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: function f (  ) is called purpose (object) function. Finding of the global minimum belongs to difficult tasks.

Optimizing problem a1a1 b1b1 f y=f(x) R b2b2 a2a2 x D Let the function f: D  R portrays n-dimensional cube D of closed intervals [a i,b i ]) to the real number y  R

Optimization – basic ideas Optimizing problems consist 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 data Output data (response) Constrains Initialization

Optimizing algorithms Metaheuristic algorithms are used in majority of software products. Here belong:  simulated annaeling  tabu search  Genetic algorithms Other algorithms that are used  Classic algorithms of stochastic optimization as random search and hill climbing algorithm  Statistical approaches e.g. the method of response? area

Optimizing algorithms The main algorithms of the module Optimizer:  All combinations  Min/Mid/Max  Hill Climb  Random solutions  Adaptive 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 optimization  Minimization 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:, where no_out_parts are number of finished parts

Solution procedure IF No_out_parts () default value of flow time Unit_Cost = SumCost / No_out_parts + constant1 RETURN Unit_Cost ELSE Unit_Cost = SumCost / No_out_parts RETURN 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 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 The dependence of the number of taken/entered pieces and running time on the change of interval of arrivals VD1

Window of the module Optimizer

The result of optimization 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 VendorPrimary search strategy OptimizerWitnessLanner Group, Inc.Simulated annealing, hill climbing OptQuestArenaOptimization Technologies, Inc. Scatter search, tabu search, neural networks OptimizSimul8 Visual Thinking International, Ltd. neural networks AutoStatAutoModAutoSimulations, Inc. genetic algorithms SimRunnerProModelProModel Corp. genetic algorithms