Strojniški vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42 © 2014 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2013.1009
Original Scientific Paper
Received for review: 2013-01-28 Received revised form: 2013-06-27 Accepted for publication: 2013-08-23
Optimization of a Product Batch Quantity
Tomaž Berlec* - Janez Kušar - Janez Žerovnik - Marko Starbek
University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Companies encounter various challenges when entering the global market, one of the most significant being the calculation of the optimal batch quantity of a product. This paper explains how to calculate the optimal batch quantity using first the basic model, and then the extended model that takes into account the tied-up capital in a production, in addition to the costs of changing the batch and storage costs. There is a case study of calculating the optimal batch quantity using the basic and extended models, together with conclusions regarding when either of the two models should be used. For optimal batch quantities we also calculated lead times, corresponding costs of tied-up capital per piece, and the difference between costs per piece when using the basic and extended models.
Keywords: optimal batch, tied-up capital, storage costs, time per unit, setup time, lead time, turnaround time, interoperation time
0 INTRODUCTION
The goal of calculating the optimal batch quantity of a product is that the product is produced in the required quantity and required quality at the lowest cost [1] to [3].
There are basically two options of planning the batch quantity [4]:
•	planning a large batch of a product in long intervals,
•	planning a small batch of a product in short intervals.
The advantages of planning a large batch of product are:
•	price advantage of ordering a large batch (low cost, protection against raising prices, volume rebate),
•	lower administrative costs,
•	lower costs of tests and shipping,
•	low risk of interruption of production because of the large stock.
The disadvantages of planning a large batch are:
•	high tied-up capital,
•	high storage costs of product inventory.
The advantages of planning a small batch of product are:
•	low tied-up capital,
•	low storage costs of product inventory,
•	high flexibility if quantities change at suppliers and buyers.
The disadvantages of planning a small batch are:
•	the costs of frequent ordering,
•	high risk of interruption of production because of a small product inventory.
Somewhere between the large and small batch quantity is the optimal batch quantity, i.e. the quantity in which the cost per product unit is the lowest.
Aggterleky [4] describes the optimal planning planes and the meaning of under- and over planning, and the influence of the reduction of total cost. Wiendahl [5] uses Harris and Andler's equation for the determination of the optimal quantity. Hardler [6] takes into account the costs of storage and delivery in determining the optimal batch quantity. Muller [7] and Piasecki [8] assert that inventory management is explained only with the basics of an optimal quantity calculation. So, in comparison to the aforementioned papers, where only the determination of the optimum quantity is given, our model is expanded to include the impact of the flow time on the batch quantity or stock.
1 ECONOMIC ORDER QUANTITY - BASIC MODEL
Changing the product batch (hereinafter referred to as 'order') causes two types of costs [9]:
•	order change costs,
•	storage costs.
Order change costs include costs for preparing documentation, costs of control and input of goods, costs of workers' wages, costs of setting up the machines, and costs of producing samples.
The order, therefore, causes annual costs, known as order change costs SMen. The higher the ordered quantity is, the lower the influence of the order change costs is (Fig. 1a). The order also causes storage costs, which include the costs of interest on the bound capital and warehouse costs.
The order therefore also causes annual storage costs SSkh which increase proportionally as the ordered quantity increases (Fig. 1b).
The sum of the annual order change costs and storage costs has the minimum value of the total costs SVsomin at the optimal batch quantity xOpt (basic model) (Fig. 1c).
*Corr. Author's Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, Ljubljana, Slovenia, tomaz.berlec@fs.uni-lj.si
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
Fig. 1. a) order change costs, b) Storage costs of order, c) Optimal batch quantity xOpt in the basic model
The optimal batch quantity in the basic model [6] and [7] can be calculated by using the following sequence of steps:
Step 1: Calculation of annual order change costs:
SMen	' SMen,
X
(1)
where SMen are annual order change costs [€/a], L annual needs [piece/a], x batch quantity [piece], and sMen order change costs [€].
Step 2: Calculation of annual storage costs: Pulse inflow and steady outflow of goods is assumed (Fig. 2). The annual storage costs depend on the warehouse inventory:
SSkl 2 ' S°bd ' P '
(2)
where SSkl are storage costs [€/a], x batch quantity [piece], sObd processing costs per piece [€/piece], and p interest rate of tied-up capital [1/a].
Step 3: Calculation of total annual order costs:
ç = ç
uVso uMm
_ L	x
+ ç m	sM„„+~ ' s,
Men ' _ sObd 'P'
x	2
(3)
Step 4: Calculation of economic order quantity
xOpt:
The economic order quantity (i.e. the minimum value of this function) can be found by differentiation.
Fig. 2. Storage inventory of goods
Differentiating SVso with respect to x and then equating to zero, we get:
d S
Vso _ L „ , SObd ' P _ 0
,	2 ' SMen + 0
d xx	2
The optimal batch quantity xOpt:
2 ' L ' S Men
SObd ' P
(4)
where xOpt are optimal batch quantity [piece], L annual needs [piece/a], sMen order change costs [€], sObd
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
processing costs per piece [€/piece], andp interest rate of tied-up capital [1/a].
The interest rate of tied-up capital p is calculated on the basis of the bank interest rate for a long-term loan with additional interest using the VDI 3330 guidelines (Table 1).
Table 1. Interest rate of tied-up capital, in [%]
A. Bank interest rate for a long-term capital loan	7.15
Additional interests:	
limitation	3 to 5
losses caused by break or rupture	2 to 4
transport	2 to 4
storage and write-off	1.5 to 2
warehouse management	1 to 2
control	1 to 2
insurance	0.5 to 1
B. SUM OF ADDITIONAL INTERESTS	11 to 20
INTEREST RATE OF TIED-UP CAPITAL p = A + B	18.15 to 27.15
the cost of order change SMen and storage costs SSkb it would also be necessary to take into account the costs of execution of operations of an order in the production SIzv and costs of disposal (transition) SPre.
Fig. 4. Tying-up the capital in the material flow from supplier to customer
2 ECONOMIC ORDER QUANTITY - EXTENDED MODEL
In our basic model for the calculation of optimal batch quantity only the order change costs SMen and storage costs SSk[ were taken into account.
Analysis of the diagram of the flow of orders through work systems [10] revealed that during the processing of the observed order in a given workplace, other orders have to wait for the release of capacities, which in turn causes additional costs related to the tied-up capital in production (Fig. 3).
Fig. 3. Flow diagram showing inventory of orders
Tying-up the capital in the material flow from a supplier through production to the customer [11] is shown in Fig. 4.
After having carried out an analysis of the flow diagram showing the status of orders, and a diagram of tying-up the capital on the path from supplier to customer, Nyhuis and Fronla [12] concluded that when calculating the economic order quantity, in addition to
The optimal batch quantity in the extended model x"Opl will be the one in which the sum of annual order change costs, the storage costs, the costs of execution of operations of an order, and the transition costs will have the minimum value (Fig. 5).
Costs [€/a]
		
