 6. the optimal base stock level in period 1, we see that they are equal. In period 2 the optimal base stock level increases above the myopic one in the anticipation of the capacity-demand mismatch in period 4. From this we can conclude that two time periods are su.cient, since the .rst build up phase starts in period 2. The same is true for the second pre-build phase, which immediately follows the peak period, starts in period 5, and build up to period 7. For comparison with the well researched models of stationary demand and capacity, we in­clude the experiment number 2 presented in Figure 3.4 (b), where the same logic of inventory build up can be noted. Observe that anticipation of future capacity shortages based on the expected values of demand and capacity patterns is not possible here. However, this is not the case if ASI is available. We see in Table 3.3 that the cost reduction can be achieved through the use of ASI, although not substantial in size. In the case of highly utilized ca­pacity (in our case the average utilization is 100%) we are looking to make use of every unit of available capacity, however when we approach to the end of the planning horizon the tendency to do so diminishes. This con.rms a well known result for the stationary unca­pacitated inventory system that the base stock level is decreasing as we approach the end of the planning horizon (Scarf, 1960). This is often the end of horizon e.ect, where when we approach the planning horizon we care less and less about the future, thus we think more in short terms -myopically. The consequence of this is that the optimal and myopic base stock levels converge. In Figures 3.6 (a) and (b), we give two additional examples where the e.ects we have described can also be observed. Table 3.2 Optimal ˆy(n=0) and myopic ˆyM base stock level, optimal system cost, and the value of ASI Exp. t 1 2 3 4 5 6 7 8 n Cost %VASI .VAS I 1 E[Dt] E[Qt] yˆ(n=0) yˆM 2 E[Dt] E[Qt] yˆ(n=0) yˆM 3 E[Dt] E[Qt] yˆ(n=0) yˆM 4 E[Dt] E[Qt] yˆ(n=0) yˆM 5 E[Dt] E[Qt] yˆ(n=0) yˆM 6 E[Dt] E[Qt] yˆ(n=0) yˆM 5 10 6 6 10 10 16 12 10 5 32 12 5 10 12 6 10 10 13 12 5 10 6 6 5 10 8 6 10 10 16 12 10 5 24 12 5 10 17 6 10 10 12 12 5 10 6 6 5 10 13 6 10 10 15 12 10 5 17 12 5 10 22 6 10 15 12 12 5 15 8 6 15 10 18 18 10 10 15 12 10 5 12 12 5 10 27 6 10 15 12 12 5 15 18 6 5 10 8 6 10 10 14 12 5 10 6 6 10 5 28 12 5 10 6 6 10 10 23 12 5 10 13 6 10 10 14 12 5 10 6 6 10 5 23 12 5 10 7 6 10 10 23 12 15 10 17 18 10 10 13 12 5 10 6 6 10 5 17 12 5 5 6 6 10 5 17 12 5 10 6 6 10 10 12 12 5 10 6 6 10 5 12 12 5 5 6 6 10 5 12 12 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 31.05 28.84 27.26 27.25 189.54 189.25 189.14 189.10 1579.28 1579.28 1579.28 1579.28 103.87 103.58 103.32 103.10 39.45 38.84 38.77 38.77 53.98 52.36 51.24 50.58 7.12 2.21 12.22 1.58 12.24 0.01 0.15 0.28 0.21 0.11 0.23 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.28 0.30 0.53 0.26 0.74 0.22 1.55 0.61 1.73 0.07 1.73 0.00 3.01 1.63 5.07 1.11 6.30 0.67 We proceed by constructing a set of four experiments (experiments 7-10) with di.erent demand and capacity patterns. The remaining parameters are set at .xed values of: T = 8, a = 0.99, h = 1 and b = 20. A discrete uniform distribution is used for demand and capacity both with a coe.cient of variation (CV )of 0.45. Through careful selection of the type of probability distribution and the parameters used, we were able to limit the number of possible space states, which allowed us to solve the underlying dynamic program. The optimal inventory costs and the optimal base-stock levels ˆy(n=0) are presented in Table 3.3, where the value of ACI, %VACI , and the marginal change in the value of ACI, .VACI , is also given. The relevance of anticipating future capacity shortages is already well-established in a de­terministic analysis of non-stationary capacitated inventory systems. To cope with periods of inadequate capacity, inventory has to be pre-built in advance by in.ating orders. Also in the stochastic variant of this problem that we are considering, some anticipation is possible through knowing the probability distributions of demand and supply capacity in future pe­riods. In this chapter we show that we can further improve this anticipative pre-building by using ACI and we can thereby attain additional cost savings. Table 3.3 Optimal base-stock level ˆy(n=0) , optimal system cost, and the value of ACI Exp. t 1 2 3 4 5 6 7 8 n Cost %VAC I .VAC I 7 E[Dt] E[Qt] ˆy(n=0) 6 3 25 6 3 21 6 3 17 6 3 14 6 9 13 6 9 13 6 9 11 6 9 10 0 1 2 3 4 1108.0 1107.6 1107.6 1107.6 1107.6 0.04 0.04 0.04 0.04 0.43 0.02 0.00 0.00 8 E[Dt] E[Qt] ˆy(n=0) 6 9 22 6 9 24 6 9 25 6 9 26 6 3 22 6 3 18 6 3 15 6 3 10 0 1 2 3 4 279.1 278.3 277.6 277.2 277.0 0.28 0.53 0.67 0.74 0.78 0.70 0.40 0.20 9 E[Dt] E[Qt] ˆy(n=0) 6 6 22 6 6 21 6 6 20 6 6 18 6 6 17 6 6 15 6 6 13 6 6 10 0 1 2 3 477.4 476.6 476.2 476.0 0.16 0.25 0.28 0.78 0.41 0.15 4 476.0 0.29 0.04 10 E[Dt] E[Qt] ˆy(n=0) 3 9 7 3 9 12 3 9 17 12 9 20 3 9 11 3 9 16 12 9 19 3 9 5 0 1 2 3 4 78.8 70.6 66.7 65.7 65.4 10.37 15.36 16.62 17.00 8.16 3.93 0.99 0.30 We .rst consider three highly utilized settings (experiments 7, 8 and 9) with average uti­lization4 of 100% and di.erent expected demand/capacity patterns. Due to high utilization, shortages occur frequently and ACI is therefore needed for anticipation. However, in experi­ment 7, where in the starting periods we are faced with multiple periods of high demand and inadequate capacity, the over-utilized system prevents inventory pre-building the value of 4De.ned as the sum of average demand over the sum of average supply capacity over the whole planning horizon. ACI is almost zero. If we reverse this setting in experiment 8, the costs are greatly reduced and the value of ACI is higher due to having early excess capacity available to build up the necessary inventory to cope with the subsequent capacity shortages. We see that the value of ACI depends strongly on whether there exists an opportunity to increase orders prior to a capacity shortage, which is a.ected both by the system’s utilization and the nature of the demand/capacity pattern. The results of experiment 10 support this insight, where we study a system with lower average utilization (58%) and expected capacity shortages in periods 4 and 7. By having an insight into the next period’s available capacity, we can lower the inventory cost by 10.37%, while an additional period of ACI data gives an additional 4.99% cost reduction. However, prolonging the ACI horizon further does not improve the performance greatly. Additional information on future supply conditions helps in making more e.ective ordering decisions as %VACI increases with the length of information horizon n, however the bene.ts are diminishing. 3.4.2 E.ect of volatility We proceed by investigating the in.uence demand uncertainty, supply capacity uncertainty, and cost structure on the value of ACI. The new base scenario is characterized by the following parameters: T = 8, a = 0.99 and h = 1. A discrete uniform distribution is used to model demand and supply capacity where the expected demand is given as E[D1..8] = (3,3,3,3,3,9,3,3) and the expected supply capacity as E[Q1..8] = (6,6,6,6,6,6,6,6). We vary: (1) the cost structure by changing the backorder cost b = {5,20,100}; and (2) the coe.cient of variation of demand CVD = {0,0.25,0.45,0.65} and supply capacity CVQ = {0,0.25,0.45,0.65}, where the CVs do not change over time5 . To observe the e.ect of the demand and the supply capacity uncertainty on the value of ACI, we .rst analyze the deterministic demand case presented in experiments 12, 13 and 14. Observe that the costs can be substantially decreased, even by up to almost 70% (experiment 12). Since there is no uncertainty in demand, we can attribute the decrease in costs solely to the fact that ACI resolves some of the remaining supply capacity volatility, which greatly increases the relative bene.t of ACI. In particular, when CVQ is low and the ACI horizon is long, it is likely that shortages are fully anticipated and the backorders can be fully avoided. When CVQ increases, the absolute savings denoted by .VACI increase for all analyzed experiments. However, this is not the case with the relative savings denoted by %VACI , which decrease for b = 20 and b = 100. Although ACI resolves some uncertainty in the system, the remaining uncertainty drives the costs up and the relative value of ACI 5Since it is not possible to come up with the exact same CVs for discrete uniform distributions with di.erent means, we give the approximate average CVs for demand and supply capacity distributions with means E[D1..8]and E[Q1..8]. actually decreases. When we introduce demand uncertainty into the system, we observe that when CVD in­creases, both %VACI and .VACI are decreasing, looking at experiments 13, 15 and 16. The latter is intuitively clear since the bene.ts of a more precise alignment of the base-stock level, made possible by the revealed ACI, are greatly diminished because the volatile de­mand causes the inventory position to deviate from the planned level. In addition, due to carrying a lot of safety stock to prevent stockouts, the total costs increase, which in turn reduces the relative savings obtained through ACI. However, when both demand and supply uncertainty are increased, we observe the non-monotone behavior of %VACI (experiments 16, 18 and 19). We see that %VACI is increasing with an increased uncertainty in the system for low b/h ratio (b = 5), while the opposite is true for b = 100. To study the e.ect of the cost structure on the value of ACI, we vary the b/h ratio. For all experiments in Table 3.4 we observe that .VACI is increasing with an increase in b/h. However, this is not the case with %VACI , where the behavior is highly non-monotone, depending strongly on other system parameters. For low b/h ratios, we would anticipate low values of ACI, due to the fact that shortage costs are lower and thus there is less need to anticipate possible shortages. However, when there is higher capacity uncertainty in the system, relative savings are higher for low than for high b/h ratio, which can be observed in experiments 13, 14 and 19. We attribute this to the prevalent e.ect of high CVQ as described above, where ACI is not able to resolve much of the capacity uncertainty, which in the case of high b/h ratio causes higher increase in total backorder costs and leads to correspondingly low value of ACI. The opposite can be noted for high b/h ratio, looking at experiments 12 and 18, where higher %VACI is attained under low capacity uncertainty. In the case of high b/h ratio, holding extra inventory is relatively inexpensive and one would assume that ACI is not needed. Although the probability of a shortage occurring is now very small, ACI allows us to more or less completely avoid costly backorders, thus relatively higher savings are possible particularly when b/h is high. The observed combined e.ect of the analyzed system parameters, CVD, CVQ and b/h, can be attributed to the complex interaction between them, which clearly suggests that they need to be considered in an integrated manner. Table 3.4 Optimal system cost and the value of ACI under varying ACI horizon n b 5 20 100 5 20 100 5 20 100 Exp. 11 12 CVD 0 0 CVQ 0 0.25 n all n 0 1 2 3 4 2.88 6.71 4.82 3.95 3.79 3.79 Cost 2.88 9.44 5.77 4.15 3.82 3.79 2.88 11.51 6.40 4.23 3.82 3.79 0.00 28.25 41.13 43.48 43.62 %VAC I 0.00 38.95 56.08 59.56 59.90 0.00 44.39 63.27 66.80 67.11 0.00 1.90 0.86 0.16 0.01 .VAC I 0.00 3.68 1.62 0.33 0.03 0.00 5.11 2.18 0.41 0.04 13 0 0.45 0 1 2 3 4 16.10 10.90 8.68 7.78 7.46 28.88 20.21 15.34 12.93 11.97 59.22 48.08 40.33 36.39 34.87 32.32 46.08 51.70 53.65 30.03 46.88 55.21 58.57 18.81 31.90 38.56 41.11 5.20 2.21 0.91 0.31 8.67 4.87 2.41 0.97 11.14 7.75 3.94 1.51 14 15 16 0 0.25 0.45 0.65 0.45 0.45 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 35.67 26.71 22.71 20.84 20.05 23.23 20.43 19.25 18.82 18.65 34.35 32.37 31.44 31.09 30.98 77.84 66.12 58.62 54.10 51.84 41.28 35.19 32.04 30.51 29.94 61.50 56.89 54.33 53.02 52.50 248.87 237.45 227.39 220.59 217.10 86.60 78.13 72.53 69.77 68.69 139.65 133.95 130.49 128.29 127.31 25.10 36.34 41.58 43.80 12.07 17.16 19.00 19.72 5.79 8.49 9.49 9.82 15.06 24.69 30.50 33.41 14.76 22.39 26.08 27.46 7.50 11.66 13.79 14.63 4.59 8.63 11.36 12.77 9.78 16.24 19.43 20.68 4.08 6.55 8.13 8.83 8.95 4.01 1.87 0.79 2.81 1.18 0.43 0.17 1.99 0.93 0.34 0.11 11.72 7.50 4.53 2.26 6.09 3.15 1.52 0.57 4.61 2.55 1.31 0.52 11.42 10.06 6.80 3.49 8.47 5.60 2.76 1.08 5.69 3.46 2.21 0.97 17 18 19 0.65 0.25 0.65 0.45 0.25 0.65 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 47.58 45.85 45.04 44.73 44.64 15.81 15.02 14.69 14.62 14.61 62.49 58.52 56.92 56.27 56.03 87.57 84.22 82.18 81.05 80.59 22.05 20.28 19.20 18.84 18.78 135.97 131.15 127.94 126.15 125.37 224.55 221.12 219.22 217.90 217.20 28.98 25.55 23.38 22.28 21.96 455.85 450.80 447.64 445.45 444.33 3.63 5.34 5.98 6.19 5.01 7.10 7.54 7.58 6.35 8.91 9.96 10.34 3.82 6.16 7.44 7.98 8.06 12.96 14.58 14.86 3.55 5.91 7.22 7.80 1.53 2.37 2.96 3.27 11.87 19.34 23.12 24.22 1.11 1.80 2.28 2.53 1.73 0.82 0.30 0.10 0.79 0.33 0.07 0.01 3.97 1.60 0.65 0.24 3.35 2.04 1.13 0.46 1.78 1.08 0.36 0.06 4.83 3.21 1.79 0.78 3.43 1.90 1.32 0.70 3.44 2.17 1.10 0.32 5.06 3.15 2.20 1.11 3.4.3 E.ect of utilization To generalize some of the above results and to explore the e.ect of the utilization on the value of ACI, we .nally perform an extensive experiment by randomly generating non-stationary demand and capacity patterns with the following distribution parameters, the planning horizon T = 8, the set of expected demand values E[Dt] = {3,6,9,12} and the expected supply capacity values E[Qt]= {3,6,9,12}, for capacity uncertainty CVQ = 0.45 and two demand uncertainty scenarios CVD = {0,0.45}. For each of the two experiments we perform, 10,000 demand/capacity patterns are generated. We keep the cost structure constant with b = 20 and h = 1, and determine the relative value of ACI, %VACI (n = 3), (n=0) (n=3) and absolute value of ACI, ft -ft , based on a comparison between the n = 0 and n = 3 settings. The results are presented in Figures 3.7 and 3.8. Relative value of ACI Absolute value of ACI 20% 15 15% 10 10% 5 5% 0% 0 Utilization Utilization 90th perc. 10th perc. Mean 90th perc. 10th perc. Mean Figure 3.7 The relative and the absolute value of ACI for CVD = 0.45 and CVQ = 0.45 Relative value of ACI Absolute value of ACI 100% 30 25 80% 20 60% 15 40% 10 20% 5 0% 0 Utilization Utilization90th perc. 10th perc. Mean90th perc. 10th perc. Mean Figure 3.8 The relative and the absolute value of ACI for CVD = 0 and CVQ = 0.45 One of the biggest determinants of the extent of cost savings that can be achieved by using ACI is the average expected capacity utilization of the system through the planning horizon. Observe that both the relative value of ACI and the absolute cost savings vary considerably with a change in utilization. When both demand and capacity are uncertain, we see that 0,300,300,350,350,400,400,450,45 0,500,500,550,550,600,600,650,650,700,700,750,750,800,800,850,850,900,900,950,951,001,001,051,051,101,101,151,151,20 0,301,20 0,300,350,350,400,400,450,450,500,500,550,550,600,600,650,650,700,700,750,750,800,800,850,850,900,900,950,951,001,001,051,051,101,101,151,151,20 1,20 the peak relative cost saving of around 12% is achieved when utilization is around 45% (Figure 3.7). While for high utilization there is no surplus capacity available to pre-build the inventory when a capacity shortfall is expected, the value of ACI is also limited for the system with low utilization as there is usually enough capacity available in every period and early anticipation is not required. For absolute cost savings the peak occurs at a higher 60% utilization, which we attribute to the fact that costs rise considerably with greater utilization. In the case of no demand uncertainty, we see in Figure 3.8 that the relative cost savings are substantial, achieving on average well above 60% in the case of low utilization. However, since the costs in such a setting are low, the absolute savings are small. Similar to the above, the peak absolute savings happen at 70% utilization. This is an important observation as the focus in practice is often solely on relative savings, while the perspective of absolute cost savings is unjusti.ably neglected. Finally, we can conclude that the results show considerable cost savings can be achieved through the use of ACI. Further, the uncertain capacitated settings for which we have rec­ognized the greatest bene.ts are a good representation of the business environment in which many companies actually operate in practice. 3.5 “Zero-full” supply capacity availability In this section, we model the supply capacity as a Bernoulli processes, meaning that there are randomly interchanging periods of complete capacity unavailability and full availability -denoted as “zero-full” supply capacity availability. A supply setting described above can also be observed in practice. We give two examples in which a retailer would be facing periods of full or zero supply capacity availability and could take use of ACI if available. When a supplier/manufacturer is facing high setup costs in his production process, which are due to setting up the production process to meet particular customers/retailers needs, he would be reluctant to frequently change production program. A situation like this one can be observed in many capital-intensive industries, where the production resources are usually quite in.exible, resulting in high production switching costs (metal industry, food processing etc.). This results in a highly irregular supply to a particular retailer. As it would be too costly for a manufacturer to ful.ll each separate order from a retailer directly from a production line, while at the same time the option of storing high inventories of products is also not possible for longer periods (particularly in food processing industry), a relatively low service level to the retailer is acceptable. However, to optimize his production costs, manufacturer already considers high setup costs in his master production scheduling by determining the production plan for several production periods in advance. Thus, he can communicate the information about the future supply availability to the retailer and while he can not lower the irregularity of supply, he can lower its uncertainty in a number of near-future periods. Advance information about future suppliers capacity availability enables a retailer to prepare for periods of supply unavailability in advance, and by doing so signi.cantly decrease both the inventory holding costs (resulting from high inventory levels needed to cope with supply uncertainty) and the backorder costs (mainly a consequence of high supply variability). A similar practical setting is when the manufacturers transportation costs of supplying the products to the retailer are high and manufacturer resorts to less frequent supplies to the retailer. We introduce the parameter pt that describes the probability of full capacity availability in period t, 0 = pt = 1 for every t. To model ACI, we characterize each of the future periods for which ACI is already revealed with the state description at. State at attains the value of 0 for the zero availability case and 1 in the full availability case. In period t, the manager obtains ACI at+n on the supply capacity availability in period t + n, where n denotes the length of the ACI horizon. Thus, in period t the supply capacity availability for n future periods is known and is record in the ACI state vector zat = (at+1, at+2,. . . , at+n). Note that the capacity availabilities in periods t + n + 1 and later are still uncertain. We present the results of the numerical analysis which was carried out to quantify the value of ACI and to gain insights into how the value of ACI changes with a change in the relevant system parameters. In Figure 1 we present an example of an end-customer demand and supply pattern that a retailer is facing. A weekly demand roughly varies between 0 and 100, with an average of 45 and a coe.cient of variation of 0.6. The actual supply process is highly irregular with random periods of zero supply and a probability of full supply availability of p= 0.6. A gross packaging size from a manufacturer to the retailer contains 20 units. Units 140 120 100 80 Actual Demand 60 Actual Supply 40 20 0 Weeks Figure 3.9 Example of end-customer demand and retailers supply pattern. In the numerical experiment we selected the following set of input parameters: T = 20, n = {0,1,2,3,4,5}, p = {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1} is assumed to be constant over the whole planning horizon, the cost structure h = 1 and b = {5,20,100}, discount factor a = 0.99, uniformly distributed demand with the expected value of E[D] = 5 and coe.cient of variation CVD = {0,0.3,0.6}. The results are presented in Figures 3.10-3.11 and Tables 3.5-3.7. Let us .rst observe the e.ect of changing the system parameters on the total cost. Obvi­ously, decreasing the extent of supply capacity availability by decreasing the value of p will increase the costs. Due to the increased probability of multiple consecutive periods of zero capacity, the likelihood of backorders occurring will rise dramatically and the costs will grow particularly in the case of a high b/h cost ratio. In addition, costs rise when one has to deal with the e.ect of growing demand uncertainty. The demand uncertainty causes deviations in the actual inventory levels from the desired target levels set by the manager. This re­sults in frequent mismatches between the demand and the available on-hand inventory and, consequently, high costs. a) b=5, CVD=0 b) b=100, CVD=0 (%) (%) VACI VACI (%) (%) VACI VACI 100 90 n 80 70 70 1 60 60 250 50 40 40 3 30 30 4 20 20 10 10 5 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p . p . 