Batch Size Optimization Based on Production Part Cost
Batch Size Optimization Based on Production Part Cost
Christina Windmark*, Kathrine Spang, Fredrik Schultheiss, and JanEric Ståhl
Division of Production and Materials Engineering
Lund University
Lund, 223 63, Sweden
ABSTRACT
When investigating different location and/or system designs the possible variables to take into consideration can
differs between the alternatives. Different production system will have different optimal working conditions and
hence should be compared with parameters suitable for the actual production system. When planning production
and calculating production costs the batch size is of high interest. Based on a manufacturing part cost model, this
paper will present a new model, close connected to the production system, integrating production performance,
setup times, material costs, material handling costs and tied capital, giving the production economic optimal batch
size. The aim is to give companies a model for determining the economic optimal batch size in order to use this
knowledge to make strategic decisions regarding production planning. Mathematical simulations are performed to
analyse the differences in result from the developed model and Wilsons existing standard method for calculating
the economic order quantity, hence to verify the importance of making an indepth analysis, taking the production
system into consideration. The advantage of the developed model is the usage of production costs based on variable
batch sizes, giving a more accurate outcome.
1. INTRODUCTION
Analysis of production costs is generally an essential part while evaluating products, production systems and
production location. The costs of producing one product depend on different parameters as quality, cycle time, lead
time and batch size. When planning and executing manufacturing operations the batch size can in many cases be one
of the most important parameters to consider. When having small batch sizes the time for setup has to be divided over
a small number of products, which could make the costs considerably higher than when producing large batches. On
the other side having large a batch size the setup costs can be low per product but storage costs, costs for tied capital
and costs for product obsolescence considerably will be higher. Besides the most obvious difference in cost connected
to batch size the downtime rate, quality rate and material losses could be connected to the batch size, especially if the
production lines/machines have long runin time and require a lot of adjustments at the beginning of production. The
costs connected to these scenarios have to be taken into consideration to attain a comprehensive analysis regarding
batch size. In the case of strictly customer order driven manufacturing, the batch size is determined mainly by the level
of yearly demand and the times at which orders are received and delivery takes place. When many of the orders during
a given year are of contractual character, one may well decide to combine several batches and, in connection to this,
manufacture products that are to be stocked.
This paper will present a model for analysing the economic optimal batch size based on manufacturing and
inventory costs. The model will be compared with the model of economic order quantity (EOQ) first presented by
Ford [1] and later on Hadley and Whitin [2]. The presented model have parameters depending on batch size, which
makes it possible to generate the economic optimal batch size without using fixed production costs, in contrast to
Fords model also known as Wilson’s model. Both models will be mathematical simulated and the results compared.
2. BACKGROUND
There are a numerous number of literature works where the importance of production volume and batch size are
raised. When searching for the wording “economic order quantity” on scholar.google.com and www.scopus.com over
1 300 000 respectively 2 400 hits on articles and books are obtained, showing the scale of the field. The optimal level
of production order quantity depends on many aspects such as time, economic and synergy effects of producing
different products after each other, making setups and tool changes more efficient and therefore makes smaller or
* Corresponding author: Tel.: (46) 46222 43 88 Fax: (46) 46222 85 04; Email: christina.windmark@iprod.lth.se
Flexible Automation and Intelligent Manufacturing, FAIM2014
larger batches dependent on order from costumer. Models for estimating the economic order quantity have been used
by a number of different authors, with different views and perspective and different level of generalizations and
simplifications. In this paper the authors take the economic aspect into consideration. The model presented in this
paper is based on the model developed by Ståhl [3]. In contrast to [3], this paper present the optimal batch size with the
total production part cost making the result more visually for the user, when deciding upon batch size. Table 1 presents
an extract of the published models within batch size optimization and the parameters used by the authors.
Table 1. Parameters used in batch size optimaztion models.
Authors
Setup costs
Order cost/shipment costs
Stockholding cost vendor
Stockholding costs buyer
Production rate
Demand
Batch time
Number of shipments per batch
