Exam 5: Advanced Linear Programming Applications

The composite unit in DEA

In data envelopment analysis, the percentage of an individual operating unit's resources that are available to the composite operating unit is the

To develop a portfolio that provides the best return possible with a minimum risk, the linear programming model will have an objective function which

The Eastern Washington County School Corporation is interested in comparing educational performance at four elementary schools and has hired you to prepare a DEA model to do so. After detailed conversations with the corporation administrative staff and the building principals, you have isolated the following input and output measurements: Input Measures Output Measures Average classroom size Percent of children in remedial classes Percent of students on reduced price lunch Average first grade score on standard test Average number of parent volunteer hours per week Average fifth grade score on standard test Data is collected for each school on each measure School Archer Hayes Ralston Creekside Inputs Classroom size 21 28 32 20 \% reduced lunch 10 8 25 2 Avg. vol. hours 30 46 15 64 Outputs \% remediation 15 12 19 3 1st grade score 83 84 62 85 5th grade score 76 72 53 79 Develop the DEA model that would evaluate the efficiency of Ralston Elementary School.

If a pure strategy solution exists for a two-person, zero-sum game, it is the optimal solution to the game.

Hervis Car Rental in Austin, TX has 50 high-performance Shelby-H Mustangs in its rental fleet. These cars will be in greater demand than usual during the last weekend in July when the Central Texas Mustang Club holds its annual rally in Austin. At times like this, Hervis uses a revenue management system to determine the optimal number of reservations to have available for the Shelby-H cars. Hervis has agreed to have at least 60% of its Shelby-H Mustangs available for rally attendees at a special rate. Although many of the rally attendees will request a Saturday and Sunday two-day package, some attendees may select a Saturday only or a Sunday only reservation. Customers not attending the rally may also request a Saturday and Sunday two-day package, or make a Saturday only or Sunday only reservation. Thus, six types of reservations are possible. The cost for each type of reservation is shown here. Two-Day Package Saturday Only Sunday Only Rally 125 \ 75 \ 65 Regular \ 150 \ 85 \ 75 The anticipated demand for each type of reservation is as follows: Two-Day Package Saturday Only Sunday Only Rally 20 10 15 Regular 10 20 25 Hervis Car Rental would like to determine how many Shelby-H Mustangs to make available for each type of reservation in order to maximize total revenue. a.Define the decision variables. b.Formulate a linear programming model for this revenue management application.

DEA will show all but one operating unit to be relatively inefficient in the case where an operating unit producing the most of every output and also consumes the least of every input.

Write a summary of the DEA approach and explain how you would interpret the solution.

In portfolio models, risk is minimized by diversification.

DEA is used to measure the relative efficiency of two (or more) units in different industries, such as a bank and a school.

In a two-person, zero-sum, pure-strategy game, there is no advantage to either player to switch from its strategy even if one of the players discovers the other player's strategy in advance.

Describe the steps used to determine when a two-person, zero-sum game has a pure-strategy solution.

It is possible for DEA to show all operating units to be relatively inefficient.

List several industries in which revenue management has been applied and list several different types of decisions being made with the aid of the methodology.

Portfolio manager Max Gaines needs to develop an investment portfolio for his conservative clients. His task is to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the portfolio provides the best return possible with a minimum risk. Formulate the maximin linear program. $\text { Annual Returns (Flanning Scenarios) }$ Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 22.37 26.73 6.46 -3.19 Large-Cap Blend 14.88 18.61 10.52 5.25 Mid-Cap Blend 19.45 18.04 5.91 -1.94 Small-Cap Blend 13.79 11.33 -2.07 6.85 Intermediate Bond 7.29 8.05 9.18 3.92

For a two-person, zero-sum, mixed-strategy game, it is necessary to solve the LP for each player in order to learn both players' optimal strategies.

Consider a two-person, zero-sum game where the payoffs listed below are the winnings for Company X. Identify the pure strategy solution. What is the value of the game? Company Y Strategies Company X Strategies 3 5 9 8 4 3 7 6 7

If it is optimal for both players in a two-person, zero-sum game to select one strategy and stay with that strategy regardless of what the other player does, the game

Mountainside State Park has four visitor centers. To study the operation of these centers, a DEA model has been developed that compares inputs (size, number of staff, weekly hours of operation) and outputs (% of visitors attending educational program, daily sales in gift shop). The computer solution is shown below. What can you conclude about the efficiency of the North center? LINEAR PROGRAMMING PROBLEM MIN 1E+0wn+0ws+0we+0ww S.T. 1) -400E+400wn+1200ws+2400we+1500ww<0 2) -3E+3wn+6ws+10we+7ww<0 3) -56E+56wn+108ws+92we+108ww<0 4) -49E+49wn+83ws+56we+72ww>0 5) -38E+38wn+425ws+1200we+630ww>0 6) +1wn+1ws+1we+1ww=1 OPTIMAL SOLUTION Objective Function Value = 1.000 Variable Value Reduced Cost E 1.000 0.000 wn 1.000 0.000 ws 0.000 0.000 we 0.000 0.000 ww 0.000 0.000 Constraint Slack/Surplus Dual Price 1 0.000 0.008 2 0.000 0.000 3 0.000 0.001 4 0.000 -0.034 5 0.000 -0.013 6 0.000 -1.000

Revenue management methodology can be applied in the case of nonperishable assets.