Modeling and simulation of systems

Simulation optimization and example of its usage in flexible production system control

Parts of lecture

Simulation optimization

Determination of optimization problem for

flexible production system

Demonstration of solution by the usage of

simulator Witness

What is simulation optimization ? Simulation optimization is characterized as

optimization of outputs from simulation models. (M.C. Fu, profesor, University of Maryland)

Simulation optimization provides structural approach to determination of optimal values of input parameters in which optimum is measured by the function of output parameters from simulatiom model. (L.W.Schruben, profesor, University of California, Berkeley)

Simulation optimization – description of problem

Think of discrete event simulation model with p deterministic input parameters =(1, 2,... n) and q stochastic outputs variables Y, which are the function .

Assume that input parameters are defined on? permissible are . Define the real function of variable y, C(Y), which combines output variables q into the simple output stochastic variable. The goal is to determine the values , as well as functions F(), which is optimized.

F() is the objective function (it is also called simulation response function)

Simulation optimization

The optimal value of goal function cannot be established directly but must be determined as output of simulation operations.

Simulation model is understood as mechanism which transforms input parameters to output.

In other words – simulation model is function (which explicit form is not known), which evaluates set inputs

Optimization by means of simulation

Simulation can be used for optimization of chosen parameters in pretended (e.g. production) system.

The goal is improvement of values of objective function.

Advantages and disadvantages of simulation optimization

The definition of objective function is very simple. Complicated mathematical device is not needed.

Simulation also records time connections

Simulation is extraordinarily computationally demanding in association with simulation.

Optimization – basic ideas

Optimization is the process of assessment of such settings of individual parameters of the object in model which maximize or minimize the purpose function.

The goal of optimization is to find maximum or minimum (generally the extreme - absolute) of purpose function at the fulfilment of several restricted conditions.

Optimizing problem

Global minimum of the function f in the area is given by relation:

fopt minarg

function f() is called purpose (object) function.

Finding of the global minimum belongs to difficult tasks.

Optimizing problem

a1 b1

f y=f(x)R

b2

a2

x

D

Let the function f: DR

nn

n

iii babababaD ,...,,, 22

111

portrays n-dimensional cube D of closed intervals [ai,bi])

to the real number yR

Optimization – basic ideas

Optimizing problemsconsist of three basic components:

Objective function;

Complex of variables which influence the value of purpose function;

Complex of constrains for variables.

Support of simulation by optimizing module

Software packages are solved as additional modules (plug-in) to the basic simulation platform

Example - module Optimizer to simulator Witness.

Simulation model

Optimizer

Input dataOutput data (response)

Constrains

Initialization

Optimizing algorithms

Metaheuristic algorithms are used in majority of software products. Here belong:simulated annaeling tabu searchGenetic algorithms

Other algorithms that are usedClassic algorithms of stochastic optimization

as random search and hill climbing algorithmStatistical approaches e.g. the method of

response? area

Optimizing algorithms

The main algorithms of the module Optimizer:All combinationsMin/Mid/MaxHill Climb Random solutionsAdaptive Thermostatical SA

Example of the usage of simulation optimization in control of PVS

The modern flexible production systems are complicated, highly automated, integrated systems controlled by computer.

Often demands for the changes of the number and types of products demand to change the production strategies. The controlled strategies have to respect the basic goals of production.

These goals are opposed and it is difficult to reach them.

Simulation is appropriate method for solution of problems connected with production control.

Subsequent simulation optimization can find optimal values of chosen goal in dependence on random (meaningful) input parameters

Example of optimization

The goal of optimizationMinimization of production costs per one

piece in dependence on the size of intervals of ordering at fulfilment of other production goals:

Demanded number of produced pieces( max.) Short running production time (min.) High usage of production capacities (max.)

The goals of production are opposed!

Objective function

The function SumCost is growing up with rising of number of finished parts.

Therefore it is not proper to use it as an objective function.