Size of shipment
Total stock in system
Total production cost
Unit production cost
Price per unit
Raw material carrying costs
Carrying cost finished goods
Quantity raw material ordered
Batch size
Number of shipments
Time between shipments
Quantity produced between shipments
Manufacturing time
Cycle time
Downtime/downtime rate
Average inventory
Inventory built up at end of shipment
Ford (1913) [1], Hadley and Whitin (1963)
[2] x x x x x x
Banerjee (1986) [4] x x x
x x
x x
x
Cheng (1989) [5] x
x
x
x x
x
Golhar and Sarker (1992) [6] x x x
x x
x
x x x x x x x x x x x x
Hill (1999) [7] x x x x x x x x x x
x
x
Model presented in this paper x x x
x x x
x x
x
x x
x
x x x
Most of the models presented in Table 1, with the exception of [5], have in common the simplification that the
production cost per part is constant and not dependent on batch size. Cheng [5] present a model where the production
cost is dependent on the yearly demand. Ståhl [3] and Persson and Svenle [8] present a model for batch size
optimization, where Ståhl [3] present the theoretical framework and Persson and Svenle [8] an extensive case study of
the theoretical model on products at a stone crusher manufacturer. The majority of the models in Table 1 do not take
account of measures of production losses such as rejection rates qQ, downtime rates qS, and the like. An exception is
Golhar and Sarker [6], where downtime rate is regarded. Ståhl [3] presents a correlation between batch size and
accumulated downtime rate qS during the production of the batch similar to that as presented in Figure 1, which
indicate that the batch size can have a considerable impact on performance parameters. Figure 1 shows five batches
where the accumulated downtime rate presented as a function of the batch size and the thick black line represent the
trend of the five batches analysed. The function is used when finding the relation between batch size and downtime to
use when estimating the optimal batch level based on variable parameters. To get reliable data, the number of batches
analysed has to increase considerably compared to the analysis in Figure 1. More studies in the field has to be
conducted and it is important to notice that the result from this kind of analysis will be very company specific due to
company specific production facilities and different quality assurance levels. This paper will compare the result from
the model presented in this paper with the model developed by Ford [1] and have the intention to answer the following
questions:
1. How large is the differences between the less elaborate model developed by Ford [1] and a model based
on theoretical framework presented by Ståhl [3] regarding batch size and total production part cost?
2. How is the total production part cost affected by the batch size?
3. What are the advantages of the two models?
Batch Size Optimization based on Production Part Cost
Figure 1. Relationship between downtime rate and batch size for five different batches of one product.
3. THE MODELS
In this section the two models are presented. All parameters used in the equations are specified in Appendix 1. Due
to the differences between the two models concerning how different parameters are dependent on the order
quantity/batch size different generalisations are made. Table 2 present the assumptions of the models.
Table 2. The simplifications and assumptions of the two models.
Simplifications and assumption common for the two models
• No products obsoletes in stock
• No deficiencies in deliveries occur
• Quantity discount from suppliers are not taken into account
• The whole batch is processed before entering finished goods inventory or is transported to costumer
• There is always a safety stock of 10 %
• The setup time is fixed and nondepending of batch size
• Costs of handling incoming orders are constant
• Internal transfers are not dependent on batch size
• External logistics are not included
• Storage holding cost is constant
• Material is ordered and received to one batch at a time
Specific for the model based on Ståhl [3] Specific for model based on Ford [1]
• Either constant or periodic demand
• Equipment hourly cost is not dependent on batch size
and market demand and is supposed to be based on
the last five years production.
• Downtime losses are dependent on batch size
• Production cost per part is constant.
• Demand is constant and continues
Figure 2 gives a general of the material flow and production conditions. The production is regarded as one
line/machine and there are only two inventories, raw material and finished goods. The model consists of four parts,
manufacturing costs, cost of tied capital, costs of inventory and inbound logistic and costs of handling incoming
orders.
The model is based on the manufacturing part cost equation presented by Ståhl et al [9], Equation 1, where index M
e t a r e m i t n w o d e h t l e d o m t s o c t r a p g n i r u t c a f u n a m l a n i g i r o e h t o t n o s i r a p m o c n I . g n i r u t c a f u n a m s t n e s e r p e r e s a c s i h t niq
S is a function of the batch size N0. Equation 2 gives an example of such a function. The function is based a
mathematical adjustment of measured data and a1, a2 and a3 is variables. When implementing the model this function
has to be adopted and adjust to the product investigated.
k
M
=
k
B
(
1