1.1 c) b=5, CVD=0.6 d) b=100, CVD=0.6 90 80 100 90 80 n 70 70 1 60 60 2 50 50 40 40 3 30 30 4 20 20 10 10 5 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p . Figure 3.10 Relative value of ACI These costs can be e.ectively reduced when ACI is available. We start by looking at the relative value of ACI as de.ned in (4.10). Extending the ACI horizon obviously also increases the extent of the cost savings. However, the marginal gain of increasing n by 1 period varies substantially depending on the particular setting. When we consider the case of low supply capacity unavailability (p close to 1), we observe a surprisingly high relative decrease in costs measured through %VACI . This can be attributed to the fact that ACI enables us to anticipate and prepare for the rare periods of complete capacity unavailability. Thus, we can avoid backorders and at the same time lower the inventory levels that we would otherwise need to compensate for the event of multiple successive periods with zero capacity. Especially in the case of low demand uncertainty, and also a high b/h ratio, %VACI can reach levels above 80%, even close to 90% (Figure 3.10a and 3.10b, where we denote the settings with the biggest relative value of ACI by denoting the numbers in Italic). When the manager wants to gain the most from anticipating future supply capacity unavailability, it would be helpful if no additional uncertainties were present that would prevent him meeting the desired target inventory level. We also observe that these high relative cost savings are already gained with a short ACI horizon. Extending n above 1 only leads to small further cost reductions. This is an important insight regarding the practical use of ACI, when the majority of gains are already possible with limited future visibility it is more likely that the manager will be able to obtain ACI (possibly also more accurate information) from his supplier. While the short ACI horizon is su.cient in the case of p being close to 1, we see that a longer ACI horizon is needed in a setting with high supply capacity unavailability. Observe that for low p values the relative marginal savings are actually increasing. When multiple periods of zero capacity can occur one after the other it is particularly important to anticipate the extent of future capacity unavailability. In such a setting, it is very important if one can have an additional period of future visibility. Several researchers who have studied a conceptually similar problem of sharing advance demand information suggest that prolonging ¨ the information horizon has diminishing returns ( Ozer and Wei, 2004). Although we consider a special case of zero or full supply availability in this section, this result actually shows that this does not hold in general. While we have observed a large relative decrease in costs in some settings, it may be more important for a particular company to determine the potential decrease in absolute cost .gures. Therefore we continue by studying the absolute marginal change in the value of ACI as de.ned in (4.5). Intuitively, we would expect that the biggest absolute gains would occur in a setting where the uncertainty of supply is high, and the possible shortage anticipation through ACI would be the most bene.cial. We con.rm this in Tables 3.5-3.7 (where we denote the settings with the biggest total absolute value of ACI by denoting the numbers in Italic; for each setting we also show at which n the majority of the savings are gained by denoting the marginal value of ACI in bold), where we see that the biggest absolute savings are attained for pbetween 0.2 and 0.4. Here, the cost decrease is bigger for a higher n (Figure 3.11). In fact, the lower the availability the more we gain by prolonging the ACI horizon. In the case of extremely low levels of capacity availability the system becomes too hard to manage due to an extremely long inventory pre-build phase, and the gains from using ACI Table 3.5 Value of ACI for b = 5 n 1 0.9 0.8 0.7 0.6 p 0.5 0.4 0.3 0.2 0.1 CVD = 0 Cost 0 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 0.00 50.27 15.35 12.80 12.63 12.62 12.62 96.94 41.34 30.64 29.13 28.94 28.93 125.47 81.37 57.70 52.10 50.95 50.75 179.06 124.98 99.81 86.90 82.67 81.53 237.05 195.20 158.87 141.55 132.92 128.60 330.42 294.66 257.85 231.73 216.10 208.02 483.98 459.92 423.57 393.87 371.69 357.55 788.55 774.45 750.38 724.36 700.84 682.65 1605.02 1600.15 1591.42 1580.07 1568.04 1556.62 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 69.47 74.54 74.87 74.89 74.89 57.35 68.39 69.95 70.14 70.16 35.15 54.01 58.47 59.39 59.55 30.20 44.26 51.47 53.83 54.47 17.65 32.98 40.29 43.93 45.75 10.82 21.96 29.87 34.60 37.04 4.97 12.48 18.62 23.20 26.12 1.79 4.84 8.14 11.12 13.43 0.30 0.85 1.55 2.30 3.02 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 34.93 2.55 0.17 0.01 0.00 55.59 10.71 1.51 0.19 0.02 44.10 23.67 5.60 1.15 0.20 54.08 25.17 12.91 4.23 1.14 41.85 36.34 17.31 8.64 4.31 35.76 36.81 26.12 15.63 8.08 24.07 36.35 29.70 22.18 14.14 14.10 24.07 26.01 23.52 18.19 4.87 8.73 11.35 12.03 11.43 CVD = 0.3 Cost 0 1 2 3 4 5 36.42 36.42 36.42 36.42 36.42 36.42 66.32 51.05 49.49 49.34 49.32 49.32 106.95 72.54 66.80 65.69 65.47 65.43 143.63 103.78 91.44 87.86 86.80 86.50 188.38 148.55 128.42 120.28 117.10 115.89 250.56 212.64 185.71 171.24 164.06 160.60 341.67 310.01 278.23 257.05 243.97 236.26 493.68 470.80 439.62 413.51 394.21 380.68 797.66 784.39 762.32 738.37 717.10 699.76 1612.92 1608.29 1600.09 1589.57 1578.25 1567.28 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 23.02 25.38 25.61 25.64 25.64 32.17 37.54 38.58 38.78 38.82 27.74 36.34 38.83 39.56 39.77 21.14 31.83 36.15 37.84 38.48 15.13 25.88 31.66 34.52 35.90 9.27 18.57 24.77 28.60 30.85 4.64 10.95 16.24 20.15 22.89 1.66 4.43 7.43 10.10 12.27 0.29 0.80 1.45 2.15 2.83 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 15.27 1.56 0.16 0.01 0.00 34.41 5.74 1.11 0.22 0.04 39.85 12.34 3.58 1.05 0.30 39.83 20.13 8.14 3.18 1.21 37.92 26.93 14.47 7.18 3.45 31.66 31.78 21.18 13.08 7.71 22.89 31.18 26.11 19.30 13.53 13.27 22.07 23.95 21.27 17.34 4.64 8.20 10.52 11.32 10.97 CVD = 0.6 Cost 0 1 2 3 4 5 82.77 82.77 82.77 82.77 82.77 82.77 103.57 97.48 96.36 96.27 96.25 96.25 131.77 118.05 113.87 113.09 112.93 112.90 170.95 146.71 138.02 135.46 134.67 134.45 218.26 187.68 173.37 167.47 165.06 164.14 278.82 248.42 227.97 216.94 211.39 208.71 369.59 342.67 316.89 299.75 289.30 283.18 521.15 500.58 474.04 451.86 435.63 424.42 824.76 812.40 792.54 771.42 752.82 737.78 1638.21 1633.80 1626.11 1616.46 1606.21 1596.43 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 5.89 6.96 7.05 7.07 7.07 10.41 13.58 14.17 14.29 14.32 14.18 19.27 20.76 21.22 21.35 14.01 20.57 23.27 24.37 24.80 10.90 18.24 22.19 24.18 25.14 7.28 14.26 18.90 21.73 23.38 3.95 9.04 13.30 16.41 18.56 1.50 3.91 6.47 8.72 10.55 0.27 0.74 1.33 1.95 2.55 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 6.10 1.11 0.10 0.01 0.00 13.72 4.17 0.78 0.16 0.03 24.24 8.70 2.56 0.79 0.22 30.59 14.30 5.90 2.41 0.92 30.40 20.45 11.04 5.55 2.67 26.92 25.78 17.14 10.45 6.11 20.57 26.54 22.18 16.23 11.21 12.35 19.86 21.12 18.60 15.05 4.41 7.69 9.66 10.25 9.78 51 Table 3.6 Value of ACI for b = 20 n 1 0.9 0.8 0.7 0.6 p 0.5 0.4 0.3 0.2 0.1 CVD = 0 Cost 0 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 0.00 108.32 36.62 23.29 21.99 21.87 21.86 179.28 86.10 60.49 54.15 52.65 52.32 252.26 169.13 119.59 103.61 98.54 96.96 358.94 277.04 215.45 186.30 172.62 167.01 508.71 442.65 371.59 324.44 299.17 286.37 751.09 704.27 630.89 572.82 532.90 508.52 1202.76 1171.85 1115.29 1055.23 1005.40 968.87 2258.01 2241.12 2208.71 2166.73 2123.84 2085.40 5569.65 5563.93 5553.19 5538.60 5521.71 5504.13 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 66.20 78.50 79.70 79.81 79.82 51.97 66.26 69.80 70.63 70.82 32.96 52.59 58.93 60.94 61.56 22.82 39.98 48.10 51.91 53.47 12.98 26.95 36.22 41.19 43.71 6.23 16.00 23.73 29.05 32.29 2.57 7.27 12.27 16.41 19.45 0.75 2.18 4.04 5.94 7.64 0.10 0.30 0.56 0.86 1.18 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 71.70 13.33 1.30 0.12 0.01 93.18 25.61 6.35 1.50 0.33 83.13 49.54 15.98 5.06 1.58 81.90 61.59 29.15 13.68 5.61 66.06 71.07 47.15 25.27 12.80 46.82 73.38 58.07 39.92 24.37 30.91 56.56 60.06 49.83 36.53 16.90 32.40 41.99 42.89 38.43 5.72 10.73 14.59 16.90 17.57 CVD = 0.3 Cost %VAC I 0 1 2 3 4 5 1 2 3 4 5 36.42 36.42 36.42 36.42 36.42 36.42 0.00 0.00 0.00 0.00 0.00 135.85 68.81 63.45 62.96 62.91 62.91 49.35 53.29 53.66 53.69 53.69 197.09 120.85 102.20 98.60 97.90 97.77 38.68 48.15 49.97 50.33 50.39 275.72 195.27 161.33 150.50 147.27 146.31 29.18 41.49 45.41 46.59 46.93 378.34 304.22 254.05 232.04 222.98 219.35 19.59 32.85 38.67 41.06 42.02 527.83 466.15 403.82 367.95 349.05 339.39 11.69 23.49 30.29 33.87 35.70 778.85 733.12 669.31 621.02 590.23 571.58 5.87 14.06 20.26 24.22 26.61 1222.38 1192.13 1139.93 1086.73 1044.31 1013.44 2.47 6.75 11.10 14.57 17.09 2279.71 2263.06 2232.48 2193.87 2155.37 2121.60 0.73 2.07 3.77 5.45 6.94 5593.61 5587.95 5577.69 5564.05 5548.48 5532.50 0.10 0.28 0.53 0.81 1.09 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 67.04 5.36 0.50 0.04 0.00 76.24 18.65 3.60 0.70 0.13 80.45 33.93 10.83 3.23 0.96 74.12 50.17 22.01 9.06 3.63 61.68 62.33 35.87 18.90 9.65 45.73 63.81 48.29 30.79 18.66 30.25 52.20 53.20 42.42 30.87 16.65 30.58 38.61 38.50 33.77 5.66 10.25 13.64 15.57 15.98 CVD = 0.6 Cost 0 1 2 3 4 5 91.05 91.05 91.05 91.05 91.05 91.05 172.61 125.66 122.83 122.53 122.50 122.50 244.32 176.03 164.82 162.49 162.04 161.96 319.62 249.67 225.10 217.56 215.35 214.72 423.43 356.90 317.03 300.44 293.76 291.19 574.03 516.59 464.83 435.73 420.89 413.61 817.42 773.42 717.15 675.66 649.29 633.51 1272.77 1243.09 1195.14 1148.12 1111.34 1085.03 2335.62 2319.01 2289.68 2253.91 2219.02 2189.11 5652.68 5646.91 5636.73 5623.40 5608.53 5593.55 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 27.20 28.84 29.02 29.03 29.04 27.95 32.54 33.49 33.68 33.71 21.88 29.57 31.93 32.62 32.82 15.71 25.13 29.04 30.62 31.23 10.01 19.02 24.09 26.68 27.95 5.38 12.27 17.34 20.57 22.50 2.33 6.10 9.79 12.68 14.75 0.71 1.97 3.50 4.99 6.27 0.10 0.28 0.52 0.78 1.05 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 46.95 2.83 0.31 0.03 0.00 68.29 11.21 2.33 0.45 0.09 69.95 24.57 7.53 2.21 0.63 66.53 39.87 16.58 6.69 2.56 57.44 51.77 29.10 14.84 7.27 44.01 56.27 41.49 26.37 15.78 29.68 47.95 47.02 36.78 26.31 16.61 29.33 35.77 34.88 29.91 5.77 10.19 13.33 14.87 14.99 52 Table 3.7 Value of ACI for b = 100 n 1 0.9 0.8 0.7 0.6 p 0.5 0.4 0.3 0.2 0.1 CVD = 0 Cost 0 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 0.00 228.71 92.58 73.78 71.44 71.17 71.14 378.08 244.86 193.67 180.58 177.43 176.79 593.46 474.87 386.03 354.33 343.94 340.62 917.66 814.32 701.78 645.25 620.31 609.62 1428.07 1352.24 1239.56 1155.18 1108.20 1083.50 2357.22 2305.11 2207.25 2109.67 2040.57 1995.91 4301.25 4268.18 4201.86 4118.46 4041.37 3980.49 9380.07 9362.27 9327.24 9278.93 9225.11 9173.66 26309.55 26303.63 26292.43 26277.00 26258.77 26239.36 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 59.52 67.74 68.76 68.88 68.90 35.24 48.77 52.24 53.07 53.24 19.98 34.95 40.29 42.04 42.61 11.26 23.52 29.69 32.40 33.57 5.31 13.20 19.11 22.40 24.13 2.21 6.36 10.50 13.43 15.33 0.77 2.31 4.25 6.04 7.46 0.19 0.56 1.08 1.65 2.20 0.02 0.07 0.12 0.19 0.27 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 136.14 18.80 2.33 0.27 0.03 133.22 51.19 13.09 3.15 0.64 118.59 88.84 31.70 10.39 3.33 103.34 112.54 56.53 24.93 10.70 75.83 112.68 84.38 46.98 24.71 52.11 97.86 97.58 69.10 44.66 33.07 66.33 83.40 77.08 60.88 17.79 35.04 48.31 53.81 51.46 5.93 11.20 15.43 18.23 19.40 CVD = 0.3 Cost 0 1 2 3 4 5 36.42 36.42 36.42 36.42 36.42 36.42 242.76 131.39 116.14 114.73 114.59 114.58 405.88 279.88 238.30 229.71 227.95 227.61 623.04 503.45 431.75 408.36 400.95 398.74 943.58 843.04 746.22 700.99 681.75 673.90 1461.26 1385.07 1278.49 1208.93 1171.12 1151.68 2392.68 2339.59 2245.67 2159.98 2100.90 2063.92 4343.79 4309.98 4244.88 4167.18 4098.11 4045.61 9435.50 9417.12 9382.14 9335.43 9285.20 9238.70 26375.73 26369.44 26357.98 26342.61 26324.81 26306.24 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 45.88 52.16 52.74 52.80 52.80 31.04 41.29 43.40 43.84 43.92 19.20 30.70 34.46 35.65 36.00 10.66 20.92 25.71 27.75 28.58 5.21 12.51 17.27 19.86 21.19 2.22 6.14 9.73 12.20 13.74 0.78 2.28 4.07 5.66 6.86 0.19 0.57 1.06 1.59 2.09 0.02 0.07 0.13 0.19 0.26 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 111.37 15.25 1.41 0.13 0.01 126.00 41.58 8.58 1.77 0.34 119.60 71.70 23.39 7.40 2.22 100.54 96.82 45.24 19.24 7.85 76.19 106.58 69.55 37.81 19.44 53.09 93.92 85.70 59.08 36.98 33.82 65.10 77.70 69.07 52.51 18.38 34.98 46.71 50.23 46.49 6.28 11.47 15.36 17.80 18.57 CVD = 0.6 Cost 0 1 2 3 4 5 91.05 91.05 91.05 91.05 91.05 91.05 305.27 190.60 180.61 179.79 179.71 179.70 466.78 344.85 310.01 303.93 302.77 302.56 687.14 569.93 508.33 489.69 484.30 482.77 1010.66 911.29 825.00 787.16 771.89 765.97 1532.47 1455.87 1357.91 1297.26 1265.60 1249.83 2470.47 2416.16 2326.60 2249.08 2197.44 2165.92 4432.75 4397.50 4332.98 4259.50 4196.44 4149.50 9543.15 9523.54 9487.67 9441.23 9392.55 9348.54 26482.59 26475.69 26463.49 26447.36 26428.97 26410.08 %VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 37.56 40.84 41.10 41.13 41.13 26.12 33.59 34.89 35.14 35.18 17.06 26.02 28.73 29.52 29.74 9.83 18.37 22.11 23.62 24.21 5.00 11.39 15.35 17.41 18.44 2.20 5.82 8.96 11.05 12.33 0.80 2.25 3.91 5.33 6.39 0.21 0.58 1.07 1.58 2.04 0.03 0.07 0.13 0.20 0.27 .VAC I 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 114.66 10.00 0.81 0.08 0.01 121.94 34.84 6.08 1.15 0.21 117.21 61.60 18.64 5.39 1.53 99.37 86.29 37.84 15.26 5.93 76.60 97.96 60.65 31.65 15.77 54.31 89.55 77.52 51.64 31.52 35.26 64.52 73.48 63.06 46.93 19.61 35.87 46.44 48.69 44.00 6.90 12.19 16.13 18.39 18.88 53 a) b=5, CVD=0 b) b=100, CVD=0 150 140 130 .V .V .V .V n 80 120 110 70 1 60 100 90 80 2 50 70 40 60 3 50 30 40 4 20 30 20 510 10 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p . p . c) b=5, CVD=0.6 d) b=100, CVD=0.6 100 150 140 130 80 120 n 70 100 110 1 60 90 50 70 80 2 40 60 3 30 40 50 4 20 30 10 10 20 5 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 p . Figure 3.11 Absolute change in the value of ACI are limited. The e.ect of demand uncertainty on an absolute decrease in costs is not as obvious as it was in the relative case. This can be attributed to the interaction of two factors. While stronger demand uncertainty intensi.es the di.culties of managing inventories as described above, it also contributes to higher costs and thus provides more potential for savings through ACI. To summarize, while the relative cost savings are highest for the case of close to full avail­ability due to the fact that one can completely avoid backorders with only a small extra inventory, the cost reduction in absolute terms is greater for cases with medium to low sup­ply capacity availability. Further, we show that in most cases having only a little future visibility already o.ers considerable savings, although when one faces the possibility of con­secutive periods of supply unavailability it can be very bene.cial to extend the ACI horizon. In general, managers should recognize that the extent of savings shown clearly indicates that sharing ACI should be encouraged in supply chains with unstable supply conditions. In our experience, the current dynamic programming cost formulation is manageable in terms of the complexity of the calculations and can also be used for larger practical problems. 3.6 Relation between ACI and ADI The use of advance demand information (ADI) has been widely studied in the literature. We refer the reader to the review in the introductory section. In this section, we construct a capacitated ADI model with structural similarities to our ACI model in order to explore the key di.erences and possible similarities between ACI and ADI. In particular, we show that the models cannot be equivalent because, for equivalence to hold, the available capacity should always be able to cover at least that part of demand that is revealed by ADI and hence should never be binding. We therefore conclude that the model we propose in this chapter clearly poses a new problem and can serve as the basis for a new line of research. We .rst construct a capacitated ADI model. In this model, we assume that demand is generated by two independent customer groups: customers that place their orders N periods ahead, and those that place their orders for the current period in a traditional way without ADI. At the end of period t, we record the demand from the ”traditional” customers that had not been observed prior to its materialization, which we refer to as the unobserved part of demand dt and we also record the observed part of demand ot+N for the future period t + N. Parameter N represents the length of the information horizon over which ADI is available, assuming perfect ADI. Also, we assume that there is a .xed supply capacity Qc limiting the size of an order in each period. The idea behind construction of the capacitated ADI model is as follows: The capacity Qc available in each period is used both to cover the observed part of demand and the unobserved part of demand. If the observed part of demand is modeled as realization of a random variable and it is met .rst due to the .rst-come-.rst-served principle, the remaining capacity to cover the unobserved part is also a random variable. This suggests that we have a random capacity available to satisfy the unobserved part of demand dt, which is conceptually similar to our stochastic demand ACI inventory model. To derive a formal ADI model description we start by writing xt in a modi.ed form ˜xt, as x˜t = xˆt -ot. (3.8) Observe that ˜xt already accounts for the observed part of the current period’s demand. By ordering, we raise ˜xt to the corresponding modi.ed inventory position after ordering y˜t, given as ˜yt = x˜t + ˜zt, where ˜zt = Qc . Due to the update of the inventory position with observed demand, ˜yt only needs to cover the remaining, unobserved part of demand. From (3.8), we can show that at the start of period t + 1, ˜xt is updated in the following manner: x˜t+1 = x˜t + ˜zt -dt -ot+1. We also need to track the demand observations, given through ADI, that will a.ect future orders. These constitute the ADI vector zot = (ot+1,. . . , ot+N-1) that gets updated at the start of period t + 1 zot+1 = (ot+2,. . . , ot+N), by including the new information ot+N collected through period t and purging the oldest data point ot+1, with which we update ˜xt+1. The state space is described by an N -dimensional vector (˜xt, zot). The minimal discounted expected cost function of the proposed ADI model, optimizing the cost over a .nite planning horizon T from time t onward and starting in the initial state (˜xt, zot), can be written as: ft(˜xt, zot)= min {Ct(˜yt)+ x˜t=y˜t=x˜t+Qc (3.9) aEDt ft+1(˜yt -Dt, zot+1), if T -N -1 = t = T , + aEDt,Ot+N ft+1(˜yt -Dt, zot+1), if 1 = t = T -N, where fT+1(·) = 0. Note that the ADI model given by (3.9) is a special case of the model ¨ introduced by Ozer and Wei (2004). Consequently, it can be shown that the optimal ordering policy solving (3.9) is a state-dependent modi.ed base-stock policy, where the optimal base-stock level is a function of the ADI vector zot. Now that we have constructed the ADI model (3.9), we can write the formulations which establish the relationships between the ACI and ADI model dynamics. For the models to be equivalent in their base-stock levels we need to assume that ˜yt = yt holds. This is intuitively clear since both need to cover the same demand. Note that we assume the unobserved part of demand in the ADI model is modeled in the same way as the ACI model’s demand given Section 3.2. By substituting the relevant ACI dynamics formulations into the corresponding ADI formulations, we show that the ACI model’s xt and zt are related to their ADI model counterparts ˜xt and ˜zt in the following way: xt = x˜t + ot, zt = z˜t -ot. By updating x˜t at the start of period t, we already use some of total capacity Qc to cover the observed part of demand ot. Now we only have remaining capacity Qc-ot to su.ciently raise yt to cover the unobserved part of demand. This directly relates to having .nite capacity qt, which limits the extent to which we can raise yt in the ACI model. We can write the following relationship: qt = Qc -ot. (3.10) By comparing the above ADI model with our ACI model, we .rst derive the relationship between the two information horizon parameters. Observe that at the start of period t the ADI vector zot gives observed demand for N -1 future periods, which will a.ect the size of the following N -1 orders. This corresponds to having ACI for future n periods given by the ACI vector zqt. The ACI and ADI horizon parameters n and N are therefore related in the following way: n = N -1. We make our most important observation by substituting the ADI dynamics formulations into (3.4). Comparing the constructed new ADI dynamic programming formulation with the one proposed by (3.9) shows an inconsistency at the lower bound over which the mini­mization is made. Instead of looking for the optimal ˜yt by minimizing the cost function over x˜t = y˜t = x˜t + Qc, minimization is made over ˜xt + ot = y˜t = x˜t + Qc . For direct equivalence of the ACI model and the ADI model to hold, we always have to be able to raise ˜yt above x˜t + ot for all t. This means that the available capacity Qc should be su.cient to cover at least the observed part of demand. This ”negativity issue” also stems directly from (3.10) where, for qt = 0 to hold, Qc -ot = 0 has to hold. The .xed capacity therefore has to exceed all possible realizations of the observed part of demand, Qc = ot. This assumption only holds when Qc = max ot, which happens either when an uncapacitated system, Qc . 8, is considered or in a more realistic scenario where the observed part of demand only represents a small portion of the total demand. However, this implies that the remaining capacity that covers the unobserved part of demand cannot vary considerably. Thus, the equivalence of the two models would only hold if we consider our ACI model with a near-stationary capacity distribution with low uncertainty. As we have learned from the numerical analysis this in fact describes the ”less interesting” setting where the value of ACI is small. Based on these .ndings we conclude that restrictive assumptions are needed to guarantee direct equivalence between the capacitated ADI model and the proposed ACI model in terms of optimal base-stock levels and optimal costs. Hence, we conclude that the model we propose in this chapter is essentially di.erent from capacitated ADI models. 3.7 Summary In this chapter, we develop a model that incorporates advance capacity information (ACI) into inventory decision making and explore its e.ect on making e.ective ordering decisions within a periodic review inventory planning system facing limited stochastic supply. Based on the convexity of the relevant cost functions, we are able to show that the form of the optimal policy is a modi.ed base-stock policy with a single state-dependent base-stock level. Essentially the base-stock level depends on realizations of future supply capacities revealed by ACI, and is a decreasing function of the ACI extent. We complement this result by showing additional monotonicity properties of the optimal policy. Another contribution of this work lies in establishing a link with advance demand information modeling by derivation of a capacitated ADI model. We show that the ACI model is only equivalent to the ADI model under a very speci.c and restrictive assumption. This implies that, although there are structural similarities, research on ACI will require new analyses of extensions of the model proposed in this chapter since ADI insights will not necessarily hold for the ACI model. The ACI is structurally di.erent from ADI due to the capacity restriction. By means of a numerical analysis we develop some additional managerial insights. In par­ticular, we establish the following conditions in which inventory costs can be decreased by using ACI: (1) when there is a mismatch between demand and supply capacity, which can be anticipated through ACI, and there is an opportunity to pre-build inventory in an ade­quate manner; (2) when uncertainty in future supply capacity is high and ACI is used to reduce it e.ectively; and (3) in the case of high backorder costs, which further emphasizes the importance of avoiding stock outs. In these, managers should seek to establish long-term contractual agreements which encourage the sharing of ACI. Such relationships may bring considerable operational cost savings. There are multiple ways to extend the work presented in this chapter. While the proposed model assumes perfect ACI, the ACI model can be extended to describe the situation where the communicated supply limit might not be completely accurate. This information can be denoted as imperfect ACI. The consequence of ACI not being exact is that there is still some uncertainty in the actual supply capacity availability. This leads to a situation where the inventory position does not re.ect the actual realization of orders in the pipeline and anticipating future supply conditions is made harder due to the remaining share of the uncertainty. The present ACI model assumes that ACI reveals the supply capacity availability for current and n future orders, meaning that when placing an order exact supply capacity realization is known. An interesting idea for future research would be that supply information is received only after an order has been placed, in case there is a positive supply lead time. In this instance, the order has to be placed not knowing the available supply capacity; however, we observe advance supply information for the order that is already given and is currently still in the pipeline. Advance supply information indicates whether the order will be .lled fully or just partially before the actual delivery, and thus enables the decision maker to react if necessary. Chapter 4 Inventory management with advance supply information It has been shown in numerous situations that sharing information between the companies leads to improved performance of the supply chain. In this chapter, we study a positive lead time periodic-review inventory system of a retailer facing stochastic demand from his customer and stochastic limited supply capacity of the manufacturer supplying the products to him. The manufacturer is willing to share the advance supply information (ASI) about the actual future replenishments of the pipeline orders placed by the retailer. ASI is provided at a certain time after the order has been placed and the retailer can now use this information to decrease the uncertainty of the supply, and thus improve its inventory policy. For this model we develop a dynamic programming formulation, and characterize the optimal ordering policy as a state-dependent base-stock policy. In addition, we show some properties of the base-stock level. While the optimal policy is highly complex, we obtain some additional insights by comparing it to the state-dependent myopic inventory policy. Our numerical results show the bene.ts of using ASI and based on them we provide useful managerial insights. 