Spreadsheet Simulation

Decision Making under Uncertainty: Typical Situation The decision maker often has no control over the random event Financial Risks: stock prices, interest rates, currency rates, commodity prices, defaults, etc. Market Risks: demand, input costs, prices Political Risks (strikes (Venezuela oil fields), nationalization) Natural Risks: weather, natural disasters Make a decision, x Random event, r Profit, f(x,r) Today Future date Time

Random Variables & Risk A random variable is any variable whose value cannot be predicted or set with certainty. Many “input cells” in spreadsheet models are actually random variables. the future cost of raw materials future interest rates future number of employees in a firm expected product demand Decisions made on the basis of uncertain information often involve risk. “Risk” implies the potential for loss.

Best-Case/Worst-Case Analysis Best case - plug in the most optimistic values for each of the uncertain cells. Worst case - plug in the most pessimistic values for each of the uncertain cells. This is easy to do but tells us nothing about the distribution of possible outcomes within the best-case and worst-case limits.

Possible Performance Measure Distributions Within a Range worst case best case worst case best case worst case best case worst case best case

Monte Carlo Simulation Develop a model that “behaves like” the real problem, with a special consideration of the random or probabilistic input variables Conduct a series of computer runs (called trials) to learn the behavior of the simulation model Compute the summary (output) statistics and make inferences about the real problem

Origins of Monte Carlo Physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus or how much energy the neutron was likely to give off following a collision, the problem could not be solved with analytical calculations. John von Neumann and Stanislaw Ulam suggested that the problem be solved by modeling the experiment on a computer using chance. Being secret, their work required a code name. Von Neumann chose the name "Monte Carlo". The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble.

Monte Carlo Simulation A model that mimics the behavior of a (random) system Probabilistic Input (random number) Analysis r1, r2, …, rn Controllable Input Model Outcome

Random event r Analysis r1, r2, …, rn Simulation Model Choose x (decision variable) Simulate some number, say n, random outcomes from a given distribution, r1, r2,…, rn Compute simulated profits: f(x, r1), f(x, r2), …, f(x, rn) (objective function) Evaluate results, e.g. estimate the expected profit or standard deviation by: Decision x Profit

Monte Carlo Simulation Advantages Allows statistical “sampling” from a large number of potential scenarios. Can simulate years of real time in seconds of computer time Compresses time Can modify a simulation and observe the results without changing the real system (e.g., effect of an expensive new machine, changes in traffic light timing) Experimental tool Can get results for very complex systems

Monte Carlo Simulation Advantages (cont’d) Can represent random events succinctly, via high level parameters e.g. modeling stock prices as Normally distributed random variables Need only two parameters, mean (μ) and standard deviation (σ) to specify distribution, while allowing for infinitely many possible outcomes Disadvantages May be difficult to characterize risk and uncertainty accurately Programming and experimental effort can be costly ($ and time) compared to other analytical methods Not an optimization technique Simulation results are not exact Statistical analysis is useful to interpret simulation output

Simulation 1) Identify the uncertain cells in the model. To properly assess the risk inherent in the model we need to use simulation. Simulation is a 4 step process: 1) Identify the uncertain cells in the model. 2) Implement appropriate RNGs for each uncertain cell. 3) Replicate the model n times, and record the value of the bottom-line performance measure. 4) Analyze the sample values collected on the performance measure.

Random Number Generators A RNG is a mathematical function that randomly generates (returns) a value from a particular probability distribution. We can implement RNGs for uncertain cells to allow us to sample from the distribution of values expected for different cells.

Examples of Discrete Probability Distributions

Examples of Continuous Probability Distributions

Discrete vs. Continuous Random Variables A discrete random variable may assume one of a fixed set of (usually integer) values. Example: The number of defective tires on a new car can be 0, 1, 2, 3, or 4. A continuous random variable may assume one of an infinite number of values in a specified range. Example: The amount of gasoline in a new car can be any value between 0 and the maximum capacity of the fuel tank.

How RNGs Work =IF(RAND()<0.5,1,2) The RAND() function returns uniformly distributed random numbers between 0.0 and 0.9999999. Suppose we want to simulate the act of tossing a fair coin. Let 1 represent “heads” and 2 represent “tails”. Consider the following RNG: =IF(RAND()<0.5,1,2)

Simulating the Roll of a Die We want the values 1, 2, 3, 4, 5 & 6 to occur randomly with equal probability of occurrence. Consider the following RNG: =INT(6*RAND())+1 If 6*RAND( ) falls INT(6*RAND( ))+1 in the interval: returns the value: 0.0 to 0.999 1 1.0 to 1.999 2 2.0 to 2.999 3 3.0 to 3.999 4 4.0 to 4.999 5 5.0 to 5.999 6

Background Information In August, Walton Bookstore must decide how many of next year’s nature calendars to order. Each calendar costs the bookstore $7.50 and is sold for $10. After February 1 all unsold calendars are returned to the publisher for a refund of $2.50 per calendar. Walton believes that the number of calendars it can sell by February 1 follows this probability distribution. Walton wants to maximize the expected profit from calendar sales. Open “Walton1.xls”

WALTON1.XLS For a fixed order quantity, we will show how Excel can be used to simulate 50 replications (or any other number of replications). Each replication is an independent replay of the events that occur. To illustrate, suppose we want to estimate the expected profit if Walton orders 200 calendars. To do this we need to simulate 50 independent simulations. This file contains the setup needed to begin the simulation.