q
B)
1

q
Q
B
k
CP
MD∙
60
t
0
MD
1

q
Q(
1

q
P)
CP
k
CS
MD∙
60
t
0
MD
1

q
Q(
1

q
P)
∙
q
S
(
1

q
S
0
)
T
su
∙
MD
N
0
CS
k
D
MD∙
60
t
0
MD
1

q
Q(
1

q
P)(
1

q
S
0
)
T
su
∙
MD
N
0
D
(1)
q
S(
N
0)
a
1
a
2
∙
N
0
3 (2)
0 2000 4000 6000 8000
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Batch size
Downtime rate
Model
Flexible Automation and Intelligent Manufacturing, FAIM2014
Figure 2. Schematic depiction of the material flow in production, some products go direct to the costumer and other
to the finished goods inventory for later shipments to costumer.
Included in the model is cost for tied capital including work in progress, tied capital in raw material inventory and
finished goods inventory. Equation 3 to 6 presents the calculations connected to the cost of tied capital where kCap is
the total part cost of tied capital. The cost of tied capital in work in progress is calculated by the mean value between
the raw material cost, kB and the part cost after manufacturing kM multiplied with the ratio between the time for
producing the batch and the total number of minutes in one year multiplied with the interest rate p. In the cost equation
of tied capital in raw material inventory, kRMI and finished goods inventory, kFGI, NC is the customer order quantity
direct delivered to the customer and MD the market demand in one year. NSS is the number of part in safety stock.
There is always a 10 % safety stock included in these calculations.
k
WIP
k
B
k
M
2
∙
p
∙
t
0
∙
N
0
365
∙
24
∙
60
(3)
k
RMI
k
B
∙
p
∙
N
0
2MD
0
(4)
k
FGI
k
M
∙
p
∙
(
N
0
N
C)
2MD
0
.
1
(5)
k
Cap
k
PIA
k
RMI
k
FGI (6)
The cost equation for inventory buildings and internal transportation is presented in Equation 7, where pe is the
pallet equivalent as presented in [10], kS the surface area cost/year, A, the total area used for handling and storing
products, pptot the total number of pallet places, tt the time for transporting the batch and kG the hurly cost for
transportation inhouse.
k
GIL
p
e
k
S
A
p
ptot
∙
N
0
2MD
t
∙
k
G
60
(7)
The costs of new order, kNO is the cost of ordering new parts to the raw material inventory, KOP and the costs
connected to receiving and handling new orders, KHNO multiplied with number of batches/year divided with market
demand, MD. The cost is presented in Equation 8.
k
NO
MD
N
0
∙ (
K
HNO
K
OP)
MD
(8)
The total part cost equation is presented in Equation 9 and in Figure 3 examples of the four cost functions and the total
part cost function are shown.
Batch Size Optimization based on Production Part Cost
k
N
0
k
M
k
Cap
k
GIL
k
NO (9)
Figure 3. Example of the cost functions included in the model assuming MD = 100 000 and p = 10 %.
The economic order quantity model based on Ford´s model is presented below. Equation 10 presents the ordering
cost KCO and Equation 11 the cost of handling and storing a product, h. In equation 12 the economic order quantity,
EOQ is calculated using the result from Equation 10 and 11.
K
OC
(
k
D
k
CS)
60
∙
T
su
(10)
h
k
B
∙
p
∙
1
.
1
∙
∙
1
.
1
(11)
∙
∙
2
(12)
4. RESULTS
In the following section the result from the mathematical analysis is presented. The models has been analysed
using the following values presented in Table 3. An extraction of the results from the analyses is presented in Table 4.
Table 4 presents the optimal batch sizes Nopt and EOQ and the total part cost based on the batch sizes obtained, for
different market demands and interest rates. In the table k(Nopt) is the total production part cost connected to Nopt and
k(EOQ) to EOQ. Figure 4 presents four of the simulations of the cost function k(N0) regarding market demand of 10
000 parts/year and 100 000 parts/year respectively. The optimal batch sizes from the two models are marked with
vertical lines and cost increase with 2.5, 5.0, 7.5 and 10 % are marked with horizontal lines. In order to attain a more
accurate view of the cost correlation between the batch size and production part cost the raw material cost kB is
subtracted.
Table 3. The values on the paramets used in the simulations. Estimated from experiense, optained in Swedish industry.
Parameter kB kCP kCS kD qQ qP qB t0 NC Tsu pe ka pptot tt kG KHNO KOP
Values 350 600 550 350 0.05 0 0.03 3 400 360 0.05 1 400 2 500 50 250 200 2 000
0 1×104 2×104 3×104
450
500
550
Batch size
Costs
0 1×104 2×104 3×104
0
10
20
Batch size
Costs
0 1×104 2×104 3×104
400
420
440
460
480
Batch size
Costs
kM
kCap
0 1×104 2×104 3×104
0
10
20
30
Batch size
Costs
0 1×104 2×104 3×104
0
2
4
6
Batch size
Costs
k kProd MNO
kGIL
