Many situations do, in fact, involve several decision makers who compete with one another to arrive at the best outcome. These types of competitive decision-making situations are the subject of game theory. Anyone who has played such games as card games or board games is familiar with situations in which competing participants develop plans of action in order to win. Game theory encompasses similar situations in which competing decision makers develop plans of action in order to win.
Types of Game Situations
Competitive game situations can be subdivided into several categories. One classification is based on the number of competitive decision makers, called players, involved in the game. A game situation consisting of two players is referred to as two-person game. When there are more than two players, the game situation is known as an n-person game.
Games are also classified according to their outcomes in terms of each player's gains and losses. If the sum of the players' gains and losses equals zero, the game is referred to as zero-sum game. In a two-person game, one player's gains represent at another's losses. For example, if one player wins 100, then the other player loses 100; the two values sum to zero. Alternatively, if the sum of the players' gains and losses does not equal zero, the game is known as a non-zero sum game.
The two-person, zero-sum game is the one most frequently used to demonstrate the principles of game theory because it is the simplest mathetimatically.
Example of competitive situations that can be organized into two-person, zero-sum games include
1. a union negotiating a new contract with management
2. two armies participating in a war game
3. two politicians in conflict over a proposed legislative bill, one attempting to secure its passage and the other attempting to defeat it
4. a retail firm trying to increase its market share with a new product and a competitor attempting to minimize the firm's gains
5. a contractor negotiating with a government agent for a contract on a project
Game Strategies
A strategy is a plan of action to be followed by a player. Each player in a game has two or more strategies, only one of which is selected for each playing of a game.
1. A Pure Strategy when each player in the game adopts a single strategy as an optimal strategy, then the game is a pure strategy game. The value of a pure strategy game is the same for both the offensive player and the defensive player. In contrast, in a mixed strategy game, the players adopt a mixture of strategies if the game is played many times.
A pure strategy game can be solved according to the minimax decision criterion. According to this principle, each player plays the game in order to minimize the maximum possible losses. The offensive player will select the strategy with the largest of the minimum payoffs (called the maximin strategy), and the defensive player will select the strategy with the smallest of the maximum payoffs (called the minimax strategy).
2. Dominant Strategies. Dominance occurs when all the payoffs for one strategy are better than the corresponding payoffs for another strategy.
3. A mixed strategy occurs when each player selects an optimal strategy and they do not result in an equilibrium point( the same outcome) when the maximin and mnimax decision criteria are applied.
Practice exercises
1. Payoff Table for Camera Companies
Camera Company 1 Strategies Camera Company 2 Strategies
A B C
1 9 7 2
2 11 8 4
3 4 1 7
2. Two fast foods chains, MacBurger and Burger Doodle, dominate the fast food market. MacBurger is currently the market leader, and Burger Doodle has developed three marketing strategies, encompassing advertising and new product lines, to gain a percentage of the market now belonging to MacBurger. The following payoff table shows the gains for Burger Doodle and the losses for MacBurger given the strategies of each company.
Burger Doodle Strategies MacBurger Strategies
A B C
1 4 3 6
2 -2 5 1
3 3 2 4
Determine the mixed strategy for each company and the expected market share gains for Burger Doodle and losses for Macburger.
3. Consider the following payoff table for a mixed strategy game between two players
Player 1 Strategies Player 2 Strategies
A B C
1 50 60 30
2 10 32 25
3 20 55 4
Determine the mixed strategy for each player and the expected gains and losses that result.
4. Given the following payoff table for a mixed strategy game between two players, determine the strategy and the gains and losses for each player.
Player 1 Strategies Player 2 Strategies
A B C D
1 40 30 20 80
2 90 50 60 65
3 80 75 52 90
4 60 40 35 50
5. Consider the following payoff table for two game players.
Player 1 Strategies Player 2 Strategies
A B C D
1 6 25 18 10
2 12 14 19 11
3 20 15 7 9
4 15 30 21 16
Thursday, August 15, 2013
Tuesday, August 6, 2013
THE DECISION THEORY
A decision is the act of deciding what single act among all alternatives is to be taken into account. Successful decision making comprises a number of steps to process:
1. Clearly define the problem.
2. Identify the possible alternatives and their outcomes.
3. List the profit for each combination of alternatives and outcomes.
4. Select one mathematica; decision theory model.
5. Apply the model and make your decision.
The success or failure that persons, companies, and management experience depends on the decision they make. Making use of quantities in decision-making helps a great deal in minimizing mistakes and failures. Some methods of computation should therefore be learned.