4.1 Introduction Nowadays companies are facing di.culties in e.ectively managing their inventories mainly due to the highly volatile and uncertain business environment. While they are trying their best to ful.ll the demand of their customers by using more or less sophisticated inventory control policies, their e.orts can be severely hindered by the unreliable and limited deliv­eries from their suppliers. Due to the widespread trend of establishing plants overseas or outsourcing to specialists it has become increasingly more di.cult for the companies to re­tain control over their procurement process. As the complexity of supply networks grows, so do the challenges and ine.ciencies the companies are facing; orders get lost or are not delivered in full, shipment are late or don’t arrive at all. It has been well acknowledged both in the research community as well as by practitioners that these uncertainties can be reduced and better supply chain coordination can be achieved through the improved provision of information (Lee and Padmanabhan, 1997; Chen, 2003). While the performance of a supply chain depends critically on how its members coordinate their decisions, sharing information can be considered as the most basic form of coordination in supply chains. In light of this, the concept of achieving so-called Supply Chain Visibility is gaining on importance, as it provides accurate and timely information throughout the supply chain processes and networks. This enables companies to share the information through often already established B2B communication channels and ERP solutions. EDI formatted electronic noti.cations on the status of the order ful.llment process, such as order acknowledgements, inventory status, Advance Shipment Notices (ASN), and Shipment Status Messages (SSM) are shared, enabling companies to track and verify the status of their order and consequently foresee supply shortages before they happen (Choi, 2010). There are also multiple examples of companies like UPS, FedEx and others in shipping industry, and Internet retailers like eBay and Amazon, that are o.ering real time order ful.llment information also on the B2C level. Real visibility in the supply chain can be regarded as a prerequisite for the companies to reach new levels of operating e.ciency, service capabilities, and pro.tability for suppliers, logistics partners, as well as their customers. However, while the technological barriers to information sharing are being dismantled, the abundance of the available information by itself is not a guarantee for improved performance. Therefore the focus now is on developing new tools and technologies that will use this information to improve on current state of the inventory management practices. In this chapter, we investigate the bene.ts of advance information sharing (ASI). We consider a retailer facing stochastic demand from the end customer and procuring the products from a single manufacturer with stochastic limited supply capacity. We assume that the order is replenished after a given .xed lead time, which constitutes of order processing, production and shipping delay. However it can happen that the quantity received by the retailer is less than what he ordered originally. This supply uncertainty can be due to, for instance, the allocation policy of the manufacturer, which results in variable capacity allocations to her customers or to an overall capacity shortage at certain times. This stochastic nature of capacity itself may be due to multiple causes, such as variations in the workforce (e.g. holiday leaves), unavailability of machinery or multiple products sharing the total capacity. We assume that the manufacturer tracks the retailer’s order evolution and at certain point, when she can assess the extent to which the order will be ful.lled, she shares ASI with the retailer, giving him feedback on the actual replenishment quantity ahead of the time of the physical delivery of products. ASI enables the retailer to respond to the possible shortage by adjusting his future order decisions, and by doing this possibly o.set the negative impact of the shortage. Based on this rationale we pose the following two research questions: (1) How can we integrate ASI into inventory decision model, or, more speci.cally can we characterize the optimal policy that would account for the availability of ASI, and describe its properties? (2) Can we quantify the value of ASI and establish the system settings where utilizing ASI is of the most importance? The practical setting in which the above modeling assumptions could be observed is food processing industry, where the food processing facilities/manufacturers are being supplied with the agricultural products. The products are harvested periodically and the product availability is changing through time depending on a variety of factors: weather, harvesting capacity etc. Also it is reasonable to assume that supply capacity cannot be backordered as in a lot of cases harvested products cannot be stored for longer periods. The supply process taking a number of time periods is done in two phases. After an order is made by the retailer, the harvesting part of the production process commences, where the production outcome is uncertain. Then the products are delivered to the food processing facility. At this point the product availability is revealed and is no longer uncertain, and ASI is communicated to the customer in the form of ASN. The actual replenishment follows after the product is fully processed. It is also worthwhile to shed some light on the di.erences between the ASI model proposed in this chapter and the ACI model from Chapter 3. We .rst describe the practical setting and the relevant di.erences in the way these two proposed types of upstream information are shared. Later within this chapter we give more insights into the modeling and structural di.erences of the two inventory models. t-L …t-m … t … t+n Figure 4.1 The time perspective of sharing ASI and ACI. The main di.erence in the way information is shared in the case of ASI and ACI model is in the time delay between the placement of the order and the time the information on the available supply capacity is revealed. In the ASI case, the supply capacity information is revealed after the order has been placed and the lack of supply capacity availability results in the replenishment below the initial order. In the case of ACI, the information is available for future supply capacity availability, thus the order placed is aligned with this availability. ASI thus only allows the decision maker to respond to the actual realized shortages in a more timely manner. While based on ACI, the decision maker can anticipate the potential future shortages and accordingly adopt his ordering strategy. Below we give some additional di.erences between the two inventory models in terms of the level of reliability and availability of information, and the extent of the potential savings: • Reliability: ASI can be considered as more reliable/perfect information as it is com­municated after the order has been placed with the manufacturer. Therefore it is reasonable to assume that the manufacturer has already put the order on the list of orders for execution within the Manufacturing Execution System, which also means the capacity availability has been thoroughly checked. In the case of ACI, the order has not been placed yet. The supply capacity availability will depend on the manu­facturer’s plans within the Master Production Scheduling. It is reasonable to assume that the MPS will be less reliable than MES. • Availability: Due to the above argument, it is also reasonable to assume that the manufacturer is more willing to share ASI than ACI. In addition to this, manufacturer may not be willing to share the future capacity availability due to strategic reasons. While sharing ASI particularly in the case when the manufacturer cannot deliver the full order can be considered as a required business practice. • Savings potential: Through sharing ACI the supply capacity availability can be anticipated and consequently the supply shortages can be avoided. Thus it is reasonable to assume that the potential savings will be higher than in the case of sharing ASI. We proceed with a short literature review. Our work builds on the broad research stream of papers assuming uncertainty in the supply processes. In the literature the supply uncertainty is commonly attributed to one of the two sources: yield randomness and randomness of the available capacity. Our focus lies within the second group of problems, where Federgruen and Zipkin (1986a,b) were the .rst to address the capacitated stationary inventory problem with a .xed capacity constraint and have proven the optimality of the modi.ed base-stock policy. Kapuscinski and Tayur (1998) extend this result by studying the non-stationary version of the model, where they assume periodic demand. Later, a line of research extends the focus to capture the uncertainty in capacity, by analyzing models with limited stochastic production capacity (Ciarallo et al., 1994; G¨ull¨u et al., 1997; Khang and Fujiwara, 2000; Iida, 2002). Ciarallo et al. (1994) explore di.erent cases of the stochastic capacity constraint in a single and multiple periods setting. In the analysis of a single period problem, they show that stochastic capacity does not a.ect the order policy. The myopic policy of newsvendor type is optimal, meaning that the decision maker is not better o. by asking for a quantity higher than that of the uncapacitated case. For a .nite horizon stationary inventory model they show that the optimal policy remains to be a base-stock policy, where the optimal base-stock level is increased to account for the possible, however uncertain, capacity shortfalls in the future periods. Iida (2002) extends this result for the non-stationary environment. Although a lot of attention in recent decades has been put in assessing the bene.ts of sharing information in the supply chains, the majority of the research is focused on studying the e.ect of sharing downstream information, in particular demand information (Gallego and ¨ ¨ Ozer, 2001; Karaesmen et al., 2003; Wijngaard, 2004; Tan et al., 2007; Ozer and Wei, 2004). Review papers by Chen (2003), Lau (2007) and Choi (2010) show that sharing upstream information has been considered in the literature in the form of sharing lead time information, production cost information, production yield, and sharing capacity information. Capacity information sharing is of particular interest to us, where several papers have been discussing sharing information on future capacity availability (Jak.si.c et al., 2011; Altu.g and Muharremo.glu, 2011; C¸ inar and G¨ull¨u, 2012; Atasoy et al., 2012). Jak.si.c et al. (2011) study the bene.ts of sharing perfect information on future supply capacity available for orders to be placed in future periods. They show that the optimal ordering policy is a state-dependent base-stock policy characterized by the base-stock level that is a function of advance capacity information. Altu.gand Muharremo.glu (2011) work on a similar model; however they assume that the evolution of the capacity availability forecasts is done via the Martingale Method of Forecast Evolution (MMFE). Zhang et al. (2006) discuss the bene.ts of sharing advance shipment information. A setting in which a company receives the exact shipment quantity information is closely related to the one proposed in this chapter, however they assume that inventory is controlled through a simple non-optimal base-stock policy and as such it fails to capture the uncertainty of supply. Our model can be considered as a generalization of the model by Zhang et al. (2006), as we allow for both, demand and supply capacity, to be stochastic, and more importantly we model the optimal system behavior by considering considering the optimal inventory policy that is able to account for the supply uncertainty by setting appropriate safety stock levels. We propose that by having timely feedback on actual replenishment quantities through ASI, we can re.ne the inventory policy and improve its performance. To our knowledge the exploration of the relationship between the proposed way of modeling ASI and the optimal policy parameters has not yet received any attention in the literature. The remainder of the chapter is organized as follows. We present a model incorporating ASI and its dynamic cost formulation in Section 4.2. The optimal policy and its properties are discussed in Section 4.3. We proceed by the study of the approximate inventory policy based on the state-dependent myopic policy in Section 4.4. In Section 4.5 we present the results of a numerical study and point out additional managerial insights. Finally, we summarize our .ndings and suggest directions for future research in Section 4.6. 4.2 Model formulation In this section, we introduce the notation and the model of advance supply information for orders that were already placed, but are currently still in the pipeline. The model un­der consideration assumes periodic-review, stationary stochastic demand, limited stationary stochastic supply with .xed supply lead time, .nite planning horizon inventory control sys­tem. Unmet demand is fully backlogged. However, the retailer is able to obtain ASI on supply shortages a.ecting the future replenishment of the orders in the pipeline from the manufacturer. We introduce the ASI parameter m that represents the time delay in which ASI is communicated with the retailer. The parameter m e.ectively denotes the number of periods between the time the order has been placed with the manufacturer and the time ASI is revealed. More speci.cally, ASI on the order zt-m placed m periods ago is revealed in period t after the order zt is placed in the current period (Figure 4.2). Depending on the available supply capacity qt-m, ASI reveals the actual replenishment quantity, determined as the minimum of the two, min(zt-m, qt-m). We assume perfect ASI. Observe that the longer the m, the larger is the share of the pipeline orders for which the exact replenishment is still uncertain. Furthermore, we assume that the un.lled part of the retailer’s order is not backlogged at the manufacturer, but it is lost. We give the summary of the notation in Table 4.1 and we introduce some later upon need. Table 4.1 Summary of the notation T : number of periods in the .nite planning horizon L : constant nonnegative supply lead time, a multiple of review periods (L = 0); m : advance supply information parameter, 0 = m = L h : inventory holding cost per unit per period b : backorder cost per unit per period a : discount factor (0 = a = 1) xt : inventory position in period t before ordering yt : inventory position in period t after ordering ˜xt : starting on-hand inventory in period t zt : order size in period t Dt : random variable denoting the demand in period t dt : actual demand in period t Qt : random variable denoting the available supply capacity at time t qt : actual available supply capacity limiting order zt given at time t, for which ASI is revealed m periods later We assume the following sequence of events. (1) At the start of period t, the decision maker reviews the current inventory position xt. (2) The ordering decision zt is made up to uncertain supply capacity and correspondingly the inventory position is raised to yt = xt+zt. (3) Order placed in period t -L is replenished in the extent of min(zt-L, qt-L), depending ASI Uncertain Pipeline orders on the available supply capacity. ASI on the order placed in period t-m is revealed, which enables the decision maker to update the inventory position by correcting it downward in the case of insu.cient supply capacity, yt -(zt-m -qt-m)+ (where (x)+ = max(x, 0)). (4) At the end of the period previously backordered demand and demand dt are observed and satis.ed from on-hand inventory; unsatis.ed demand is backordered. Inventory holding and backorder costs are incurred based on the end-of-period on-hand inventory. Due to the positive supply lead time, each of the orders remains in the pipeline stock for L periods. For orders placed m periods ago or earlier we have already obtained ASI, while for more recent orders the supply information is not available yet. Therefore we can express the inventory position before ordering xt as the sum of net inventory and the certain and uncertain pipeline orders: t-m-1 t-1 xt = x˜t + min(zs, qs) + zs. (4.1) s=t-L s=t-m Note, that due to perfect ASI the inventory position xt re.ects the actual quantities that will be replenished for the orders for which ASI is already revealed, while there is still uncertainty in the actual replenishment sizes for recent orders for which ASI is not known yet. Observe also that m denotes the number of uncertain pipeline orders. Therefore, m lies within 0 = m = L, and the two extreme cases can be characterized as: • m = L, or so-called “No information case”, which corresponds to the most uncertain setting as the actual replenishment quantity is revealed no sooner than at the moment of actual arrival. This setting is a positive lead time generalization of of the Ciarallo et al. (1994) model. • m = 0, or so-called “Full information case”, which corresponds to the full information case, where before placing the new order, we know the exact delivery quantities for all pipeline orders. This is the case with the least uncertainty within the context of our model. Observe however that the current order is still placed up to uncertain supply capacity. When movingfrom period t to t+1, we obtain ASI for the order zt-m placed in period t-m. Correspondingly, the inventory gets corrected downwards if the order exceeds the available supply capacity, thus inventory position xt is updated in the following manner: xt+1 = xt + zt -(zt-m -qt-m)+ -dt. (4.2) Note, that there is dependency between the order quantity and the size of the correction to xt. If zt is high, it is more probable that the available supply capacity will restrict the replenishment of the order, thus the correction will be bigger, and vice versa for low zt. To fully describe the system behavior, we do not only need to keep track of xt, but also have to track the pipeline orders for which we do not have ASI yet. We denote the stream of uncertain pipeline orders with the vector z t = (zt-m,zt-m+1,. . . ,zt-2,zt-1). In period t + 1, z t+1 gets updated by the inclusion of the new order zt, and the order zt-m is dropped out as its uncertainty is resolved through the received ASI. A single period expected cost function is a function of xt and all uncertain orders, including ˜ the most recent order zt, given in period t. Cost charged in period t + L, Ct+L(˜xt+L+1), reassigned to period t when ordering decision is made, can be expressed as: t Ct(yt,z t,zt)= aLE.C˜t+L(yt - (zs )+ -DL), (4.3) Qt,Qt,DL -Qst t s=t-m where the inventory position after ordering accounted for the possible future supply short­ .t .t+L ages, yt - (zs -Qs)+, is used to cover the lead time demand, DL = s=t-m ts=t Ds The minimal discounted expected cost function, optimizing the cost over a .nite planning horizon T, from time t onward, and starting in the initial state (xt,z t), can be written as: ft(xt,z t)= min{Ct(yt,z t,zt)+ aEDt,Qt-m ft+1(yt -(zt-m -Qt-m)+ -Dt,z t+1)}, for t = T, xt=yt (4.4) where fT+1(·)= 0. The cost function ft is a function of inventory position before ordering and orders given in last m periods, for which ASI has not yet been revealed. 4.3 Analysis of the optimal policy In this section, we show the necessary convexity results of the relevant cost functions. This allows us to establish the structure of the optimal policy and show some of its properties. Lets de.ne Jt as the cost-to-go function of period t: Jt(yt,z t,zt)= Ct(yt,z t,zt)+aEDt,Qt-m ft+1(yt-(zt-m-Qt-m)+-Dt,z t+1)}, for t = T. (4.5) The minimum cost function f de.ned in (4.4)can now be expressed as: ft(xt,z t)= min Jt(yt,z t,zt), for t = T, (4.6) xt=yt We proceed by establishing the necessary convexity results that allow us to establish the structure of the optimal policy. Observe that the single period cost function Ct(yt,z t,zt) is not convex already for the zero lead time case as was originally shown by Ciarallo et al. (1994), where we elaborate on this in detail in Lemma C.1 in the Appendix B. Ct(yt,z t,zt) is shown to be convex in yt and quasiconvex in zt (Figure 4.3), which however still su.ce for the optimal policy to exhibit the structure of the base-stock policy. for a particular yt. We show that the results ofthe zero leadtime case can be generalizedto the positive leadtime case, where the convexity of the costs functions in the inventoryposition is established given a more comprehensive system’s state description (xt,z t). In Lemma B.1 in the Appendix B, we show that the single period cost function Ct(yt,z t,zt)is not a convex function in general, but it exhibits a unique although state-dependent minimum. Based on this result one can show that the related multi-period cost functions Jt(yt,z t,zt)and ft(xt,z t)are convex in the inventory position yt and xt respectively (we show in Appendix B that the convexity holds also for other characterizations of the inventory position), as shown in the next Lemma: Lemma 4.1 For any arbitrary value of information horizon m, value of the ASI vector z t and the order zt, the following holds for all t: 1. Jt(yt,z t,zt)is convex in yt, 2. ft(xt,z t)is convex in xt. Based on the results following Lemma 4.1, we establish a structure of the optimal policy in the following Theorem: Theorem 4.1 Assuming that the system is in the state (xt,z t), let yˆt(z t) be the smallest minimizer of the function Jt(yt,z t,zt). For any z t, the following holds for all t: 1. The optimal ordering policy under ASI is the state-dependent base-stock policy with the optimal base-stock level yˆt(z t). 