Chapter 18 If mathematical analysis is too difficult, we can try each possibility out on paper. That way we can find which alternative appears to work best over a series of hypothetical futures. Simulation

Monte Carlo Simulation Simulation is a trial and error approach. Possible future cases are generated in accordance with underlying probabilities. Reality is duplicated using simple book-keeping type record keeping. No complex mathematics is needed. Numerical solutions are provided because: Analytic solutions may be difficult to obtain. Models require unrealistic assumptions.

Queuing Simulation of Sammy Lee’s Barbershop Probability distributions are obtained for times between arrival and for service. Random numbers generate time events. A log is kept. Reality is duplicated just as if real customers were arriving for haircuts. Times are not real. Customers are not real. The log entries are analyzed for key summary statistics. These may be analogous to what would be instead obtained from a mathematical model.

Queuing Simulation of Sammy Lee’s Barbershop

PERT Simulation Simulation is a valuable tool in PERT because: Activity completion times are uncertain. The three-time-estimate approach gives useful probability distributions. Traditional methods erroneously focus on single paths. Simulation simply runs the project on paper many times. Project completion times may be evaluated statistically.

Probabilities for Time to Construct a House The following data apply.

Simulating House Construction with PERT QuickQuant provided these simulation results.

Simulating House Construction with PERT The second QuickQuant screen tells us about unexpected longest paths. The a priori critical path was of longest duration only 437 times out of 500. In some projects, it may be longest as little as 1% of the time, or less.

Simulation Spreadsheet Templates and Software Simulation Templates Palisade Decision Tools @RISK 4.0

Simulation Spreadsheet Templates M/M/1 discrete arrivals and service M/M/1 exponential arrivals and service M/M/1 repeated simulation with Excel’s data table option Inventory, Discrete Demand, Backordering Forecasting two parameter exponential smoothing Risk Analysis Palisade Decision Tools @RISK 4.0

Generating Random Numbers (Figure 18-6) Entering =RAND() in a cell returns a random number between zero and one. =100*RAND() returns a two digit random number between 00 and 99

Simulation of Sammy Lee’s Barbershop (Figure 18-7) 3. Depress the F9 key to get a new simulation. 1. Enter the arrival distribution in A29:B34 and the service distribution in D29:E34 (shown next). 4. If more than 20 trials are desired, copy the formulas down until the desired number is obtained. The ranges in the three average formulas must be adjusted to take this into account. 2. Average results, Wq, W, Lq, and L are in cells G28:H31 (shown after the arrival and service distributions).

Arrival and Service Distributions (Figure 18-8) If the arrival and service distributions have more than 6 probabilities then the table ranges in columns B and E must be adjusted to take this into account.

Summary Results (Figure 18-9) Average results: Wq, W, Lq, and L.

Inventory Simulation Data (Figure 18-15 top portion) Discrete Demand, Backordering 1. Enter the problem name in C3. 2. Enter the problem information in G6:G15. 3. Enter the demands in D18:E25. If you have more than 8 periods, expand the table down to include all your demands.

Inventory Simulation Results (Figure 18-15 bottom portion) 2. The simulation details for each period. 3. If more than 8 demands are entered in D18:E25 (shown previously) copy row 50 down the same number of rows and adjust the AVERAGE formulas in row 51 to include all the periods.

Inventory Simulation Formulas Simulation results formulas.

Inventory Simulation Formulas Simulation detail formulas.

Other Inventory Simulation Templates Discrete Demand, Lost Sales Normal Demand, Backordering Normal Demand, Lost Sales

Discrete Demand, Lost Sales Inventory Simulation Results 1. The data section (top portion) of the spreadsheet is identical with the backorders case seen previously. Discrete Demand, Lost Sales Inventory Simulation Results Bottom Portion of Spreadsheet 2. Modifications are in columns G, H, and K. The: ending inventory cannot be negative, lost sales are computed, and the lost sales shortage cost is utilized. 3. Formulas not shown are the same as for the backorders case.

Inventory Simulations with Normal Demand For normally distributed demands, the spreadsheets are similar to the discrete demand cases. The only modifications are two new rows in the data section containing m and s, the mean and standard deviation of the normal distribution, and the demands in column F are generated according to a normal distribution =NORMINV(RAND(),m,s) The templates for these cases are on the CD-ROM.

Forecasting Simulation (Figure 18-16) 1. Enter the problem name in C3. 2. Enter the problem parameters in E7:E11 3. Depress the F9 key to get a new simulation. 4. Periods 12 - 96 are hidden so the results fit on one page. 5. Mean Squared Error

Repeating Simulations with Excel’s Data Table Option (Figure 18-17) 1. To repeat the barbershop simulation 100 times, enter the numbers 1, 2, . . . , 100 in cells A44:A143. 2. Enter the formulas shown in cells B43:D43 (they refer back to Fig 18-8 shown previously). 3. Highlight cells A43:E143, click on Data on the menu bar and select Table to get the Table dialog box shown next. 4. Click the cursor in the Column input cell line, then on an empty cell, and then click the OK button. Cells A44:E143 will fill with the results of 100 simulations. The numbers you obtain will be different because of the random nature of the simulation process.

Repeating Simulations with Excel’s Data Table Option (Figure 18-17) Clicking on Data and selecting the Data Table option yields the Table dialog box. Click in the Column input cell line, then click on any empty cell, and finally click the OK button. The result is 100 repetitions of the barbershop simulation. If a different number of repetitions is desired, highlight a different number of rows (the number of repetitions is equal to the number of rows highlighted).

Frequency Distribution (Figure 18-18) 1. Using the Chart Wizard to graph the results of repeated simulation trials makes it easy to see how simulations results vary. 2. Here the 100 repetitions of L are graphed for the barbershop simulation. The L from the one simulation in Fig 18-9 is 1.72. It appears to be a rather untypical value.