Decision theory is a general approach to decision making that is useful in many different aspects of operations management. It provides a framework for analysis of decision. It includes a number of different techniques that can classify according to the degree of uncertainty.
Decision under Certainty - when one knows with certainty which of the possible future conditions will actually happen, he simply chooses the alternative with the highest payoff under the state of nature.
Decision under Uncertainty - from certainty, the opposite extreme is uncertainty where there is no available information on how likely the various states of nature are. Under this condition, there are four possible decision criterions, namely :
a) Maximin - takes into account only the worst possible outcome for each alternative. This approach establishes a guaratee minimum.
b) Maximax - takes into account only the best possible outcome for each alternative.
c) Laplace - takes into account only the best average possible outcome for each alternative.
d) Minimax Regret - determines the worst regret for ach alternative, and chooses the alternative with the best worst.
DECISION TREES
Another useful technique for analyzing a decision situation is a decion tree. A decision tree, like the probability tree is a graphical diagram consisting of nodes and branches. However, rather than determining the probability of each branch as in a probability tree, in a decision tree the user computes the expected value of each outcome and makes a decision based on these expected values. The primary benefit of a decision tree is that it provides an illustration (or picture) of the decision-making process. This makes it easier to correctly compute the necessary expected values and to understand the process of making the decision.
The decision tree has two types of nodes: a square represents a decision point and a circle stands for a chance event. The tree is read left to right, and analyzed from right to left (starting with the last decision that might be made). For each decision, choose the alternative that will yield the greatest return. If chance events follow a decision, choose the alternative that has the highest expected value or the lowest expected loss.
1. T. Bone Puckett, a corporate raider, has acquired a textile company and is contemplating the future of one of its major plants located in South Carolina. Three alternative decisions are being considered: (1) expand the plant and produce lightweight, durable materials for possible sales to the military, a market with little foreigh competition; (2) maintain the status quo at the plant, continuing production of textile goods that are subject to heavy foreign competition; or (3) sell the plant now. If one of the first two alternatives is chosen, the plant will still be sold at the end of a year. The amount of profit that could be earned by selling the plant in a year depends on foreign market conditions, including the status of a trade embargo bill in Congress. The following payoff table describes this decision situation.
Decision States of Nature
Good Foreign Competitive Conditions Poor Foreign Competitive Condition
Expand 800,000 500,000
Maintain status quo 1,300, 000 -150, 000
Sell now 320,000 320, 000
a. Determine the best decision using the following decision criteria.
1. Maximax
2. Maximin
3. Minimaz regret
4. Hurwicz (alpha =0.3)
5. Equal likelihood
b. Assume that it is now possible to estimate a probability of 0.70 that good foreign competitive conditions will exist and a probability of 0.30 that poor conditions will exist. Determine the dest decision using the expected value and expected opportunity loss.
c. Compute the expected value of perfect information.
d. Develop a decision tree for this decision situation, with expected values at the probability nodes.
2. A farmer in Iowa is considering either leasing some extra land or investing in savings certificates at the local bank. If weather conditions are good next year, the extra land will allow the farmer to have an excellent harvest. However, if weather conditions are bad, the farmer will lose money. The savings certificates will result in the same return regardless of the weather conditions. The return for each investment given each type of weather condition is hown in the following payoff table.
Decision Weather
Good Bad
Lease land 90,000 -40,000
Buy savings certificate 10,000 10,000
Hurwicz (alpha = 0.4)
3. The owner of the Burger Doodle Restaurant is considering two ways to expand operations: operning a drive-up windom or serving breakfast. The increase in profits resulting from these proposed expansions depends on whether a competitor opens a franchise down the street. The possible profits from each expansion in operations given in both future competitive situations are shown in the following payoff table.
Decision Competitor
Open Not Open
Drive-up window -6,000 20,000
Breakfast 4,000 8,000
Hurwicz (alpha = 0.10)
4. Stevie Stone, a bellhop at the Royal Sundown Hotel in Atlanta, has been offered a management position. Although accepting the offer would assure him a job if there were a recession, if good economic conditionsprevailed he would actually make less money as a manager than as a bellhop (because of the large tips he gets as a bellhop). His salary during the next five years for each job given each economic condition is shown in the following payoff table.