2. Under the optimal policy, the inventory position after ordering yt(xt,z t)is given by xt, yˆt(z t)= xt, yt(xt,z t)= (4.7) yˆt(z t), xt < yˆt(z t). The proof is by induction, where we provide the details in the Appendix B. The optimal inventory policy is characterized by a single optimal base-stock level ˆyt(z t)that determines the optimal level of the inventory after ordering. The optimal base-stock level however is state-dependent as it depends on uncertain pipeline orders z t, for which ASI has not yet been revealed. Observe that due to not knowing the current period’s capacity, we are not limited in how high we set the inventory position after ordering. The logic of the optimal policy is such that yt should be raised to the optimal base-stock level ˆyt, although in fact yt does not re.ect the actual inventory position as it is possible that the order will not be delivered in its full size. We proceed by studying the properties of the optimal base-stock level in relation to the outstanding uncertain pipeline order status. In the following theorem we show that due to convexity results of Lemma 4.1 the optimal base-stock level is higher in the case of larger uncertain pipeline orders (Part 3 of Theorem 4.2). In the followingformulations we suppress the subscript t in state variables for clarityreasons. We de.ne the .rst derivative offunctions ft(x,·)and Jt(x,·)with respect to x as ft ' (x,·)and Jt ' (x,·). Observe that for any two ASI vectors z 1 and z 2 in period t, z 1 = z 2 holds if and only if each element of z 1 is smaller than or equal to the corresponding element of z 2. In Parts 1 and 2 of Theorem 4.2, we show that the rate of the change in costs is higher in the case where there are lower uncertain pipeline orders in the system. Theorem 4.2 For any z 1 =z 2 and z1 = z2, the following holds for all t: 1. Jt ' (y,z 1,z1)= Jt ' (y,z 2,z2)for all y, 2. ft ' (x,z 1)= ft ' (x,z 2)for all x, 3. yˆt(z 1)= yˆt(z 2). The dependency of the optimal base-stock level on z t can be intuitively attributed to the following; if we were placinghigh orders (with regards to expected supplycapacityavailable) in past periods, it is likelythat a lot ofthe orders willnot be realizedin its entirety. This leads to probable replenishment shortages and demand backordering due to insu.cient inventory availability. Therefore it is rational to set the optimal base-stock level higher with a goal of taking advantage of every bit of available supply capacity in the current period. By setting high targets, we aim to get the most out of the capacity, that is, we want to exploit the opportunity of the possibility of a large supply availability, although the chances that it will be actually realized can be small. If currently, we are not facing supply shortages the tendency to use the above logic diminishes. The result is also con.rmed in Table 4.2. This observation is equivalent to what we observed in ACI model in Chapter 3, where the optimal base-stock level is aligned with the future supply capacity availability revealed through sharing ACI. Table 4.2 Optimal base-stock levels ˆyt(zt-2,zt-1) (L = 3, m = 2, E[D] = 5, CVD = 0, E[Q] = 6, CVQ = 0.33) zt-2 0 1 2 3 4 5 6 7 8 9 zt-1 0 1 2 3 4 5 6 7 8 9 20 20 20 20 20 20 21 22 23 24 20 20 20 20 20 20 21 22 23 24 20 20 20 20 20 20 21 22 23 24 20 20 20 20 20 20 21 22 23 24 20 20 20 20 20 21 21 22 23 24 20 20 20 20 21 21 22 23 23 24 21 21 21 21 21 22 22 23 24 25 22 22 22 22 22 23 23 24 25 25 23 23 23 23 23 24 24 25 25 26 24 24 24 24 24 24 25 25 26 27 In addition to the monotonicity properties of the optimal base-stock level, we study the sensitivity of the base-stock level to a change in the vector of outstanding uncertain pipeline orders z . In the case of ACI, we point out in Conjecture 3.1 in Chapter 3, that taking away a unit of future supply capacity in period t+i a.ects the optimal base-stock level more than taking away a unit of supply capacity which is available in later periods, t+i+1 and beyond. One could suggest that in the case of ASI increasing the size of the older pipeline order will a.ect the optimal base-stock level more than doing the same with one of the more recent pipeline orders. It can be quickly checked that this intuition is wrong. In Table 4.3, we calculate the di.erences in the base-stock levels for the setting presented in Table 4.2 above. Observe that increasing the pipeline order by 1 unit for any of the two orders, results in all possible scenarios, where none, both or either of the two base stock levels, ˆyt(zt-2 + 1, zt-1) and ˆyt(zt-2, zt-1 + 1), are increased. The above suggestion does not hold due to the fact that yt includes all the pipeline orders. If any of the uncertain pipeline orders is increased, the exposure to a potential shortage increases. Due to the fact that the shortage cannot be avoided, the best one can do is to immediately increase the optimal base-stock level to try to assure su.cient inventory at least L periods from now. In the ACI case such an immediate reaction is not necessary if future capacity availability allows the postponement of the decision to increase the inventories. Thus, it is not a surprise that the Table 4.2 is close to being symmetric (there are exceptions to this, where we point one below), which can be helpful in reducing the computational e.orts in determining the optimal base-stock levels (again, this is not the case with the ACI model). Table 4.3 The di.erence in the base-stock levels, ˆyt(zt-2 + 1, zt-1)-yˆt(zt-2, zt-1 + 1), and vice versa. zt-2 + 1 1 2 3 4 5 6 7 8 9 zt-1 + 1 1 2 3 4 5 6 7 8 9 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 -1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 While the above sensitivity property does not hold in general for the ASI case, we point out that it might hold for the special case where all the initial uncertain pipeline orders are equal. For the case denoted in bold in Table 4.2, it holds that ˆyt(8,5) = yˆt(5,8), where the initial state is (5,5) and the orders were increased by 3 units. Observe. also that the change in the base-stock level is always at most equal to the change in the size of the uncertain pipeline order. We expect that if the pipeline order is increased by ., the corresponding increase in the base-stock level is at most equal to . or lower. Based on the above observations, we form the following two conjectures: Conjecture 4.1 The following holds for all t: 1. If zt-m = zt-m+1 = · · · = zt-1, then yˆt(z + .ei)= yˆt(z + .ei+1)for i=t-m,. . . ,t-1. 2. yˆt(z + .ei)-yˆt(z )= . for i=t-m,. . . ,t-1. 4.4 Insights from the myopic policy We proceedbyestablishingthe approximate inventorypolicythat wouldcapture the relation­ship between the uncertain pipeline orders and the target inventory position. Approximate policy is an extension of a solution to a single-period problem (myopic solution). The rele­vant inventorycosts that are in.uenced bythe orderingdecision zt in period t are reassigned costs charged in period t + L based on the on-hand inventory ˆxt+L+1: t x˜t+L+1(:,zt)= yt -Dt L - (zt-m -qt-m)+ . (4.8) s=t-m The myopic solution satis.es the following relationship for the non-stockout probability: P{x˜t+L+1(:,zt)= 0}= b/b+ h. (4.9) The steps ofthe approximate policyalgorithm are as follows: (1)Determine the distributions .t Vt(z t,zt)= DL + (zt-m -qt-m)+ . (2) Determine the approximate state-dependent t s=t-m base-stock levels y˜ˆt(z t). (3) Assuming that the approximate policy instructs the decision maker to place the order withan accordance to (2), simulate the orderingpolicyand evaluate the relevant costs. While it would be great if myopic policy would provide a reliable estimate of the optimal costs (which we show for the dual sourcing model in Chapter 5), one can easily see that this is not the case for ASI model (and similarly for the ACI model). The myopic policy cannot account for future supply shortages (that is for the orders that are still to be placed in the future). Particularly in the case of a highly utilized system and in the case of high demand/supplycapacityuncertaintythat leads to probable demandand supplymismatches. Thus, the following study is primarily concerned with capturing the state-dependency of the optimal base-stock levels by exploring the relationship between the vector of uncertain pipeline orders and the corresponding approximate base-stock levels. In Table 4.4 we present the base-stock levels for the optimal policy, myopic policy and the di.erences between the two. Note that the myopic policy is optimal when there is no supply capacity uncertainty (which corresponds to a base-stock level of 23). Looking at the di.erences between the base-stock levels we see that the myopic base-stock levels are always lower and thus can be regarded as a lower bound for the optimal base-stock levels. The di.erences are decreasing with increasing uncertain pipeline orders. This can be attributed to the fact that myopic policy accounts for the potential shortages in the replenishment of the pipeline orders, but fails to account for future supply unavailability. For high uncertain pipeline orders, the additional inventory to cover the supply shortage is also su.cient to cover the risk of future shortages. In fact, we observe a risk pooling e.ect, where the base-stock level of 29 is su.cient to cover both risks. Table 4.4 The optimal and myopic base-stock levels, (L = 3, m = 2, E[D] = 5, CVD = 0.5, E[Q] = 6, CVQ = 0.33) Myopic zt-2 0 1 2 3 zt-1 4 5 6 7 8 9 0 23 23 23 23 23 23 24 24 25 26 1 23 23 23 23 23 23 24 24 25 26 2 23 23 23 23 23 23 24 24 25 26 3 23 23 23 23 23 23 24 24 25 26 4 23 23 23 23 23 23 24 25 25 26 5 23 23 23 23 23 24 24 25 26 27 6 24 24 24 24 24 24 25 25 26 27 7 24 24 24 24 25 25 25 26 27 28 8 25 25 25 25 25 26 26 27 28 29 9 26 26 26 26 26 27 27 28 29 29 Optimal zt-2 0 1 2 3 zt-1 4 5 6 7 8 9 0 27 27 27 27 27 27 27 28 29 29 1 27 27 27 27 27 27 27 28 29 29 2 27 27 27 27 27 27 27 28 29 29 3 27 27 27 27 27 27 27 28 29 29 4 27 27 27 27 27 27 28 28 29 29 5 27 27 27 27 27 27 28 29 29 29 6 27 27 27 27 28 28 28 29 29 29 7 28 28 28 28 28 29 29 29 29 29 8 29 29 29 29 29 29 29 29 29 29 9 29 29 29 29 29 29 29 29 29 29 Di.erence zt-2 0 1 2 3 zt-1 4 5 6 7 8 9 0 4 4 4 4 4 4 3 4 4 3 1 4 4 4 4 4 4 3 4 4 3 2 4 4 4 4 4 4 3 4 4 3 3 4 4 4 4 4 4 3 4 4 3 4 4 4 4 4 4 4 4 3 4 3 5 4 4 4 4 4 3 4 4 3 2 6 3 3 3 3 4 4 3 4 3 2 7 4 4 4 4 3 4 4 3 2 1 8 4 4 4 4 4 3 3 2 1 0 9 3 3 3 3 3 2 2 1 0 0 4.5 Value of ASI In this section we estimate the extent of the savings gained through incorporating ASI into the inventory system. We perform a numerical numerical analysis to quantify the value of ASI and assess the in.uence of the relevant system parameters. Numerical calculations were done by solving the dynamic programming formulation given in (4.4). To determine the in.uence of ASI parameter m, demand uncertainty, supply capacity un­certainty, and system utilization on the value of ASI, we set up the base scenario that is characterized by the following parameters: T = 10, L = 3, a = 0.99 and h = 1 and b = 20. A discrete uniform distribution is used to model demand and supply capacity where the expected demand is given as E[D]= 4 and the expected supply capacity varies E[Q]= {4,6,8}, which means that the utilization of the system is Util = {1,0.75,0.5}. In addition we vary the coe.cient of variation of demand CVD = {0,0.65}and supply capac­ity CVQ = {0,0.33,0.65}, and the ASI parameter m = {3,2,1,0}, covering both the No information and Full information case. We de.ne the relative value of ASI for m = L, %VASI , as the relative di.erence between the optimal expected cost of managing the system in the No information case (m = L), and the system where we have obtained ASI on a number of pipeline orders (m = L): (m=L) (m=L) -f ft t %VASI (m = L)= . (4.10) (m=L)ft We also de.ne the marginal change in the value of ASI, .VASI . With this we measure the extra bene.t gained by decreasing the number of uncertain pipeline orders by obtaining ASI sooner, from m to m -1: (m) (m-1) .VASI (m -1) = ft -ft We present the results in Figures 4.4 and 4.5. Not surprisingly, the insights about the value of ASI very much correspond to the ones for ACI model. The interplay of system parameters is relatively complex, which is exhibited in the fact that the value of ASI changes in a non-monotone manner. This is particularly the case in the e.ect of the system’s utilization, where the majority of the gains are made at (in our case) moderate utilizations. Increasing capacity uncertainty is where we would anticipate that the value of ASI will be increasing and majority of the gains would be made. This can be observed in .VASI for both low and high demand uncertainty scenario, while this only partially holds for %VASI . The increasing demand uncertainty decreases both the relative and the absolute value of ASI. While the relative value of ASI extends over 30%, and for most of the scenarios above 10%, it drops below 4% for all scenarios under high demand uncertainty. It is expected that the bene.ts of ASI fall short of the ones observed when studying the ACI model. The interesting observation is made when studying the in.uence of the ASI parameter m. While we have observed that with a limited insight into future supply capacity availability the majority of the bene.ts can be gained in most of the scenarios under ACI model, this is not the case with ASI model. When m is decreased from m = L in the No information case, we observe the increasing returns with m approaching 0. As ASI only allows for a response to shortages in replenishment, it is the scenario in which ASI is revealed soon after the order is placed that allows the decision maker to promptly respond to the shortages. 4.6 Summary In this chapter, we analyze a periodic review inventory system with positive lead time and stationary stochastic demand and supply capacity. As the No information case of our model can be considered as a positive lead time generalization of the paper by Ciarallo et al. (1994), we also extend the scope of the model by incorporating the possibility to obtain information about the available supply capacity for the pipeline orders. ASI is revealed after the order has been placed, but before it is replenished. We show that the optimal policy is highly complex due to the extensive system’s state description, where apart from the inventory position, the stream of uncertain pipeline orders has to be monitored and adapted constantly. However, despite this complexity, we show that the optimal policy is a state-dependent base-stock policy. We show that the base-stock levels should be increased to compensate for the increased replenishment uncertainty. Despite the fact that the myopic policy does not provide a good approximation for the optimal base-stock levels (and optimal costs), we show that by inclusion of the safety factor that would compensate for the future supply capacity uncertainty, the myopic policy adequately captures the risk of shortages in pipeline orders. Numerical calculations show that the bene.ts obtained through ASI can be relatively big (although also highly dependent on other system parameters), however in this case ASI should be revealed for the most of the uncertain pipeline orders as we observe the increasing returns with the increasing ASI availability. The analysis in the future could explore di.erent alternatives to the presented ASI model. Additional insights could be gained by studying simpli.ed settings (for instance constant demand, Bernoulli distributed supply capacity etc.), where explicit expressions could be obtained that would better capture the supply uncertainty structure in the system. Chapter 5 Dual sourcing: trading o. stochastic capacity limitations and long lead times In this chapter, we model a periodic review inventory system with non-stationary stochas­tic demand, in which a manufacturer is procuring a component from two available supply sources. The faster supply source is modeled as stochastic capacitated with immediate de­livery, while the slower supply source is modeled as uncapacitated with a longer .xed lead time. The manufacturer’s objective is to choose how the order should be split between the two supply sources in each period, where the slower supply source is used to compensate for the supply capacity unavailability of the faster supply source. Note that this is di.erent from the conventional dual-sourcing problem, and motivated by the new reality of near-shoring options. We derive the optimal dynamic programming formulation that minimizes the total expected inventory holding and backorder costs over a .nite planning horizon and show that the optimal policy is relatively complex. We extend our study to show that the myopic two-level base-stock policy provides a nearly perfect estimate of the optimal costs. Our numerical results reveal the bene.ts of the dual sourcing and explain the trade-o. between the replenishment responsiveness and its reliability. We show that in most cases the man­ufacturer should develop a hybrid procurement strategy, taking advantage of both supply sources to minimize its expected total cost. 5.1 Introduction With the rise of reshoring options and growth of B2B supply platforms, the complexity of sourcing management in the supply chain is increasing. Recent studies by BCG (2014b) and others suggest that the cost of sourcing in the U.S.A. may be approaching the cost of producing in China due to matters such as the rising wages in coastal China, the increase in transportation costs from more remote regions in China, the exchange rate developments of the RMB vs the USD, and the growth in productivity in the U.S.A. due to extensive automation. The realization that the economics of manufacturing are swinging in favor of the U.S.A. has seen companies becoming interested in shifting manufacturing back to the U.S.A., for both goods to be sold at home and those intended for major export markets. The BCG’s survey found that the number of respondents saying that their companies are already bringing production back from China to the United States had risen by 20 %, from roughly 13% to 16% in the past year (BCG, 2014a). Similar developments are taking place in Europe. Although the comparative attractiveness of the U.S.A. and other developed economies has increased, the current nearshoring trend is facing several challenges. The traditionally pre­sumed unlimited capacity availability of nearshore manufacturing options may be decreasing due to the fact that so much capacity has been taken out of the system over the past two decades and that widespread automation is reducing the ability to respond to short-term changes in demand. Companies would have to rebuild their supply chains and identify people with the right skills to handle the increasingly sophisticated automated operations. In an executive survey by Alix Partners (2014), the respondents mention that the challenges center on the availability and capability of the local workforce, and the capability and .exibility of suppliers. In addition, with one-third of U.S. 3PLs reporting increased volumes and revenues as a result of nearshoring, their CEOs are also reporting capacity shortages across transport modes, including truckload, LTL, intermodal and rail, with the outcome that “The capacity crunch has led to higher rates and longer transit times with 3PLs struggling to meet on-time service goals and cost targets” (Penske Logistics, 2014). The fact that manufacturing jobs are coming back to developed economies due to reshoring, does not mean that they are not continuing to develop in Asia and other developing countries. In fact, the number of global sourcing options is still growing, with the global supply chains e.ectively pooling the global manufacturing capacities and making them available to com­panies worldwide. For instance, the rise of B2B platforms such as Alibaba, enables suppliers from further away to .exibly respond to market needs, albeit with longer (transportation) lead times. As a consequence, the trade-o.s in dual sourcing are changing. Most research in dual sourcing studies the trade-o.s between a nearshoring option that is more expensive and faster against an o.shoring option that is cheaper but requires a longer lead time. Yet the developments described above suggest that at least for part of the product portfolio the trade-o. may be less about product cost, and nearby suppliers may be faster but not necessarily more .exible. They may su.er from limited capacity availability. In this chapter, we focus on such a newly developing trade-o.. We study the problem of a manufacturer procuring a component for the production of a .nished product, where two sourcing options are considered: a faster, yet partially available source, and a slower, yet fully available source. Both sources supply at the same landed cost. In this case, it is unlikely that the faster source is always favorable. In any case, supplying from the faster nearshore source will involve uncertainties related to the current capacity availability, which could result in uncertain replenishment to the manufacturer. This uncertainty would need to be hedged using the slower, presumably uncapacitated, o.shore source. When a manufacturer experiences or anticipates a supply shortage of the nearshore source, a decision has to be made concerning the extent to which the slower o.shore source should also be utilized. Due to the sourcing lead time, the ordering decisions are made before demand for the .nished product is realized. Thus, the decision maker is facing both, the uncertain replenishment from the fast supply source, and the uncertainty of demand for the .nished product, and needs to allocate the replenishment between the two supply sources. To study this, we model a zero-lead-time supply source that is stochastic capacitated, where the supply capacity is exogenous to the manufacturer and the actual capacity realization is only revealed upon replenishment. The reliable positive lead time supply source is mod­eled as uncapacitated with a .xed one-period lead time. Both the supply capacity of the nearshore source and the demand are assumed to be stochastic and non-stationary with known distributions in each time period. Unmet demand is backordered. In each period the manufacturer places the order with the fast source, the slow source, or with both sources. Our goal is to .nd an optimal policy that minimizes the inventory holding costs and back-order costs over a .nite planning horizon. See Figure 5.1 for a sketch of the supply chain under study. We proceed with a review of the relevant literature on supply uncertainty models in a single-stage setting, where our interest lies in two research tracks: single sourcing inventory models with random capacity and dual-sourcing models with suppliers that di.er in their delivery times and/or supply capacity availability. The way we model the supply capacity of the faster supply source is in line with the work of Ciarallo et al. (1994); Khang and Fujiwara (2000); Iida (2002); Jak.si.c et al. (2011) and Jak.si.c and Fransoo (2015), where the random supply/production capacity determines a random upper bound on the supply availability in each period. For a .nite horizon stationary inventory model they show that the optimal policy is a base-stock policy (or a modi.ed base-stock policy if capacity is known prior to placing an order), where the optimal base-stock level is increased to account for possible, albeit uncertain, capacity shortfalls in future periods. An important observation for our work is the insight provided by Ciarallo et al. (1994) in their analysis of a single-period problem, where they show that stochastic capacity does not a.ect the order policy. The myopic policy of the newsvendor type is optimal to cover the demand uncertainty, meaning that the decision maker is not better o. by asking for a quantity higher than that of an uncapacitated case. For a general review of multiple supplier inventory models we refer the interested reader to Minner (2003). The review is based on the important criteria for the supplier choice, mainly the price and the supplier’s service. A more focused review of multiple sourcing inventory models when supply components are uncertain by Tajbakhsh et al. (2007) reveals that most of these models consider uncertainty either in supply lead time, supply yield, or supplier availability. More speci.cally, the review of dual-sourcing literature shows that a series of papers shares some basic modeling assumptions with our model: dual-sourcing periodic review with deterministic lead times that are di.erent for the two supply sources. These papers can be divided into two streams depending on the assumptions on supply capacity availability, where for the .rst stream the assumption of unconstrained suppliers holds, while for the second some sort of capacity constraint is introduced at the faster supplier or both suppliers. Most of these papers rely on the price di.erence between the two suppliers that stimulates the manufacturer to partially source from the cheaper, slower supplier. However, we argue that in the case where the supply capacity availability at the faster supply source is limited, variable, and potentially stochastic, the above-mentioned price incentive is not needed. Thus, in our case the incentive to .nd the optimal dual-sourcing strategy lies in .nding the right balance between the responsiveness of the faster supply source and the reliability of the slower supply source. In the .rst of the two above-mentioned streams that assumes unconstrained suppliers, the search for the best dual-sourcing strategy revolves around the dilemma of when to use a faster and necessarily more expensive supplier to compensate for a slow response by a cheaper supplier. Several papers discuss the setting in which the lead times of the two suppliers di.er by a .xed number of periods (Daniel, 1963; Fukuda, 1964; Bulinskaya, 1964; Whittemore and Saunders, 1977). More speci.cally, Fukuda (1964) shows the optimality of the two-level base-stock policy for the case where the two suppliers’ lead times di.er by one period (so­called consecutive lead times). The policy instructs that .rst the order with the fast supply source is placed so that the inventory position is raised to the .rst base-stock level, and then the slow supply source is used to raise the inventory position to the second base-stock level. For general, nonconsecutive, lead times, Whittemore and Saunders (1977) found the optimal policy to be quite complex and lacking structure. Within the second stream of papers, the .rst to assume a capacitated faster supplier is Daniel (1963). He showed that a modi.ed two-level base-stock policy is also optimal in this case, where ordering with the faster supplier is up to the .rst base-stock level if the .xed capacity limit allows it. Yazlali and Erhun (2009) impose minimum and maximum .xed capacity limits on the availability of the two supply sources, and again show that the two-level base-stock policy is optimal when leadtimes are consecutive. Similarly, Xu (2011) assumes .xed capacity bounds on the cumulative order quantity to the two suppliers. He proves the optimality of a myopic ordering policy for a setting where the faster supply mode provides instant delivery, while the slower delivers an order one period later. While the modeling perspective of the papers referred to above is similar to ours, they all still primarily focus on exploring the trade-o. between the price di.erence and responsiveness. An exception to this is the work by Yang et al. (2005) who explore a trade-o. between variable capacity restrictions and responsiveness. They consider a Markovian capacity constraint on the fast supply mode, and show that the base-stock type policy is optimal in the situation where lead times di.er by one period and there are no .xed ordering costs. While their paper comes closest to our focus on exploring the e.ect of reliability of a faster supply mode on the optimal dual sourcing strategy, there are still relevant di.erences in our work. They assume that the capacity is known at the time orders are placed with the two supply channels, and also that the current period’s demand is realized before an order with a longer lead time supply channel is placed. The remainder of the chapter is organized as follows. We present the model formulation in Section 5.2. In Section 5.3, the general structure of the optimal policy is characterized and the myopic policy is developed. In Section 5.4, we present the results of a numerical study to determine the optimal costs of dual sourcing, to study the order allocation between the supply modes and quantify the bene.ts of dual sourcing over single sourcing. Additionally, we explore the option of obtaining the advance capacity information on future supply capacity availability of the faster supplier in Section 5.5. Finally, we summarize our .ndings and suggest possible extensions in Section 5.6. 