The Simulation Inputs: Enter the cost data in the range B4:B6, the probability distribution of demand in the range D5:F9, and the proposed order quantity, 200, in cell B9. Create a cumulative probability column in column E by entering the value 0 in E5 and then the formula =D5+E5 and copy it down column E. Generate Random Number: Enter a random number in cell B19 with the formula =RAND( ) and copy it to the range B19:B68. Then freeze the random numbers in this range.

The Simulation -- continued Generate demands: The key to the simulation is the generation of the customers demands in the range C19:C68 from the random numbers in column B and the probability distribution of demand. To do this we: Divide the interval from 0 to 1 into five segments. The lengths of the segments relate to the probabilities of various demands. Then we associate a demand with each random number depending on which interval the random number falls into.

Simulation -- continued To accomplish this we can follow one of two ways: The first is to use a nested IF statement in cell C19 (and copy it down C). The second and simpler way is to use the VLOOKUP function. To do this we create a “lookup table” in the range E5:F9 and name it Lookup. Then enter the formula =VLOOKUP(B19,Lookup,2) in cell C19 and copy it to the range C19:C68. The function compares the random number to the values in E5:E9 and returns the appropriate demand in F5:F9. Revenue: Once the demand is known, the number of calendars sold is the smaller of the demand and the order quantity. To calculate revenue for the first replication in D19 we enter =$B$5*MIN(C19,$B$9).

Simulation -- continued Ordering Cost: The cost of ordering the calendars does not depend on the demand; it is the unit cost multiplied by the number ordered. Calculate this in cell E19 with the formula =$B$4*$B$9. Refund: If the order quantity is greater than the demand, there is a refund of $2.50 for each calendar left over, otherwise there is no refund. Therefore, enter the total refund for the first replication in cell F19 with the formula =$B$6*MAX($B$9-C19,0). Profit: Calculate the profit for this replication in G19 with the formula =D19-E19+F19.

Simulation -- continued Copy to other rows: Do the same bookkeeping for the other 49 replications by copying the range D19:G19 to the range D20:G68. Summary Measures: Each profit value in column G corresponds to one randomly generated demand. First, calculate the average and standard deviation of the 50 profits in cells B12 and B13 with the formulas =AVERAGE(Profits) and =STDEV(Profits). Similarly, calculate the smallest and largest profit with the MIN and MAX functions.

Simulation -- continued Confidence Interval for expected profit: Finally, calculate a 95% confidence interval for the expected profit in cells E13 and E14 with the formulas =AvgProfit-TINV(0.05,49)*StDevProfit/SQRT(50) =AvgProfit+TINV(0.05,49)*StDevProfit/SQRT(50) At this point we need to look and see what we have accomplished. Let’s look at the results of the simulation.

Simulation for Walton Bookstore

Accomplishments So here is what we have accomplished: In the body of the simulation rows 19-68, we randomly generated 50 possible demands and the corresponding profits. There are only five possible demand values and also for our order quantity, 200, the profit is $500 regardless of whether demand is 200, 250, or 300. There are 14 trials with profit equal to - $250, 9 trials with profit equal to $125, and 27 trials with profit equal to $500. The average of the 50 profits is $222.50 and their standard deviation is $328.58. (Answers may differ because of the random numbers.)

Probability Distributions The probability distribution of profit is as follows: P(Profit = -$250) = 14/50 P(Profit = -$125) = 9/50 P(Profit = -$500) = 27/50 We also estimate the mean of this distribution to be $222.50 and its standard deviation to be $328.58. It is important to be aware that with computer simulation each time it is run the answers will be slightly different. This is the reason for the confidence interval.

Confidence Interval The confidence intervals can be found in cells E13 and E14. This interval expresses our uncertainty about the mean of the profit distribution. Our best guess is the value we observed but because the corresponding confidence interval is very wide, from $129.12 to $315.88, we are not sure of the true mean of the profit distribution.

Simulation Simulation: The act of reproducing the behavior of a system using a model that describes the processes of the system. Time Compression: The feature of simulations that allows them to obtain operating characteristic estimates in much less time than is required to gather the same operating data from a real system. Monte Carlo simulation: A simulation process that uses random numbers to generate simulation events.