Decision Economic Conditions
Good Recession
Bellhop 120,000 60,000
Manager 85,000 85,000
Hurwicz (alpha = 0.4)
5. Consider the following payoff table for three alternative investments, A, B, and C, under two future states of the economy, good and bad.
Investment Economic Conditions
Good Bad
A 70,000 25,000
B 120,000 -60,000
C 40,000 40,000
Hurwicz (0.3)
6. A farmer in Georgia must decide which crop to plant next year on his land: corn, peanuts, or soybeans. The return from each crop will be determined by whether a new trade bill with the USSR passes the Senate. The profit the farmer will realize from each crop given the two possible results on the trade bill is shown in the following payoff table.
Crop Trade Bill
Pass Fail
Corn 35,000 8,000
Peanuts 18,000 12,000
Soybeans 22,000 20,000
Hurwicz (0.3)
7. The owner of the Columbia Construction Company must decide among building a housing development, constructing a shopping center, or leasing all the company's equipment to another company. The profit that will result from each alternative will be determined by whether material costs remain stable or increase. The profit from each alternative given the two possibilities for material costs is shown in the following payoff table.
Decision Material Costs
Stable Increase
Houses 70,000 30,000
Shopping center 105,000 20,000
Leasing 40,000 40,000
Hurwicz (0.2)
8. An investor is considering investing in stocks, real estate, or bonds under uncertain economic conditions. The payoff table of returns for the investor's decision situation is shown below.
Investment Economic Conditions
Good Stable Poor
Stocks 5,000 7,000 3,000
Real estate -2,000 10,000 6,000
Bonds 4,000 4,000 4,000
Hurwicz (0.3)
9. A concessions manager at the Tech vs A and M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance if rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates the following profits will result from each decision given each set of weather conditions.
Decision Weather Conditions
Rain Overcast Sunshine
0.30 0.15 0.55
Sun visors -500 -200 1,500
Umbrellas 2000 0 -900
10. The Blitzkrieg Banking House in Berlin speculates in the money market. The status of the U.S dollar determines the return from investments in other currencies. The banking house will invest in the dollar, the yen, or other currencies. The banking house will invest in the dollar, the yen, or the mark. The return from each is shown in the following payoff table.
Currency Value of the Dollar
Increases (0.3) Remains stable(0.50) Declines(0.20)
Dollar 210,000 0 -170,000
Yen -10,000 20,000 80,000
Mark -40,000 35,000 150,000
1. Clearly define the problem.
2. Identify the possible alternatives and their outcomes.
3. List the profit for each combination of alternatives and outcomes.
4. Select one mathematica; decision theory model.
5. Apply the model and make your decision.
The success or failure that persons, companies, and management experience depends on the decision they make. Making use of quantities in decision-making helps a great deal in minimizing mistakes and failures. Some methods of computation should therefore be learned.
Decision theory is a general approach to decision making that is useful in many different aspects of operations management. It provides a framework for analysis of decision. It includes a number of different techniques that can classify according to the degree of uncertainty.
Decision under Certainty - when one knows with certainty which of the possible future conditions will actually happen, he simply chooses the alternative with the highest payoff under the state of nature.
Decision under Uncertainty - from certainty, the opposite extreme is uncertainty where there is no available information on how likely the various states of nature are. Under this condition, there are four possible decision criterions, namely :
a) Maximin - takes into account only the worst possible outcome for each alternative. This approach establishes a guaratee minimum.
b) Maximax - takes into account only the best possible outcome for each alternative.
c) Laplace - takes into account only the best average possible outcome for each alternative.
d) Minimax Regret - determines the worst regret for ach alternative, and chooses the alternative with the best worst.
DECISION TREES
Another useful technique for analyzing a decision situation is a decion tree. A decision tree, like the probability tree is a graphical diagram consisting of nodes and branches. However, rather than determining the probability of each branch as in a probability tree, in a decision tree the user computes the expected value of each outcome and makes a decision based on these expected values. The primary benefit of a decision tree is that it provides an illustration (or picture) of the decision-making process. This makes it easier to correctly compute the necessary expected values and to understand the process of making the decision.