Flexible Automation and Intelligent Manufacturing, FAIM2014
Table 4. The optimal batch size from the two models with market demand (MD) 10 000 to 250 000/year for intrest 1, 5, 10, 20, 30
% and the part cost connected to the batch size.
MD 1 % 5 % 10 % 20 % 30 %
10 000 Nopt 7 425 4 410 2 935 1 883 1 435
EOQ 2 679 1 643 1 232 899 742
k(Nopt) 473.2 485.5 496.7 513.9 528.2
k(EOQ) 479.9 492.5 503.1 519.6 533.5
40 000 N0 23 468 1 2913 8 528 5 278 3 928
EOQ 5 358 3 287 2 464 1 799 1 485
kProd1 507.7 494.2 478.3 468.1 453.8
kProd2 476.3 488.6 498.7 514.5 527.9
100 000 N0 48 375 24 675 15 825 9 568 7 028
EOQ 8 472 5 196 3 895 2 844 2 348
kProd1 448.0 459.4 469.3 484.7 497.7
kProd2 475.6 487.8 497.9 513.4 526.8
250 000 N0 92 625 41 255 26 375 14 935 10 840
EOQ 13 400 8 216 6 159 4 496 3 712
kProd1 441.1 453.1 463.0 478.2 490.9
kProd2 475.3 487.5 497.5 513.0 526.3
Figure 4. To the left, market demand 10 000, to the right market demand 100 000, the diagrams in the top have intrest rate 10 % and
in the bottom interset rate 1 %. The function is k (N0). The marking to the left is optimal batchsize according to Fords model and the
right according to the presented model in this paper and also the minimum value of the function k(N0).
Batch Size Optimization based on Production Part Cost
5. DISCUSSIONS AND CONCLUSIONS
The results presented in Figure 4 indicate that a rather large span of batch sizes gives an economic optimal
production cost, allowing a company to use a large variety of batch sizes without compromising costs. When studying
Figure 4 it is clear that the derivate of the cost function is larger for smaller batch sizes than for larger. This can also be
seen in results from Ford [1]. The result indicates that the economic risk in both models is higher when producing to
small batches than to large. In [11] Liker describes the benefits of lowering buffers and stock level, hence decreasing
batch sizes in order to see problems in the production, symbolically called the Japanese sea. When a company is active
in production development and continually works with improvements but cannot in a near future solve problems
concerning production rate and performance, a higher batch size can be advantageous. Especially if the production
cost is not considerably higher. However the choice of having larger batch sizes in order to deliver products on time to
costumers requires constant update and transparency by varying the batch size to see the actual performance of the
production system. This is illustrated by variation in sea level to both get a transparent production where continues
improvements can be made and at the same time make the company more profitable and competitive by delivering
products on time. The motive in producing parts that are to be kept in stock should not be to avoid the high costs of
startup times and setup times this can involve, but rather to improve manufacturing technology and profitability.
This paper shows the difference in result using the basic formula of Ford [1] and the more indepth model based on
previously presented by Ståhl [3]. The model based on Ståhl [3] gives a much higher economic order quantity/batch
size than the model developed by Ford [1]. However the cost difference is not near the relative differences in batch
size. Other authors have already mentioned the low impact of the changes in batch size on the part cost, for example
[12]. The economic batch size could be a large span of number of parts. Yet the differences in cost is not negligible
between the models, especially not when larger market demand is involved. Figure 4 shows that the mathematical
simulation performed in this study gives a cost increase with about 10 % as compared to when using Ford’s model
instead of the model presented, assuming that market demand is 100 000 parts/year. Important to note is that this result
is valid only for the values simulated.
The main advantage of the model developed by Ford is the simplicity of the model and according to the result
presented in this paper the economic outcome is close to optimal for smaller market demands. The benefits of using
the model presented is the possibility to use production cost dependent on batch size and to easy illustrate the part cost
together with the batch size. A problem occurring using the model based on Ford is to know the value of
manufacturing cost per part, which is dependent on batch size. It is common that the only manufacturing costs
connected to batch size is the setup costs and the current model makes it possible to take other cost correlations into
