2. Building and Running a FTIM n 1. Define the system of interest. Identify the DVs, IRVs, DRVs, and Objective. n 2. Develop an objective function of these DVs, IRVs, and DRVs. n 3. For each IRVs, use stat. analysis to determine the appropriate distr. – a formula, or a data table. n 4. Find an inverse of the cdf of each dist. or use EXCEL function or VLOOKUP

3. building continued... n Based on the formulas developed in steps 2- 4, build a spreadsheet with following sections. – Input parameter area – DV area; – Prob. Distr. area (optional) – Random variates generation and simulation area – Performance measure and summary statistics

4. An EXAMPLE: n pages 17-19 (Fresh Foods example)

5. An Example: Perishable Product Inventory Control Decisions n Step 1: The system of interest: Inventory control system. – Goal: Maximize the expected weekly profit. – DV: Order Quantity -- Q. – IRV: weekly demand -- D. – DRV: an indicator variable for stockout: I=0: No stockout, I=1: Stockout. In this case, this DRV may not be relevant.

6. Example continued... n Step 2: Develop the weekly profit function. n Step 3 & 4: Determine the distri. for each IRV--D. This is a discrete RV with only 5 possible values => a VLOOKUP table should be used (This step may require some statistical analysis, details to be discussed later). n Step 5: Develop a spreadsheet model according to the guidelines of building an FTIM.

7. Output Data Analysis n The output of N simulation replications => a data sample with N observations. n SM actually generates the sample data for the performance measure. n Summary Statistics should be presented – mean, standard deviation. – confidence intervals with certain c.l.s – histogram or other distribution related graph

8. output continued... n Major EXCEL functions that do the job: – =AVERAGE(data range); – =STDEV(data range); – =MIN(data range); =MAX(data range). – =NORMSINV(c.l.) -- returns the z-score for a specific confidence level. – lower limit (upper limit)=mean - (+) NORMSINV(c.l.)*STDEV/SQRT(N)

9. output continued... n The statistical analysis performed on the output data is valid only when all these replications are independent. n For an FTIM, the Quality of the simulation results depends on the quality of the EXCEL’s 0-1 random number generator (=RAND()). n “=RAND()” has been tested extensively. It is a good random number generator.

10. Complicating Factors n Lack of independence between consecutive data points => autocorrelation => naïve application of conventional statistical techniques => misleading results. n In a queueing system (VTIM), the customer waiting times tend to have some positive autocorrelation. (due to the queueing discipline such as FCFS).

11. Complicating Factor continued.. n The monthly profits may also have some positive autocorrelation due to the fact that demand may have that autocorrelation. (Due to the property of input RVs). n Risks of using simulation models: – Statistical inference based on simulated data which do not satisfy the critical assumptions => – Misleading information => – expensive errors in policy. n This issue will be discussed in later section.

12. How accurate performance measure (expected weekly profit) an SM can generate? n The accuracy is determined by the width of the confidence interval of the performance measure. n The larger the number of replications => the narrower the CI => The more accurate the result is.

13. Determining the number of replications. n A procedure : – Try an arbitrary number (>=50) of replications. – Calculate s  an estimate for the standard deviation based on the first set of simulation replications (>=50). – For |CI| <=D, we use n=z  2 n 2 s 2  D 2

14. How to Generate a large number of replications? n Using EXCEL’s recalculate key “F9”. n Using EXCEL’s Copy command. n Using EXCEL’s DATA/TABLE function. n Using EXCEL’s Add-in programs for simulations such as @RISK or Crystal Ball

15. “What if” Analysis in SMs n What happens to the performance measure of the system if the value of a DV changes? n It is more tricky than the “what if” analysis in an AM. Because – the difference between the performance measures for different DV values may be due to the randomness of the simulation runs. This is true in particular for the slight differences between the performance of different decisions.

16. “What if” continued... n “Quick and Dirty” solution: Just increase the number of replications. If the difference is very significant and consistent, then we may make a conclusion. However, this is NOT scientific and reliable. n Use Hypothesis testing to solve this problem. (To be discussed later).

17. Simulating the Number of Occurrences of Random Events n Occurrences of Random Events: – arrivals of cars to a gas station. – machine breakdowns. – new competitors entering the market. – accidents during the rush hour. – insurance claims during a year. n the number of “events” that occur during some time period => Discrete RV.

18. What distribution we usually use for this RV? n Poisson Distribution: For a unit time Where x is the number of events occurring in a unit of time and is the mean (average arrival rate per unit time)

19. For a time period t, we have

20. Poisson continued... n all non-negative integers. completely characterized by a single parameter, its mean. Its variance is also equal to mean   t can be used to approximate Binomial distribution and can be approximated by normal distribution when certain conditions are met.

21. How to simulate the Poisson RVs? n Determine the time period in which a certain # of random events occur Determine the mean ( ) n Create a Vlookup table: using EXCEL function: “=POISSON(x, mean, 1). n Use VLOOKUP function to generate the number of event occurrences in one time period.

22. Poisson Dist. is closely related to Exponential Dist. n Another way of looking at the process of random event occurrences: The interarrival time (TBA) is exponentially distributed if the number of arrivals for a fixed time period has a Poisson distribution. The mean of the TBA is 1/ 

23. Applications of Poisson Distribution n Market Share Model n Machine Maintenance Model n Waiting-line (queueing) Model

24. Basic Structure of the Market Share Model n Total market capacity at time 0 is estimated as M. n Several dominant companies and a number of small companies as a single class. n Initial market share distribution is known. n A random change in the market share due to customers randomly switch among the companies, and entering of new or exiting of existing small companies.

25. Structure continued... n The most difficult part in modeling is to get the switching probability matrix. n Consider an example: – Assume that we have K players where K-1 of them are dominant companies and the Kth player represents all small companies. – S 1, S 2, …, S k : market share dist. at time 0.

26. Structure continued... n New entries: Poisson Distribution. n Exiters: Certain Chance of exiting for small firms (in %). n Market Share Loss Percentages (Show in the class). n These SLPs can also be random! Some distributions may be selected for these SLPs

27. Objective of Market Share Model n To forecast the future market share of the company (in particular, Large firms are more interested in this issue). n Start with the initial (current) market share distribution, market share loss %s, and the process of new entries and exiters. n Choose a planning horizon. n Build the Model using EXCEL.

28. An Example: n Use a flowchart to design the model. n Show the example in class.

29. FTIM -SMs Applications n Bidding Models n Inventory Models n Project management Models n Details of these models will be discussed in class.