The decision tree has two types of nodes: a square represents a decision point and a circle stands for a chance event. The tree is read left to right, and analyzed from right to left (starting with the last decision that might be made). For each decision, choose the alternative that will yield the greatest return. If chance events follow a decision, choose the alternative that has the highest expected value or the lowest expected loss.
1. T. Bone Puckett, a corporate raider, has acquired a textile company and is contemplating the future of one of its major plants located in South Carolina. Three alternative decisions are being considered: (1) expand the plant and produce lightweight, durable materials for possible sales to the military, a market with little foreigh competition; (2) maintain the status quo at the plant, continuing production of textile goods that are subject to heavy foreign competition; or (3) sell the plant now. If one of the first two alternatives is chosen, the plant will still be sold at the end of a year. The amount of profit that could be earned by selling the plant in a year depends on foreign market conditions, including the status of a trade embargo bill in Congress. The following payoff table describes this decision situation.
Decision States of Nature
Good Foreign Competitive Conditions Poor Foreign Competitive Condition
Expand 800,000 500,000
Maintain status quo 1,300, 000 -150, 000
Sell now 320,000 320, 000
a. Determine the best decision using the following decision criteria.
1. Maximax
2. Maximin
3. Minimaz regret
4. Hurwicz (alpha =0.3)
5. Equal likelihood
b. Assume that it is now possible to estimate a probability of 0.70 that good foreign competitive conditions will exist and a probability of 0.30 that poor conditions will exist. Determine the dest decision using the expected value and expected opportunity loss.
c. Compute the expected value of perfect information.
d. Develop a decision tree for this decision situation, with expected values at the probability nodes.
2. A farmer in Iowa is considering either leasing some extra land or investing in savings certificates at the local bank. If weather conditions are good next year, the extra land will allow the farmer to have an excellent harvest. However, if weather conditions are bad, the farmer will lose money. The savings certificates will result in the same return regardless of the weather conditions. The return for each investment given each type of weather condition is hown in the following payoff table.
Decision Weather
Good Bad
Lease land 90,000 -40,000
Buy savings certificate 10,000 10,000
Hurwicz (alpha = 0.4)
3. The owner of the Burger Doodle Restaurant is considering two ways to expand operations: operning a drive-up windom or serving breakfast. The increase in profits resulting from these proposed expansions depends on whether a competitor opens a franchise down the street. The possible profits from each expansion in operations given in both future competitive situations are shown in the following payoff table.
Decision Competitor
Open Not Open
Drive-up window -6,000 20,000
Breakfast 4,000 8,000
Hurwicz (alpha = 0.10)
4. Stevie Stone, a bellhop at the Royal Sundown Hotel in Atlanta, has been offered a management position. Although accepting the offer would assure him a job if there were a recession, if good economic conditionsprevailed he would actually make less money as a manager than as a bellhop (because of the large tips he gets as a bellhop). His salary during the next five years for each job given each economic condition is shown in the following payoff table.
Decision Economic Conditions
Good Recession
Bellhop 120,000 60,000
Manager 85,000 85,000
Hurwicz (alpha = 0.4)
5. Consider the following payoff table for three alternative investments, A, B, and C, under two future states of the economy, good and bad.
Investment Economic Conditions
Good Bad
A 70,000 25,000
B 120,000 -60,000
C 40,000 40,000
Hurwicz (0.3)
6. A farmer in Georgia must decide which crop to plant next year on his land: corn, peanuts, or soybeans. The return from each crop will be determined by whether a new trade bill with the USSR passes the Senate. The profit the farmer will realize from each crop given the two possible results on the trade bill is shown in the following payoff table.
Crop Trade Bill
Pass Fail
Corn 35,000 8,000
Peanuts 18,000 12,000
Soybeans 22,000 20,000
Hurwicz (0.3)
7. The owner of the Columbia Construction Company must decide among building a housing development, constructing a shopping center, or leasing all the company's equipment to another company. The profit that will result from each alternative will be determined by whether material costs remain stable or increase. The profit from each alternative given the two possibilities for material costs is shown in the following payoff table.
Decision Material Costs
Stable Increase
Houses 70,000 30,000
Shopping center 105,000 20,000
Leasing 40,000 40,000
Hurwicz (0.2)
8. An investor is considering investing in stocks, real estate, or bonds under uncertain economic conditions. The payoff table of returns for the investor's decision situation is shown below.
