Maximin strategy A strategy where the player seeking to maximize the value of thegame selects the strategy that maximizes the minimum payoff obtainable by the otherplayer.Minimax strategy A strategy where the player seeking to minimize the value of thegame selects the strategy that minimizes the maximum payoff obtainable by the otherplayer.Pure strategy When one of the available strategies is optimal and the player alwaysselects this strategy regardless of the strategy selected by the other player.Saddle point A condition that exists when pure strategies are optimal for both players.Neither player can improve the value of the game by changing from the optimal purestrategy.Mixed strategy When a player randomly selects its strategy based on a probability dis-tribution. The strategy selected can vary each time the game is played.Expected value In a mixed strategy game, a value computed by multiplying each payoffby its probability and summing. It can be interpreted as the long-run average payoff for themixed strategy.

PROBLEMS

Note: The following problems have been designed to give you an understanding and apprecia-tion of the broad range of problems that can be formulated as linear programs. You should beable to formulate a linear programming model for each of the problems. However, you will needaccess to a linear programming computer package to develop the solutions and make therequested interpretations.1. In Section 5.1 data envelopment analysis was used to evaluate the relative efficiencies of

four hospitals. Data for three input measures and four output measures were provided inTables 5.1 and 5.2.a. Use these data to develop a linear programming model that could be used to evaluate

the performance of General Hospital.b. The following solution is optimal. Does the solution indicate that General Hospital is

relatively inefficient?

248 Chapter 5 Advanced Linear Programming Applications

Objective Function Value 5 1.000

Variable Value Reduced Costs

E 1.000 0.000WG 1.000 0.000WU 0.000 0.000WC 0.000 0.331WS 0.000 0.215

c. Explain which hospital or hospitals make up the composite unit used to evaluate Gen-eral Hospital and why.

2. Data envelopment analysis can measure the relative efficiency of a group of hospitals. Thefollowing data from a particular study involving seven teaching hospitals include threeinput 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 objec-

tive function?

Problems 249

Input Measures

Full-Time Supply Bed-DaysEquivalent Expense Available

Hospital Nonphysicians (1000s) (1000s)

A 310.0 134.60 116.00B 278.5 114.30 106.80C 165.6 131.30 65.52D 250.0 316.00 94.40E 206.4 151.20 102.10F 384.0 217.00 153.70G 530.1 770.80 215.00

Output Measures

Patient-Days Patient-Days(65 or older) (under 65) Nurses Interns

Hospital (1000s) (1000s) Trained Trained

A 55.31 49.52 291 47B 37.64 55.63 156 3C 32.91 25.77 141 26D 33.53 41.99 160 21E 32.48 55.30 157 82F 48.78 81.92 285 92G 58.41 119.70 111 89

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?3. Refer again to the data presented in Problem 2.

a. Formulate a linear programming model that can be used to perform data envelopmentanalysis for hospital E.

b. Solve the model.c. Is hospital E relatively inefficient? What is the interpretation of the value of the objec-

tive 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 associatedwith a unit that is not inefficient?

4. The Ranch House, Inc., operates five fast-food restaurants. Input measures for the restau-rants include weekly hours of operation, full-time equivalent staff, and weekly supplyexpenses. Output measures of performance include average weekly contribution to profit,

a. Develop a linear programming model that can be used to evaluate the performance ofthe Clarksville Ranch House restaurant.

b. Solve the model.c. Is the Clarksville Ranch House restaurant relatively inefficient? Discuss.d. Where does the composite restaurant have more output than the Clarksville restau-

rant? How much less of each input resource does the composite restaurant requirewhen compared to the Clarksville restaurant?

e. What other restaurants should be studied to find suggested ways for the Clarksvillerestaurant to improve its efficiency?

