Advances in Production Engineering & Management
Volume 12 | Number 3 | September 2017 | pp 213-220 https://doi.Org/10.14743/apem2017.3.252
ISSN 1854-6250
Journal home: apem-journal.org Original scientific paper
Production lot-sizing decision making considering bottleneck drift in multi-stage manufacturing system
Zhou, F.L.a*, Wang, X.a,b *, He, Y.D.a, Goh, M.c
department of Industrial Engineering, Chongqing University (CQU), Chongqing, P.R. China
bState Key Laboratory of Mechanical Transmission, Chongqing University (CQU), Chongqing, P.R. China
cDepartment of Analytics and Operations, NUS Business School and The Logistics Institute-Asia Pacific, National University
of Singapore (NUS), Singapore
A B S T R A C T
A R T I C L E I N F O
Under a capacity constrained multi-product manufacturing system, the products are usually prepared and produced in lots. As a lot-sizing strategy is critical for effective production and high productivity, this encourages practical and research interest in the strategic batch sizing decision for a minimum procedure time in an order-to-delivery (OTD) operating environment. While the lot-sizing plan can be formed by studying the manufacturing parameters of the established bottleneck procedure, for a multi-stage manufacturing system, the bottleneck is not fixed and fluctuates with the production rate or batch size. This paper proposes a lot-sizing strategy to determine the optimal lot-size for each class of products taking bottleneck drifting into consideration. A queuing network analyser (QNA) method is employed to deal with the non-linear mixed integer programming model targeting at the total flow time minimization of the system. A practical case is presented and solved using the proposed method, and the results are validated with Flexsim, a simulation model.
© 2017 PEI, University of Maribor. All rights reserved.
Keywords:
Manufacturing system Multistage manufacturing system Lot-sizing decision making Lead time
Queuing network analyser (QNA)
*Corresponding authors: deepbreath@outlook.com (Zhou, F.L.) wx921@163.com (Wang, X.)
Article history: Received 18 January 2017 Revised 24 May 2017 Accepted 11 June 2017
1. Introduction
In the modern manufacturing era, enterprises are increasingly focusing on optimizing the total flow time under an order-to-delivery (OTD) environment [1]. With the proliferation in product variety, many products are usually processed and produced in the same production system with a view to improving the production efficiency and reducing cost. In the multi-product production system, the productivity and lead time of the manufacturing workshop is directly affected by the batch size of the product [2, 3]. With mass production, the productivity can be increased albeit with a longer lead time for the lot production; with low-volume production, the effect is reversed. In the literature, this phenomenon of large lot sizes will cause long lead times known as the batching effect. As the lot size decreases (the jobshop context), the lead time will also decrease, but once a minimal lot size is reached a further reduction in lot size will cause high traffic intensities resulting in longer lead times, called the saturation effect [4]. Therefore, to meet the shortest delivery lead time possible, it is often necessary to determine the right type and produce products in an appropriate quantity.
There are many models, in the literature, focusing on the complex relationship between batch, lead time, and work-in-process (WIP) [5-7]. Karmarkar [5] had studied the influences of batch (lot-size) on manufacturing lead times, and reported that the best lot sizes, high capacity
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utilization exact a high price in lead time and WIP. The relationship between batch size and lead-time variability was investigated by Kuik and Tielemans [6], who concluded that a minimization of the average queue delay or the average time in system performance measure, would not result in minimum lead-time variability. Vaughan [7] developed a comprehensive process lot sizing/order point model, and found that it was more efficient to adjust the safety stock and lead time than to change the shortage penalty parameter solely through adjusting the safety stock. Recently, some scholars argued that the lead time and productivity levels of the production system were determined by the bottleneck equipment, and the optimal processing of a single batch was studied and analysed [8-12]. Koo and Koh [8] proposed a batch decision optimization model that included a variety of product lines with the same preparation time by targeting maximum profit as the objective. Similarly, an extended model for optimizing the lot size in various production lines with different process and preparation times was formulated [9]. Liu [13] analysed the possibility of resources becoming the bottleneck through the bottleneck drift index, and calculated the optimal lot size and lead time of the bottleneck resource targeting the non-value added time and the difference in the manufacturing unit processing rate minimum as the optimization goal. However, they did not consider the bottleneck drifting caused by lot sizes.
