HW05

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Western Michigan University *

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6100

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Management

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Feb 20, 2024

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Uploaded by MasterWaterBuffalo3592

Homework #5 Problem 1. Data envelopment analysis can measure the relative efficiency of a group of hospitals. The following data from a particular study involving seven teaching hospitals include three input measures and four output measures: a. Formulate a linear programming model so that data envelopment analysis can be used to evaluate the performance of hospital D. b. Solve the model. c. Is hospital D relatively inefficient? What is the interpretation of the value of the objective function? d. How many patient-days of each type are produced by the composite hospital? e. Which hospitals would you recommend hospital D consider emulating to improve the efficiency of its operation? Problem 2. With the data in Problem 1, a. Formulate a linear programming model that can be used to perform data envelopment analysis for hospital E. b. Solve the model. c. Is hospital E relatively inefficient? What is the interpretation of the value of the objective function? d. Which hospitals are involved in making up the composite hospital? Can you make a general statement about which hospitals will make up the composite unit associated with a unit that is not inefficient?

Problem 3. Hanson Inn is a 96-room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. The Classic Corvette Owners Association scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least 50% of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two-night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are possible: Convention customers/two-night package; convention customers/Friday night only; convention customers/Saturday night only; regular customers/two-night package; regular customers/ Friday night only; and regular customers/Saturday night only. The cost for each type of reservation is shown here: The anticipated demand for each type of reservation is as follows: Hanson Inn would like to determine how many rooms to make available for each type of reservation in order to maximize total revenue. a. Define the decision variables and state the objective function. b. Formulate a linear programming model for this revenue management application. c. What are the optimal allocation and the anticipated total revenue? d. d. Suppose that one week before the convention the number of regular customers/ Saturday night only rooms that were made available sell out. If another nonconvention customer calls and requests a Saturday night only room, what is the value of accepting this additional reservation?

Problem 4. Two opposing armies, Red and Blue, must each decide whether to attack or defend. These decisions are made without knowledge of the opposing army’s decision. The payoff table, in terms of value of property gained or lost for the Red Army, appears below. Any gains for the Red Army are losses for the Blue Army. a. What is the optimal mixed strategy for the Red Army? b. What is the optimal mixed strategy for the Blue Army? Problem 5. Two television stations compete with each other for viewing audience. Local programming options for the 5:00 p.m. weekday time slot include a sitcom rerun, an early news program, or a home improvement show. Each station has the same programming options and must make its preseason program selection before knowing what the other television station will do. The viewing audience gains in thousands of viewers for Station A are shown in the payoff table. Determine the optimal strategy for each station. What is the value of the game? Problem 6. In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B’s bill. If the bills match, Player B wins Player A’s bill. a. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. b. Is there a pure strategy? Why or why not? c. Determine the optimal strategies and the value of this game. Does the game favor one player over the other? d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.

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Problem 7. The offensive coordinator for the Chicago Bears professional football team is preparing a game plan for the upcoming game against the Green Bay Packers. A review of game tapes from previous Bears – Packers games provides data on the yardage gained for run plays and pass plays. Data show that when the Bears run against the Packers’ run defense, the Bears gain an average of 2 yards. However, when the Bears run against the Packers’ pass defense, the Bears gain an average of 6 yards. A similar analysis of pass plays reveals that if the Bears pass against the Packers’ run defense, the Bears gain an average of 11 yards. However, if the Bears pass against the Packers’ pass defense, the Bears average a loss of 1 yard. This loss, or ne gative gain of – 1, includes the lost yardage due to quarterback sacks and interceptions. Develop a payoff table that shows the Bears’ average yardage gain for each combination of the Bears’ offensive strategy to run or pass and the Packers’ strategy of usi ng a run defense or a pass defense. What is the optimal strategy for the Chicago Bears during the upcoming game against the Green Bay Packers? What is the expected value of this strategy?

HW05

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