5. Reconsider the Leisure Airlines problem from Section 5.2. The demand forecasts shownin Table 5.3 represent Leisure Airs best estimates of demand. But because demand cannotbe forecasted perfectly, the number of seats actually sold for each origin-destination-itinerary fare (ODIF) may turn out to be smaller or larger than forecasted. Suppose thatLeisure Air believes that economic conditions have improved and that their original fore-cast may be too low. To account for this possibility, Leisure Air is considering switchingthe Boeing 737-400 airplanes that are based in Pittsburgh and Newark with Boeing 757-200 airplanes that Leisure Air has available in other markets. The Boeing 757-200 airplanehas a seating capacity of 158 in the coach section.a. Because of scheduling conflicts in other markets, suppose that Leisure Air is only able

to obtain one Boeing 757-200. Should the larger plane be based in Pittsburgh or inNewark? Explain.

b. Based upon your answer in part (a), determine a new allocation for the ODIFs. Brieflysummarize the major differences between the new allocation using one Boeing 757-200 and the original allocation summarized in Figure 5.5.

c. Suppose that two Boeing 757-200 airplanes are available. Determine a new allocationfor the ODIFs using the two larger airplanes. Briefly summarize the major differencesbetween the new allocation using two Boeing 757-200 airplanes and the original allo-cation shown in Figure 5.5.

d. Consider the new solution obtained in part (b). Which ODIF has the highest bid price?What is the interpretation for this bid price?

250 Chapter 5 Advanced Linear Programming Applications

Input Measures

Restaurant Hours of Operation FTE Staff Supplies ($)

Bardstown 96 16 850Clarksville 110 22 1400Jeffersonville 100 18 1200New Albany 125 25 1500St. Matthews 120 24 1600

Output Measures

Restaurant Weekly Profit Market Share (%) Growth Rate (%)

Bardstown $3800 25 8.0Clarksville $4600 32 8.5Jeffersonville $4400 35 8.0New Albany $6500 30 10.0St. Matthews $6000 28 9.0

market share, and annual growth rate. Data for the input and output measures are shown inthe following tables:

6. Reconsider the Leisure Airlines problem from Section 5.2. Suppose that as of May 1 thefollowing number of seats have been sold:ODIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Seats Sold 25 44 18 12 5 9 20 33 37 11 5 8 27 6 35 7a. Determine how many seats are still available for sale on each flight leg.b. Using the original demand forecasted for each ODIF, determine the remaining de-

mand for each ODIF.c. Revise the linear programming model presented in Section 5.2 to account for the

number of seats currently sold and a demand of one additional seat for the PittsburghMyrtle Beach Q class ODIF. Resolve the linear programming model to determine anew allocation schedule for the ODIFs.

7. 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 normalroom rates and takes reservations based on a revenue management system. The ClassicCorvette Owners Association scheduled its annual convention in Louisville for the firstweekend in June. Hanson Inn agreed to make at least 50% of its rooms available for con-vention attendees at a special convention rate in order to be listed as a recommended hotelfor the convention. Although the majority of attendees at the annual meeting typically re-quest a Friday and Saturday two-night package, some attendees may select a Friday nightonly or a Saturday night only reservation. Customers not attending the convention mayalso request a Friday and Saturday two-night package, or make a Friday night only or Sat-urday night only reservation. Thus, six types of reservations are possible: Conventioncustomers/two-night package; convention customers/Friday night only; convention cus-tomers/Saturday night only; regular customers/two-night package; regular customers/Friday night only; and regular customers/Saturday night only. The cost for each type ofreservation is shown here:

Problems 251

Two-Night Friday Night Saturday NightPackage Only Only

Convention $225 $123 $130Regular $295 $146 $152

Two-Night Friday Night Saturday NightPackage Only Only

Convention 40 20 15Regular 20 30 25

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 ofreservation 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. Suppose that one week before the convention, the number of regular customers/

Saturday night only rooms that were made available sell out. If another nonconventioncustomer calls and requests a Saturday night only room, what is the value of accept-ing this additional reservation?