Business analytics methods, models and decisions evans analytics2e ppt SC b

Supplementary Chapter B Optimization Models with Uncertainty Risk Analysis in Optimization  In the chapters on linear, integer, and nonlinear optimization, we used deterministic models  In most situations, some of the data will be uncertain, which implies inherent risk  Stochastic models incorporate uncertainty  If an optimization model has uncertain variables, we might first solve it deterministically and then use Monte Carlo simulation to analyze the results Example B.1: Uncertainty in the Sklenka Ski Model  The Sklenka Ski model (Chapter 13), seeks to maximize profit subject to constraints on: - Fabrication labor hours - Finishing labor hours - Market mixture  Suppose the labor hours required for finishing is stochastic; then overtime will be needed if more than 21 hours of finishing time are required  Finishing time will be modeled by triangular distributions  How often will overtime be needed if the optimal solution of 5.25 Jordanelle and 10.5 Deercrest skis are scheduled each day? Example B.1 Continued  Spreadsheet model Specify the finishing time input distributions in cells B7 and C7 Specify the available finishing hours as an uncertain output cell D16 Example B.1 Continued  Analytic Solver Platform Simulation Results The likelihood of needing overtime is about 85% Chance Constraints  A chance constraint is one that specifies the fraction of trials in a simulation that must satisfy a constraint  Suppose that the company wants to determine a daily schedule so that the probability of overtime —that is, requiring more than 21 hours of finishing time—is less than 0.1, or 10% of the time ◦ In this case, we would want to specify that the percentage of trials requiring less than 21 hours of finishing time is at least 90% Defining Chance Constraints  Chance constraints are defined by a percentile, or value at risk (VaR), measure  A VaR constraints with chance p% requires that the constraint be satisfied p% of the time ◦ This does not consider the magnitude of the violation when the constraint is not satisfied  Conditional at risk (CVaR) constraints place bounds on the average magnitude of all violations of the constraint that may occur (1−p)% of the time ◦ CVaR is more conservative than VaR Example B.2: Solving the SSC Model with a Chance Constraint  Sklenka Skis wants to determine a production schedule that has no more than a 10% probability of overtime being required That is, they want a 90% probability of needing 21 or fewer hours of finishing labor Example B.2 Continued  Solution with chance constraint Example B.2 Continued  Simulation results with chance constraint Example B.7: Using Multiple Parameterized Simulations   Newsvendor Model with Historical Data First, set the demand in cell B11 =PsiDisUniform(D2:D21) Then select cell B12 and set a lower limit of 40 and upper limit of 51 in the Function Arguments dialog (see text for further implementation details) Analytic Solver Platform will run 12 simulations for each purchase quantity Example B.7 Continued  Now we want to find the optimal purchase quantity by varying purchase quantity between 40 and 51 Select cell B12 Risk Solver Parameters Simulation Values or Lower: 40 Upper: 51 Options All Options Simulation Simulations to Run: 12 Example B.8: Optimizing the Hotel Overbooking Model  Hotel Overbooking Monte Carlo Simulation Model with Custom Demand Example B.9: Optimizing the Hotel Overbooking Model  See text for implementation details Solver identifies 313 reservations as the best solution, just as we found using the multiple parameterized simulation approach A Portfolio Allocation Model  An investor has $100,000 to invest in four assets The expected annual returns and minimum and maximum amounts with which the investor will be comfortable allocating to each investment are:  Arbitrate pricing theory provides estimates of the sensitivity of investments to risk factors such as inflation, industrial production, interest rates, etc  Risk factors Optimization Model  Determine how much to invest in each asset to maximize the total expected annual return, remain within the minimum and maximum limits for each investment, and meet the limitation on the weighted risk per dollar invested (assumed to be 1.0)  Define X as the amount invested in asset i i Maximize 0.05X1 + 0.07X2 + 0.11X3 + 0.04X4 X1 + X2 + X3 + X4 ≤ 100,000 − 0.5X1 + 1.8X2 + 2.1X3 − 0.3X4 ≤ 1.0(X1 + X2 + X3 + X4) 2,500 ≤ X1 ≤ 5,000 X2 ≥ 30,000 X3 ≥ 15,000 X4 ≥ Example B.10: Setting Up the Spreadsheet Model Example B.11: Setting Up the Simulation Model  Assume that annual returns are uncertain for all but the savings account  Life insurance returns are uniformly distributed  Cell B6: =PsiUniform(4%, 6%)  Bond mutual fund returns are normally distributed  Cell B7: = PsiNormal(7%, 1%)  Stock fund returns are lognormally distributed  Cell B8: = PsiLogNormal(11%, 4%)  Also, define Cell D20 (total expected return) as an uncertain output cell by adding +PsiOutput() Example B.12: Setting Up the Optimization Model Example B.12 Continued  Simulation of the expected return Project Selection  Project-selection and capital-budgeting projects typically have many uncertainties because they involve future events  Returns and resource requirements are often uncertain estimates  Implementing a project is not guarantee of successful completion  Analytic Solver Platform allows for the incorporation of uncertainties in project selection models Example B.13: A Project-Selection Model with Uncertainty  Example 15.5 (Hahn Engineering Project Selection)  Expected returns are uncertain and can be modeled using lognormal distributions  Also, assume that some projects are riskier than others and have different probabilities of being completed successfully Example B.13 Continued  To model whether a project is successful, use a binomial probability distribution with n =  Use IF statements to apply the returns and success probabilities only to “selected” projects  Specify total return as a changing output cell  See text for implementation details Example B.13 Continued  Model and solution Example B.13 Continued  Simulation results ... 2000) In cell B2 2, we calculate the lead-time demand by multiplying the annual demand rate (B1 3) by the lead time in B2 1 and define it as an output cell Example B. 3 Continued  Distribution of lead-time... Assume demand is normally distributed with a mean of 15,000 units and a standard deviation of 2,000 units Example B. 3 Continued  Spreadsheet model Cell B5 is defined to be normally distributed... production schedule that has no more than a 10% probability of overtime being required That is, they want a 90% probability of needing 21 or fewer hours of finishing labor Example B. 2 Continued