. order change costs		
	SprL'+Sohl+Ssitl /	
\ * ^^ Sobj+Ssu		
ySvr''	0 s Ski	
¿¿a		—►
x'0p, x0p, Batch quantity [piece]
Fig. 5. Optimal batch quantity x0pt in the extended model
The optimal batch quantity x0pt in the extended model is calculated in the following sequence:
Step 1: Calculation ofannual order change costs: s* —L
S Men — * ' SMen ,
X
where S*Men are annual order change costs [€/a], L annual needs [piece/a], x* batch quantity [piece], sMen order change costs [€].
Optimization of a Product Batch Quantity
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
Step 2: Calculation of annual storage costs: Pulse inflow and steady outflow of goods is
assumed (Fig. 3). The annual storage costs are:
*
* _ x
SSkl = 2 ' SObd ' p ,
where S*Sk! are annual storage costs [€/a], x* batch quantity [piece], sObd processing costs per piece [€/ piece], p interest rate of tied-up capital [1/a].
Step 3: Calculation of annual costs of order-processing times:
The one-dimensional lead time of an order operation, consisting of the turnaround time of the operation TIzv and the interoperation time Tpre is shown in Fig. 6 [10].
		
	T 1p	tObd
Tpre		Tjzv
To		
		
Fig. 6. Lead time of operation; Tp setup time [min], tObd manufacturing time [min], TIzv turnaround time [day], TPre interoperation time [day], TO lead time of operation [day]
Known order-processing times allow for a calculation of annual costs due to operation-execution
times Sz
(Sm
) ■ L ■ p
2 ■ RC

(5)
where S*Izv are annual costs arising from the operation-execution of an order [€/a], sObd processing costs per piece [€/piece], sMat material costs per piece [€/piece], L annual needs [piece/a], p interest rate of tied-up capital [1/a], RC available time [day/a], ETIzv total operation-execution time [day].
Step 4: Calculation of annual costs due to interoperation time:
_ (SMat + SObd ) • L • P
SPr e
2 • RC
•IT,
(6)
where SPr e are annual costs due to interoperation time [€/a], and ETPre sum of interoperation times [Wd].
Step 5: Calculation of total annual costs S*Vso:
SVso * ' SMen + ~ ' SObd ' P +
x	2
+ (SMat + SObd ) ' L ' P '^T
2' RC
(SMat + SObd ) ' L ' P 2' RC
L	x
SVso = ~ ' SMen + "TT ' SObd ' P '
x	2
ZT»r
(8)

