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Chapter 12 Monte Carlo Simulation and Risk Analysis Model Uncertainty and Risk Analysis Many situations dictate that randomness be explicitly incorporated into our models This is usually done by specifying probability distributions for the appropriate uncontrollable inputs ◦ Such models are called stochastic, or probabilistic Risk is the likelihood of an undesirable outcome It can be assessed by evaluating the probability that the outcome will occur along with the severity of the outcome Risk analysis seeks to examine the impact of uncertain inputs on various outputs Example 12.1: Incorporating Uncertainty in the Outsourcing Decision Model Production volume is uncertain; assume normal with a mean of 1000 and standard deviation of 100 Replace cell B12 with =ROUND(NORM.INV(RAND(), 1000, 100, true), 0) Whenever F9 key or Formula > Calculation > Calculate Now is clicked, the value of demand will change randomly Monte Carlo Simulation Monte Carlo simulation is the process of generating random values for uncertain inputs in a model, computing the output variables of interest, and repeating this process for many trials to understand the distribution of the output results Monte Carlo simulation can easily be accomplished on a spreadsheet using a data table Example 12.2: Using Data Tables for Monte Carlo Spreadsheet Simulation Excel file Outsourcing Decision Monte Carlo Simulation Model Enter the trial number (1 to 20) in column D Reference the cells associated with model outputs in row 3: (E3, F3, G3) (=B12, =B19, =B20) Select the range for the data table (D3:G23) In the Data Table dialog, enter any blank cell for the Column Input Cell Monte Carlo Simulation Using Analytic Solver Platform Develop a spreadsheet model Determine probability distributions for uncertain input variables Identify output variables you want to predict Choose the number of trials for the simulation Run the simulation Interpret the results Defining Uncertain Model Inputs For many decision models, empirical data may be available, either in historical records or collected through special efforts In other situations, historical data are not available, and we can draw upon the properties of common probability distributions to help choose a representative distribution that has the shape that would most reasonably represent the analyst’s understanding about the uncertain variable Uniform or triangular distributions are often used in the absence of data In Analytic Solver Platform, use custom Excel functions or the Distributions button in the ribbon Example 12.3: Using Analytic Solver Platform Probability Distribution Functions Outsourcing Decision Model Demand (production volume) is normally distributed with a mean of 1000 and standard deviation of 100 units ◦ ◦ =PsiNormal(1000, 100) in cell B12 Use ROUND function to ensure that the result is a whole number: =ROUND(PsiNormal(1000,100),0) Unit cost has a triangular distribution with a minimum of $160, most likely value of $175, and a maximum of $200 ◦ =PsiTriangular(160, 175, 200) in cell B10 Example 12.4: Using the Distributions Button in Analytic Solver Platform For demand, select cell B12 Click the Distributions button in the Analytic Solver Platform ribbon and select the normal distribution from the Common category Example 12.4 Continued Normal distribution dialog Change the parameters to mean = 1000, stdev = 100 Interactive Simulation Whenever the Simulate button is clicked, you will notice that the lightbulb in the icon turns bright If you change any number in the model, Analytic Solver Platform will automatically run the simulation for that quantity; this makes it easy to conduct what-if analyses ◦ Example: change the purchase quantity to 50; mean profit is less than if purchase quantity is 44 Overbooking Model with Custom Demand Historical demand data shown in columns D and E Assume that each reservation has a constant probability p = 0.04 of being canceled; therefore, the number of cancellations (cell B14) can be modeled using a binomial distribution with n = number of reservations made and p = probability of cancellation Example 12.16: Defining a Custom Distribution in Analytic Solver Platform Select cell B12 and then click on the Distributions button in the ribbon and choose Discrete from the Custom category Edit the range for “values” and “weights” in the Parameters section ◦ Values correspond to the range of demand in cells D2:D13, and weights are the relative frequencies or probabilities in cells E2:E13 Or, use the function in cell B12: =PsiDiscrete($D$2:$D$13,$E$2:$E$13) Example 12.16 Continued To model the number of cancellations in cell B14, choose the binomial distribution from the Discrete category in the Distributions list The number of trials is the value in cell B13 and is referenced in the Parameters section Or, use the function =PsiBinomial(B13, 0.04) in cell B14 Overbooking Model Results Frequency charts for number of overbooked customers and net revenue if 310 reservations are accepted You can use Interactive Simulation to quickly change the number of reservations to find the best solution Cash Budget Model Cash Budgeting is the process of projecting and summarizing a company’s cash inflows and outflows expected during a planning horizon Most cash budgets are based on sales forecasts Because of the inherent uncertainty in sales forecasts, Monte Carlo simulation is an appropriate tool for modeling cash budgets Example Cash Budget Model Highlighted cells are uncertain variables (blue) and uncertain functions (green) Cash Budget Model Assumptions Sales are normally distributed with a standard deviation of 10% of the mean Sales in adjacent months are correlated with one another, with a correlation coefficient of 0.6 On average, 20% of sales are collected in the month of sale, 50%, in the month following the sale, and 30%, in the second month following the sale ◦ These figures are uncertain, so a uniform distribution is used to model the first two values (15% to 20% and 40% to 50%, respectively), with the assumption that all remaining revenues are collected in the second month following the sale Example 12.17: Simulating the Cash Budget Model Without Correlations Define distributions for all uncertain variables ◦ ◦ ◦ Example for April sales (cell E5): =PsiNormal(600000,60000) Cell B7: =PsiUniform(15%, 20%) Cell B8: =PsiUniform(40%, 50%) Define the available balances in row 25 as output variables Example 12.17 Continued Trend chart Example 12.17 Continued Likelihood of not meeting minimum balance in April Correlating Uncertain Variables Unless you specify otherwise, Monte Carlo simulation assumes that each of the uncertain variables is independent of all the others Analytic Solver Platform allows you to specify correlations between uncertain variables Example 12.18: Incorporating Correlations in Analytic Solver Platform Cash Budget Monte Carlo Simulation Model Click the Correlations button in the Simulation Model group in the ribbon ◦ In this example we are only correlating the variables in the range E5:K5 Move these to the right pane Example 12.18 Continued Initial correlation matrix The numerical values show the correlations (initially set to zero) ◦ The green distributions are those used in the uncertain cells ◦ The blue scatterplots show visual representations of the correlations between the variables Replace the zeros by the correlations you want in the model Example 12.18 Continued Analytic Solver Platform will check that the correlations are mathematically consistent; if not, it will ask you to adjust the correlations Always choose Yes Click the Update Matrix button and then Accept Update Adjusted correlation matrix: Analytic Solver Platform then adds these to the simulation model ... 2,000,000 units and standard deviation of 400,000 units R&D costs: uniform between $600,000,000 and $800,000,000 Clinical trial costs: lognormal with mean of $150,000,000 and standard deviation... made and p = probability of cancellation Example 12. 16: Defining a Custom Distribution in Analytic Solver Platform Select cell B12 and then click on the Distributions button in the ribbon and. .. B10 Example 12. 4: Using the Distributions Button in Analytic Solver Platform For demand, select cell B12 Click the Distributions button in the Analytic Solver Platform ribbon and select the
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