account. The essential application area of the model is the possibility to have manufacturing cost dependent on batch
size when estimating the optimal batch size. The model presented in this paper could be useful when pricing, making
the company aware of the part cost for a certain order quantity ordered by the costumer.
Further research should be intended to investigate further correlations between the batch size and performance
parameters, company studies have shown that there could be a strong correlation with start of production and quality
losses. More activities as external logistics and batch size dependent cost in internal logistics should be added to the
model investigated to give better analyses.
ACKNOWLEDGMENT
The authors would like to acknowledge the support provided by the Sustainable Production Initiative (SPI).
REFERENCES
[1] H. W. Ford: “How many part to make at ones”, Factory, The Magazine of Management, Vol. 10, No. 2, pp 135–136, 152,
1913.
[2] G. Hadley and T. M. Whitin, Analysis of inventory systems, PrenticeHall international series in management – and
quantitative methods series, 1963.
[3] J.E. Ståhl: " NEXT STEP in Costbased Sustainable Production Development – Cases from different production operations”,
Key note speak, ICAMME International conference on advances in manufacturing and materials engineering 3–5 July,
Kuala Lumpur, Malaysia, 2012.
[4] A. Banerjee: "A joint economiclotsize model for purchaser and vendor", Decision sciences, Vol. 17, No. 3, pp. 292–311,
1986.
Flexible Automation and Intelligent Manufacturing, FAIM2014
[5] T.C.E Cheng: "An economic order quantity mode with demanddependent unit cost”, European Journal of Operational
Research, Vol. 40, pp. 252–256, 1989.
[6] D. Y. Golhar and B. R. Sarker: “Economic manufacturing quantity in a justintime delivery system”, International journal
of production research, Vol. 30, No. 5, pp. 961–972, 1992.
[7] R. M. Hill: "The optimal production and shipment policy for the singlevendor singlebuyer integrated productioninventory
problem”, International journal of production research, Vol. 37, No. 11, pp. 2463–2475, 1999.
[8] N. Persson and V. Svenle: "Beräkning av orderkvantiteten i ny produktionsanläggning  En tillverkningsekonomisk analys åt
Sandvik Rock Processing”, Master thesi Lund University, Division of production and material engineering, 2013.
[9] J.E. Ståhl, C. Andersson and M. Jönsson, “A basic economic model for judging production development”, Paper presented
at the 1st Swedish Production Symposium, 28–30 August 2007. Gothenburg, Sweden.
[10] C. Windmark and C. Andersson: "Costs of Inbound Logistics: Towards a Decision Support Model for Production System
Design", Advances in Sustainable and Competitive Manufacturing Systems. Springer International Publishing, pp. 1049–
1061, 2013.
[11] J. Liker: "The Toyota Way 14 Management Principles from the World’s Greatest Manufacturer”, McGrawHill
Professional, 2004.
[12] S. Axsäter, Inventory control, Springer, 2006.
APPENDIX 1: LIST OF SYMBOLS
Parameter Description Value
A Area used when handling and storing products m2
a1, a2, a3 Constants 
EOQ Economic order quantity (Wilson´s formula) unit
h Handling and inventory costs per part currency/unit
kM Manufacturing costs per part currency/unit
kB Material costs per part currency/unit
kCap Costs of tied capital per part currency/unit
kCP Hourly machine costs during production currency/h
kCS Hourly machine costs during downtimes and adjustments currency/h
kD Salary costs currency/h
kFGI Tied capital in finished goods inventory per part currency/unit
KHNO Costs of handling a new order in production currency/batch
kGIL Costs per part of inbound logistics currency/unit
kNO Costs of new orders currency/unit
KOC Ordering costs currency/batch
KOP Costs for order processing min
kProd Total production part costs currency/unit
kRMI Tied capital in raw material inventory per part currency/unit
kS Costs of surface area currency/m2
kWIP Tied capital in work in progress per part currency/unit
MD Market demand unit/year
N0 Nominal batch size unit
NC Customer order quantity unit/order
Nopt Optimal batch size unit
NSS Number of parts in safety stock unit
p Interest rate 
pe Pallet equivalent pallets/unit
pptot Total number of pallet places number
qP Relative rate reduction 
qQ Rejection rate 
qS Downtime proportion 
t0 Nominal cycle time per part (for line production the throughput time) min
tt Time for transportation min
Tsu Setup time min