Investment Economic Conditions
Good Stable Poor
Stocks 5,000 7,000 3,000
Real estate -2,000 10,000 6,000
Bonds 4,000 4,000 4,000
Hurwicz (0.3)
9. A concessions manager at the Tech vs A and M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance if rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates the following profits will result from each decision given each set of weather conditions.
Decision Weather Conditions
Rain Overcast Sunshine
0.30 0.15 0.55
Sun visors -500 -200 1,500
Umbrellas 2000 0 -900
10. The Blitzkrieg Banking House in Berlin speculates in the money market. The status of the U.S dollar determines the return from investments in other currencies. The banking house will invest in the dollar, the yen, or other currencies. The banking house will invest in the dollar, the yen, or the mark. The return from each is shown in the following payoff table.
Currency Value of the Dollar
Increases (0.3) Remains stable(0.50) Declines(0.20)
Dollar 210,000 0 -170,000
Yen -10,000 20,000 80,000
Mark -40,000 35,000 150,000
Sunday, August 4, 2013
ASSIGNMENT MODEL
The Assignment is a special type of transportation model where the resources (rows) are collected to the destinations (columns) on a one-to-one correspondence so as to minimize cost. This may refer, in particular, to teams assigned to projects; jobs assigned to workers; machines assigned to laborers; etc. It should be observed that the number of rows is equal to the number of columns in the Assignment Table to ensure that no row cannot be assigned to a column, or vice-versa on a one-to-one correspondence.
1. The Atlantic Coast Conference has four basketball games on a particular night. The conference office wants to assign four teams of officials to the four games in a way that will minimize the total distance traveled by the officials. The distances in miles for each team of officials to each game location are shown below:
Officials Game Sites
Raleigh Atlanta Durham Clemson
A 210 90 180 160
B 100 70 130 200
C 175 105 140 170
D 80 65 105 120
2. A metal parts shop has three operators and three machines: a drill press, a lathe, and a grinder. Each operator is qualified to operate each machine. The following table shows the time ( in minutes) required by each operator to produce a part on each machine.
Operator Machine
Press Lathe Grinder
1 22 18 35
2 41 30 28
3 25 36 18
Determine the optimal assignment that will minimize the total machine time.
3. A plant has four operators to be assigned to four machines. The time (minutes) required by each worker to produce a product on each machine is shown below.
Operator Machine
A B C D
1 10 12 9 11
2 5 10 7 8
3 12 14 13 11
4 8 15 11 9
4. A shop has four machinists to be assigned to four machines. The hourly cost of having each machine operated by each machinist is as follows.
Machinist Machine
A B C D
1 12 11 8 14
2 10 9 10 8
3 14 8 7 11
4 6 8 10 9
5. The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown below.
Salesperson Region
A B C D E
1 17 10 15 16 20
2 12 9 16 9 14
3 11 16 14 15 12
4 14 10 10 18 17
5 13 12 9 15 11
6. The Bunker Manufacturing firm has five employees and six machines, and wants to assign the employees to the machines so as to minimize cost. A cost table showing the cost incurred by each employee on each machine is given below.
Employee Machine
A B C D E F
1 12 7 20 14 8 10
2 10 14 13 20 9 11
3 5 3 6 9 7 10
4 9 11 7 16 9 10
5 10 6 14 8 10 12
7. Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost. Identify all alternative solutions if there are multiple optimal solutions,
Operator Machine
A B C D
1 10 2 8 6
2 9 5 11 9
3 12 7 14 14
4 3 1 4 2
8. An electronics firm produces electronic components, which it supplies to various electrical manufacturers. Quality control records indicate that different employees produce different numbers of defective items. The average number of defects produced by each employee for each of six components is given in the following table.
Employee Component
A B C D E F
1 30 24 16 26 30 22
2 22 28 14 30 20 13
3 18 16 25 14 12 22
4 14 22 18 23 21 30
5 25 18 14 16 16 28
6 32 14 10 14 18 20
9. A dispatcher for the Citywide Taxi Company presently has six taxicabs at different locations and five customers who have called for service. The mileage from each taxi's present location to each customer is shown below.
