Graduate Program in Engineering and Technology Management Introduction to Simulation Aslı Sencer

2 Simulation – Very broad term – methods and applications to imitate or mimic real systems, usually via computer Applies in many fields and industries Very popular and powerful method

3 Advantages of Simulation Simulation can tolerate complex systems where analytical solution is not available. Allows uncertainty, nonstationarity in modeling unlike analytical models Allows working with hazardous systems Often cheaper to work with the simulated system Can be quicker to get results when simulated system is experimented. Aslı Sencer

4 The Bad News Don’t get exact answers, only approximations, estimates Requires statistical design and analysis of simulation experiments Requires simulation expert and compatibility with a simulation software Softwares and required hardware might be costly Simulation modeling can sometimes be time consuming. Aslı Sencer

5 Different Kinds of Simulation Static vs. Dynamic  Does time have a role in the model? Continuous-change vs. Discrete-change  Can the “state” change continuously or only at discrete points in time? Deterministic vs. Stochastic  Is everything for sure or is there uncertainty? Aslı Sencer

6 Using Computers to Simulate General-purpose languages (C, C++, Visual Basic) Simulation softwares, simulators  Subroutines for list processing, bookkeeping, time advance  Widely distributed, widely modified Spreadsheets  Usually static models  Financial scenarios, distribution sampling, etc. Aslı Sencer

7 Simulation Languages and Simulators Simulation languages  GPSS, SIMSCRIPT, SLAM, SIMAN  Provides flexibility in programming  Syntax knowledge is required High-level simulators  GPSS/H, Automod, Slamsystem, ARENA, Promodel  Limited flexibility — model validity?  Very easy, graphical interface, no syntax required  Domain-restricted (manufacturing, communications) Aslı Sencer

8 When Simulations are Used The early years (1950s-1960s)  Very expensive, specialized tool to use  Required big computers, special training  Mostly in FORTRAN (or even Assembler) The formative years (1970s-early 1980s)  Computers got faster, cheaper  Value of simulation more widely recognized  Simulation software improved, but they were still languages to be learned, typed, batch processed Aslı Sencer

9 When Simulations are Used (cont’d.) The recent past (late 1980s-1990s)  Microcomputer power, developments in softwares  Wider acceptance across more areas Traditional manufacturing applications Services Health care “Business processes”  Still mostly in large firms  Often a simulation is part of the “specs” Aslı Sencer

10 When Simulations are Used (cont’d.) The present  Proliferating into smaller firms  Becoming a standard tool  Being used earlier in design phase  Real-time control The future  Exploiting interoperability of operating systems  Specialized “templates” for industries, firms  Automated statistical design, analysis Aslı Sencer

11 Popularity of Simulation Consistently ranked as the most useful, popular tool in the broader area of operations research / management science  1979: Survey 137 large firms, which methods used? 1. Statistical analysis (93% used it) 2. Simulation (84%) 3. Followed by LP, PERT/CPM, inventory theory, NLP,  1980: (A)IIE O.R. division members First in utility and interest — simulation First in familiarity — LP (simulation was second)  1983, 1989, 1993: Heavy use of simulation consistently reported 1. Statistical analysis 2. Simulation Aslı Sencer

12 Today: Popular Topics  Real time simulation  Web based simulation  Optimization using simulation Aslı Sencer

13 Simulation Process Develop a conceptual model of the system  Define the system, goals, objectives, decision variables, output measures, input variables and parameters. Input data analysis:  Collect data from the real system, obtain probability distributions of the input parameters by statistical analysis Build the simulation model:  Develop the model in the computer using a HLPL, a simulation language or a simulation software Aslı Sencer

14 Simulation Process (cont’d.) Output Data Analysis:  Run the simulation several times and apply statistical analysis of the ouput data to estimate the performance measures Verification and Validation of the Model:  Verification: Ensuring that the model is free from logical errors. It does what it is intended to do.  Validation: Ensuring that the model is a valid representation of the whole system. Model outputs are compared with the real system outputs. Aslı Sencer

