The lot size is one of the directions of production which markedly
influences production costs. The lot size influences a production
flexibility, an amount of parts in process, flow time, capacity
utilization etc. The goal is to determine the lot size so that
production costs will be minimal (Tomek & Vavrova,2000).
There are several known methods for determination of the lot size
in the world. One of them is called economically optimal lot size. This
method defines the mathematical formula which minimizes the set-up costs
and storage costs. This lot size is expressed by mathematical model that
solves the compromise between the reduction of fixed costs per piece and
increasing of lot size, and on the other side increasing of the storage
costs (Habchi & Labrune, 1995).
The optimal lot size is determined as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[D.sub.o]--optimal lot size in pieces, [Q.sub.p]--planned number of
parts in pieces, [N.sub.pz]--batch set-up costs, [N.sub.j]--costs per
one piece, [n.sub.s]--annual storage costs including credit interest,
t--fraction period of the year (Gregor et al, 2000).
Another approach is based on a search of minimal lot size which is
needed for effective utilization usually as bottleneck or
capital-intensive capacity unit. The lot size, defined in this way,
provides economical utilization of chosen capacity units (bottlenecks).
The minimal lot size is determined as (Gregor et al, 2000):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[t.sub.pz]--time for set-up in min., [t.sub.k]--part time in min.,
D--lot size in pieces, a--coefficient. a = 0,04 for complicated parts
and a = 0,1 for production with automatic machines.
2. THE NEED OF ALTERNATIVE SOLUTION
The classic calculation methods of lot size consider just few
factors which influence lot size. That is why the attribute
"optimal" is applied only for strictly defined conditions. The
necessary input values of set up costs or storage costs are in fact
qualified approximations but not exact values. Such calculated optimal
lot size is at least approached to optimal values.
The authors assume that lot size is influenced by other factors not
only by considered classic methods of calculation. Here belong:
production type, orientation of material flow, system flexibility,
organization of manufacturing process etc. The authors have designed the
alternative method that can determine lot size more accurate than
standard methods. The method uses simulation and simulation optimization
as the base procedures.
3. METHOD STAGES
The method for determination of lot size includes these basic
* Creation of model
The method assumes the building of a very detail simulation model
of the production system. The simulation model has to respect the number
and types of machines and also their interchange ability, definition of
storage subsystem, definition of transport subsystem, number of human
resources. The material flow orientation is implemented into the
simulation model. Every part enters the model into the batch that has
defined its lot size. The model has to be validated (Masar et al, 2009).
* Calculation of production costs
The model of costs calculation is very important stage of this
method. We propose only variable costs in the model for calculation. The
variable costs depend on produced quantity. Then the total production
costs include operation costs, set up costs, transport costs, labour
costs and storage costs. The production cost is computed according to
the following function in the realized model.
Total _cos ts = [p.summation over (I=1)] O[C.sub.i] + S[C.sub.i] +
L[C.sub.i] + T[C.sub.i] + ST[C.sub.i] (3)
P is the number of entered parts, OC--operation costs, SC setup
costs, LC--labour costs, TC--transport costs and STC storage costs.
The individual costs items are calculated in entities of the
discrete-event simulation model. Therefore the costs will be calculated
only if the defined event occurs; for example when the technological
operation is finished. In this way the costs calculation is very
accurate. The value of total production costs is growing up with rising
of the number of finished parts. Therefore it is not proper to use it as
an objective function. The objective function will calculate the
production cost per finished part.
* Definition of the objective function
As it was mentioned, the lot size is influenced not only by costs
but by more factors. The authors in the process of definition of the
objective function went out of production goals. There were also
included the number of finished parts, machine utilization and flow time
in objective function besides the costs which represents the important
goals of production. The objective function was defined as follows:
IF No_out_parts 0 > default value of finished parts AND Machine
utilisation 0 > default value of machine utilisation AND Flow time 0
< default value of flow time Unit|_Costs = SumCosts / No_out_parts
RETURN Unit Costs ELSE Unit_Costs = SumCosts / No_out_parts + constant
RETURN Unit Costs ENDIF
Default value represents quantitatively evaluated production goals.
The constant should be about order higher than Unit_Costs value. Partial
values of the objective function are always calculated when the specific
element of production system finishes its activity.