Specialty Steel Products Co. Example B.1 Specialty Steel Products Company produces items such as machine tools, gears, automobile parts, and other specialty items in small quantities to customer order. Demand is measured in machine hours. Orders are translated into required machine-hours. Management is concerned about capacity in the lathe department. Assemble the data necessary to analyze the addition of one more lathe machine and operator.

Specialty Steel Products Co. Example B.1 Historical records indicate that lathe department demand varies from week to week as follows: Weekly Production Relative Requirements (hr) Frequency 200 0.05 250 0.06 300 0.17 350 0.05 400 0.30 450 0.15 500 0.06 550 0.14 600 0.02 Total 1.00

Specialty Steel Products Co. Example B.1 Average weekly production is determined by multiplying each production requirement by its frequency of occurrence. Weekly Production Relative Requirements (hr) Frequency 200 0.05 250 0.06 300 0.17 350 0.05 400 0.30 450 0.15 500 0.06 550 0.14 600 0.02 Total 1.00 Average weekly production requirements = 200(0.05) + 250(0.06) + 300(0.17) + … + 600(0.02) = 400 hours

Specialty Steel Products Co. Example B.1 Weekly Production Relative Requirements (hr) Frequency 200 0.05 250 0.06 300 0.17 350 0.05 400 0.30 450 0.15 500 0.06 550 0.14 600 0.02 Total 1.00 Regular Relative Capacity (hr) Frequency 320 (8 machines) 0.30 360 (9 machines) 0.40 400 (10 machines) 0.30 The average number of operating machine-hours in a week is: 320(0.30) + 360(0.40) + 400(0.30) = 360 hours Average weekly production requirements = 400 hours

Specialty Steel Products Co. © 2007 Pearson Education Specialty Steel Products Co. Weekly Production Relative Requirements (hr) Frequency 200 0.05 250 0.06 300 0.17 350 0.05 400 0.30 450 0.15 500 0.06 550 0.14 600 0.02 Total 1.00 Regular Relative Capacity (hr) Frequency 320 (8 machines) 0.30 360 (9 machines) 0.40 400 (10 machines) 0.30 The average number of operating machine-hours in a week = 360 Hrs. Experience shows that with 11 machines, the distribution would be: Regular Relative Capacity (hr) Frequency 360 (9 machines) 0.30 400 (10 machines) 0.40 440 (11 machines) 0.30 Average weekly production requirements = 400 hours Example B.1

Specialty Steel Products Co. Assigning Random Numbers Random numbers must now be assigned to represent the probability of each demand event. Random Number: A number that has the same probability of being selected as any other number. Since the probabilities for all demand events add up to 100 percent, we use random numbers between (and including) 00 and 99. Within this range, a random number in the range of 0 to 4 has a 5% chance of selection. We can use this to represent our first weekly demand of 200 which has a 5% probability.

Specialty Steel Products Co. Assigning Random Numbers Random numbers in the range of 0-4 have a 5% chance of occurrence. Event Weekly Demand (hr) Probability 200 0.05 250 0.06 300 0.17 350 0.05 400 0.30 450 0.15 500 0.06 550 0.14 600 0.02 Random numbers in the range of 5-10 have a 6% chance of occurrence. Random numbers in the range of 11-27 have a 17% chance of occurrence. Random numbers in the range of 28-32 have a 5% chance of occurrence.

Specialty Steel Products Co. Assigning Random Numbers If we randomly choose numbers in the range of 00-99 enough times, 5 percent of the time they will fall in the range of 00-04, 6% of the time they will fall in the range of 05-10, and so forth. Event Existing Weekly Random Weekly Random Demand (hr) Probability Numbers Capacity (hr) Probability Numbers 200 0.05 00–04 320 0.30 00–29 250 0.06 05–10 360 0.40 30–69 300 0.17 11–27 400 0.30 70–99 350 0.05 28–32 400 0.30 33–62 450 0.15 63–77 500 0.06 78–83 550 0.14 84–97 600 0.02 98–99

Specialty Steel Products Co. Model Formulation Formulating a simulation model entails specifying the relationship among the variables. Simulation models consist of decision variables, uncontrollable variables and dependent variables. Decision variables: Variables that are controlled by the decision maker and will change from one run to the next as different events are simulated. Uncontrollable variables are random events that the decision maker cannot control.

Specialty Steel Products Co. Example B.2 Simulating a particular capacity level Using the Appendix 2 random number table, draw a random number from the first two rows of the table. Start with the first number in the first row, then go to the second number in the first row. Find the random-number interval for production requirements associated with the random number. Record the production hours (PROD) required for the current week. Draw another random number from row three or four of the table. Find the random-number interval for capacity (CAP) associated with the random number. Record the capacity hours available for the current week.