5.2 Model formulation In this section, we give the notation and the model description. The faster, zero-lead­time, supply source is stochastic capacitated where the supply capacity is exogenous to the manufacturer and the actual capacity realization is only revealed upon replenishment. The slower supply source is modeled as uncapacitated with a .xed one period lead time. The demand and supply capacity of the faster supply source are assumed to be stochastic non-stationary with known distributions in each time period, although independent from period to period. In each period, the customer places an order with either an unreliable, or a reliable supply mode, or both. Presuming that unmet demand is fully backordered, the goal is to .nd the optimal policy that would minimize the inventory holding costs and backorder costs over .nite planning horizon T. We intentionally do not consider any product unit price di.erence and .xed ordering costs as we are primarily interested in studying the trade-o. between the capacity uncertainty associated with ordering from a faster supply source and the delay in the replenishment from a slower source. Any .xed costs would make the dual-sourcing strategy less favorable, and the di.erence in the .xed costs related to any of the two ordering channels would result in a relative preference of one channel over the other. The notation used throughout the chapter is summarized in Table 5.1 and some is introduced when needed. Table 5.1 Summary of the notation. T : number of periods in the .nite planning horizon ch : inventory holding cost per unit per period cb : backorder cost per unit per period a : discount factor (0 = a = 1) ˜xt : on-hand inventory before ordering in period t xt : inventory position before ordering in period t yt : inventory position after ordering from a faster capacitated supply source in period t wt : inventory position after ordering from a slower uncapacitated supply source in period t zt : order placed with the faster supply source in period t vt : order placed with the slower supply source in period t dt, Dt : actual realization and random variable denoting demand in period t gt(Dt) : probability density function of demand in period t Gt(Dt) : cumulative distribution function of demand in period t qt, Qt : actual realization and random variable denoting the available supply capacity of the faster capacitated supply source in period t rt(Qt) : probability density function of the supply capacity of the faster supply source in period t Rt(Qt) : cumulative distribution function of the supply capacity of the faster supply in period t We assume the following sequence of events. (1) At the start of the period, the manager reviews the inventory position before ordering xt, where xt = x˜t + vt-1 is the sum of the on-hand stock ˜xt and the order vt-1 to a slower supply source made in the previous period. (2) Order zt to a faster supply source and order vt to a slower supply source are placed. For the purpose of the subsequent analysis, we de.ne two inventory positions after the order placement, yt and wt. First, after placing order zt the inventory position is raised to yt, yt = xt+zt, and order vt is raised to wt, wt = xt+zt+vt. Observe that it makes no di.erence in which sequence the orders are actually placed as long as both are placed before the current period’s capacity of the fast supply source qt and demand dt are revealed. (3) The order with the slower supply source from the previous period vt-1 and the current period’s order zt are replenished. The inventory position can now be corrected according to the actual supply capacity realization wt -(zt -qt)+ = xt +min(zt, qt)+vt, where (zt -qt)+ = max(zt -qt,0). (4) At the end of the period, demand dt is observed and satis.ed through on-hand inventory; otherwise, it is backordered. Inventory holding and backorder costs are incurred based on the end-of-period on-hand inventory, ˜xt+1 = yt -(zt -qt)+ -dt. Correspondingly, the expected ˜˜ single-period cost function is de.ned as Ct(yt, zt)= aEQt,Dt Ct(˜xt+1)= aEQt,Dt Ct(yt -(zt - Qt)+ - Dt), where C˜t(˜xt+1) = ch(˜xt+1)+ + cb(-x˜t+1)+ . The minimal discounted expected cost function that optimizes the cost over a .nite planning horizon T from period t onward, starting in the initial state xt, can be written as: ft(xt)= min Ct(yt, zt) + aEQt,Dt ft+1(wt -(zt -Qt)+ -Dt) , for 1 = t = T (5.1) xt=yt=wt and the ending condition is de.ned as fT+1(·)= 0. 5.3 Characterization of the near-optimal myopic policy In this section, we .rst characterize the optimal policy for the non-stationary demand and supply capacity setting. We show that the structure of the optimal policy is relatively complex, where the order with the faster supply source depends on the inventory position before ordering, while the order with the slower supply source is raised to a state-dependent base-stock level. We continue by studying the stationary setting by introducing the myopic policy where the myopic orders are the solutions to the extended single period problem, and show the properties of the two myopic base-stock levels1 . We conduct the numerical analysis to show that the costs of the myopic policy provide a very accurate estimate of the optimal costs. See the Appendix D for proofs of the following propositions. In the literature review, we refer to a series of papers studying the dual sourcing inventory problem with consecutive lead times. The two-level base-stock policy characterizes the struc­ture of the optimal policy in all of them, both in the case of uncapacitated supply sources and in the case where one or both supply sources exhibit the .xed capacity limit. However, by studying the convexity properties of the cost functions given in (5.2)-(5.4), we show that these are not convex in general. In addition, we show that the optimal inventory position after ordering with the faster supplier ˆyt is not independent of the inventory position before ordering xt, and therefore cannot be characterized as the optimal base-stock level. As single-period costs Ct in period t are not in.uenced by order vt, we can rewrite (5.1) in the following way: ft(xt)= min Ct(yt, zt) + min aEQt,Dt ft+1(wt -(zt -Qt)+ -Dt) , for 1 = t = T , xt=yt yt=wt (5.2) 1With the phrase “myopic policy” we denote the policy that optimizes the single period problem. As we later demonstrate that the myopic policy is not optimal in a multiperiod setting, we avoid using the phrase “optimal myopic policy” to avoid confusion. which now enables us to introduce auxiliary cost functions Jt(yt, zt)and Ht(wt, zt): Jt(yt, zt)= Ct(yt, zt) + min aEQt,Dt ft+1(wt -(zt -Qt)+ -Dt), for 1 = t = T (5.3) yt=wt Ht(wt, zt)= aEQt,Dt ft+1(wt -(zt -Qt)+ -Dt), for 1 = t = T (5.4) Based on the above observations, we give the structure of the optimal policy under stochastic non-stationary demand and supply capacity in the following proposition: Proposition 5.1 Let yˆt(xt) be the smal lest minimizer of the function Jt(yt, zt) and wˆt(ˆzt) be the smal lest minimizer of the function Ht(wt,zˆt). The fol lowing holds for all t: 1. The optimal inventory position after placing the order with the faster supplier yˆt(xt)is a function of the inventory position before ordering xt. 2. The optimal inventory position after placing the order with the slower supplier wˆt(ˆzt) is a state-dependent base-stock level. 3. The inventory position wt(xt)after placing the order with the slower supply source is: . . xt, wˆt(ˆzt = 0) = xt, zˆt = 0,vˆt = 0, . wt(xt)= wˆt(ˆzt = 0), yˆt(xt)= xt < wˆt(ˆzt = 0), zˆt = 0,vˆt = wˆt(ˆzt = 0) -xt, . . wˆt(ˆzt > 0), xt < yˆt(xt), zˆt = yˆt(xt)-xt,vˆt = wˆt(ˆzt > 0) -yˆt(xt). (5.5) For a clearer representation of the optimal policy as given in (5.5), we depict the optimal order sizes depending on the initial inventory position xt and the corresponding inventory positions after ordering in Figure 5.2. For comparison, we also plot the myopic base-stock level ˆyt M that optimizes the single period cost function Ct (Part 3 of Lemma C.1 in Appendix D). For xt < yˆt M we order up to ˆyt M for small optimal orders with a faster supplier ˆzt, while for higher ˆzt, ˆyt(xt) lies below ˆyt M . For xt = yˆt M no ˆzt is placed. Similarly, looking at the optimal order with the slower supply source ˆvt, it is only placed if xt = wˆt(ˆzt = 0), and at yˆt(xt)= xt < wˆt(ˆzt = 0) the inventory position xt is increased to a constant base-stock level wˆt(zt = 0). For xt < yˆt(xt)the size of the order with the slower supply source depends on the anticipated supply shortage of the faster supply source, thus the state-dependent base-stock level ˆwt(ˆzt)depends on the size of the order ˆzt placed with the faster supply source. Based on the above results, the optimal policy instructs that the order with the slower supply source should compensate for the probable supply shortage of the faster supply source. The goal here is to bring the next period’s starting inventory position to the optimal level (where the demand uncertainty also needs to be taken into account). In the next period, this enables the decision-maker to place the order with the faster supply source in such a way that the period’s inventory costs are minimized given the supply capacity uncertainty and demand uncertainty. Next, we focus on a stationary demand and supply capacity setting. Based on the insights obtained studying the optimal inventory policy, one might suggest that the myopic base-stock level ˆyM is a good approximation for ˆyt(xt). To study this, we introduce the notion of the myopic policy, which optimizes the so-called extended single period problem in every period. As the model under consideration assumes di.erent lead times for the faster and the slower supplier, the extended period covers the time interval in which both ordering decisions are made that directly in.uence the single period costs Ct in period t. It e.ectively starts by placing an order vt-1 with the slower supplier in period t-1, followed by placing an order zt with the faster supplier in period t. The ability to cover the demand Dt at the end of period t depends on the replenishment of the two orders and the realization of the demand (Figure 5.3). Observe also that with the order vt we cannot in.uence the costs in period t. The myopic equivalent to the optimal cost function as given in (5.2) can therefore be written in the following way: fM(xt)= min Ct(yt, zt) + min aEQt,Dt fM (wt -(zt -Qt)+ -Dt), for 1 = t = T , (5.6) t t+1 xt=yt yt=wt where the search for the optimal yt resorts to minimizing the single period cost function Ct. In the following proposition, we show that the myopic policy can be characterized as the two-level base-stock policy, where the optimal order ˆzt is placed up to the .rst base-stock level ˆyM, and the optimal order ˆvt is placed up to the second state-dependent base-stock level ˆw(ˆz). For clarity reasons we omit the subscript t in parts of the following text. Proposition 5.2 Under stationary stochastic demand and supply capacity, the fol lowing results hold for all t: 1. ft M(xt)is convex in xt. 2. The optimal myopic policy is the two-level base-stock policy with the constant base-stock levels yˆM and wˆM(ˆz), where: ( ) MM M G-1 cb yˆ= yˆt is the solution to minxt=yt Ct(yt, zt): yˆ= , and cb+ch wˆM(ˆz)= wˆt-1 M(ˆzt-1)is the solution to minyt-1=wt-1 aEQt-1,Dt-1 minxt=yt Ct(yt, zt). In Proposition 5.3 we derive the properties of the state-dependent myopic base-stock level wˆM(ˆz). Part 1 suggests that for the pair of optimal orders ˆz = {zˆ1,zˆ2}placed with the faster supply source in any period, the decision-maker has to raise the base-stock level ˆwM(ˆz1)above wˆM(ˆz2), when ˆz1 = zˆ2. From this ˆvM(ˆz1) = vˆM(ˆz2) follows directly given the fact that ˆz1 and ˆz2 were placed up to the constant ˆyM . With an increase in order ˆz, the probability of a supply shortage at the faster supply source increases. To compensate for this, it is optimal that a higher order ˆv is placed with the slower supply source. In Part 2, we show that the level of compensation to account for the additional supply uncertainty (due to the higher ˆz placed) should at most be equal to the di.erence between the optimal order sizes .. Proposition 5.3 The fol lowing holds for all t: 1. For any pair zˆ= {zˆ1,zˆ2}, where zˆ1 = zˆ2: wˆ(ˆz1)= wˆ(ˆz2). 2. For . = 0: wˆ(žw(ˆ z + .)- ˆzt)= .. Observe that the constant base-stock level ˆyM is the solution to the multiperiod uncapac­itated single source inventory model. For the capacitated single supplier model, Ciarallo et al. (1994) show that while ˆyM optimizes a single period problem, it is far from optimal in a multiperiod setting. However, we show that in the dual sourcing model under considera­tion the appropriate combination of the two myopic base-stock levels provides a very good substitute for the optimal base-stock levels. In Table 5.2, we present the optimal and the myopic inventory positions. Numbers in bold are used to denote the cases, where the myopic inventory positions and orders di.er from the optimal ones. The results for ˆy(x) con.rm that ˆy is a function of x. We see that with increasing x, ˆy(x)is increasing, approaching the myopic ˆyM . For lower x, the optimal policy suggests that it is not optimal anymore to place zˆup to ˆyM . This can be attributed to the increased uncertainty about the replenishment of ˆz. To compensate for the relatively smaller ˆz placed, the optimal decision is to rely more heavily on the slower supplier, by increasing ˆv above ˆvM . Table 5.2 The optimal and the myopic inventory positions and orders. x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ˆy(x) ˆyM ˆz ˆzM 14 16 14 16 14 16 13 15 14 16 12 14 14 16 11 13 15 16 11 12 15 16 10 11 15 16 9 10 15 16 8 9 15 16 7 8 15 16 6 7 15 16 5 6 15 16 4 5 15 16 3 4 16 16 3 3 16 16 2 2 16 16 1 1 16 16 0 0 ˆw(ˆz) 29 28 ˆwM (ˆzM ) 31 30 ˆv(ˆz) 15 14 ˆvM (ˆzM ) 15 14 27 29 13 13 27 28 13 12 27 27 12 11 26 27 11 11 25 26 10 10 25 25 10 9 25 25 10 9 24 25 9 9 24 24 9 8 24 24 9 8 23 24 8 8 23 23 7 7 23 23 7 7 23 23 7 7 23 23 7 7 23 23 6 6 23 23 5 5 23 23 4 4 23 23 3 3 23 23 2 2 23 23 1 1 23 23 0 0 Despite the fact that the myopic base-stock levels generally di.er from the optimal inventory positions, we now proceed to show that the costs of the myopic policy provide a nearly perfect estimate of the optimal costs. In Figure 5.4, we provide the histogram of the relative di.erences in costs of the optimal and myopic policy. The accuracy of the myopic policy is tested on 300 scenarios, where in almost 60% of the scenarios the costs of the myopic policy are equal to the optimal costs. In only 4 cases the myopic costs di.ered for around 0.3%, which is also the highest cost di.erence observed. A careful study of suboptimal scenarios has not revealed a clear pattern that would point out the characteristics of these scenarios. In general, the accuracy is lower when both the demand and supply capacity uncertainty are high, which is not unexpected. !!bb !b bb ! !!bb !b bb ! !!bb !b bb ! =·=8 = ·=6 =·=4 =·=2 =·=8 5 2= 1== =·== =·== =·2= =·4= =·6= =·8= 1·== vv =·== 5== v v 8== bbbbbb b b =·=6 = ·=4 =·=2 /ë5 7== /ëv 6== /ë =·=8 =·=6 =·=4 =·=2 =·5 =·67 1 2 4== == 2== =·== 1== = Figure 5.4 The relative di.erence in costs between the optimal and the myopic policy. We conclude this section with the discussion on the optimality of the myopic policy. The high accuracy of the myopic policy might suggest that myopic policy could be optimal under certain assumptions. However, the policy does not exhibit the necessary properties of the myopic optimality. Heyman and Sobel (1984) point out the following two conditions (among others) for the myopic policy to be optimal: single period cost function needs to be additively separable on the state and action, and that the myopic policy guarantees that the set of consistent states (from which the optimal action can be taken) are visited in the next period. It is clear that the single period cost function Ct does not satisfy the .rst condition, as it does not depend additively on xt and yt. While for ˆwM it does hold that the optimal action will again be feasible in the next period, this is not the case for ˆyM as the system can easily end up in xt > yˆM from which the optimal action is infeasible. Despite the fact that the myopic base-stock levels di.er from the optimal inventory positions after ordering, the myopic policy provides a nearly perfect estimate of the optimal costs. The accuracy can be attributed to the possibility of balancing the orders placed with the two supply sources, which in most situations leads to the optimal costs. 5.4 Value of dual sourcing In this section, we present the results of the numerical analysis, where we investigate the e.ect of di.erent system parameters on: (1) the optimal costs of dual sourcing; (2) the relative utilization of the two supply sources; and (3) the bene.ts of dual sourcing compared to single sourcing from either supply source. Numerical calculations were carried out by solving the dynamic programming formulation given in (5.1). We used the following set of input parameters: T = 12, ch = 1, a = 0.99, and a discrete uniform distribution to model stochastic demand and supply capacity. Throughout the experiments we varied the utilization2 of the faster supply source, Util = (0,0.5,0.67,1,2,8), the per unit backorder cost cb = {5,20,100}, and the coe.cient of variation of demand, CVD = {0,0.14,0.26,0.37,0.49,0.61}, and the supply capacity of the faster supply source, CVQ = {0,0.14,0.26,0.37,0.49,0.61}, where the CV s do not change over time3 . 5.4.1 Optimal costs of dual sourcing We start by studying the optimal costs under di.erent system parameters and by comparing the performance of the dual sourcing model with the two base cases: the worst case in which 2De.ned as the expected demand over the available capacity of the faster supply source. 3We give the approximate average CVs for demand and supply capacity distributions since it is impossible to come up with the exact same CVs for discrete uniform distributions with di.erent means. the supply is only available through the slower supply source (the faster supply source is unavailable, Util = 8), and the best case in which the faster supply source is assumed to be uncapacitated, and thus becomes fully available (Util = 0). The results are presented in Figure 5.5 and Table 5.3. 0.00 0.20 0.40 0.60 0.00 0.20 0.40 0.60 /ëv /ëv Figure 5.5 Optimal system costs. As expected, the system costs rise with increasing demand and capacity uncertainty, as well as with increasing utilization of the faster supply source. The sensitivity of costs is highest in the case of increasing CVD. Moreover, it is at a high CVD where the costs are the most sensitive to an increase in CVQ and Util. For relatively low CVD and up to moderate CVQ, the system works with minimal costs even for the cases of high utilization. In this setting, the capacity and demand realizations are easy to anticipate, and sourcing from the slower supply source to compensate for the lack of supply capacity availability with the faster supply source does not increase the costs compared to the best case. However, for a high CVD, due to the longer lead time, ordering with the slower supply source becomes riskier as the exposure to demand uncertainty increases. This leads to a relatively high increase in costs already for moderate utilizations. 5.4.2 Relative utilization of the two supply sources To investigate to what extent either of the two supply sources is utilized under di.erent system parameters, we present the calculations for the share of inventories replenished from a faster capacitated supply source in Figure 5.6. The higher the CVQ and Util, the lower the share of inventories replenished from the faster supply source. A high CVQ leads to more probable shortages in replenishment from the faster source and there is thus a greater need to compensate for these shortages by placing a bigger share of orders with the slower source. However, as an order with the slower supply source needs to be placed one period earlier, one cannot take advantage of learning about the demand realization in the current period. The accuracy of this compensation strategy su.ers when CVD is increasing, which is re.ected in a high increase in costs. /ë=0.61 =0.14 5 /ë 5 .SA.GŽIAIA.lG...ÐÐu..Ž..