In reality, in a relatively balanced production system with a large variety of products, the fluctuation of the product portfolio and the lot sizes can alter the load level of the resources, which might lead to bottleneck drifting. As a result, the lot size and lead time found in previous studies may not be optimal. Adacher and Cassandras [4] studied the optimal lot sizing problem with the example of two products and two processing operations, using the substitution method and the random comparison algorithm. Glock [14] solved the optimal lot size problem for a multi-stage manufacturing system by studying the relationship between the quantity and the total cost. Amy [15] proposed a mixed nonlinear integer programming model to solve the lot-sizing optimization problem with multiple suppliers in multiple periods considering quantity discounts, and a Genetic Algorithm (GA) is developed to minimize the total related cost. Therefore, this paper investigates the lot-sizing problem with bottleneck drifting. As the lot-sizing decision making is related to productivity, which influences the efficient outputof the manufacturing system [1], this paper seeks to examine the lot-sizing problem for different experimental productivity scenarios.
In this paper, we apply QNA (queuing network analysis) [16] to establish a lot-size optimization model on a multi process production system, to decide the optimal lot sizes in the production of multiple products. The objective of the batch optimization model is to produce products with the shortest lead time according to the demand. QNA is an accurate queuing network analysis model developed by Bell Laboratories in the US [16]. QNA is used to analyze the queuing network by estimating the distribution and variation coefficient of each arrival process and each service time. QNA is suitable for solving complex queuing network problems, given its low computational complexity. Given the relationship between the processes of a multi operation system, we apply the QNA method to determine the waiting time of the tandem process queue, with the total process time as the shortest objective function, such that the changes caused by the bottleneck of the product batch will not affect the optimization objective.
The next section presents the studied problem and the assumptions thereof. In Section 3, the non-linear programing model considering bottleneck drifting is formulated and a four-step algorithm is designed by a traversal operation and implemented to deal with the lot-sizing optimization model. A practical case is presented in Section 4 to validate the proposed model. Lastly, we conclude with some remarks.
2. Problem description and assumptions
As shown in Fig. 1, the production line is a system with many work procedures. Each service station represents a processing or assembly operation. All kinds of products enter and leave independently with a minimum unit. Each batch of products follows a first-come-first-served (FCFS) queuing strategy.
The assumptions are as follows:
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Production lot-sizing decision making considering bottleneck drift in multi-stage manufacturing system
•	The equipment utilization rate is less than 1, and productivity cannot exceed the production capacity;
•	Each service station shows the same product capacity for the same products;
•	The proportion of each product is known, and each class of products has a fixed manufacturing procedure.
•	The processing times of various products are different and independent;
•	Each lot of products arrives as a Poisson process;
•	The preparation time before lot-size production, and the preparation time of each product is different and independent.
w,	WBN	wn
— - m©—-
Fig. 1 Production system with multi-stage procedures The parameters are denoted and described as follows:
Notation	Interpretation
F	Total processing time, units/period
Pj	Average processing time in process j, units/period
Wj	Average queuing time on equipment j, units/period
Sj	Average processing time on equipment j , units/period
Xi	Productivity of product item i, (i = 1,2,3,...,m); X =
Ri	Lot size of product item i
n	Proportion of product itemi, (£iri = 1)
Pij	Processing time on equipment j of product item i
Ta	Setup time on equipment j of product item i
su	Total time required for product item i (stj =qiPij +tî7-)
tu	Service rate of a certain batch item i = 1 /s^)
pj	Logistics intensity of equipment j
	Average number of arrivals of product i per time unit (arrival rate)
CSj	Variation coefficient of processing time on equipment j
CClj	Variation coefficient of product inter-arrival time on equipment j
cdj	Variation coefficient of product inter-left time on equipment j
3. Model and algorithm
The non-linear mixed integer programming (MIP) model is formulated and constructed by minimizing the total flow time with the QNA technique. The algorithm procedures through the traversal operation are designed and implemented in this section.
3.1 Objective function
The optimization objective of the described problem is targeted as the total production time minimization considering production capacity and other common constraints. The model is formulated as follows.
n	n
minF = ^ = ^(w,-+s,-)	(1)
7 = 1	7 = 1
m
s. t. Jix íPíj + xJij/qd< 1	(2)
í=i
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= ^Uij,i = l,2,...,m	(3)
Uij = xipiJ	(4)
Vj = ^uij,i = l,2,...,m	(5)
Vij = xiTiJ/qi	(6)
Ii ^ Vi> RieZ+	(7)
x¿,q¿>0	(8)
Uj
where parameters u and v are the processing utilization rate and setup utilization rate respectively. Eq. 1 is the objective function for the total production time minimization. Eqs. 2 to 8 are the relevant constraints on productivity, order, and practical production.
3.2 Estimation of service time
In the production system, the arrival of a variety of products can be regarded as a Poisson process. It is reasonable to simulate the multi-process production system with the tandem queuing model, where Fj denotes the mean flow time of a product in processing j, which includes the waiting and processing times. The mean processing time of a unit product in process j is found from Eq. 9.