Specialty Steel Products Co. Example B.2 Simulating a particular capacity level If CAP > PROD, then IDLE HR = CAP – PROD If CAP < PROD, then SHORT = PROD – CAP If SHORT < 100 then OVERTIME HR = SHORT and SUBCONTRACT HR = 0 If SHORT > 100 then OVERTIME HR = 100 and SUBCONTRACT HR = SHORT – 100 Repeat steps 1 - 8 until you have simulated 20 weeks.

Specialty Steel Products Co. 20-week simulation © 2007 Pearson Education Specialty Steel Products Co. 20-week simulation 10 Machines Existing Demand Weekly Capacity Weekly Sub- Random Production Random Capacity Idle Overtime contract Week Number (hr) Number (hr) Hours Hours Hours 1 71 450 50 360 90 2 68 450 54 360 90 3 48 400 11 320 80 4 99 600 36 360 100 140 5 64 450 82 400 50 6 13 300 87 400 100 7 36 400 41 360 40 . . . . . . . . . . . . . . . . . . . . . . . . 20 37 400 19 320 80 Total 490 830 360 Weekly average 24.5 41.5 18.0

Specialty Steel Products Co. 1000-week simulation A steady state occurs when the simulation is repeated over enough time that the average results for performance measures remain constant. Comparison of 1000-week Simulations 10 Machines 11 Machines Idle hours 26.0 42.2 Overtime hours 48.3 34.2 Subcontract hours 18.4 8.7 Cost $1,851.50 $1,159.50

Capital Bank An example of queuing system simulation Capital Bank is considering opening the bank on Saturdays morning from 9:00 a.m. Management would like to determine the waiting time on Saturday morning based on the following data: Data: There are 5 teller positions of which only three will be staffed. Ann Doss is the head teller, experienced, and fast. Bill Lee and Carla Dominguez are associate tellers less experienced and slower.

Capital Bank Data: Service time distributions: Ann’s Bill and Carla’s Service Time Probability Service Time Probability 0.5 minutes 0.05 1 minute 0.05 1 0.10 1.5 0.15 1.5 0.20 2 0.20 2 0.30 2.5 0.30 2.5 0.20 3 0.10 3 0.10 3.5 0.10 3.5 0.05 4 0.05 4.5 0.05

Capital Bank Data: Customer inter-arrival time distribution: Inter-arrival time Probability 0.5 Minutes 0.65 1 0.15 1.5 0.15 2 0.05 Service priority rule is first come first served A simulation model is required to analyze the service.

Capital Bank – Solution Calculating expected values: E (inter-arrival time) = 0.5 (0.65) + 1 (0.15) + 1.5 (0.15) + 2 (0.05) = 0.80 minutes [75 customers arrive per hour on the average, (60 / 0.8 = 75)] E (service time for Ann) = 0.1 (0.05) + 1 (0.10) +…+ 3.5 (0.05) = 2 minutes [Ann can serve 60 / 2 = 30 customers per hour on the average] E (Service time for Bill and Carla) = 1 (0.05) + 1.5 (0.15) +…+ 4.5 (0.05) = 2.5 minutes [Bill and Carla can serve 60 / 2.5 = 24 customers per hour on the average]

Capital Bank – Solution To reach a steady state the bank needs to employ all the three tellers (30 + 2 (24) = 78 > 75)

Capital Bank – Simulation Logic If no customer waits in line, an arriving customer seeks service by a free teller in the following order: Ann, Bill, Carla. If all the tellers are busy the customer waits in line and takes then the next available teller. The waiting time is the time a customer spends in line, and is calculated by [Time service begins] minus [Arrival Time]

CAPITAL – Simulation Demonstration 1.5 1.5 1.5 1.5 1.5 1.5 Bill 1.5 Ann 1.5 1.5 1.5 3.5 Mapping Interarrival time 80 – 94 1.5 minutes Mapping Ann’s Service time 35 – 64 2 minutes

CAPITAL – Simulation Demonstration Bill 3 5.5 Ann 1.5 3.5 Mapping Interarrival time 80 – 94 1.5 minutes Mapping Bill’s Service time 40 – 69 2.5 minutes

CAPITAL – Simulation Demonstration Waiting time 3 3.5

CAPITAL – 1000 Customer Simulation

CAPITAL – 1000 Customer Simulation This simulation estimates two performance measures: Average waiting time in line (Wq) = 1.67 minutes Average waiting time in the system W = 3.993 minutes Average inter-arrival time = .80 minutes. To determine the other performance measures, we can use Little’s formulas: Average number of customers in line Lq =(1/.80)(1.67) = 2.0875 customers Average number of customers in the system = (1/.80)(3.993) = 4.99 customers.

WALTON4.XLS The company does not know exactly how many calendars its customers will demand, but it does have historical data on demands for similar calendars in previous years. Walton wants to use these historical data to determine a reasonable probability distribution for next year’s demand for calendars. Then it wants to use this probability distribution, together with @Risk, to simulate the profit for any particular order quantity. It eventually wants to find the “best” order quantity.