ÐG 1.00 0.80 0.60 0.40 0.20 ÜÜÜÜÜLLL ÜÜÜÜÜ2 ÜÜÜÜÜl ÜÜÜÜÜ0.6L ÜÜÜÜÜ0.L ÜÜÜÜÜ0 0.00 0.00 0.20 0.40 0.60 /ëv /ëv Figure 5.6 The share of inventories replenished from a faster capacitated supply source. Although utilization of the faster supply source represents a hard limit on the size of possible replenishment, the actual utilization falls well below its maximum (theoretical) availability. For instance, for Util = 1, the share of inventories sourced from the faster source is at 71%, and decreases further to 41% as demand uncertainty increases (even when CVQ = 0). Still, observe that up to moderate Util and low CVQ the sourcing is done almost exclusively through the faster source (100% for CVD = 0.14), which suggests that the faster supply source is actually su.ciently available in this setting. While we would expect increasing demand uncertainty to make sourcing from the slower supply source less favorable, somewhat counter-intuitively the opposite holds when CVQ is relatively low. In a setting with high CVD more safety stock is required to avoid backorders. To keep inventories at this relatively higher level, occasionally a large order with the faster supply source needs to be placed as a response to the high demand realization (which is more probable due to the high CVD). However, this might lead to a supply shortage even for relatively low system utilizations, and consequently to costly backorders. It therefore makes sense to replenish some inventories from the slower supply source, so that we start the next period with a higher starting inventory position. While this can lead to higher inventory holding costs, it is still less costly than incurring backorder costs. Next, we study the e.ect of the cost structure on the actual utilization of the two supply sources. When backorder costs increase relative to holding costs, increasing the share of sourcing from the slower supply source is optimal (Figure 5.7). This is in line with the above reasoning where, through sourcing from the slower supplier, one can avoid backorders while incurring higher inventory holding costs instead. The share of inventories replenished from a slower supplier also increases relatively more for higher CVQ, particularly when CVD is also high, as the need for a more reliable supply source increases. This e.ect is higher for lower utilizations, which can be attributed to the fact that at high Util ordering through the slower source is already extensively used (around 80% of inventories is replenished from /ë5=0.14 /ë5= 0.00 0.20 0.40 0.60 0 00 0 20 0 40 0 60 /ë/ëv v Figure 5.7 The share of inventories replenished from the faster capacitated supply source. the slower source) and despite the increased backordering costs, increasing the exposure to the slower supply source further only marginally increases the bene.ts. 5.4.3 Bene.ts of dual sourcing Lastly, the bene.ts of dual sourcing are assessed relative to the performance of the two single sourcing cases. To quantify the bene.ts of dual sourcing, we de.ne the relative value of dual sourcing, %VDS , as the relative cost savings over a single-source setting in which either a faster (FS )or slower (SS)supply source is used: {F S,SS} -fDS %VDS/{F S,SS} = ft t , (5.7) {F S,SS}ft , fSS where fF S and fDS represent the corresponding cost functions from (5.1) that apply tt t to a chosen decision policy. We present the results on the relative value of dual sourcing compared to single sourcing in Table 5.3. In the table, numbers in bold are used to denote the single supply source with the lower costs in each of the settings. Looking at the bene.ts of dual sourcing over sourcing from a faster capacitated supply source, it is expected that its utilization largely in.uences the extent of the savings that can be achieved through the dual sourcing. For an overutilized system, %VDS/F S approaches 100%, while for moderate utilizations the relative savings still range from 20% to over 50%. The lack of availability and reliability of supply from a faster source inherently exposes the system to supply shortages, and leads to substantial backorder costs. For up to moderate Util with high CVQ, the average supply availability is su.cient, although the system is exposed to supply shortages due to the highly variable supply capacity of the faster supply source. This again results in considerable bene.ts from utilizing the slower uncapacitated supply source as an alternative. It is the settings with up to moderate Util and low CVQ where supply solely through the faster supply source is adequate to run the system at close to minimum costs. Table 5.3 The optimal costs and the relative value of dual sourcing. Costs (fDS t ) %VDS/F S %VDS/S S Util 8 CVQ CVD 0.00 0.0 0.14 41.8 0.37 112.1 0.61 182.7 0.00 100.0 0.14 93.8 0.37 85.3 0.61 79.1 0.00 - 0.14 0.0 0.37 0.0 0.61 0.0 2 2 2 2 0.00 0.14 0.37 0.61 0.0 0.0 0.0 0.0 22.7 22.7 25.8 32.5 82.9 84.1 89.5 96.8 150.6 151.6 155.7 162.3 100.0 100.0 100.0 100.0 93.6 93.8 93.4 92.4 81.6 81.5 80.9 80.4 73.0 72.9 72.7 72.5 ---- 45.6 45.6 38.2 22.2 26.1 25.0 20.1 13.6 17.6 17.0 14.8 11.1 1 1 1 1 0.00 0.14 0.37 0.61 0.0 0.0 0.0 0.0 22.7 22.7 22.7 30.5 69.5 71.3 78.3 91.1 128.9 131.1 140.1 153.7 -100.0 100.0 100.0 66.8 75.7 87.2 88.8 62.0 63.3 68.5 71.9 56.5 56.8 59.1 61.9 ---- 45.6 45.6 45.6 27.1 38.1 36.5 30.2 18.8 29.4 28.3 23.3 15.8 0.67 0.67 0.67 0.67 0.00 0.14 0.37 0.61 0.0 0.0 0.0 0.0 22.7 22.7 22.7 28.7 68.2 68.2 72.2 87.6 117.1 119.2 129.9 148.5 --100.0 100.0 0.0 0.0 68.7 84.7 1.0 8.5 44.8 62.1 15.3 19.6 36.5 49.8 ---- 45.6 45.6 45.6 31.3 39.2 39.2 35.6 21.9 35.9 34.8 28.9 18.7 0.5 0.5 0.5 0.5 0.00 0.14 0.37 0.61 0.0 0.0 0.0 0.0 22.7 22.7 22.7 27.7 68.2 68.2 69.4 85.2 113.6 114.4 123.1 144.8 --100.0 100.0 0.0 0.0 37.1 81.4 0.0 0.0 20.5 54.7 0.0 0.2 16.5 40.7 ---- 45.6 45.6 45.6 33.8 39.2 39.2 38.1 24.0 37.8 37.4 32.6 20.7 0 0.0 22.7 68.2 113.6 0.0 0.0 0.0 0.0 - 45.6 39.2 37.8 Sourcing exclusively through the slower supply source is optimal in the setting where there is no demand uncertainty (for CVD = 0, the cases denoted with ”-” represent the settings in which the costs of using a single supply mode are zero), and obviously when the faster supply source is completely unavailable (Util = 8). For the remaining settings, we observe that %VDS/SS ranges from 15% to 45%, and is less sensitive to the changes in Util and CVQ than in the case of sourcing solely through the faster supply source, as expected. While utilizing the slower supplier successfully resolves the CVQ and Util related problems in the settings where the faster supplier’s replenishment is inadequate (when %VDS/F S is high), we see that considerable savings are still to be gained by partially sourcing through the faster supply source. As already noted, demand uncertainty negatively a.ects the accuracy of using the slower yet reliable supply source to compensate for potential supply shortages of the faster source. It is therefore somewhat surprising that %VDS/SS are decreasing with increasing CVD. However, the opposite holds when looking at the absolute cost savings. These are increasing consid­erably with an increase of CVD, particularly for up to moderate Util and low CVQ. This is in-line with our previous observations where such a set of parameter values characterizes the settings in which sourcing is performed almost exclusively through the faster supply source. %VDS/F S is in general also decreasing with increasing CVD, which is in agreement with our initial observation that a slower supply source is inherently more exposed to the risk of demand uncertainty. Here the absolute cost savings exhibit non-monotonic behavior with increasing CVD. For settings with high CVQ the slower reliable source is already extensively used and the absolute savings are decreasing with CVD (more so for lower Util). However, this is not the case for low CVQ (particularly at moderate Util), where the absolute savings are increasing with CVD. This is the setting we already addressed in which limited sourcing from the slower supply source is bene.cial for tackling instances of supply unavailability due to the placement of big orders in response to large demand variations. Obviously, the absolute cost reduction is signi.cant in the case where we need to compensate for a general lack of supply availability (high Util and high CVQ), and smaller when the system only faces occasional supply shortages due to large demand uncertainty. Finally, we may conclude with the observation that, besides the setting in which the faster supply source is close to being fully available and essentially reliable, the relative cost savings are considerable, ranging from just above 10% to close to 50%. Based on these results, we can conclude that dual sourcing can successfully resolve the issues related to an overutilized and uncertain availability of a faster supply source. In addition, a smart decision on how to split the orders between the two supply sources depending on the system setting, also substantially reduces the costs of demand uncertainty. 5.5 Value of advance capacity information In this section, we study a situation in which the faster supplier provides an upfront in­formation on exact future supply capacity availability. This information is modeled in the same way as in Jak.si.c et al. (2011) in a single supplier setting (Chapter 3). When ACI is available, the supply capacity availability is known prior to placing the order with the two suppliers, either as ACI on current period’s capacity or on capacity of future periods. Observe that due to the near-optimality of the myopic policy, it is su.cient to obtain the information on current period’s capacity and next period’s capacity, while further future insight into supply capacity availability of the faster supplier will not bring additional ben­e.ts. Due to the myopic nature of the policy it is su.cient to resolve the supply capacity uncertainty for the time span in which .rst the order with the slower supply source, followed by the order with the faster supply source in the next period. We present the results of the numerical analysis, which was carried out to con.rm the above insight and determine the value of sharing ACI of the faster supply channel. We show that ACI can e.ectively help the company to reduce the negative e.ects of the uncertain replenishment from the faster supply channel. In Figure 5.8 we present the cost comparison between the base setting without ACI and the settings in which ACI is available for the current and for the future period, for di.erent utilizations and levels of supply capacity uncertainty of the faster supply source. The cost curve depicted as Util = 0 represents the scenario with the lowest costs, where ordering is done solely with the faster supplier with in.nite capacity. The worst case scenario is the case where the faster supplier has no capacity and all orders are placed to with the slower supplier, and is depicted with a Util = Inf cost curve. Our interest lies in studying the intermediate scenarios, where we are predominantly interested in the extent of visibility needed to obtain the majority of the bene.ts of ACI, and the e.ect the supply capacity uncertainty CVQ has on the costs under both, the situation without and with ACI. //// 190 180 170 1 0 1S0 140 1 0 120 110 100 /ë5///// /ë v In Chapter 3, we have shown that generally the additional bene.t of extending the supply capacity visibility diminishes with revealing ACI for future periods. However, we quickly see that this is not the case in a dual sourcing model under consideration. The value of ACI, de.ned as the relative di.erence in costs between a situation without and with ACI, is up to 1.5% in the case of ACI on current period’s capacity. For the case of future ACI availability the cost savings become considerable, approaching 20%. Thus, we only get the majority of the bene.ts if we extend the information horizon to the following period. Observe .rst that ACI will not a.ect the order placed with the faster supplier as it is always placed up to the myopic base-stock level. It is the possibility that the order with the slower supplier can be modi.ed in accordance to ACI, which results in reduced costs. Relatively small bene.ts obtained through learning only about the current supply capacity availability, can be explained by the fact that with the order with the slower supplier one can only in.uence the inventory levels in the following period. While unfavourable ACI may result in the capacity shortage already in the current period, the order with the slower supplier can only be used to in.uence the costs in the following period. However, in the case of future ACI the capacity shortage in the following period can be anticipated and the order with the slower supplier can be increased accordingly. Looking at the e.ect of supply capacity uncertainty of the faster supplier, the costs of the situation without and with ACI are equal for CVQ = 0. It is expected that the reduction in costs through ACI is increasing when CVQ is increasing, as ACI e.ectively reduces the uncertainty of the supply capacity availability. The largest savings are attained at settings with high CVQ, no matter what the utilization of the faster supplier is. While the supply capacity uncertainty of the faster supplier has the predominant e.ect on the bene.ts of ACI, the demand uncertainty also in.uences the possible savings attained through ACI. In the case of a highly utilized faster supplier, the slower supplier is used more heavily. While ACI helps the decision maker to come up with a better ordering decision with the slower supplier, the delayed replenishment together with demand uncertainty results in high costs due to demand and supply mismatches, thus the value of ACI diminishes. This e.ect is lower for the case of low utilization of the faster supplier. Here ACI is helpful in covering the mismatches mainly through the faster supplier, therefore the exposure to the demand uncertainty risk faced by a supply through the slower supplier is lower. 5.6 Summary In this chapter, we analyze a sourcing situation in which both sourcing from a nearshore and o.shore supply source may be used. As the product purchase costs of sourcing from the two sources are more or less similar in various industry sectors nowadays, we argue that the trade-o. is leaning towards .nding the right balance between the limited supply availability of the nearshore supply options and the longer lead times of o.shore sourcing. Accordingly, we model a dual-sourcing inventory model with a stochastic capacitated, zero-lead-time nearshore supplier and an uncapacitated o.shore supplier with a longer lead time. We derive the dynamic programming formulation for the optimal inventory holding and backorder costs, where we show that the structure of the optimal inventory policy is not a two-level base-stock policy as it is the case in both unlimited and .xed capacity dual sourcing models studied in the literature. The optimal inventory position to which we order by placing an order with the stochastic capacitated faster supplier depends on the inventory position before ordering. To reduce the complexity of the policy we study the myopic policy, which proves to be a two-level base-stock policy, where the .rst myopic base-stock level corresponds to the solution of the single period newsvendor problem. We show that the myopic policy is not optimal, however it still provides a very accurate approximation of the optimal costs. The numerical analysis reveals several managerial insights. The results show that the costs increase substantially when the capacity of the faster supply source is stochastic (already for moderate utilization of the faster supply source). The extent to which we rely on the faster supply source will therefore depend on the trade-o. between its current availability and the bene.ts we obtain by delaying the order decision, and thus taking advantage of the latest demand information. However, somewhat counter-intuitively, we observe that in some settings the increasing demand uncertainty leads to an increase in sourcing from the slower supplier. We attribute this to the high costs related to incurring backorders, which results in a strategy primarily oriented to avoiding supply shortages, namely a strategy that instructs the decision-maker to prebuild inventories by utilizing the slower supply source. In addition, we show that the relative cost savings of dual sourcing over single sourcing range from 10% to close to 50%. Exceptions to this are cases where the faster supply source is only moderately utilized and the demand uncertainty is low. This shows that, despite the lack of a price incentive for placing orders with the slower o.shore supplier (which is a common assumption in the literature), the o.shore supply option is used to obtain considerable operational cost savings. Given the limitations and uncertainties that the move towards nearshoring brings in terms of supply availability, we show that a mixed strategy which allocates the orders between the two sourcing options is optimal. Our results provide a potential justi.cation for continuing to use o.shore sources following the commencement of nearshoring operations, despite the fact that the costs are similar and the lead times are longer. Managers will need to take the capacity .exibility of the nearshore option into account when moving an operation nearshore. Future research could relax the assumptions about the two delivery lead times. However, assuming a non-zero-lead time of the faster supply source greatly increases the complexity of the model as the inventory position becomes a function of multiple uncertain pipeline orders. Accordingly, also the myopic policy could be very complex (state-dependent .rst base-stock level) and probably less accurate, even in the case where the lead times of the two supply sources di.er by one period. Chapter 6 Conclusions We now present the conclusions of the research presented in this monograph. We provide answers to the relevant research questions formulated in Section 1.4. Note, that this is a gen­eral overview of the insights we have gained throughout our study, as the individual model’s speci.cs have already been discussed extensively in conclusions of each of the chapters. The work presented in this monograph is aimed at giving insights into managing inventories in a stochastic demand and limited stochastic supply capacity setting. We have developed four inventory models with di.erent information sharing and supply options aimed at im­proving the inventory system’s performance. The theoretical contribution of the monograph revolves around the .rst two relevant research questions: Q1. How can we incorporate supply capacity information and alternative supply options into the underlying stochastic capacitated inventory model? The road towards developing the proposed inventory models started through derivation of a discrete model formulation, identi.cation of the system control variables, description of the state space, and representation of the evolution of the system by giving the relevant func­tional equations. In the case of ACI and ASI models in Chapters 3 and 4, the state space needs to be expanded to incorporate supply capacity information. In the case of ACI model, the state of the system is described by the starting inventory position and the information on supply capacity availability in future periods. In the case of ASI model, advance replen­ishment information is used to correct the inventory position in the case of supply shortage. For the remaining pipeline orders the replenishment remains uncertain, which needs to be accounted for by keeping track of these orders in the state space description. In the dual sourcing model in Chapter 5 the orders to the two supply sources are interdependent. Apart from ACI model, the inventory position before replenishment is uncertain in the remaining two models, which presents a di.culty in terms of estimating the resulting on-hand stock availability to cover demand. This is subsequently re.ected in increased computational com­plexity associated with the quanti.cation of the relevant inventory costs. Finally, we derive a dynamic programming formulation for the .nite horizon optimal discounted expected in­ventory costs. Q2. What is the structure of the optimal ordering policy and can we derive its properties? By studying the convexity properties of the relevant cost functions, we are able to characterize the structure of the optimal policy for all the models under study. We .rst show that in the case of ACI model in Chapter 3 the auxiliary cost function is convex, thus there exists a single minimizer that minimizes the system costs. We show that the optimal inventory position after ordering does not depend on the inventory position before ordering, which characterizes the base-stock policy. We show that the optimal policy is the modi.ed base-stock policy with a single base-stock level, which is a function of ACI, where the order in each period is placed up to the base-stock level if su.cient supply capacity is available. The ASI model in Chapter 4 is an extension of the underlying stochastic inventory model studied in Ciarallo et al. (1994). Despite the fact that the single period cost function (and consequently the multiperiod cost functions) exhibits the quasiconvex shape, we show that the minimum costs are attained at the single state-dependent inventory position. In line with this, we show that the optimal policy is the base-stock policy, where the optimal base-stock level depends on the remaining uncertain pipeline orders. For both advance information models we derive additional structural properties that describe the state dependency of the optimal policy parameters. Favourable information about future supply capacity availability decreases the optimal base-stock level in the ACI case. In the case of ASI, the exposure to the potential supply shortages depends on the size of the remaining uncertain pipeline orders. Thus, the optimal base-stock level needs to be increased in the case when higher orders were placed in the recent past. The properties needed for the optimality of the base-stock policy cannot be proven for the dual sourcing model in Chapter 5. One would expect that the optimal policy would be a two-level base-stock policy observed in the uncapacitated case and the case with .xed capacity. We show that the inventory position after placing the order with the faster supplier in fact depends on the inventory position before ordering, and thus does not behave as a base-stock policy. The inventory position after placing the order with the slower supplier can be characterized as the secondary base stock level, however it is state-dependent on the order placed with the faster supplier. Study of the myopic policy showed that it accurately approximates the cost of the optimal policy despite the simpli.ed structure of the policy. Myopic policy is a two-level base-stock policy, where the .rst base-stock level corresponds to the newsvendor base-stock level of an uncapacitated single period inventory problem. Even in the cases where the target inventory positions of the optimal policy and the myopic policy di.er a lot, we observe nearly perfect accuracy of the myopic policy. In general, it holds that for an e.ective inventory control it is not practical to derive the optimal action for each of the possible system states, but rather an e.ort should be put in .nding an optimal policy, a structured rule that describes the optimal behavior. However, while we have shown that the optimal policies have a relatively simple structure, the calcu­lation of the optimal base-stock levels remains a di.cult task. The reason for this is twofold. Firstly, due to the state-dependency, the base-stock level generally needs to be determined for each system state. The second reason is that due to the complex state-description, and the stochastic nature of both the demand and supply process, the problem complexity becomes large and eventually too large to enable the evaluation of the optimal costs, the problem commonly referred to as ”Curse of dimensionality” (Puterman, 1994). This greatly reduces the likelihood that a realistic inventory problem can be solved. It turns out that the numer­ical analysis of the optimal policies is not computationally e.cient and even not feasible for larger