= ^jri(pijqi + Tij)	(9)
In a serial queuing system, the arrival processes are determined by the output of the previous process in addition to the first process. In this paper, the QNA method is used to estimate the arrival process variation coefficient of each node, which is regarded as an update process. Therefore, the queuing time in a steady state can be obtained [17]. Suresh and Whitt (1990) proposed the estimation method for the arrival process variation coefficient, and improved the accuracy of the estimation of the queuing time when the traffic intensity is 0.9 [18]. The improved coefficient of the arrival process is presented in Eq. 10.
cdf = (1-p,2(1 - pjiO) caf + p2(1 - p^)«/
2 _ 2 [10) CGLj — Ciij_^
The queuing time by the QNA technique is estimated from Eq. 11 and Eq. 12.
If ca2 < 1
_ _ Sj (caf +csf) Zi Xjjpjj +rij/qi) f 2[1 - YJÍxi(pij +Tij/qi)](1 - caf)] Wj~ 2[1-Zixi(pij + xiTij/qi-)] eXP{ 3Ylixi(pij + Tij/qi){caf+csj) ' ( )
If caf >1
_ _ SjPjjcaj + csf) _ Sjjcaf + csf)YJÍxi(pi¡ + TlJ/ql) 1 2(1 -pj)	2[1-1Zixi(pij + xiTij/qi)]
3.3 Productivity analysis
As the proportion of products rt is known, the productivity xt of an item i can be found from the total productivity X. The capacity constraints are illustrated in Eq. 13. From the upper bound on the lot-size of each class of products ql, we deduce the upper bound of X as shown in Eq. 14.
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xY4i(riPij+riTij/qi)<1	(13)
X=[l/YJ.(.riPij+riTij/clj\	(14)
where [ZJ is the largest integer less than or equal to X. 3.4 Algorithm
To establish the optimized lot-sizing of each kind of product with different productivity levels, we targeted the total production time as the objective function and formulated a non-linear MIP model. To determine the optimal lot size, we introduce the algorithm for dealing with this issue. First, the upper bound on productivity is analysed based on the previous section. Then, the optimal solution is searched by a traversal operation. Fig. 2 shows the steps of the algorithm. As Fig. 2 shows, there are four steps in the proposed algorithm.
Four stages
Satege 1: Initialization. Initialize objective value setting: F* = +<», qr* = 0. We obtain the upper bound of total productivity X for different kinds of products by Eq. 14.
Stage 2: Total production time F (including processing and waiting) computed by Eqs. 9 to 12.
Stage 3: If F < F*, then q* =qt. If F> F*, then for i = k (fc = 1,2, ...,m),ql =qt — 1, and return to stage 2.
Stage 4: If qt = 1, stop. The best lot-sizing solution is qr*, and the minimum total production time is denoted as F*. If qt ^ 1, then return to stage 3.
Fig. 2 Two-step optimization algorithm for multi-stage production system
4. Case study and simulation comparison 4.1 Case solutions
We consider a production system with five main procedures, and four classes of products need to be processed (Fig. 3). The information on the different parameters are presented in Table 1 such as processing time, upper bound of each class of products and product ratio. Suppose the arrival times of all products are regarded as Poisson, then ca1 = 1, csf = 0.5. The best solution of the lot-sizing for each class of products is calculated using the proposed algorithm procedure.
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W1	W2	W3	W4	W5
Fig. 3 Serial production system with five procedures
Table 1 Parameter values of four products in serial manufacturing system
Parameters		Product 1	Product 2	Product 3	Product 4
	T¡1	0.018	0.012	0.015	0.01
	T¡2	0.01	0.02	0.02	0.01
Waiting time	T¡3	0.02	0.01	0.02	0.01
	t14	0.015	0.005	0.015	0.01
	T¡5	0.012	0.012	0.015	0.01
	Pu	0.008	0.01	0.006	0.006
	Pi2	0.008	0.01	0.006	0.006
Processing time	Pi3	0.01	0.008	0.005	0.005
	Pi 4	0.01	0.01	0.006	0.005
	Pis	0.01	0.009	0.005	0.006
Product ratio	ri	0.3	0.2	0.1	0.4
Upper bound	Ri	30	30	50	50
From the algorithm, we obtain the best lot sizes under different productivity conditions. For instance, with a productivity of 90 %, the best solution of the four products is 7, 6, 10 and 7 units respectively, and with minimum total production time of 1.230 units. The fifth procedure is the bottleneck with the highest utilization rates of 84.02 %, as shown by Table 2. The results are illustrated in Table 2 for the various productivity situations.
