Computer SimulationHenry C. CoTechnology and Operations Management, California Polytechnic and State University

Simulation Models (Henry C. Co)

Simulation Models (Henry C. Co)

Simulation ModelSimulation: a descriptive technique that enables a decision maker to evaluate the behavior of a model under various conditions.Simulation models complex situationsModels are simple to use and understandModels can play what if experimentsExtensive software packages available

Simulation Models (Henry C. Co)

Analytic models: values of decision variables are the outputs.Simulation models: values of decision variables are the inputs. Investigate the impacts on certain parameters when these values change.

Simulation Models (Henry C. Co)

Why Simulation?

Simulation Models (Henry C. Co)

Analytic modelsMay be difficult or impossible to obtain. Typically predict only average or steady-state behavior. Simulation modelsWide availability of software and more powerful PCs make implementation much easier than before. More realistic random factors can be incorporated.Easier to understand.

Simulation Models (Henry C. Co)

Simulation Process

Simulation Models (Henry C. Co)

Identify the problemDevelop the simulation modelTest the modelDevelop the experimentsRun the simulation and evaluate resultsRepeat until results are satisfactory

Simulation Models (Henry C. Co)

ImplementationIdentify the boundaries of the system of interest. Identify the random variables, decision variables, parameters, and the performance measure(s). Develop an objective function for the performance measure(s) in terms of random variables, decision variables, and parameters. Use computer to generate the simulated values of these random variables.Compute the values of the objective function using these simulated values of random variables and values of decision variables. Statistical analysis.

Simulation Models (Henry C. Co)

Monte Carlo Simulation

Simulation Models (Henry C. Co)

Monte Carlo method: Probabilistic simulation technique used when a process has a random componentIdentify a probability distributionSetup intervals of random numbers to match probability distributionObtain the random numbers Interpret the results

Simulation Models (Henry C. Co)

Major Components of Models

Simulation Models (Henry C. Co)

Random input factors: sales, demand, stock prices, interest rates, the length of time required to perform a task.Random performance measures:Business profit within a time interval. Average waiting time of a customer in a queuing system.Random input factors random performance measures.

Simulation Models (Henry C. Co)

An Analog Approach

Simulation Models (Henry C. Co)

Game Spinner for uniform random variable on the interval 0 to 1.

Every point on the circumference corresponds to a number between 0 and 1.For example, when the pointer is in the 3 Oclock position, it is pointing to the number 0.25.

Simulation Models (Henry C. Co)

Simulating a Discrete Distribution

Simulation Models (Henry C. Co)

10% of the interval (0.0 to 0.09999) is mapped (assigned) to a demand d= 8.20% of the interval (0.1 to 0.29999) is mapped to d =9.30% of the interval (0.3 to 0.59999) is mapped to d =10.etc., etc.

Simulation Models (Henry C. Co)

Excel Functions Useful in SimulationRAND(): a volatile Excel FunctionFunction =RAND() generates a uniformly-distributed random number between 0 -1.VLOOKUP

Simulation Models (Henry C. Co)

Use function =RAND() to generate a uniformly-distributed random number between 0 and 1.

Simulation Models (Henry C. Co)

Simulation Models (Henry C. Co)

F4=RAND() ; copy and paste F5:F13G4=VLOOKUP(F4,$B$4:$C$10,2,1); copy and paste G5:G13

Simulation Models (Henry C. Co)

A Machine Breakdown Example

Simulation Models (Henry C. Co)

F4=RAND() ; copy and paste F5:F13G4=VLOOKUP(F4,$B$4:$C$10,2,1); copy and paste G5:G13

Simulation Models (Henry C. Co)

Simulating a Continuous Distribution

Simulation Models (Henry C. Co)

The inverse transformation method To transform this random number into a sample value of the random variable. F(w) is the CDF F(x)=Prob. {W x}.

Simulation Models (Henry C. Co)

Inverse Transformation MethodDefine F(x)=Prob. {W x} = the probability that random variable W is less than or equal to a specific value w.Denote the 0-1 random number by u and let u = F(x).Use =RAND() to generate a value for u, substitute it into x= F-1(u) which in turn gives a value of x.

Simulation Models (Henry C. Co)

EXCEL Implementation Exponential Distributionu = RAND()For example, if arrival rate = 0.05, and RAND()=.75, the observation from the exponential distribution is (-1/0.05)ln(1-.75) = 23.73.Normal Distn: Function NORMINVFor example, NORMINV(RAND(),1000,100) returns a normally distributed random number with mean 1000 and standard deviation 100.

Simulation Models (Henry C. Co)

Using an EXCEL Simulation ModelInformation obtained from a Simulation model:Summary statistics about the performance measuresDownside Risk and Upside RiskDistribution of outcomesBased on the simulation results (Output), several alternatives (decisions) can be evaluated.

Simulation Models (Henry C. Co)

How Reliable is the Simulation?

Simulation Models (Henry C. Co)

The more trials we run, the higher the confidence we have in our results (just like any statistical analysis with real data sample).The confidence intervals about the parameters (or any other estimated parameters) can be computed. Given sample size and significant level confidence intervals can be computed, or given the half width of the confidence interval and significance level compute the minimum number of replications we have to run.

Simulation Models (Henry C. Co)

Advantages

Simulation Models (Henry C. Co)

Solves problems that are difficult or impossible to solve mathematicallyAllows experimentation without risk to actual systemCompresses time to show long-term effectsServes as training tool for decision makers

Simulation Models (Henry C. Co)

Limitations

Simulation Models (Henry C. Co)

Does not produce optimum solutionModel development may be difficultComputer run time may be substantialMonte Carlo simulation only applicable to random systems

Simulation Models (Henry C. Co)