Modified objective function:

partsoutnoSumCostCostUnit ___ , where

no_out_parts are number of finished parts

Solution procedure IF No_out_parts () < default value of finished

parts OR Machine utilisation () < default value of machine utilisation OR Flow time () > default value of flow time Unit_Cost = SumCost / No_out_parts + constant1RETURN Unit_Cost

ELSEUnit_Cost = SumCost / No_out_partsRETURN Unit_Cost

ENDIF

The chosen production system

Notes to objective function We look for the minimum of the objective function

(costs) It is necessary to maximize some goals at the same

time (usage of capacities, number of taken products) That is why quantitative evaluations of production

goals were determined (usage of capacities, number of taken products, running time)

If these determined goals were not reached in given experiment then constant was added to the value of the objective function

All the necessary values were obtained at preparatory experiments

Preparatory experiments Verification of the functionality of the model; Determination of the length of warm-up period;

Period under which the system get into normal operation During this period characteristics of the system are not

registered Discovery of upper and lower limits of the system

loading; These experiments serve as determination of restriction

of independent quantities Determination of quantitative characteristics of

production system. The concrete values of other goals are in the process of

discovery

Example – the group of preparatory experiments

Experiments x - 35

0

10

20

30

40

50

60

70

80

90

100

110

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Input interval VD1 (min)

Ave

rag

e u

tili

zati

on

of

wo

rkp

lace

s (%

)

0

100

200

300

400

500

600

Co

sts

(Sk)

Avertage worplaces utilization Costs per piece Minimal worplaces utilization

The dependence of the average usage of workplaces and costs per 1piece on the change of interval of arrivals VD1

Example – the group of preparatory experiments

Experiments x - 35

0

100

200

300

400

500

600

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Input interval VD1 (min)

Nu

mb

er

of

pa

rts

050100150200250300350400450

Tim

e (

min

)

Input V1 Output V1 Input V2

Output V2 Flow time of V1 Flow time of V2

The dependence of the number of taken/entered piecesand running time on the change of interval of arrivals VD1

Window of the module Optimizer

The result of optimization

The best results of objective function

176,424

172

174

176

178

180

182

184

186

49 50 51 53 54 55 56 57 59 45 53 55 56 46 49 50 53 44 47 48 49 45 47 44 37 38 39 40 38 39

30 30 30 30 30 30 30 30 30 31 31 31 31 32 32 32 32 33 33 33 33 34 34 35 37 37 37 37 38 38

Input intervals VD1/VD2 (min.)

Co

sts

per

par

t (S

k)

The values of other studied goals :· running time of production (69,261 min);· the average usage of workplaces

(76,365%);· the total number of produced pieces (632).

Synthesis of knowledges Determined goals of production are opposed.

The improvement of one or more goals of production is showed in degradation of other goals

Very strict set of restrictions can cause the location of one result or smaller number of solutions which do not have to be the optimal solution but only local extreme;

The effort of management to increase the production by bigger loading of system leads to the fact that the average flow time is increased, the whole costs for production are increased, elaboration of the production is bigger though the average usage of workplaces is increased

The usage of capacities is very different

Synthesis of knowledges

It is possible partly compensate the contrast between residence of capacities usage and increase of running time of production by reduction of lot sizes. The reduction of lot sizes favourably influences or decreases the size of running time of production in dependence on the usage; It is more profitable in technologically related products when the proportion of lot size comes to one at simultaneous processing of several lot sizes;

Running time of lot sizes as well as the usage of capacities are dependent on the orientation of material flow;

The software for simulation optimization

Optimization package

Simulation platform

Vendor Primary search strategy

Optimizer Witness Lanner Group, Inc. Simulated annealing, hill climbing

OptQuest Arena Optimization Technologies, Inc.

Scatter search, tabu

search, neural networks Optimiz Simul8 Visual Thinking

International, Ltd.neural networks

AutoStat AutoMod AutoSimulations, Inc.

genetic algorithms

SimRunner ProModel ProModel Corp. genetic algorithms