15 Simulation Process (cont’d.) Analyze alternative strategies on the validated simulation model. Use features like  Animation  Optimization  Experimental Design Sensitivity analysis:  How sensitive is the performance measure to the changes in the input parameters? Is the model robust? Aslı Sencer

16 Static Simulation: Monte-Carlo Simulation Static Simulation with no time dimension. Experiments are made by a simulation model to estimate the probability distribution of an outcome variable, that depends on several input variables. Used the evaluate the expected impact of policy changes and risk involved in decision making. Ex: What is the probability that 3-year profit will be less than a required amount? Ex: If the daily order quantity is 100 in a newsboy problem, what is his expected daily cost? (actually we learned how to answer this question analytically) Aslı Sencer

Example: Estimation of the Area 17 How do we estimate the area of a lake? a b Estimate p by shooting arrows! Consider the experiement: Shoot an arrow into the rectangle Estimate of p = # hits in the lake / # shoots Let p=area of lake/area of rectangle Area of lake= p. (ab) Aslı Sencer

Why Shoot Arrows? Shooting arrows seems silly now, but it has important simulation features:  Experiment to estimate something hard to compute exactly (in 1733)  Randomness, so estimate will not be exact; estimate the error in the estimate  Replication (the more the better) to reduce error  Sequential sampling to control error — keep tossing until probable error in estimate is “small enough”  Variance reduction Aslı Sencer

19 Ex1: Simulation for Dave’s Candies Dave’s Candies is a small family owned business that offers gourmet chocolates and ice cream fountain service. For special occasions such as Valentine’s day, the store must place orders for special packaging several weeks in advance from their supplier. One product, Valentine’s day chocolate massacre, is bought for $7,50 a box and sells for $ Any boxes that are not sold by February 14 are discounted by 50% and can always be sold easily. Historically Dave’s candies has sold between boxes each year with no apparent trend. Dave’s dilemma is deciding how many boxes to order for the Valentine’s day customers. Aslı Sencer

20 Ex1: Dave's Candies Simulation If the order quantity, Q is 70, what is the expected profit? Selling price=$12 Cost=$7.50 Discount price=$6 If D<Q Profit=selling price*D - cost*Q + discount price*(Q-D) D>Q Profit=selling price*Q-cost*Q Aslı Sencer

Probability Distribution for Demand YearDemand Demand Distribution Demand (x i, i=1,...,6 ) Probability P(Demand=x i ) 401/6 501/6 601/6 701/6 801/6 901/6 Aslı Sencer

Generating Demands Using Random Numbers During simulation we need to generate demands so that the long run frequencies are identical to the probability distribution found. Random numbers are used for this purpose. Each random number is used to generate a demand. Excel generates random numbers between 0-1. These numbers are uniformly distributed between 0-1. Random numbers Aslı Sencer

Generating random demands: Inverse transformation technique 23 P(demand=x i ) (x i ) /6 P(demand<=x i ) (x i ) /6 4/6 3/6 2/6 1/6 U1U1 D 1 =80 U2U2 D 2 =50 1.Generate U~UNIFORM(0,1) 2.Let U=P(Demand<=D) then D=P -1 (U) Aslı Sencer

24 Generating Demands Demand (x i ) Probability P(Demand=x i ) Cumulative Probability P(Demand<=x i ) Random numbers 401/6 [0-1/6] 501/62/6(1/6-2/6] 601/63/6(2/6-3/6] 701/64/6(3/6-4/6] 801/65/6(4/6-5/6] 901/61(5/6-1] Aslı Sencer

25 Ex1: Simulation in Excel for Dave’s Candies Use the following excel functions to generate a random demand with a given distribution function.  RAND(): Generates a random number which is uniformly distributed between 0-1.  VLOOKUP(value, table range, column #): looks up a value in a table to detremine a random demand.  IF(condition, value if true, value if false): Used to calculate the total profit according to the random demand. Aslı Sencer

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