From Table 2, the best lot-sizing solution for each class of products obtained from our algorithm suggests the following findings:
•	As the productivity rate declines, the best lot-size for each class of products and the total production time decrease.
•	In a relatively balanced manufacturing system, with a variation in lot-sizing, the bottleneck of the manufacturing system drifts as we imagined.
•	As the productivity rate declines, the utilization rate of the machine fluctuates slightly, rather than declining. The reason for the utilization rate increases is that the waiting time has been increased with the much more preparation operations.
Table 2 Results for different productivity situations
Produc-		Utilization rate of the machine (%)				Total time		Lot size (piece)		
tivity (%)	Machine 1 Machine 2 Machine 3 Machine 4 Machine 5					(days)	Prod 1	Prod 2	Prod 3	Prod 4
90	83.64	83.40	81.56	82.18	84.02	1.230	7	6	10	7
89	82.85	82.67	80.85	81.41	83.24	1.173	7	6	9	7
88	81.92	81.74	79.94	80.50	82.30	1.121	7	6	9	7
87	82.11	81.43	80.27	80.52	82.11	1.073	6	6	9	7
86	81.35	80.74	79.59	79.77	81.35	1.028	6	6	8	7
85	81.89	81.74	80.04	79.94	81.89	0.983	6	5	8	6
84	80.93	80.78	79.10	79.00	80.93	0.941	6	5	8	6
83	79.96	79.82	78.16	78.05	79.96	0.903	6	5	8	6
82	79.22	79.15	77.51	77.33	79.22	0.869	6	5	7	6
81	79.71	78.99	78.18	77.61	79.23	0.835	5	5	7	6
80	79.79	79.09	78.29	77.71	79.31	0.803	5	5	7	5
79	78.80	78.10	77.31	76.74	78.32	0.772	5	5	7	5
78	78.74	78.67	77.11	76.16	78.27	0.743	5	4	7	5
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4.2 Simulation verification
A simulation test is implemented to validate the effectiveness and validity of the proposed model. When the variation coefficient of the processing time is csj > 1, we supposed the processing time complies with the H2 distribution. When csj = 1, we obtain the exponential distribution. When cs^ > 1, the processing time can be treated as an Erlang distribution [18]. The manufacturing process is simulated by the Flexsim software with the original data information in Table 1, and the simulation is run for ten thousand days (10 times). Fig. 4 compares the simulated total production time and the proposed model.
From Fig. 4, the deviation result between the simulation model and programming model is about 12%. From Wu's research [19], when cs2 < 1, the estimation variation of waiting time for multi-stage queuing system by the QNA method is 6.3%. As there are five procedures in the manufacturing system, the variation of the serial system is much more than the single machine (6.3%) due to the deviation accumulation. As the waiting time estimation method focuses on a single machine, the accuracy of the estimation of waiting time directly influences the performance of the programming model. As for the multi-stage serial manufacturing system with limited procedures, the best solution accuracy of the proposed model is satisfactory and tolerable based on the above comparison analysis.
■ Simulation data Model data
1,4
.........1111111
o 0,6
Q.
a 0,4 £
K 0,2
0IIIIIIIIIIIII
78 79 80 81 82 83 84 85 86 87 88 89 90
Productivity rate
Fig. 4 Comparison of total time for simulation model and programming model
5. Conclusion
In this paper, the non-linear MIP model by the QNA method is proposed to deal with the lot-sizing problem for a multi-stage manufacturing system with multiple classes of products. Not only can the model be applied into a lot-sizing optimization problem of a manufacturing system with fixed bottlenecks, but also this model is applicable to a sensitive production system whose bottleneck fluctuates with the variation of the product combination and lot-sizing. The model is validated against a simulation model run by the Flexsim software, and it demonstrates excellent performance. However, from the comparison results, the model shows tolerable variation, which stimulates us to improve the accuracy of the model by revising the waiting time parameters. In future, the lot-sizing decision making problem which jointly considers bottleneck drift and unpredictable events (machine malfunction or failure, and stochastic occurrences) can be studied.
Acknowledgements
The authors are grateful to the anonymous referees for their valuable comments and constructive suggestions on our manuscript. The research is sponsored by the following scientific programmes: Research Fund for Doctoral Program of Higher Education of China (RFDP, 20130191110045), National Key Technology Research and Development Program of China (2015BAH46F01 and 2015BAF05B03). We also wish to thank the editorial team and the reviewers for the fast review process, and the Automotive Collaborative Innovation Centre (ACIC) in Chongqing University (CQU) and CA automotive Co. Ltd. for their assistance and support.
Simulation data Model data
..■i

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