The material flow rate ST is defined by the Eq (9):
, _	+ ^TPie
ST =
(9)
Therefore, Eq. (9) can be transformed to:
YTv + ZTpr. = ST TIZV.	(10)
The total time of the operation-execution of an order is defined as:

Tp + X ■ teX
60 ■ KAP '
(11)
where ETIzv is total time of operation execution of an order [day], Tp setup time [min], te1 time per unit [min/ piece], KAP daily capacities [h/day], x* batch quantity [piece].
If Eqs. (10) and (11) are inserted in Eq. (8), the function for calculating the total order-dependent costs is transformed to:
T	*
SVso * ' SMen + ~ ' S Obd ' P +
x	2
(S.
Mat ' •'Obd
snlJ ' L ' p ' ST y Tp + x ' tei 2' RC	y 60' KAP '
Step 6: Calculation of the economic order
*
quantity x0pt \
The minimum value of the function S**so can be found by differentiation. Differentiating S**so with respect to x* and then equating to zero, we get:
dSVso _ L _ s + SObd ' P
J *	/ * \ 2 Men
dx (x*)	2
, (SMa, + SObd ) ■ L ■ P
2 ■ RC
ST
t„,
60 ■ KAP
= 0,
SVso SMen + SSkl + SIzv + SPr e ,	(7)
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
2 • L
(* )
2 Men J Obd
(SMat + SObd ) • L • P • ST • ^ fe1
60 • RC
KAP
The economic order quantity (i.e. optimal batch quantity) is therefore:
2 • L • s.
Is	p i (sMat + SObd )'
I *Obd ' P "r 60^C
^ ^ ST ^
KAP
. (12)
where xOpt is optimal batch quantity [piece], L annual needs [piece/a], sMen order change costs [€], sObd processing costs per piece [€/piece], p interest rate of tied-up capital [1/a], sMat material costs per piece [€/ piece], RC available time [day/a], ST material flow rate, te1 time per unit [min/piece], KAP daily capacities [h/day].
These models are applicable when there is no fluctuation on the relation market-producer. There are neither distributions of production and demand processes considered nor the stochastic character of the mentioned processes.
3 CASE STUDY OF CALCULATING THE OPTIMAL BATCH QUANTITY OF A PRODUCT
Company X, which is a supplier to a car components manufacturer, found it increasingly difficult to be competitive on the global market due to excessively long manufacturing lead times and too high product prices.
The company's management organised a creativity workshop [13] to [15] in order to identify urgent measures, whose implementation would improve their market competitiveness.
The results of the creativity workshop showed that it would be necessary to do the following in the company:
•	significantly reduce the inventory in entry and exit warehouses, and on disposal locations within the production,
•	significantly reduce the lead times of orders. After the presentation of the results of the
creativity workshop, management decided that they would first solve the problem of large stocks (i.e. tied-up capital), which significantly raise the price of products (i.e. non-competitiveness) [16] to [18].
A project team was established in the company, in order to analyse the causes of large stocks and to propose possible solutions.
Team members analysed inventories in all warehouses. Together with the heads of warehouses
and the planners of production, they concluded that the batch quantities are defined on the basis of the experience of planners (i.e. estimates), which lead either to excessive stocks or a shortage of goods.
Product 1: Shield		
Operations:	Time per unit: te1 [min/piece]	Setup time: Tp [min]
10-CNC milling	1.40	120
20-washing	2.22	15
30-examination	2.05	25
40-assembly of the tube	0.30	10
TOTAL	5.97	170
Picture of the product
Picture of the machine
Product 2: Left suspension support		
Operations:	Time per unit: te1 [min/piece]	Setup time: Tp [min]
10-CNC milling I	27.50	210
20-CNC milling II	6.50	30
30-examination-measurements	0.50	10
40-assembly of bearings	1.05	15
TOTAL	35.55	265
Picture of the product
Picture of the machine
Fig. 7. te1 and Tp times for shield and suspension support
The project team contacted the experts for help in solving the problem of over- and understocking of goods. Experts suggested that project team could use Eqs. (4) and (12) in order to calculate the optimal batch quantities of products.
It was agreed that the project team would calculate the optimal batch quantities by using both equations and find differences between the results of Eq. (4) and Eq. (12), and finally calculate additional annual costs to thus determine the advantages of using the Eq. (4) or Eq. (12).
XOpt ~
Optimization of a Product Batch Quantity
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
The project team decided that the first experimental calculation of the optimal batch quantity would be carried out for two products:
•	Product 1: Shield;
•	Product 2: Left suspension support.
The project team obtained the following data from the technology routings for both products:
•	Data on times per unit te1 and setup times Tp (Fig. 7).
•	Other data, required for calculation of optimal product batch (Table 2), were obtained from various departments in the company.
Table 2. Data from company departments
■ basic model:
PRODUCT	Product 1	Product 2
Sum of order changes SsMen [€]	226	196
Costs of execution per piece Ss/zv [€/piece]	3.65	45.5
Costs of material per piece [€/piece] (incorporated in the costs of execution)	1.52	29
Annual needs L [piece/a]	15000	1500
Available time RC [day/a]	250	250
Interest rate of tied-up capital p [1/a]	20	20
Daily capacities KAP [h/day]	16	16
Material flow rate ST [-]	3	3
In order to assess the usability of the basic and extended model for the calculation of optimal batch quantity for both products, the project team decided that it would carry out the following: • Calculation of optimal batch quantity for Products 1 and 2 using the basic and extended model: • basic model:
~Opt
1 extended model:
2 • L • s.
2 • L • s,.
((SMat +SObd )'L'P
60 •RC
ST •Z w'
Calculation of batch lead times for Products 1 and 2 using the basic and extended model: • basic model:
T
Izv ' * Izv
TO = ST • T, ■ extended model:
To* = ST • TIzv*