Solution We will use this example to illustrate many of @Risk’s features. We first see how it helps use to choose an appropriate “input” distribution for demand. Then we will use it to build a simulation model for a specific order quantity and generate outputs from this model. Finally we will see how the RISKSIMTABLE function enables us to simultaneously generate outputs from several order quantities so that we can chose a “best” order quantity.

Loading @Risk The first step, if you have not already done it, is to install Palisade Decision tools suite. Once @Risk is loaded, you will see two new toolbars, the Decision Tools toolbar shown here and the @Risk toolbar shown on the next slide.

Fitting a Probability Distribution Some of the historical demand data appears on the next slide. As the text box indicates, Walton believes the probability distribution of demand for next year’s calendars should closely match the histogram for the historical data. To see which probability distributions match the histogram well, we can use @Risk’s fitting ability, using the following steps.

Fitting a Probability Distribution -- continued Model window. Click on the Show @Risk-Model Window toolbar button. @Risk has two windows that get you outside of Excel: The Model and Results windows. The former helps in setting up the model; the latter shows results from running a simulation. For now, we require the Model window. Insert a Fit Tab. Once the Model window is showing, select the Insert/Fit Tab menu item. This brings up a one-column “spreadsheet” on the left. Copy and paste data. We want to copy the historical data to this mini-spreadsheet. To do so, go back to the Excel windows, copy the historical data, go back to the @Risk Model window, and paste the data – copy and paste work in the usual way.

Fitting a Probability Distribution -- continued Select candidate distributions. @Risk has many probability distributions from which to select. To see the candidates, select the Fitting/Specify Distributions to Fit menu item. This brings up the dialog box shown on the next slide. You can check as many of the candidates as you like. Some are undoubtedly unfamiliar to you so you might want to stick with familiar distributions such as the normal and triangular. However, we clicked on the OK to accept the defaults shown in the figure.

Fitting a Probability Distribution -- continued

Fitting a Probability Distribution -- continued Do the fitting. Select the Fitting/Run Fit Now menu item to see which of the candidate distributions most closely match the historical data. @Risk evaluates the fits in several different ways, and it also allows you to check the fits visually. After it runs, you will see a screen as shown on the next slide. This screen shows one of the candidate distributions superimposed on the histogram of the data. Examine the selected distribution. To do so, select the Insert/Distribution Window menu item, and fill it out as shown on the slide after the next. Specifically, select Fit Results in the Source box, select By Name in the Choose box and click on Normal. @Risk provides a very friendly interface for examining the resulting normal distribution.

Fitting a Probability Distribution -- continued

Fitting a Probability Distribution -- continued It has two “sliders” that you can drag in either direction to see probabilities of various areas under the curve. Also you can enter X values” or P values” directly into the boxes in the right column to obtain equivalent information. A caution about negative values. We should point out that there is a potential drawback to using this normal distribution. Although the mean demand in this example is approximately three standard deviations to the right of 0, so that a negative demand is very unlikely there is still some chance that one can occur – which would not make physical sense in our model. To ensure that negative demand do not occur, there are two possibilities.

Fitting a Probability Distribution -- continued First, we could use a truncated normal distribution of the form =RISKNORMAL(Meandem,StDev,0,1000). The function disallows values below the third argument or above the fourth argument. The other possibility is to choose a probability distribution that, by its very definition, does not allow negative values. On such distribution is the Weibull distribution, which provides one of the best fits to the historical data.

Developing The Simulation Model Now that we have chosen a probability distribution for demand, the spreadsheet model for profit is essentially the same as we developed earlier without @Risk. It appears on the next slide. The only new things to be aware of are as follows. Input distribution. We want to use the normal distribution for demand found from @Risk’s fitting procedure. To do this, enter the fitted mean and standard deviation in cells E4 and E5. Then enter the formula =ROUND(RISKNORMAL(MeanDem,StdevDem),0) in cell A13 for the random demand. This uses the RISKNORMAL function to generate a normally distributed demand with the fitted mean and standard deviation. Because demands should be integers, we use Excel’s ROUND function, with second argument 0, to round this value to 0 decimals.

Developing The Simulation Model -- continued Output cell. When we run the simulation, we want @Risk to keep track of profit. In @Risk’s terminology, we need to designate the Profit cell, E13, as an output cell. There are two ways to designate a cell as an output cell. One way is to highlight it and then click on the Add Output Cell button on the @Risk toolbar. An equivalent way is to add RISKOUTPUT( )+ to the cell’s formula. Either way, the formula in cell E13 changes from =B13+D13-C13 to =RISKOUTPUT( )+B13+D13-C13. The plus sign following RISKOUTPUT ( ) does not indicate addition. It is simply @Risk’s way of saying: Keep track of the value in this cell as the simulation progresses. Any number of cells can be designated in this way as output cells. They are typically “bottom line values of primary interest.”