problems. At the same time, we show that oversimpli.cation and disregarding the stochastic nature of the model leads to poor results. While, these observations may limit the applicability of the optimal policy in a practical setting, we now proceed to review the managerial insights gained through the numerical analysis of the proposed inventory policies. By providing answers to the second group of research questions we aim to reveal the extent of the possible gains obtained through integrating information or alternative supply options into the inventory system with uncertain capacity. As the underlying model is inherently di.cult to manage, both due to the limited as well as uncertain supply availability, we show that the potential for the cost reduction is considerable. This should motivate the companies to explore the insights gained through the study of the proposed four inventory models, and correspondingly adapt their existing inventory control policies. Q3. What is the value of supply capacity information and alternative supply options in terms of inventory cost reduction? Foreknowledge of future uncertain events is always useful in managing an inventory system. We argued that through advance information about supply availability one can better an­ticipate the potential mismatches between demand and supply capacity in the future. This anticipation of future supply shortages is bene.cial to make timely and correct ordering de­cisions, which result in either building up stock to prevent future stockouts, or reducing the stock in the case the supply conditions in the future might be favorable. Thus, system costs can be reduced by carrying less safety stock while still achieving the same level of perfor­mance. In the case of ACI model in Chapter 3, we observe the reduction of inventory costs of up to 30% and more, particularly for certain settings in the special case of “zero-full” supply capacity availability in Section 3.5. The cost savings in the case of ASI model in Chapter 4 are lower, going up to around 10%. This is expected as the supply capacity visibility is limited to the orders already placed, and therefore only allows the decision maker to react to the supply shortage revealed through ASI. While advance information will not increase the supply availability, it improves either the anticipation of the future supply shortages or enables a more timely reaction to inadequate replenishment of pipeline orders. The idea behind the dual sourcing model in Chapter 5 is that the exposure to the supply shortages of the primary supply channel can be reduced by relying to alternative supply option. The option to place a part of the order with the slower reliable supplier reduces the costs substantially compared to sourcing solely from the stochastic capacitated supplier, as well as relying fully to the slower supplier. To further elaborate the setting in which the proposed inventory policies provide the majority of the bene.ts, we review the last research question: Q4. Which are the determining factors of the magnitude of the expected cost bene.ts, or more speci.cally, what is the in.uence of the relevant system parameters? The numerical study of the value of information sharing and alternative supply options re­volves around analyzing the e.ect of the relevant system parameters on the inventory costs: utilization of the primary stochastic capacitated supply channel, the level of demand and supply capacity uncertainty, and the ratio between the inventory holding and the backorder costs. We show that the utilization has the predominant e.ect on the inventory cost reduc­tion. A highly utilized or even overutilized system is in most of the settings the primary candidate in which one should try to take advantage of the proposed supply alternatives. In general, we observe that the remaining parameters e.ect the inventory cost reduction in a complex way. Due to the interaction between the multiple parameters, we show that each cannot be treated independently and therefore need to be considered in an integrated manner. In the case of ACI in ASI models, interesting conclusions were drawn based on the analysis of the e.ect of changes in uncertainty of demand and supply capacity. Increasing the uncertainty of demand caused a decrease in the value of advance information. This can be attributed to the fact that advance information is of limited use when the system gets more and more chaotic due to increased demand variability. However the opposite is true when supply capacity becomes more stochastic. We observe that the value of advance information signi.cantly increases. We attributed this to the fact that when one has to deal with highly variable and uncertain supply capacity availability every bit of information on the capacity realization helps in reducing the uncertainty. In the case of the alternative supply option, the relationship between the demand and supply capacity uncertainty and the inventory cost reduction is nontrivial. We show that a hybrid strategy of employing both the primary uncertain supplier and the alternative slower reliable supplier needs to be implemented. The share of the order placed with each of the suppliers will heavily depend on the particular system characteristics. By showing the potential gains that can be made by incorporating advance information and alternative supply options, we provide an incentive to companies to more e.ectively incorporate already available information into their inventory policies, establish the necessary mechanisms that would facilitate the additional information exchange, and proactively look for the possible alternative supply channels to improve the supply availability. Appendix A Chapter 3 Proofs Preliminaries for Theorem 3.1 Lemma A.1 Let b . B, where B is a convex set, and y . Y(b), where Y(b)is a nonempty set for every b, assume that g(y, e) is convex in y and e. If f(b, e) := minAy=b g(y, e) and f(b, e)= -8 for every b and e, then f is also convex in b and e. Proof: Let 0 = . = 1. Then .f(b, e) + (1 -.)f(¯b, e¯) = .min g(y, e) + (1 -.)min g(y, e¯) Ay=b Ay=¯b = .g(y0* , e) + (1 -.)g(y1* ,e¯) = g(.(y0* , e) + (1 -.)(y1 * ,e¯)) (A.1) = min g(y, .e + (1 -.)¯e) (A.2) Ay=.b+(1-.)¯b = f(.(b, e) + (1 -.)(¯b, e¯)) ¯ (A.1) is due to convexity of g and (A.2) is because Ay0 * = b and Ay1 * = b, which implies * A(.y0 * + (1 -.)y1)= .b + (1 -.)¯b. . Lemma A.2 If J(y, e) is convex in y and e, then f(x, q, e)= minx=y=x+q J(y, e) is convex in x. Proof: Let h(b, e):= minAy=b J(y, e)where A = [-1,1] and b = [-x, x +q]. By Lemma A.1, we conclude that h(b, e)is convex in b and e. Let b = x+z, where 0 = z = q for any q, then we have h(b, e)= f(x, q, e)and since this is a.ne mapping, f is also convex in x. . Proof of Theorem 3.1: The theorem can be proven by regular inductive arguments and the results of Lemmas A.1 and A.2. . Proof of Theorem 3.1: Convexity results of Theorem 3.1 directly imply the proposed structure of the optimal policy. . Proof of Theorem 3.2: In the case with no ACI, the decision maker has to decide for the order size without knowing what the available supply capacity for the current period will be. Inventory position after ordering yt only gets updated after the order is actually received, where it can happen that the actual delivery size is less than the order size due to the limited supply capacity. The single-period cost c(x, z)for q = 0 is: 18 1x+z c(x, z) = (1 -R(z))b (d-x -z)g(d)dd+ h (x + z -d)g(d)dd x+z 0 1 18 z + b (d-x -q)g(d)dd r(q)dq 0 x+q 1 1 z x+q + h (x + q-d)g(d)dd r(q)dq. (A.3) 0 0 The .rst line in (A.3) represents the case where the capacity q is higher than the order size z, which means that the capacity is not limiting the order placed. While the rest accounts for the case of the capacity constrained system, where the order size is reduced due to limited supply capacity. We proceed by writing the optimal discounted expected cost function from period t to T: 1 8 Ht(xt, zt) = ct(xt, zt) + a(1 -Rt(zt)) Ht+1(xt + zt -dt)g(dt)ddt 0 1 zt 1 8 + a Ht+1(xt + qt -dt)g(dt)ddt rt(q)dq, for 1 = t = T (A.4) 0 0 where HT+1(·)= 0. The .rst-order condition for optimality is obtained by taking .rst partial derivative of (A.4), where zt * is the optimal order size in period t: 1 8 ** * (1 -Rt(z ))(h+ b)G(xt + z )- b + a H * ' (xt + z -dt)g(dt)ddt= 0. tt t+1 t h+ b0 (A.5) The optimal ordering policy is a base-stock policy with the optimal base-stock level yt * . We are now ready to show the equivalence of the two models. By using (3.5) and (A.7) we write the optimal discounted expected cost function ft for n = 0 as: . . Ct(xt) + aEDt,Qt+1 yˆt = xt, . ft+1(xt -Dt), ft(xt, qt)= Ct(ˆyt) + aEDt,Qt+1 ft+1(ˆyt -Dt), yˆt -qt = xt < yˆt, (A.6) . . Ct(xt + qt) + aEDt,Qt+1 ft+1(xt + qt -Dt), xt < yˆt -qt, Assuming that the single-period cost function Ct(yt)and the optimal discounted cost function ft(xt, qt)are di.erentiable, we write the following three possible cases: Case 1: When ˆyt = xt, ft(xt, qt) = EDt [b(xt -dt)- + h(xt -dt)+] + aEDt ft+1(xt -dt) 11 1 8 xt 8 = b (dt -xt)gt(dt)ddt + h (xt -dt)gt(dt)ddt + a ft+1(xt -dt)gt(dt)ddt. xt 0 0 Case 2: When ˆyt -qt = xt < yˆt, 1 8 ft(xt, qt) = EDt [b(ˆyt -dt)- + h(ˆyt -dt)+] + aEDt ft+1(ˆy-dt) rt(qt)dqt yˆt-xt 1 8 1 zt xt+ˆ= (1 -Rt(ˆzt)) b (dt -xt -zˆt)gt(dt)ddt + h (xt + ˆzt -dt)gt(dt)ddt xt+ˆzt 0 1 8 + a(1 -Rt(ˆzt)) ft+1(xt + ˆzt -dt)gt(dt)ddt. 0 Case 3: When xt < yˆt -qt, 1 yˆt-xt ft(xt, qt) = EDt [b(xt + qt -dt)- + h(xt + qt -dt)+] + aEDt ft+1(xt + qt -dt) rt(qt)dqt 0 11 1 zˆt 8 xt+qt = b (dt -xt -qt)gt(dt)ddt + h (xt + qt -dt)gt(dt)ddt rt(qt)dqt 0 xt+qt 0 1 1 zˆt 8 + a ft+1(xt + qt -dt)gt(dt)ddt. 0 0 The proof is by backward induction starting in time period T. Observe that for period T, yT * = yˆT holds, since both optimal base-stock levels, since both optimal order sizes are solutions of the same single-period newsvendor problem (Ciarallo et al. (1994), Proposition 1). Based on this we conclude that for period T the equivalence of the models holds obviously for Case 1 and if we sum up the Cases 2 and 3 (xT < yˆT), we see the sum is equivalent to the de.nition of cT(xT,zˆT)given in (A.3). From here it holds fT(xT)= HT(xT). Assuming that ft+1(xt+1)= Ht+1(xt+1). The optimal base-stock level ˆyt is a solution of the following equation: b (h+ b) G(ˆyt)- + aEDt f ' (ˆyt -dt)= 0. h+ b t+1 Thus the equivalence ˆyt = yt * holds from (A.5) and the induction assumption. Again we analyze the three possible cases given through (A.6). Adding up the Cases 2 and 3 (xt < yˆt), leads to ft(xt) = Ht(xt) from the de.nition of ct(xt, zt) given in (A.3), and the induction argument is concluded. . Lemma A.3 Let f(x) and g(x) be convex, and xf and xg be their smal lest minimizers. If f ' (x)= g ' (x)for all x, then xf = xg. Proof: Following Heyman and Sobel (1984), lets assume for a contradiction that xf < xg, then we can write 0 < d = xg -xf and f ' (xg -d)= 0 holds. From f ' (xg -d)= 0 and initial assumption f ' (x)= g ' (x)we have 0 = f ' (xg -d)= g ' (xg -d). Observe that g ' (xg -d)= 0 contradicts the de.nition of xg being smallest minimizer of g(·), hence xf = xg. . Proof of Theorem 3.3: We .rst write the optimal cost function ft(x, q, zq) using the de.nition given in (3.4) and following the results of Theorem 3.1, as . . Jt(x, zq), yˆt(zq)= x, . ft(x, q, zq)= Jt(ˆyt(zq), zq), yˆt(qz)-q = x < yˆt(qz), (A.7) . . Jt(x + q, zq), x < yˆt(zq)-q. From this, the de.nition of the .rst derivative and the convexity of Jt in y proven in Theorem 3.1, we can write ft ' (x, q, zq)in the following manner . . = 0, yˆt(qz)= x, . ft ' (x, q, zq)= = 0, yˆt(qz)-q = x < yˆt(qz), (A.8) . . < 0, x < yˆt(zq)-q. Starting the induction argument at time T we .rst observe that zqT has no components due ' '' to (3.1). Part 1 holds as an equality since JT(y, zq)= JT(y)= CT(y), and using (A.7) and (A.8) it is easy to show that Part 2, fT ' (x, q2)= fT ' (x, q1), holds. Part 3 holds as an equality since ˆyT and is independent of q. Assuming that Jt ' (y, zq2)= Jt ' (y, zq1)also holds for t we have yˆt(z= ˆq1) by using Lemma A.3. To prove that this implies f ' (x, q2, z= f ' (x, q1, z q2) yt(zt q2) t q1) for t, we need to analyze the following nine cases: ' ''' Case 1: If x = yˆt(zq2)and x = yˆt(qz1), then f (x, q2, zq2)= J (x, zq2)= J (x, zq1)= f (x, q1, zq1), t ttt where the inequality is due to Part 1. Case 2: If ˆyt(qz2)-q2 = x < yˆt(zq2)and x = yˆt(qz1), then ft ' (x, q2, zq2)= 0 = ft ' (x, q1, zq1). Case 3: If x < yˆt(zq2)-q2 and x = yˆt(zq1), then ft ' (x, q2, zq2)< 0 = ft ' (x, q1, zq1). Case 4: x = yˆt(zq2)and ˆyt(zq1)-q1 = x < yˆt(zq1)is not possible due to Part 3. Case 5: If ˆyt(qz2)- q2 = x < yˆt(qz2) and ˆyt(zq1)- q1 = x < yˆt(zq1), then ft ' (x, q2, zq2) = 0 = ft ' (x, q1, zq1). Case 6: If x < yˆt(zq2)-q2 and ˆyt(qz1)-q1 = x < yˆt(zq1), then ft ' (x, q2, zq2)< 0 = ft ' (x, q1, zq1). Case 7: x = yˆt(zq2)and x < yˆt(zq1)-q1 is not possible due to Part 3. Case 8: ˆyt(zq2)- q2 = x < yˆt(zq2) and x < yˆt(zq1)- q1 is not possible due to ˆyt(qz2)-q2 > yˆt(zq1)-q1, which is due to Part 3 and q2 = q1. '' ' Case 9: If x < yˆt(zq2)-q2 and x < yˆt(zq1)-q1, then ft(x, q2, zq2)= Jt(x + q2, zq2)= Jt(x + q1, zq2)= Jt ' (x+q1, zq1)= ft ' (x, q1, zq1), where the .rst inequality is due to q2 = q1 and convexity of Jt in y, and the second inequality is due to Part 1. Going from t to t - 1 we conclude the induction argument by showing that ft ' (x, q2, zq2)= ''' '' ft(x, q1, zq1) implies Jt-1(y, zq2)= Jt-1(y, zq1). We write Jt-1(yt-1, zq2,t-1)= Jt-1(xt-1, qt-1, zq2,t-1), where (xt-1, qt-1, zq2,t-1) denotes the state space in t-1and (x2,t, q2,t, q2,t+1,. . . , q2,t+n-1, Qt+n) refers to the resulting next period’s state space, where the last element of ACI vector, Qt+n, is random. Inventory position x2,t is updated through (3.2), where the order size zt-1 is limited by the available supply capacity qt-1 in period t -1. Using (3.5) we write: ' '' z Jt-1(yt-1, zq2,t-1) = Ct-1(yt-1) + aEf t(x2,t, q2,t,Q2,t) = Ct' -1(yt-1) + aEf t ' (x1,t, q2,t, q2,t+1,. . . , q2,t+n-1, Qt+n) = Ct' -1(yt-1) + aEf t ' (x1,t, q1,t, q1,t+1,. . . , q1,t+n-1, Qt+n) = Jt' -1(yt-1, zq1,t-1), For q2,t-1 = q1,t-1, it holds that x2,t = x1,t from the state update (3.2). The .rst inequality holds directly from convexity of ft in x. From zq2,t-1 = zq1,t-1 and the state update (3.3), it follows that (q2,t, q2,t+1,. . . , q2,t+n-1) = (q1,t, q1,t+1,. . . , q1,t+n-1). The second inequality is due to induction argument and the fact that taking expectation over the same random variable Qt+n preserves inequality. . Lemma A.4 For any zq and . > 0 and all t, we have: Q1. Jt ' (y-., zq)= Jt ' (y, zq-.e1), Q2. yˆt(zq-.e1)-yˆt(zq)= ., Q3. ft ' (x -., q, zq)= ft ' (x, q, zq-.e1). Proof: We .rst prove that ft ' (x-., q, zq)= ft ' (x, q -., zq) holds for all x and t. Observe that the smallest minimizer of Jt(x -., zq)is ˆyt(qz) + ., and we analyze 4 valid cases using (A.7) and (A.8): Case 1: If ˆyt(zq)= x-. and ˆyt(qz)= x, then ˆyt(zq)+. = x and ft ' (x-., q, zq)= Jt ' (x-., zq)= Jt ' (x, zq)= ft ' (x, q -., zq), where the equality is due to convexity of Jt in y. Case 2: If ˆyt(zq)- q = x - . < yˆt(zq) and ˆyt(zq) = x, then ˆyt(qz) = x < yˆt(zq)+ . and ft ' (x -., q, zq)= 0 = ft ' (x, q -., zq). Case 3: If ˆyt(zq)-q = x-. < yˆt(zq)and ˆyt(qz)-q+. = x < yˆt(zq), then ˆyt(zq)-q+. = x < yˆt(qz) and ft ' (x -., q, zq)= 0 = ft ' (x, q -., zq). Case 4: If x < yˆt(zq)- q + ., then ft ' (x - ., q, zq) = Jt ' (x - . + q, zq) = Jt ' (x + q - ., zq) = ft ' (x, q -., zq). To prove Part 1 we write: ' '' z Jt(y-., zq) = Ct(y) + aEf t+1(y-.-Dt, qt+1,Qt+1) z = C ' (y) + aEf ' (y-Dt, qt+1 -., tt+1Qt+1) = Jt ' (y, zq-.e1). (A.9) The .rst inequality is due to convexity of Ct(y), the second inequality is due to ft ' (x-., q, zq)= ft ' (x, q -., zq)and the equality is from de.nition of Jt in (3.5). Since the smallest minimizers of functions Jt(y-., zq)and Jt(y, zq-.e1)are ˆyt(zq) + . and ˆyt(zq-.e1), Lemma A.3 implies yˆt(zq-.e1)= yˆt(qz) + . and this proves Part 2. We continue to prove Part 3 for all x and t. We use (A.7), (A.8) and the fact that ˆyt(zq-.e1)= yˆt(zq)holds from Part 3 of Theorem 3.3, and consider nine possible cases: ' '' Case 1: If ˆyt(zq)= x-. and ˆyt(zq-.e1)= x, then f (x-., q, zq)= J (x-., zq)= J (x, zq-.e1)= t tt ft ' (x, q, zq-.e1). The inequality follows from Part 1. Case 2: If ˆyt(zq)- q = x - . < yˆt(qz) and ˆyt(zq - .e1) = x, then ft ' (x - ., q, zq) = 0 = ft ' (x, q, zq-.e1). Case 3: If x -. < yˆt(qz)-q and ˆyt(qz-.e1)= x, then ft ' (x -., q, zq)< 0 = ft ' (x, q, zq-.e1). Case 4: ˆyt(zq)= x -. and ˆyt(qz-.e1)-q = x < yˆt(zq-.e1)is not possible due to Part 2. Case 5: If ˆyt(zq)-q = x-. < yˆt(zq)and ˆyt(zq-.e1)-q = x < yˆt(zq-.e1), then ft ' (x-., q, zq)= 0 = ft ' (x, q, zq-.e1). Case 6: If x -. < yˆt(qz)-q and ˆyt(qz-.e1)-q = x < yˆt(qz-.e1), then ft ' (x -., q, zq)< 0 = ft ' (x, q, zq-.e1). Case 7: x -. < yˆt(qz)-q and ˆyt(zq-.e1)= x is not possible due to Part 2. Case 8: ˆyt(zq)-q = x -. < yˆt(qz)and ˆyt(zq-.e1)= x is not possible due to Part 2. Case 9: If x -. < yˆt(zq)-q and x < yˆt(zq-.e1)-q, then ft ' (x -., q, zq)= Jt ' (x -.+q, zq)= Jt ' (x + q, zq - .e1) = ft ' (x, q, zq - .e1). The inequality follows from Part 1. With this we conclude the proof of Part 3. . Proof of Conjecture 3.2: The proof of follows directly from Conjecture 3.1 and Part 2 of Lemma A.4 for i > 1 because ˆyt(zq - .ei)- yˆt(zq) = yˆt(zq - .ei-1)- yˆt(zq) = · · · = yˆt(zq-.e1)-yˆt(zq)= .. This proof was inspired by ¨ Ozer and Wei (2003). . Appendix B Chapter 4 Proofs Before giving the proofs, we .rst provide the needed notation and the de.nitions, which enables us to give the proofs in a concise manner. For clarity reasons, we elect to suppress the time subscripts in certain parts of the proofs. We also assume a = 1 for the same reason. For the m uncertain pipeline orders in z t, we know that any particular order zi, where i = t-m. . . t-1can either be deliveredin fullor onlypartiallydependingon the available supply - capacity revealed through ASI. Based on this we de.ne the vector z , which represents the t + set of orders zi that will not be delivered in full, zi > Qi. The vector represents the set z t -- ++ Ø of orders zi that will be fully replenished, zi = Qi. Thus, z n z t and z . z = z = t t t t holds. As it will be useful in some of the following derivations to include also the order zt - , we also de.ne the extended vectors Zzt - and Zz ++ into the two vectors z and z . The two t t t corresponding supply capacity vectors are denoted as Qz - t and Qz + t . We denote the cumulative distribution function of the demand with G(Dt), and the corre­sponding probability density function with g(Dt), and the lead time demand counterparts with GL(DL)and gL(DL). The cumulative distribution function and the probabilitydensity tt tt function of supply capacity Qt are denoted as Rt(Qt) and rt(Qt). We assume that all the distributions are stationary. In the followinglemma, we provide the convexityresults and the optimal solution to a single period cost function Ct(yt,z t,zt). Lemma B.1 Let yˆt M be the smallest minimizer of Ct(yt,z t,zt) to which the optimal order M MM zˆis placed, where zˆ= yˆ-xt: t tt Q1. Ct(yt,z t,zt)is convex in yt. Q2. Ct(yt,z t,zt)is quasiconvex in zt. Q3. yˆt M(z t)is the state-dependent optimal myopic base-stock level. Proof: C(y,z ,z)is expressed in the following way: 1 .- 1 1 Z 8 8 (.) C(y,z ,z)= b DL -y+ (Zz- -Qz-)r(Qz)dz QgL(DL)dDL .+ (.--. 0 Z y-Z Q-) 1 Q-) 1 Z.- 1 8 y-(Z.--.() z +h y- Z- - r(zQgL(DL)dDL (zQ-)-DLQ)dz.(B.1) 0 Z 0 .+ To prove Part 1, we derive the .rst partial derivative of (B.1)with respect to y, where we ()T take into account that (1-R(Zz+))R(Zz-)= 1: 1 Z.- ()r . . GL - - zQ- C(y,z ,z)= -b+ (b+ h) (1-R(z+)) y- (zQ-)r(Qz-)dz(B.2) ZZ ,.y 0 and the second partial derivative: 1 .- .2 r Z (.) z C(y,z ,z)= (b+ h) (1-R(Zz+)) g L y- (Zz- -Q-)r(Qz-)dQz- . (B.3) .y2 0 Since all terms in (B.3)are nonnegative, Part 1 holds. It is easy to see that the convexity also holds in x. To show Part 2, we obtain the .rst two partial derivatives of C(y,z ,z)with respect to z: . C(y,z ,z) = (b+ h)(1-R(z)).z 1 - r.z (.) (1-R(z +)) GL y- (z - - zr(Qz-)dQz- - b (B.4) Q-), 0 b+ h 1 - .2 r.z (.) C(y,z ,z) = -r(z)(b+ h)(1-R(z +)) GL y- (z - -Qz-)r(Qz-)dQz- - b .z2 0 b+ h 1 - r.z (.) + (b+ h)(1-R(z))(1-R(z +)) g L y- (z - -Q-)Q-)dQz-(B.5) zr(z, 0 Setting (B.4) to 0 proves Part 3. Observe that ˆyM(z ) only depends on z , and not on z. Intuitively this makes sense, as due to the potential shortages in replenishment of any of uncertain pipeline orders z we increase ˆyM accordingly. However, when placing order z, it is not rational to adjust ˆyM to account for the potential shortage in replenishment of z. One merely has to hope that by ordering z up to ˆyM , the available supply capacity will be su.cient. For z = zˆM , the bracketed part in the .rst term of (B.5)is not positive, thus the .rst part is nonnegative as a whole. Since also the second term is always nonnegative, the function C(y,z ,z)is convex on the respected interval. For z > zˆM this does not hold, however we see that (B.4)is nonnegative, thus C(y,z ,z)is nondecreasing on the respected interval, which proves Part 2. Due to this, the C(y,z ,z)has a quasiconvex form, which is su.cient for ˆyM to be its global minimizer. Note, that one can show that C(y,z ,z)is quasiconvex in anyofzi, where i = t-m. . . t-1, in the same wayas presentedabove. The above derivation can be consideredas a generalization of the derivations for the zero lead time model presented in Ciarallo et al. (1994). We have shown that the convexity properties of the single period function also holds for the positive lead time case. . ProofofLemma 4.1: The proofis byinduction on t. For periodT it holds JT(yT,z T,zT)= CT(yT,z T,zT), whichbyusingthe result ofLemma B.1proves the convexityofJT(yT,z T,zT) in yT, and using (4.6)for T, also the convexity of fT(yT,z T)in xT. Assuming that ft+1 is convex in xt+1, we now want to show that this implies convexity of Jt in yt and ft in xt. Using (4.5)we write Jt as: Jt(yt,z t,zt) = Ct(yt,z t,zt) 1 8 1 zt-m + ft+1(yt -(zt-m -Qt-m)-Dt,z t+1)r(Qt-m)dQt-mg(Dt)dDt 0 0 1 8 + (1-R(zt-m)) ft+1(yt -Dt,z t+1)g(Dt)dDt. (B.6) 0 By taking the second partial derivative of (B.6)with respect to yt we quickly see that the convexity of Jt in yt is preserved due to the convexity of Ct in yt coming from Lemma B.1, while the the remaining two terms are convex due to the induction argument. To show that this also implies convexity of ft in xt, we .rst take the .rst partial derivative of Jt with respect to zt 1: . . J(yt,z t,zt) = C(yt,z t,zt) .z .z 1 1 8 zt-m + ft' +1(yt -(zt-m -Qt-m)-Dt,z t+1)r(Qt-m)dQt-mg(Dt)dDt 0 0 1 8 + (1-R(zt-m)) ft' +1(yt -Dt,z t+1)g(Dt)dDt, (B.7) 0 Partially di.erentiating (4.6)with respect to xt twice, using (B.2)and taking into account ' 1We de.ne the .rst derivative of a function ft(x)with respect to x as f (x). t the .rst-order optimality condition by setting (B.7)to zero, yields the following: r 1 Z.t - (.) .2 + -- -- f(xt,z t) = (b+ h) (1-R(Zzt )) g L xt + ˆzt - (Zzt -Qzt )r(Qzt )dQzt.x2 t 0 1 8 zt-m 1 + ft'' +1(xt + ˆzt -(zt-m -Qt-m)-Dt,z t+1)r(Qt-m)dQt-mg(Dt)dDt 0 0 1 8 + (1-R(zt-m)) ft'' +1(xt + ˆzt -Dt,z t+1)g(Dt)dDt. (B.8) 0 While the expression does not get simpli.ed as in the zero lead time case, we can easily conclude that the convexityofft in xt is also preservedas allthe terms above are nonnegative. Proof of Theorem 4.1: The convexityresults of Lemmas 4.1 and B.1 implythe proposed optimal policy structure. . Lemma B.2 Let f(x) and g(x) be convex, and xf and xg be their smallest minimizers. If f ' (x)= g ' (x)for all x, then xf = xg. Proof: Following Heyman and Sobel (1984), lets assume for a contradiction that xf < xg, then we can write 0 < d = xg -xf and f ' (xg -d)= 0 holds. From f ' (xg -d)= 0 and initial assumption f ' (x)= g ' (x)we have 0 = f ' (xg -d)= g ' (xg -d). Observe that g ' (xg -d)= 0 contradicts the de.nition of xg being smallest minimizer of g(·), hence xf = xg. . Proof of Theorem 4.2: Using (4.4)and Part 2 of Theorem 4.1, the optimal cost function ft(x,z )can be written as: Jt(x,z ,0), yˆt(z )= x, ft(x,z )= (B.9) Jt(ˆyt(z ),z ,z), x < yˆt(z ). Due to the