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.2 Continued Solver typically finds a conservative solution to problems with chance constraints. However, Analytic Solver Platform can automatically improve the solution by adjusting the size of the uncertainty set for the chance constraint auto-adjust process.

Service Levels in the EOQ Model The standard EOQ model assumes constant (deterministic) demand. In most practical situations, demand is stochastic. If D is uncertain, then the demand during the lead time will also be uncertain. This impacts how the reorder point should be chosen. We can use Monte Carlo simulation to analyze the optimal solution.

Example B.3: Finding the Distribution of Lead-Time Demand EOQ Example A.5: Annual demand = 15,000 units. Ordering costs = $200 per order. Purchase cost = $22 per item. Carrying charge rate = 20%. 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 using the function =PsiNormal(15000, 2000). In cell B22, we calculate the lead-time demand by multiplying the annual demand rate (B13) by the lead time in B21 and define it as an output cell.

Example B.3 Continued Distribution of lead-time demand

Service Levels A service level is a constraint that represents the probability that demand can be satisfied. For example, we might want to ensure that demand can be satisfied 95% of the time. We can identify the reorder point for a particular service level from the frequency chart.

Example B.4: Finding the Reorder Point for a Service Level Requirement In the distribution of lead time demand, first set the Lower Cutoff value in the Chart Statistics pane to zero and then set the Likelihood value to 0.95. This will calculate the Upper Cutoff value so that the cumulative probability is 0.95. A 95% service level requires a reorder point of about 351 units. The additional 351 - 288 = 63 units above the mean lead-time demand is called safety stock.

Hotel Pricing Model with Uncertainty Price-demand elasticities are only estimates and most likely are quite uncertain. Assume that the true values might vary from the estimates by plus or minus 25%. Model the elasticities by uniform distributions. Using the optimal prices identified by Solver, use Monte Carlo simulation to see what happens to the prediction of the number of rooms sold under this assumption.

Example B.5: Simulating the Optimal Solution to the Hotel Pricing Model Spreadsheet model Output cell

Example B.5 Continued Simulation results for 450-room capacity

Example B.6: Identifying Hotel Capacity to Meet a Service Level Constraint By changing the Upper Cutoff value in the task pane, we could identify the likelihood of exceeding that value. the likelihood of exceeding 457 rooms is close to 10%.

Example B.6 Continued Shift the capacity constraint down by 7 rooms to 443 and find the optimal prices associated with this constraint, we would expect demand to exceed 450 at most 10% of the time.

Example B6 Continued Confirmation simulation run

Optimizing Monte Carlo Simulation Models Analytic Solver Platform provides a capability – called multiple parameterized simulations - of automatically running simulations for a range of values for decision variables. In the Newsvendor Model, for example, we can vary purchase quantities of the candy boxes to determine the optimal number to purchase. In the Hotel Overbooking Model, we can find the best number of reservations to accept.

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 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 Xi as the amount invested in asset 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 ≥ 0

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 = 1. 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