Developing The Simulation Model -- continued Inputs and outputs. @Risk keeps a list of all input cells and output cells. If you want to check the list at any time, click on the Display Inputs, Outputs button on the @Risk toolbar. It provides an Explorer-like list as shown here.

Developing The Simulation Model -- continued Summary functions. @Risk provides several functions for summarizing output values. We illustrate these in the range B16:B19. They contain the formulas =RISKMIN(Profit), =RISKMAX(Profit), RISKMEAN(Profit), and RISKSTDDEV(Profit). The values in these cells are not of any use until we run the simulation. However, once the simulation runs, these formulas capture summary statistics of profit.

Running the Simulation Now that we have developed the model for Walton, the rest if straightforward. The procedure is always the same. We specify the simulation settings and the report setting and then run the simulation. Simulation settings. We must first tell @Risk how we want the simulation to be run. To do so, click on the Simulation Settings button on the @Risk toolbar. Click on the Iterations tab and fill out the dialog box as shown on the next slide. This says that we want to replicate the simulation 1000 times, each with a new random demand.

Running the Simulation -- continued

Running the Simulation -- continued Then click on the Sampling tab and fill out the dialog box as shown here. For technical reasons it is always best to use Latin Hypercube sampling, it is more efficient.

What is Sampling? Sampling is the process by which values are randomly drawn from the selected distribution. In @Risk, during each iteration of the simulation, one observation is chosen from the input distribution. As the number of iterations increases, the sample of observations more closely resembles the input distribution. When running a simulation, it is important that all areas of the input distribution get sampled, especially the low probability (high uncertainty) areas. If not, uncertainty will seem less than it actually is.

The Concept of “Efficiency” Statisticians have developed different ways to sample (or draw) from distributions. If we could do an infinite number of iterations in our simulation, these methods would produce equal results. However, since we use a finite number of iterations, sampling methods do not produce equivalent results. A sampling method is considered more efficient than another if it approximates a distribution with fewer iterations. Two popular sampling methods: Monte Carlo Simulation Latin Hypercube

Monte Carlo vs. Latin Hypercube Monte Carlo simulation draws samples from the full range of the distribution on each draw. Is an entirely random sampling technique. Requires a large number of iterations to adequately approximate the input distribution. Why? Most observations drawn are closer to the mean. Creates clustering. The tails (areas of high uncertainty) are usually underrepresented in the sampling. Latin Hypercube samples from all parts of the distribution, reducing clustering. Not entirely random (is a “stratified” sampling method) Latin Hypercube divides a distribution into intervals (strata) of equal probability and randomly draws from each interval. Insures that all portions of the distribution are sampled, including the tails. Latin Hypercube sampling is more efficient than Monte Carlo sampling:Requires fewer iterations.

Notice that the tails (high uncertainty areas) are not adequately represented. This results in underestimating risk Notice that even with only 8 observations, the tails are much more adequately represented. This results in a truer representation of risk. The area in each strata is equal but the width of each strata varies.

Running the Simulation -- continued We also recommend checking the Monte Carlo button on the Standard Recalc group. Although this has no effect on the ultimate results, it means that you will see random numbers in the spreadsheet. Report settings. @Risk has many options for displaying the outputs from a simulation. The outputs can be placed in an @Risk Results window or on new sheets of your Excel workbook. They can also be shown in more or less detail. Click on the Report settings button on the @Risk toolbar to select some of these options. In the dialog box on the next slide we have requested a summary of the simulation and detailed statistics, and we have asked that they be shown both in the @Risk Results window and on new sheets in the current workbook.

Running the Simulation -- continued

Running the Simulation -- continued Run the simulation. We are finally ready to run the simulation! To do so, simply click on the Start Simulation button on the @Risk toolbar. At this point, @Risk repeatedly generates a random number for each random input cell, recalculates the worksheet, keeps track of all output cell values. You can watch the progress at the bottom left of the screen.

Analyzing the Output @Risk generates a large number of output measures. We discuss the most important of these now. Summary Report. Assuming that the top box was checked in the @Risk Reports dialog box, we are immediately transferred to the @Risk Results window. This window contains the summary results shown here.

Analyzing the Output -- continued Detailed Statistics. We can also request more detailed statistics within the @Risk Results window with the Insert/Detailed Statistic menu item. Some of these detailed statistics appear on the next slide. All of the information in the Summary Report is here, plus some. Target values. By scrolling to the bottom of the detailed statistics list, as shown on the slide after next, you can enter any target value or target percentile. If you enter a target value, @Risk calculates the corresponding percentile, and vice versa.

Analyzing the Output -- continued

Analyzing the Output -- continued Simulation data. The results to this point summarize the simulation. It is also possible to see the full results – the data, demands and profits, from all 1000 replications. To do this select the Insert/Data menu item. A portion of the data appears on the next slide.