Cab Customer
1 2 3 4 5
A 7 2 4 10 7
B 5 1 5 6 6
C 8 7 6 5 5
D 2 5 2 4 5
E 3 3 5 8 4
F 6 2 4 3 4
10. The Southwest Athletic Conference has six basketball officials who must be assigned to three conference games, two to each game. The conference office wants to assign the officials so that the total distance they travel will be minimized. The distance each official would have to travel to each game is given in the following table:
Official Game
Austin Houston Lubbock
1 20 45 10
2 40 90 70
3 60 70 30
4 30 60 40
5 70 15 50
6 80 25 35
11. A university department head has five instructors to be assigned to four different courses. All of the instructors have taught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table
Instructor Course
A B C D
1 80 75 90 85
2 95 90 90 97
3 85 95 88 92
4 93 91 80 84
5 91 92 93 88
1. The Atlantic Coast Conference has four basketball games on a particular night. The conference office wants to assign four teams of officials to the four games in a way that will minimize the total distance traveled by the officials. The distances in miles for each team of officials to each game location are shown below:
Officials Game Sites
Raleigh Atlanta Durham Clemson
A 210 90 180 160
B 100 70 130 200
C 175 105 140 170
D 80 65 105 120
2. A metal parts shop has three operators and three machines: a drill press, a lathe, and a grinder. Each operator is qualified to operate each machine. The following table shows the time ( in minutes) required by each operator to produce a part on each machine.
Operator Machine
Press Lathe Grinder
1 22 18 35
2 41 30 28
3 25 36 18
Determine the optimal assignment that will minimize the total machine time.
3. A plant has four operators to be assigned to four machines. The time (minutes) required by each worker to produce a product on each machine is shown below.
Operator Machine
A B C D
1 10 12 9 11
2 5 10 7 8
3 12 14 13 11
4 8 15 11 9
4. A shop has four machinists to be assigned to four machines. The hourly cost of having each machine operated by each machinist is as follows.
Machinist Machine
A B C D
1 12 11 8 14
2 10 9 10 8
3 14 8 7 11
4 6 8 10 9
5. The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown below.
Salesperson Region
A B C D E
1 17 10 15 16 20
2 12 9 16 9 14
3 11 16 14 15 12
4 14 10 10 18 17
5 13 12 9 15 11
6. The Bunker Manufacturing firm has five employees and six machines, and wants to assign the employees to the machines so as to minimize cost. A cost table showing the cost incurred by each employee on each machine is given below.
Employee Machine
A B C D E F
1 12 7 20 14 8 10
2 10 14 13 20 9 11
3 5 3 6 9 7 10
4 9 11 7 16 9 10
5 10 6 14 8 10 12
7. Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost. Identify all alternative solutions if there are multiple optimal solutions,
Operator Machine
A B C D
1 10 2 8 6
2 9 5 11 9
3 12 7 14 14
4 3 1 4 2
8. An electronics firm produces electronic components, which it supplies to various electrical manufacturers. Quality control records indicate that different employees produce different numbers of defective items. The average number of defects produced by each employee for each of six components is given in the following table.
Employee Component
A B C D E F
1 30 24 16 26 30 22
2 22 28 14 30 20 13
3 18 16 25 14 12 22
4 14 22 18 23 21 30
5 25 18 14 16 16 28
6 32 14 10 14 18 20
9. A dispatcher for the Citywide Taxi Company presently has six taxicabs at different locations and five customers who have called for service. The mileage from each taxi's present location to each customer is shown below.
Cab Customer
1 2 3 4 5
A 7 2 4 10 7
B 5 1 5 6 6
C 8 7 6 5 5
D 2 5 2 4 5
E 3 3 5 8 4
F 6 2 4 3 4
10. The Southwest Athletic Conference has six basketball officials who must be assigned to three conference games, two to each game. The conference office wants to assign the officials so that the total distance they travel will be minimized. The distance each official would have to travel to each game is given in the following table:
Official Game
Austin Houston Lubbock
1 20 45 10
2 40 90 70
3 60 70 30
4 30 60 40
5 70 15 50
6 80 25 35
11. A university department head has five instructors to be assigned to four different courses. All of the instructors have taught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table
Instructor Course
A B C D
1 80 75 90 85
2 95 90 90 97
3 85 95 88 92
4 93 91 80 84
5 91 92 93 88
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