p + tel ' XOpt i
60' KAP
T =
* Izv
Yj(TP + tel ' XOpt ) 60 • KAP '
Calculation of costs per product unit using the basic and extended model:
"■Opt
where sKos are costs per product unit [€/piece], sTo costs of tied-up capital per product unit during the lead time [€/piece], and sPre costs of tied-up capital per product unit during interoperation time [€/piece],
= {SMa, + SObd ) To ' 2
extended model:
_= SObd ' P • XOpt
RC' Spre	!
"■Opt
(SMat + SObd ) ' P Tc
2
* _ SObd ' P • XOpt
~RC' Spre 2L '
Calculation of difference of costs per product unit:
As = s — s
Kos Kos Kos'
The calculations were made with MS Excel software. The results are shown in Table 3.
The project team charted the influence of the batch quantity on the product costs (Fig. 8).
The results listed in Table 3 and Fig. 8 led the project team to the following conclusions:
•	The calculated optimal batch quantities of both products are significantly different from the current estimated batch quantities. Due to this fact, the storage costs are high.
•	Batch lead times are too long.
•	The technology routing of Product 1 defines the short times per unit te1; diagrams on Fig. 8 show that at the transition from the basic to the extended model for calculation of the optimal batch quantity, the batch quantity is only slightly reduced and the total costs of tied-up capital are only slightly higher (if the total time te1 is small, the basic model can be used for calculation of optimal batch quantity).
•	The technology routing of Product 2 defines the long times per unit te1; diagrams on Fig. 8 show that at the transition from the basic to extended model of calculation, the batch quantity is considerably reduced and the total costs of tied-up capital are much higher (if the total time te1
s
Obd + STo + SPr e +
s
S Kos S Obd + STo + SPr e + *

XOpt ~
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
is large, extended model must be used for the calculation of optimal batch quantity). • The basic model for calculation of optimal batch quantity does not take into account the tying-up of capital in production, and thus the optimal batch quantities are bigger than in the extended model.
Table 3. Results of calculations of optimal batch quantities for both products
CALCULATION
Product 1: SHIELD
Product 2: SUSPENSION SUPPORT
Model
Model
	Basic	Extended	Basic	Extended
Optimal batch quantity x [piece]	3048	1896	255	163
Batch lead time TO [day]	57.40	35.90	38.88	25.25
Costs per product unit sKos [€/piece]	3.92	3.89	48.20	47.95
Difference of costs per product unit SsKos [€/piece]		0.03		0.25
At the presentation of results, the company management agreed that the project team would continue its work in order to reduce lead times of orders.
4 CONCLUSION
This paper explains how to calculate the optimal batch quantity of a product (production is within the company) using the known basic and developed extended models; the latter, in addition to the costs of changing the batch (i.e. order) and storage costs, also takes into account the costs of interoperation time and the costs of execution of operations.
The project team in the company, which is a supplier of car components manufacturer, carried out some experiments, whose results have shown when to use the basic model and when to use the expanded model for the calculation of the optimal batch quantity. Further experiments in electro-mechanical industry will be needed for reliable decision making regarding the selection of the basic or extended model.
Fig. 8. Dependency diagrams costs vs. batch quantity for Products 1 and 2
Optimization of a Product Batch Quantity
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)1, 35-42
The model also needs to be further developed requiring a connection between the optimal quantities procuring materials in warehouses.
The company management decided for the project team to carry out also an AS-IS analysis of value flow for existing batch quantities. After a transition to optimal batches, the project team will repeat the value flow analysis for the same two products and find lead-time savings.
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