Analyzing the Output -- continued Charts. To see the results graphically, click on the Profit item in the left pane of the Results window and then select the Insert/Graph/Histogram menu item. This creates a histogram of the 1000 profits from the simulation.

Analyzing the Output -- continued The same interface is available that we saw earlier – namely, we can move the “sliders” at the top of the chart to the left or right to see various probabilities. Outputs in Excel. Often we will want the simulation outputs, including charts, in an Excel workbook. The easiest way to get the numerical information shown earlier is to fill out the Report Settings dialog box as we did. Then separate sheets are created to hold the reports. This has been a quick tour through @Risk’s report capabilities. The best way to become more familiar with @Risk is to experiment with the user-friendly interface.

Using RISKSIMTABLE Walton’s ultimate goal is to choose an order quantity that provides a large average profit. We could rerun the simulation model several times, each time with a different order quantity in the OrderQuan cell, and compare the results. However, this has two drawbacks. First, it takes a lot of time and work. Second, each time we run the simulation, we get a different set of random demands. Therefore, one of the order quantities could win the contest just by luck. For a fairer comparison, it would be better to test each order quantity on the same set of random demands.

WALTON5.XLS The RISKSIMTABLE function in @Risk enables us to obtain a fair comparison quickly and easily. This file includes the setup for this model. The next slide shows the comparison model. There are two modifications to the previous model. First, we have listed order quantities we want to test in the range names OrderQuanList. Second, instead of entering a number in cell B9, we enter the formula =RISKSIMTABLE(OrderQuanList).

The Spreadsheet

Using RISKSIMTABLE -- continued Note that the list does not need to be entered in the spreadsheet. However the model is now set up to run the simulation for all order quantities in the list. To do this, click on the Simulation Settings button on the @Risk toolbar and fill out the Iterations dialog box as shown on the next slide. Specifically, enter 1000 for the number of iterations and 5 for the number of simulations.

Using RISKSIMTABLE -- continued

Using RISKSIMTABLE -- continued After running the simulations, the Report window shows the results for all five simulations. For example, the basic summary report appears on the next slide. The first five lines show summary statistics of profit. Although we do not show them here, the same information can be seen graphically. A separate histogram of profit for each simulation is easy to obtain.

Using RISKSIMTABLE -- continued

Using RISKSIMTABLE -- continued Indeed, much of the appeal of @Risk is that we can see all of these characteristics – averages, minimums, maximums, percentiles, charts – and use them to make informed decisions.

Background Information As in the previous example, Walton needs to place an order for next year’s calendar. We continue to assume that the calendars will sell for $10 and customer demand for the calendars at this price is normally distributed with mean 168.1 and standard deviation 57.6. However, there are now two other sources of uncertainty.

Background Information -- continued First, the maximum number of calendars Walton’s supplier can supply is uncertain and is modeled with a triangular distribution. It’s parameters are 125, 250, and 200. Once Walton places and order, the supplier will charge $7.50 per calendar if he can supply the entire Walton order. Otherwise, he will charge only $7.25 per calendar. Second, unsold calendars can no longer be returned to the supplier for a refund. Instead, Walton will put them on sale for $5 a piece after February 1.

Background Information -- continued At that price, Walton believes the demand for leftover calendars is normally distributed with mean 50 and standard deviation 10. Any calendars still left over, say after March 1, will be thrown away. Walton plans to order 200 calendars and wants to use simulation to analyze the resulting profit.

Solution As before, we first need to develop the model. Then we can run the simulation with @Risk and examine the results. The completed model appears on the next slide. The model itself requires a bit more logic than the previous Walton model.

Solution WALTON6.XLS

Developing The Simulation Model The model can be developed with the following steps. Random inputs. There are three random inputs in this model: the most the supplier can supply Walton, the customer demand when the selling price is $10, and the customer demand for sale-price calendars. Generate these cells in A16, D16 and G16 with the formulas =ROUND(RiskTrian(E9,E10,E11),0), =ROUND(RiskNormal(E5,E6),0) and =ROUND(RiskNormal(F5,F6),0). Note that we generate the random potential demand for calendars at the sale price even though there might not be any calendars left to put on sale.

Developing The Simulation Model -- continued Actual supply. The number of calendars supplied to Walton is the smaller of the number ordered and the maximum the supplier is able to supply. Calculate this value in cell B16 with the formula =MIN(MaxSupply,OrderQuan). Order cost. Walton gets the reduced price, $7.25, if the supplier cannot supply the entire order. Otherwise, Walton must pay $7.50 per calendar. Therefore calculate the total order cost in cell C16 with the formula =IF(MaxSupply>=OrderQuan,UnitCost1,UntiCost2)*Supply

Developing The Simulation Model -- continued Other quantities. The rest of the model is straightforward. Calculate the revenue from regular-price sales in cell E16 with the formula =UnitPrice1*MIN(Supply,Demand1). Calculate the number left over after regular-price sales in cell H16 with the formula =UnitPrice2*MIN(Leftover, Demand2). Finally, calculate profit and designate it as an output cell for @Risk in cell I16 with the formula =RISKOUTPUT( )+E16+H16-C16.