* The selection of the optimization method The selection of method
is very important step of solution procedure. Simulator Witness was
used. This simulator provides several algorithms. Result used by
algorithm will find the global extreme of the objective function
The selection of input parameters is realized in optimizing module.
It is very important to constrain the input parameters meaningfully. We
recommend to set up the constraints of input parameters through special
designed preparatory simulation experiments.
4. METHOD VERIFICATION
The authors have prepared the simulation model of flexible
manufacturing system for verification of proposed method. This model is
a typical flexible manufacturing system for batch production in machine
industry. The manufacturing system has been designed to produce two
kinds of parts (hydraulic cylinders for hydraulic equipments) at the
same time. They are labelled as VD1 and VD2. These parts were produced
Each workstation of the FMS is defined as an independent module.
All these modules form the structure of FMS according to the following
* 2 compatible machines constitute the Group1 and the Group2,
* the workstation SRP1 and the workstation SRP2 are the components
of the Robotic cell,
Each workstation has its own input and output buffers. The
transport system consists of four automated guided vehicles (AGV). In
the given FMS, there is used a combined storage, (main storage is also
system input and system output of the FMS; each workstation as well as
FMC has own buffer and there is one emergency storage).
The objective function is defined as real function inside the
simulation model in Witness.
4.1 Results of optimization and their comparison
Minimal value of the objective function has been found according to
proposed method. We have obtained the following results according to the
proposed method for the given FMS.
The result of optimization process evaluates optimal lot size 6 for
batch VD1 and a lot size 3 for batch VD2. The optimal values of input
intervals have been calculated at the same time. These values for input
intervals are 22 minutes for VD1 and 11 minutes for VD2. The followed
parameters and production goals have been reached (see Table 1).
We have compared the results according to our method with
calculation according to economically optimal lot size method that is
defined by formula (1). We have used the values obtained from simulation
model in the best experiment. The values are in Table 1.
The gained values have been used for calculation of the lot size
value according to economically optimal lot size method:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
The computed value 15 represents the sum of the both values of lot
size. It means that the result according to our method is more accurate
because it brings lower unit costs than mathematical calculation.
The result according to formula (2) is 8 pieces in one batch. It
means that our method is more accurate. The method accepts the real
conditions of the manufacturing process. If the conditions change, it
will be possible to repeat the method.
Here it is important to notice that no classic methods determine
input intervals for batches. It is very important parameter that is
determined by our method unambiguously. We have tested and have compared
our alternative method on the next different manufacturing systems. We
can certify that the method has brought more accurate results than
It is necessary to mention some important facts that have to be
fulfilled. The simulation optimization is more accurate method for
determination of lot size than the classic methods because it is able to
respect much more factors which influence lot size. But it also requires
the existence of simulation model. On the other side the simulation
model allows research in the detail way of the real manufacturing
process. Classic methods are fast and simple. The simulation
optimization can take a long time according to the restriction of the
possible solving combinations. The simulation optimization seems as
proper method for accurate method for determination of lot sizes,
especially for flexible manufacturing systems where the set up time is
This paper has been supported as a part of a solution of projects
Gregor, M., Kosturiak, J., Micieta, B., Bubenik, P. & Ruzicka,
J. (2000) Dynamic planing and production control, KPI ZU EDIS, pp.
Habchi G.1; Labrune C. (1995) Study of lot sizes on job shop
systems performance using simulation. In Simulation Practice and Theory,
Volume 2, Number 6, 15 May 1995, Springer
Masar, A., Tanuska, P. & Masarova, Renata: Possible particular
abstract approach to validation. In: Annals of DAAAM and Proceedings of
DAAAM Symposium.--25--28th November 2009, DAAAM International Vienna,
Tomek, G. & Vavrova, V., (2000) Manufacturing control, Praha:
Grada Publishing, 2000, ISBN 80-7169-955-1
Waller, A.P.: Optimization of simulation experiments, (2004).
Optimizationpaper.pdf Accessed: 2010-05-02
Tab. 1. The gained results with optimal lot sizes of batches
Parameter Quantitative Value
Unit costs 3.134 [euro]
Average capacity utilisation 70.31%
Average flow time 54.21 min.
Number of finished parts 784 parts per day
Storage costs per day 300.40 [euro]
Set up costs per day 130.70 [euro]