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

 If an optimization model has uncertain variables, we might first solve it deterministically and then use Monte Carlo simulation to analyze the results..  The Sklenka Ski model Chapter

Supplementary Chapter B

Optimization Models with Uncertainty

 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

Risk Analysis in Optimization

 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: Uncertainty in the Sklenka Ski Model

 Analytic Solver Platform Simulation Results

Example B.1 Continued

The likelihood of needing overtime is about 85%

 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%.

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.

Defining Chance Constraints

 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: Solving the SSC Model with a Chance Constraint

 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 adjust process.

auto-Example B.2 Continued

 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.

Service Levels in the EOQ Model

 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: Finding the Distribution of Lead-Time Demand

 Distribution of lead-time demand

Example B.3 Continued

 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.

Service Levels

 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.

Example B.4: Finding the Reorder Point for a Service Level

demand is called safety stock.

 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

Hotel Pricing Model with Uncertainty

 Spreadsheet model

Example B.5: Simulating the Optimal Solution to the Hotel Pricing

Model

Output cell

 Simulation results for 450-room capacity

Example B.5 Continued

 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: Identifying Hotel Capacity to Meet a Service Level

Constraint

 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 B.6 Continued

 Confirmation simulation run

Example B6 Continued

 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.

Optimizing Monte Carlo Simulation Models

 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: Using Multiple Parameterized Simulations

 Now we want to find the optimal purchase quantity by varying purchase quantity

between 40 and 51

Example B.7 Continued

Select cell B12.

Risk Solver Parameters Simulation Values or Lower: 40 Upper: 51

Options All Options Simulation Simulations to Run: 12

 Hotel Overbooking Monte Carlo Simulation Model with Custom Demand

Example B.8: 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.

Example B.9: Optimizing the Hotel Overbooking 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

A Portfolio Allocation 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

Example B.10: Setting Up the Spreadsheet Model

 Assume that annual returns are uncertain for all but the savings account.

 Life insurance returns are uniformly distributed

Example B.12: Setting Up the Optimization Model

 Simulation of the expected return

Example B.12 Continued

 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

Project Selection

 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: A Project-Selection Model with Uncertainty

 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

Example B.13 Continued