Using @Risk As always, the next steps are to specify the simulation settings, specify the report settings and run the simulation. When there are several input cells, @Risk generates a value from each of them independently and calculates the corresponding profit on each iteration. Selected results appear on the next slide. They indicate an average profit of $255.66, a 5th percentile of - $410.50, a 95th percentile of $514.25, and a distribution of profits that is again skewed to the left.

Using @Risk

Sensitivity Analysis We now demonstrate a feature of @Risk that is particularly useful when there are several random input cells. This feature lets us see which of these inputs is most related to, or correlated with, an output cell. To perform this analysis, select the Insert/Graph/Tornado Graph menu item from the @Risk Results window.

Sensitivity Analysis -- continued In the resulting dialog box, select Profit as the output variable and click on the Correlation Sensitivity button. This produces the results shown here.

Sensitivity Analysis -- continued The “regression” option produces similar results, but we believe the correlation option is easier to understand. This figure shows graphically and numerically how each of the random inputs correlates with profit – the higher the correlation, the stronger the relationship between that input and profit. In this sense, we see that the regular-price demand has by far the strongest effect on profit.

Sensitivity Analysis -- continued The other two inputs, maximum supply and sale-price demand, are not nearly as important because they are nearly unrelated to profit. Identifying important input variables can be important for real applications. If a random input is highly correlated with an important output, then it might be worth the time and cost to learn more about this input and possibly reduce the amount of uncertainty involving it.

CORRELATED VARIABLES All previous problems have had random numbers be probabilistically independent. This means that if a random value is much larger than its mean, the other random values are unaffected. Sometimes values are correlated. If they are positively correlated, then a large number for one value will tend to produce a large number for a second value. While negatively correlated values tend to move in opposite directions.

CORRELATED VARIABLES Suppose that there are two different calendars that are sold but their demand is negatively correlated. This means that if a customer buys one calendar they are unlikely to buy another one. Assume a correlation of -0.90. The other parameters of this problem are the same as Calendar.xls. Compare the profit levels for correlation values of -0.90, 0, and 0.90.

CORRELATED VARIABLES The RISKCORRMAT (correlation matrix) function is needed. A correlation matrix has 1’s along the main diagonal since a variable is perfectly correlated with itself. The correlation values appear in the other parts of the matrix and the matrix is symmetric. See Correlated Demand template.xls. We would like to run this model for 3 different correlation values that appear in cells I9:K9 as -.90, 0, and 0.90.

CORRELATED VARIABLES The correlation matrix is entered in cells J5:K6 as 1, =RISKSIMTABLE(I9:K9), =J6, and 1. The RISKSIMTABLE command will allow @Risk to run the simulation for the three different correlation values that appear in cells I9:K9. We assume that the company orders 200 calendars of each type (cells B9 and B10). The data from Calendar.xls are entered into cells B4:B6 and E4:E6.

CORRELATED VARIABLES The formulas for Revenue, Cost, Refund, and Profit from Calendar.xls are entered in row 14 for product 1 and then copied to row 15 for product 2. All values for the two products are summed in row 16. To finish the model we must randomly generate correlated demands for the two products in cells B14 and B15.

CORRELATED VARIABLES The demand for product 1 is entered in cell B14 as: =ROUND(RISKTRIANG(E4,E5,E6,RISKCORRMAT(J5:K6,1)),0) The demand for product 2 is entered in cell B15 as: =ROUND(RISKTRIANG(E4,E5,E6,RISKCORRMAT(J5:K6,2)),0) The first argument of RISKCORRMAT is the correlation matrix range. The second is an index of the variable (1 for product 1 and 2 for product 2). Correlated demand values for the two products will now be generated.

CORRELATED VARIABLES Next specify the simulation settings (1000 runs and number of simulations 3, one for each correlation value). In the Detailed Statistics window you will see Profit results for the 3 runs. The mean values are equal at 674.995.

CORRELATED VARIABLES This may be surprising but can be explained because @Risk uses the same random numbers for each run but “shuffles” them in different orders to get the correct correlations. The means are unaffected since this is like saying the average of 30, 26, and 48 is the same as the average of 48, 30, and 26.

CORRELATED VARIABLES Notice, however, that the standard deviations for the three runs are different (157.7571, 262.881, and 365.5104). This means that the variation in profit increases as the correlation goes from negative to zero to positive.

CORRELATED VARIABLES When demands are negatively correlated, high demands for one product tend to cancel low demands for another product making extreme profit values less likely. When demands are positively correlated, high and low demands tend to go together making extreme profits more likely. This is why investors are warned to diversify their portfolio to reduce risk.