M/M/S Data Table Simulation Template 1. Enter the problem name in C3. 2. Enter the problem parameters in E7:E11 This portion of the spreadsheet calculates optimal number of servers and the corresponding minimum cost, Lq, and Wq for the M/M/S model. This information is used as input for a 100 trial simulation using Excel’s Data Table option, shown next.

M/M/S Data Table Simulation Template Each time the F9 key is depressed a new 100 trial simulation is obtained.

Exponential Arrivals and Service (Figure 18-19) 1. To redo the barbershop simulation in Fig. 18-7 with exponential interarrival and service times, the formulas in B6 and E6 are changed (as shown) and copied down to row 25 (trial 20). 3. Cell E36 in the formula in cell E6 contains o the mean service time. 2. Cell B36 in the formula in cell B6 contains o the mean interarrival time.

Four Seasons Villages (Figure 18-21) 3.Because this analysis does not take possible uncertainties in revenues and costs into account, the calculated return on investment might be misleading. 1. In financial analysis a result, such as a rate of return or return in investment, is calculated based on estimates of all the factors involved. 2. In this example, estimates of revenues and costs lead to a calculated return on investment of 19.90%

Risk Analysis Risk Analysis considers the uncertainty in all the factors that affect a result. It uses simulation to determines the result’s probability distribution. Two ways of performing repeated simulations are: Excel’s Data Table option Palisade Decision Tool’s @RISK

Four Seasons Villges Risk Analysis (Figure 18-22) 1. Revenues and some costs are assumed to be normally distributed with the means in column E and standard deviations in column F. The formulas are shown next. 3. 100 simulations using either Excel’s data table option or @RISK yields the return on investment histogram shown next.

Return on Investment Histogram (Figure 18-23) This histogram indicates that the return on investment probably will be some what higher that the 19.9% original estimate. It also indicates that the chance of a negative return on investment is zero.

Tornado Graph (Figure 18-24) 1. @RISK provides several analytical tools, including information on the sensitivity of each output variable to the input distributions. Tornado Graph (Figure 18-24) 2. As an illustration, this Tornado graph shows the correlation between each input and the return on investment. The higher the correlation the more significant is the input in determining the output’s value. 3. Here, the common area cost is the most significant factor.

Four Seasons Villges Risk Analysis Formulas 1. If Excel’s Data Table option is used to do the simulation use the formula below in cell D4. 2. If @RISK is used to do the simulation use the formula below in cell D4. 3. The formula used in cell D4 is copied down to dells D5:D14, D16:D21, and D25.

Hypothesis Testing Using Excel Figure 18-25 contains customer waiting times for 10-trial simulations of two alternative queuing organizations A and B. Hypothesis testing helps determine if one alternative is better than another. The null hypothesis is that the mean waiting times are identical under the two alternatives, under the assumption that the variances are unequal. A 5% significance level is used for the test.

Data Analysis Dialog Box Click on tools on the menu bar, select the Data Analysis option, and the Data Analysis dialog box appears. In it highlight t-Test: Two-Sample Assuming Unequal Variances, and click the OK button to get the t-Test:Two-Sample Assuming Unequal Variances dialog box shown next.

t-Test: Two Sample Assuming Unequal Variances Dialog Box (Figure 18-26) 1. Enter A4:A13 in the Variable 1 Range line (or $A$4:$A$13). 2. Enter B4:B13 in the Variable 1 Range line (or $B$4:$B$13). 3. Leave the Hypothesized Mean Difference line blank or put a zero in it. 4. Enter 0.05 in the Alpha box. 5. After selecting one of the options in the Output options section, click on the OK button.

t-Test Results (Figure 18-27) The t value of -0.5744 is much smaller than the two-tailed critical value of 2.1009. The null hypothesis that the means do not differ must be accepted. There appears to be no significant difference between the two alternatives.

Palisade Decision Tools @RISK The @RISK 4.0 software program on the CD-ROM accompanying this book can be used to perform simulations. The software permits the use of more than 30 probability distributions, it has options for analyzing results and it has the capability to incorporate correlations between input variables. A few of the common distributions it permits are beta, binomial, chi-square, exponential, gamma, geometric, hypergeometric, normal, Poisson, triangular, and uniform.

@RISK To start @RISK, click on the Windows Start button, select Programs, Palisade Decision Tools, then @RISK 4.0. Both Excel and @RISK will open. You will see the normal Excel screen with two new tool bars, one for Palisade Decision Tools and the other for @RISK. The icons on these tool bars that will be used will be explained in the following slides.

Four Seasons Villges with @RISK 1. The formula is D4 is also in D5:D14, D16:D21, and D25. Four Seasons Villges with @RISK 2. Add cell D29 to the list of outputs by highlighting the cell and clicking on the Add Outputs icon. 3. Click on the Simulation Settings icon and the Simulation Settings dialog box opens as shown next.

Simulation Settings Dialog Box Iteration Tab 3. Clicking on the Sampling tab yields the dialog box shown next. 1. Enter 100 in the # Iterations line. 2. Enter 1 in the # Simulations line.

Simulation Settings Dialog Box Sampling Tab Under Standard Recalc click in the Monte Carlo button and then click the OK button.

Clicking on the Start Simulation icon gives the Summary Statistics in the box within the @RISK-Results dialog box Summary Statistics The minimum, mean, and maximum of the return on investment plus all the input variables is given in this dialog box.

@ RISK Reports 1. In the @RISK dialog box, click on Results on the menu bar, select Report Settings, and the @RISK Reports dialog box appears. 3. Here a Tornado Graph in Excel is selected (as seen previously in Fig- 18-24). 2. A variety of reports and options can be selected.