convexity of Jt in y, we can write ft ' (x,z )in the following manner ' = 0, yˆt(z )= x, f (x,z )= (B.10) t = 0, x < yˆt(z ). We start the induction argument at time T, where JT(y,z ,z) = CT(y,z ,z) holds. Using (B.2), we show that the following holds for any z 1 =z 2 and z1 = z2: 1-R(Zz1 +)= 1-R(Zz1 +), 1 - - Z 1 .1 (.)Z.2 (.) GL (Zz- Qz-)r(zQ- = GL (z- Qz-)r(zQ- y- 1 - Q-)dzy- Z2 - Q-)dz, 0 0 - '' due to y- (Zz-Qz-)= y- (Zz- -Qz-). This proves C (y,z 1,z1)= C (y,z 2,z2), and 1 2 T T consequentlyJT ' (y,z 1,z1)= JT ' (y,z 2,z2)holds. Using (B.9)and (B.10)one can quicklyshow that fT ' (y,z 1)= fT ' (y,z 2)also holds, and due to Lemma B.2 also Part 3 holds for T. Lets assume now that Jt ' (y,z 1,z1) = Jt ' (y,z 2,z2) holds for t. From Lemma B.2, we have yˆt(z 1) = yˆt(z 2), which proves Part 3. To prove Part 2, we need to show that this implies ft ' (x,z 1)= ft ' (x,z 2). We need to analyze the following four cases: '' '' Case 1: If x = yˆt(z 1) and x = yˆt(z 2), then f (x,z 1)= J (x,z 1,0)= J (x,z 2,0)= f (x,z 2), t t t t where the inequality is due to Part 1. ' ' Case 2: If x = yˆt(z 1)and x < yˆt(z 2), then ft(x,z 1)= ft(x,z 2)comes directly from (B.10). Case 3: x < yˆt(z 1)and x = yˆt(z 2)is not possible due to Part 3. Case 4: If x < yˆt(z 1)and x < yˆt(z 2), then we have ft ' (x,z 1)= ft ' (x,z 2)= 0 from (B.10). Goingfrom t to t-1we conclude the induction argument byshowingthat ft ' (x,z 1)= ft ' (x,z 2) implies J ' (y,z 1,z1)= J ' (y,z 2,z2). Using (4.5)we write: t-1t-1 ' ' ' J (yt-1,z 1,t-1,z1,t-1) = C (yt-1,z 1,t-1,z1,t-1)+ aEf (x1,t,z 1,t) t-1t-1t = Ct' -1(yt-1,z 2,t-1,z2,t-1)+ aEft ' (x1,t,z 2,t) = Ct' -1(yt-1,z 2,t-1,z2,t-1)+ aEft ' (x2,t,z 2,t) = Jt' -1(yt-1,z 2,t-1,z2,t-1), The .rst inequalityis due to C ' (yt-1,z 1,t-1,z1,t-1)= C ' (yt-1,z 2,t-1,z2,t-1), andinduction t-1t-1argument ft ' (x,z 1)= ft ' (x,z 2)(where taking expectation over the random variables Qt-m-1 ' and Dt-1 preserves the inequality). The inequality of Ct-1 holds from convexity of C˜t+L-1 in ˜xt+L, where from the assumptions z 1 =z 2 and z1 = z2, we have ˜x1,t+L = x˜2,t+L due to the .t-1 de.nition ˜xt+L = yt-1- (zs -Qs)+-DL in (4.3)(where again takingexpectations s=t-m-1 t-1 over the random variables Qt+m-1,. . . ,Qt-1 and DtL -1 preserves the inequality). The second inequality holds from convexity of ft in x, and the fact that x1,t = x2,t holds due to the state update x1,t = yt-1 -(z1,t-m-1 -qt-m-1)+ -dt-1 = yt-1 -(z2,t-m-1 -qt-m-1)+ -dt-1 = x2,t in (4.2), and the assumption z1,t-m-1 = z2,t-m-1. . Appendix C Chapter 5 Proofs In Lemma C.1 we provide the convexity results and the optimal solution to a single period cost function Ct(yt, zt). We elect to suppress subscript t in the state variables for clarity reasons. Lemma C.1 Let yˆM be the smal lest minimizer of C(y, z)to which the optimal order zˆM is M M - placed, where zˆ= yˆx: Q1. C(y, z)is convex in y. Q2. C(y, z)is quasiconvex in z, where: .2 C(y, z)= 0, for z = zˆM , .z2 . M C(y, z)= 0, for z > zˆ. .z ( ) M G-1 cb Q3. yˆM is the optimal myopic base-stock level, where yˆ= . cb+ch Proof: We .rst rewrite the single period cost function C(y, z)as: 1 8 1 y C(y, z) = a(1 -R(z)) cb (D-y)g(D)dD+ ch (y-D)g(D)dD y 0 1 z 1 8 + acb (D-y+ z -Q)g(D)dDr(Q)dQ 0 y-z+Q 1 1 z y-z+Q + ach (y-z + Q-D)g(D)dDr(Q)dQ. (C.1) 0 0 To prove Part 1, we derive the .rst partial derivative of (C.1) with respect to y: . cb C(y, z) = a(cb + ch)(1 -R(z)) G(y)- .y cb + ch 1 z + a(cb + ch) G(y-z + Q)r(Q)dQ-acbR(z), (C.2) 0 and the second partial derivative: 1 z .2 C(y, z)= a(cb + ch) (1 -R(z))g(y) + g(y-z + Q)r(Q)dQ . (C.3) .y2 0 Since all terms in (C.3) are nonnegative, Part 1 holds. Similarly for Part 2, we obtain the .rst two partial derivatives of C(y, z)with respect to z: . cb C(y, z)= a(cb + ch)(1 -R(z)) G(y)- , (C.4) .z cb + ch .2 cb C(y, z)= a(cb + ch) (1 -R(z))g(y)-r(z) G(y)- . (C.5) .z2 cb + ch Observe .rst that setting (C.4) to 0 proves Part 3. For z = zˆM , G(y)= cb holds in the cb+ch second term of (C.5). Since the .rst term is always nonnegative, the function C(y, z) is convex on the respected interval. For z > zˆM this does not hold, although we see from (C.4) that due to G(y)> cb and 1 -R(z). 0 with z, the function C(y, z)is nondecreasing, cb+ch which proves Part 2. Due to this, the C(y, z)has a quasiconvex form, which is su.cient for yˆM to be its global minimizer. . Proof of Proposition 5.1: We start by studying the convexity properties of the optimal cost functions (5.2)-(5.4), where we show that these are not convex. Note that the single-period cost function Ct(yt, zt)is convex in yt and quasiconvex in zt for any t due to Lemma C.1. By di.erentiating (5.4) with respect to wt and setting it to zero, we derive the .rst-order optimality condition, where the optimal inventory position after ordering with the slower supplier ˆwt(zt)is the smallest minimizer of the function Ht(wt, zt)in period t1: 1 1 8 8 . .wt Ht(wt, zt) = a ft' +1( ˆwt(zt)-Dt)rt(Qt)dQtgt(Dt)dDt 0 zt 1 8 1 zt + a ft' +1( ˆ(C.6) wt(zt)-(zt -Qt)-Dt)rt(Qt)dQtgt(Dt)dDt = 0. 0 0 The second partial derivative of Ht(wt, zt)with respect to wt yields: 1 8 1 8 .2 Ht(wt, zt) = a ft'' +1( ˆwt(zt)-Dt)rt(Qt)dQtgt(Dt)dDt .wt 2 0 zt 1 8 1 zt + a ft'' +1( ˆwt(zt)-(zt -Qt)-Dt)rt(Qt)dQtgt(Dt)dDt. (C.7) 0 0 ' 1We de.ne the .rst derivative of a function ft(x)with respect to x as f (x). t In addition, we obtain the second partial derivative of Jt(yt, zt)with respect to yt, which will also be useful in the following discussions: 1 .2 zt Jt(yt, zt) = a(cb + ch) (1 -Rt(zt))gt(yt) + gt(yt -zt + Qt)rt(Qt)dQt .yt 2 0 1 1 8 8 + a ft'' +1(yt + ˆvt(zt)-Dt)rt(Qt)dQtgt(Dt)dDt 0 zt 1 8 1 zt + a ft'' +1(yt -(zt -Qt) + ˆvt(zt)-Dt)rt(Qt)dQtgt(Dt)dDt.(C.8) 0 0 Similarly as above, we derive the .rst-order optimality condition for ˆyt by di.erentiating (5.3) with respect to zt, where ˆyt is the smallest minimizer of the function Jt(yt, zt). Observe that the optimal order with the slower supplier ˆvt(ˆzt)depends on the optimal order placed with the faster supplier ˆzt. . cb Jt(xt, zt) = a(cb + ch)(1 -Rt(ˆzt)) G(xt + ˆzt)- .zt cb + ch 1 8 1 8 .vˆt(ˆzt) + a 1 + f ' (xt + ˆzt + ˆvt(ˆzt)-Dt)rt(Qt)dQtgt(Dt)dDtD .zt 0 zˆt t+1 1 1 8 zˆt .vˆt(ˆzt) + a ft' +1(xt + Qt) + ˆzt)-Dt)rt(Qt)dQtgt(Dt)dDt = 0. vt(ˆ (C.9) .zt 0 0 Finally, evaluating (5.2) at ˆzt, and partially di.erentiating it with respect to xt twice, yields: 1 ˆ zt ft '' (xt) = a(cb + ch) gt(xt + Qt)rt(Qt)dQt 0 1 8 1 8 .vˆt(ˆzt) - a ft'' +1(xt + ˆzt + ˆvt(ˆzt)-Dt)rt(Qt)dQtgt(Dt)dDtD .zt 0 zˆt 1 1 8 zˆt .vˆt(ˆzt) + a 1- ft'' +1(xt + Qt) + ˆvt(ˆ(C.10) zt)-Dt)rt(Qt)dQtgt(Dt)dDt = 0. .zt 0 0 If we would assume that ft(xt+1) is convex, using the regular inductive argument on t would require ft '' (xt)= 0 to hold for the optimal cost to be convex. While the .rst term in (C.10) is nonnegative, we cannot say the same for the sum of the other two terms. Therefore, ft(xt) is not a convex function in general. Observe that if .vˆt(ˆzt)/.zt = 0, the above condition for convexity would be satis.ed. In this case, due to (C.7), (C.8) and fT+1(·)= 0, Ht and Jt would also be convex in wt and yt, respectively. However, the dependency of ˆvt(ˆzt)on ˆzt describes the core dynamics of this dual-sourcing model, where ˆvt is used to compensate for the potential shortage in replenishment of ˆzt. One can also quickly check that by di.erentiating (C.9) with respect to xt, dzˆt/dxt = -1 does not set the obtained expression to zero. Thus, ˆyt(xt) is not independent of xt, and therefore ordering with the faster supplier is not done in the manner of a base-stock policy, which proves Part 1. By evaluating (C.6) at ˆzt and di.erentiating it with respect to yt, we quickly see that dwˆt(ˆzt)/dyt = 0 needs to hold, which means that ordering with the slower supplier is done by placing order ˆvt(ˆzt)up to the optimal base-stock level ˆwt(ˆzt)= yˆt(xt)+ˆvt(ˆzt), state-dependent on ˆzt, which proves Part 2. The above results directly imply the proposed structure of the optimal policy in Part 3. . Proof of Proposition 5.2: We prove Part 1 by regular inductive argument on t. It is clear that Part 1 holds for T + 1 due to fT+1(·) = 0. Suppose that Part 1 also holds for t + 1. We need to show that it also holds for t. We need to show that both parts of (5.6) are convex. Di.erentiating Ct(yt, zt)with respect to xt, evaluating it at ˆzt (using C.4), and di.erentiating again with respect to xt yields: .2 1 zˆt Ct(xt,zˆt)= a(cb + ch) gt(xt + Qt)rt(Qt)dQt. (C.11) .x2 t 0 The above expression is nonnegative, which proves that .rst part of (5.6) is convex. The second part is convex due to the fact that taking expectations over Qt and Dt preserves the convexity of ft+1 (as in (C.7) evaluated at ˆz, presuming that ft+1 is now convex). This concludes the inductive argument and proves Part 1. Part 2 holds directly from Part 1 and the de.nition of the myopic policy in (5.6). The optimality of the two base-stock levels ˆyM and ˆw(ˆz)can be shown by applying the steps of the analysis of Parts 1 and 2 of Proposition 5.1 to (5.6). . Lemma C.2 Let f(x) and g(x) be convex, and xf and xg be their smal lest minimizers. If f ' (x)= g ' (x)for all x, then xf = xg. Proof: Following Heyman and Sobel (1984), lets assume for a contradiction that xf < xg, then we can write 0 < d = xg -xf and f ' (xg -d)= 0 holds. From f ' (xg -d)= 0 and initial assumption f ' (x)= g ' (x)we have 0 = f ' (xg -d)= g ' (xg -d). Observe that g ' (xg -d)= 0 contradicts the de.nition of xg being smallest minimizer of g(·), hence xf = xg. . Proof of Proposition 5.3: To prove Part 1, we need to show that for the .rst derivative ' '' of Ht with respect to wt, Ht(wt,zˆt), it holds that Ht(wt,zˆ1,t)= Ht(wt,zˆ2,t)for ˆz1,t = zˆ2,t for all t: Ht' (wt,zˆ1,t) = aEf t' +1(x1,t+1) = aEf t' +1(wt -[ˆz1,t -Qt]+ -Dt) = aEf t' +1(wt -[ˆz2,t -Qt]+ -Dt) ' = aEf t+1(x2,t+1) = Ht' (wt,zˆ2,t). The .rst equality is due to the de.nition of Ht in (5.4), and the second due to the state update xt+1 = wt -[ˆz1,t -Qt]+ -Dt. Due to ˆz1,t = zˆ2,t, it follows that x1,t+1 = x2,t+1, and the inequality is due to convexity of ft in xt. As taking expectation over Qt and Dt preserves inequality, we have shown that Ht' (wt,zˆ1,t) = Ht' (wt,zˆ2,t) holds. Induction on t, together with the result of Lemma C.2, proves Part 1. To prove Part 2, we write: .(wt -[.= aEf t' +1(wt -.-[ˆzt -Qt]+ -Dt) = Ht' (wt -., zˆt). Ht' (wt,zt + .) = aEf t' +1zt + . -Qt]+ -Dt) The .rst equality is due to the de.nition of Ht in (5.4) and the state update xt+1 = wt - [.The inequality is due to [.zt -Qt]+ and the convexity zt + .-Qt]+ -Dt. zt + .-Qt]+ = .+ [ˆof ft in xt. As taking expectation over Qt and Dt preserves inequality, we have shown that Ht' (wt,zt + .)= Ht' zt) holds. Observe that the smallest minimizer of Ht(wt -., zˆt) is .(wt -., ˆwˆt(ˆwt(.wt(ˆzt) + ., which proves zt) + ., and together with Lemma C.2, this implies ˆzt + .)= ˆPart 2. . Bibliography Alix Partners. 2014. 2014 reshoring/nearshoring executive survey and outlook. URL http:// www.alixpartners.com/en/LinkClick.aspx?fileticket=d7Z0ll19_ps%3D&tabi. [On­line; accessed 22-February-2015]. Altu.g, M.S., A. Muharremo.glu. 2011. Inventory management with advance supply informa­tion. International Journal of Production Economics 129(2) 302–314. Angelus, A., E. L. Porteus. 2002. Simultaneus capacity and production management of short-life-cycle, produce-to-stock goods under stochastic demand. Management Science 48(3) 399–413. Atasoy, B., R. G¨ull¨u, T. Tan. 2012. Optimal inventory policies with non-stationary supply disruptions and advance supply information. Decision Support Systems 53(2) 269–281. Austin, T. A., C. L. Givens, H. L. Lee, L. Kopczak. 1998. Unlocking hidden value in the personal computer supply chain. Andersen Consulting 188–208. Aviv, Y., Federgruen. 1997. Stochastic inventory models with limited production capacity and periodically varying parameters. Probability in Engineering and Information Sciences 11(2) 107–135. Bassok, Y., R. Anupindi. 1997. Analysis of supply contracts with total minimum commit­ment. IIE Transactions 29(5) 373–381. BCG. 2014a. Made in america, again: Third annual survey of u.s.-based manufactur­ing executives. URL http://www.slideshare.net/TheBostonConsultingGroup/bcg­mfg-survey-key-findings-slideshare-deckoctober-2014f3. [Online; accessed 22­February-2015]. BCG. 2014b. The shifting economics of global manufacturing. URL http: //www.slideshare.net/TheBostonConsultingGroup/bcg-global-mfg-cost­competitiveness-index-final?related=1. [Online; accessed 22-February-2015]. Bollapragada, R., U. S. Rao. 2006. Replenishment planning in discrete-time, capacitated, non-stationary, stochastic inventory systems. IIE Transactions 38(7) 583–595. Bollapragada, S., T. E. Morton. 1999. Myopic heuristics for the random yield problem. Operations Research 47(5) 713–722. Bulinskaya, E. 1964. Some results concerning optimal inventory policies. Theory Prob. Appli. 9(3) 502–507. Bush, C., W. Cooper. 1988. Inventory level decision support. Production and Inventory Management Journal 29(1) 16–20. Cachon, G. P. 2004. The allocation of inventory risk in a supply chain: push, pull and advance-purchase discount contracts. Management Science 50(2) 222–238. C¸ inar, E., R. G¨ull¨u. 2012. An inventory model with capacity .exibility in the existence of advance capacity information. Decision Support Systems 53(2) 320–330. Chen, F. 2003. Information Sharing and Supply Chain Coordination. Handbooks in opera­tions research and management, Elsevier, The Netherlands. Chiang, C., G. J. Gutierrez. 1996. A periodic review inventory system with two supply modes. European Journal of Operational Research 94(3) 527–547. Chiang, C., G. J. Gutierrez. 1998. Optimal control policies for a periodic review inventory system with emergency orders. Naval Research Logistics 45(2) 187–204. Choi, H. P. 2010. Information sharing in supply chain management: A literature review on analytical research. California Journal of Operations Management 8(1) 110–116. Chopra, S., P. Meindl. 2001. Supply chain management: Strategy, Planning, and Operation . Prentice-Hall, New Jersey. Ciarallo, F. W., R. Akella, T. E. Morton. 1994. A periodic review, production planning model with uncertain capacity and uncertain demand -optimality of extended myopic policies. Management Science 40(3) 320–332. Daniel, K. 1963. A delivery-lag inventory model with emergency. Multistage Inventory Models and Techniques . Stanford University Press, Stanford, CA. Erkoc, M., SD Wu. 2005. Managing high-tech capacity expansion via reservation contracts. Production and Operations Management 14(2) 232–251. Fair, R. 1989. The production smoothing model is alive and well. Journal of Monetary Economics 24(3) 353–370. Federgruen, A., P. H. Zipkin. 1986a. An inventory model with limited production capacity and uncertain demands i. the average-cost criterion. Mathematics of Operations Research 11(2) 193–207. Federgruen, A., P. H. Zipkin. 1986b. An inventory model with limited production capacity and uncertain demands ii: The discounted cost criterion. Mathematics of Operations Research 11(2) 208–215. Fukuda, Y. 1964. Optimal policies for the inventory problem with negotiable leadtime. Management Science 10(4) 690–708. ¨ ¨ Gallego, G., O. Ozer. 2001. Integrating replenishment decisions with advance demand infor­mation. Management Science 47(10) 1344–1360. Gans, N., Y. P. Zhou. 2001. Managing learning and turnover in employee sta.ng. Operations Research 50(6) 991–1006. G¨ull¨u, R. 1996. On the value of information in dynamic production/inventory problems under forecast evolution. Naval Research Logistics 43(2) 289–303. G¨ull¨Onol, N. Erkip. 1997. Analysis of a deterministic demand production/inventory u, R., E. ¨ system under nonstationary supply uncertainty. IIE Transactions 29(8) 703–709. Heyman, D., M. Sobel. 1984. Stochastic Models in Operations Research. Vol. II . McGraw-Hill, New York, NY. Iida, T. 2002. A non-stationary periodic review production-inventory model with uncertain production capacity and uncertain demand. European Journal of Operational Research 140(3) 670–683. Jak.si.c, M., J.C. Fransoo. 2015. Optimal inventory management with supply backordering. International Journal of Production Economics 159 254–264. Jak.si.c, M., J.C. Fransoo, T. Tan, A. G. de Kok, B. Rusjan. 2011. Inventory management with advance capacity information. Naval Research Logistics 58(4) 355–369. Jak.si.c, M., B. Rusjan. 2009. The value of sharing advance capacity information under “zero-full” supply capacity availability. Economic and Business Review 11(4) 321–337. Jin, M., SD Wu. 2001. Modeling capacity reservation in high-tech manufacturing. Working paper, Lehigh University, Pennsylvania . Kapuscinski, R., S. Tayur. 1998. A capacitated production-inventory model with periodic demand. Operations Research 46(6) 899–911. Karaesmen, F., G. Liberopoulos, Y. Dallery. 2003. Production/inventory control with ad­vance demand information. J. G. Shanthikumar, D. D. Yao, W. H. M. Zijm, eds. Stochas­tic Modeling and Optimization of Manufacturing Systems and Supply Chains, Interna­tional Series in Operations Research & Management Science, Kluwer Academic Publish­ers, Boston, MA 63 243–270. Karlin, S. 1960a. Dynamic inventory policy with varying stochastic demands. Management Science 6(3) 231–258. Karlin, S. 1960b. Optimal policy for dynamic inventory process with stochastic demands subject to seasonal variations. SIAM Journal 8(4) 611–629. Karlin, S., H.E. Scarf. 1958. Optimal Inventory Policy for the Arrow Harris Marchak Dy­namic Model with a Time Lag: Arrow, Karlin and Scarf (eds.) Studies in the Mathematical Theory of Inventory and Production . Stanford University Press, Stanford. Khang, D. B., O. Fujiwara. 2000. Optimality of myopic ordering policies for inventory model with stochastic supply. Operations Research 48(1) 181–184. Krane, S.D., S.N. Braun. 1991. Production smoothing evidence from physical product data. Journal of Political Economy 99(3) 558–581. Lau, J.S.K. 2007. Information Sharing in Supply Chains: Improving the Performance of Col laboration . Erich Schmidt Verlag, Berlin, Germany. Lee, L. H., V. Padmanabhan. 1997. Information distortion in a supply chain: The bullwhip e.ect. Management Science 43(4) 546–558. Lee, L. H., S. Whang. 2000. Information sharing in a supply chain. International Journal of Manufacturing Technology and Management 1(1) 79–93. McCullen, P., D. Towill. 2002. Diagnosis and reduction of bullwhip in supply chains. Supply Chain Management: An International Journal 7(3) 164–179. Metters, R. 1997. Production planning with stochastic seasonal demand and capacitated production. IIE Transactions 29(11) 1017–1029. Metters, R. 1998. General rules for production planning with seasonal demand. International Journal of Production Research 36(5) 1387–1399. Mincsovics, G., T. Tan, O. Alp. 2006. Integrated capacity and inventory management with capacity acquisition lead times. Wp, Technische Universiteit Eindhoven. Minner, S. 2003. Multiple-supplier inventory models in supply chain management: A review. International Journal of Production Economics 81 265–279. Monczka, R., R. B. Hand.eld, L. C. Giunipero, J. L. Patterson. 2009. Purchasing and Supply Chain Management. Fourth edition . South-Western Cangage Learning, Mason, USA. Nahmias, S. 1993. Production and Operations Analysis. Second edition. Irwin, Illinois. ¨ Ozer, ¨ O., W. Wei. 2003. An on-line addendum to ”inventory control with limited capacity and advance demand information”. On-line addendum, Stanford University. ¨ Ozer, ¨ Inventory control with limited capacity and advance demand O., W. Wei. 2004. information. Operations Research 52(6) 988–1000. Penske Logistics. 2014. Surveys .nds 3pls hitting stride amid economic uncer­ tainty. URL http://http://www.penskelogistics.com/newsroom/2014_9_22_pl_ 3pl_study.html. [Online; accessed 22-February-2015]. Pinker, E. J., R. C. Larson. 2003. Optimizing the use of contingent labor when demand is uncertain. European Journal of Operational Research 144(1) 39–55. Porteus, E. L. 2002. Foundations of Stochastic Inventory Theory . Stanford University Press, Stanford. Puterman, M. L. 1994. Markov Decision Processes: Discrete Stochastic Dynamic Program­ming. John Wiley & Sons, New York, NY. Rocklin, S. M., A. Kashper, G. C. Varvaloucas. 1984. Capacity expansion/contraction of a facility with demand augmentation dynamics. Operations Research 32(1) 133–147. Ryan, S. M. 2003. Capacity expansion woth lead times and autocorrelated random demand. Naval Research Logistics 50(2) 167–183. Scarf, H. 1960. The optimality of (s,S) policies in the dynamic inventory problem, Arrow, Karlin and Scarf (eds.). Mathematical Methods in Social Sciences, Stanford University Press, Stanford. Serel, D. A. 2007. Capacity reservation under supply uncertainty. Computers & Operations Research 34(4) 1192–1220. Silver, A. E., R. Peterson. 1985. Decision systems for inventory management and production planning, 2nd Edition . John Wiley & Sons, New York, NY. Silver, A. E., D.F. Pyke, R. Peterson. 1998. Inventory Management and Production Planning and Scheduling, 3rd Edition . John Wiley & Sons, New York, NY. Simchi-Levi, D., P. Kaminsky, E. Simchi-Levi. 2007. Designing & Managing the Supply Chain, 3rd Edition . McGraw-Hill, New York, NY. Tagaras, G., D. Vlachos. 2001. A periodic review inventory system with emergency replen­ishments. Management Science 47(3) 415–429. Tajbakhsh, M. M., S. Zolfaghari, C. Lee. 2007. Supply uncertainty and diversi.cation: A review. H. Jung, F. Chen, B. Seong, eds. In Trends in Supply Chain Design and Manage­ment: Technologies and Methodologies. Springer Ltd., London, England. Tan, T., O. Alp. 2005. An integrated approach to inventory and .exible capacity management under non-stationary stochastic demand and setup costs. Wp 132, Technische Universiteit Eindhoven. Tan, T., R. G¨ull¨u, N. Erkip. 2007. Modelling imperfect advance demand information and analysis of optimal inventory policies. European Journal of Operational Research 177(2) 897–923. Tayur, S. 1993. Computing the optimal policy for capacitated inventory models. Commu­nications in Statistics -Stochastic Models 9(4) 585–598. Topkis, D. M. 1978. Minimizing a submodular function on a lattice. Operations Research 26 305–321. Topkis, D. M. 1998. Supermodularity and Complementarity . Princeton University Press, Princeton, N.J. Tsay, A. A. 1999. The quantity .exibility contract and suppliercustomer incentives. Man­agement Science 45(10) 1339–58. Tsay, A. A., W. S. Lovejoy. 1999. Quantity .exibility contracts and supply chain perfor­mance. Manufacturing & Service Operations Management 1(2) 89–111. Veinott, A. 1963. Optimal Stockage policies with non-stationary stochastic demands: Scarf, Gilford and Shel ly (eds.) Multistage inventory models and techniques . Stanford University Press, Stanford. Veinott, A. 1965. Optimal policy for a multi-product, dynamic, non-stationary inventory problem. Management Science 12 206–222. Vlachos, D., G. Tagaras. 2001. An inventory system with two supply modes and capacity constraints. International Journal of Production Economics 72(1) 41–58. Whittemore, A. S., S. C. Saunders. 1977. Optimal inventory under stochastic demand with two supply options. SIAM Journal on Applied Mathematics 32(2) 293–305. Wijngaard, J. 2004. The e.ect of foreknowledge of demand in case of a restricted capacity: The single-stage, single-product case. European Journal of Operational Research 159(1) 95–109. Xu, N. 2011. Myopic optimal policy for a multi-period, two-delivery-lead-times, stochastic inventory problem with minimum cumulative commitment and capacity. International Journal of Production Economics 133(2) 719–727. Yang, J., X. Qi, Y. Xia. 2005. A production-inventory system with markovian capacity and outsourcing option. Operations Research 53(2) 328–349. Yazlali, ¨ O., F. Erhun. 2009. Dual-supply inventory problem with capacity limits on order sizes and unrestricted ordering costs. IIE Transactions 41(8) 716–729. Zhang, C., G. W. Tan, D. J. Robb, X. Zheng Zangwill. 2006. Sharing shipment quantity information in the supply chain. Omega 34(5) 427–438. Zipkin, P. H. 2000. Foundations of Inventory Management . McGraw-Hill/Irwin, Boston, MA.