2. System Analysis

3. System – set of interdependent elements that interact in order to accomplish a one or more final outcomes. Constrained and affected by: Endogenous factors – internal limits on capabilities of elements (resources), interactions, etc. Exogenous factors – outside (“environmental”) demands, supply, prices, etc. Need ways to anticipate behavior of system in order to make informed decisions.

4. System Analysis System Experiment w/ Actual System Experiment w/ Model of System Physical Model Mathematical Model Simulation Analytical Solution Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York. Our focus!

5. Simulation

6. Analytic Model – mathematical representation of a physical system. Discrete or continuous Dynamic or static Deterministic or stochastic (random) Simulation – “running” analytical model to study operation of physical system. May use exact model or an approximation Usually computer driven

7. Simulation What are some advantages of simulation? Does not affect system itself Can be used before system is implemented May reveal critical interaction while designing Provides solution when model is infeasible or intractable Allows for compression/expansion of time Possible to vary conditions Useful for “selling” idea or concept May be inexpensive (or may not)

8. Simulation What are some disadvantages of simulation? Requires programmers/computers May be time consuming Can be abused and/or misunderstood Garbage In – Garbage Out

9. Simulation Process Define Model Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York. Collect Data Assump. Valid? Verify w/ Test Data Program Simulation Sim. Valid? No Yes No Yes Design Experiments Execute Runs Analyze & Report

10. Monte Carlo Simulation

11. Traditionally, uncertain (random) values were often replaced by their means. Missed critical consequences of high/low values Monte Carlo Simulation – uses (pseudo) random numbers to vary uncertain inputs/assumptions. Useful for risk analysis Randomly samples from known distributions In Excel, we utilize =rand( ) function. Generates value in (0,1) Used as probability w/ CDF

12. Monte Carlo Simulation Simple example: Each bulb produced on a line has a 5% chance of being defective. A single bulb may be simulated in Excel as: =if(rand()<0.05,1,0)

13. Monte Carlo Simulation Simple example: 10 rand() 10 x 0.5

14. Monte Carlo Simulation Note: your numbers will be different … It’s the nature of random values!

15. Monte Carlo Simulation

16. Another example: Monte Carlo Simulation Weekly demand for bulbs is (approx.) normally distributed with mean of 100 and std. dev. of 20. A single day’s demand may be simulated in Excel as: =norminv(rand(),100,20) Note: for integer values, can use =round(norminv(rand(),100,20),0)

17. Monte Carlo Simulation

18. Let’s look at a “system”: Monte Carlo Simulation Weekly demand for bulbs is distributed as N(100,20). Each week, we refill stock up to 105 bulbs. How many do we sell? # Sold = min{Demand, 105}

19. Monte Carlo Simulation And the standard deviation using s. But also … Plus, create confidence intervals.

20. Monte Carlo Simulation

22. And another layer: Monte Carlo Simulation Weekly demand for bulbs is distributed as N(100,20). Each week, we refill stock up to 105 bulbs. Each bulb has a 5% chance of being defective. How many do we sell? (We can use the =binom.inv() function to get a random # defects.) # Sold = min{Demand, 105 – # Defects }

23. Monte Carlo Simulation

24. Additional questions: What if defect rate is not known w/ certainty? What is optimal stocking level? What if we have two (or more) types of bulbs to stock in a fixed space, each with its own costs, demands and defect rates? We will need special software (e.g., Crystal Ball, @Risk) or to learn how to program in VBA!

25. Optimization

26. Optimization Process Source: Gosavi, A. (2003), Simulation Based Optimization, Kluwer Academic Publishers, The Netherlands. Simulator Optimizer Simulator used to value current choice Optimizer moves choice in “better” direction

27. Optimization Example Consider the machine maintenance decision Simulator: given distribution of (random) time between breakdowns, cost of breakdown, current maintenance frequency and cost of maintenance, determines average cost. Optimizer: improves choice of maintenance frequency until stopping condition is reached.

28. Optimization Process Example of optimizer step: Simulator provides estimated values (V) for x n – h n and x n + h n. Slope is estimated by: V(x n – h n ) – V(x n + h n ) 2h n If current choice is x n. d n = Choice is improved: x n+1 = x n + a n d n (Note: a n and h n get smaller as process continues.) V xnxn x n+1 V(x n + h n ) V(x n – h n )