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operations research an introduction

Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions

9 Suppose that instead of ordering the EOQ q*, we use the order quantity 0.8q*. Use Equation (3) to show that HC(q) OC(q) will have increased by 2.50%.
8 Show that for q q*, an order size of q q* will have a lower cost than an order size of q q*. What is the managerial significance of this result?
7 Suppose we are ordering computer chips. Suppose that in each order, exactly 10% of all chips are defective. As soon as the order arrives, we find out which chips are defective and return them for a complete refund. What would be the optimal ordering policy in this situation?
6 Suppose we order three types of clothing for Ceiling Mart. The optimal reorder intervals are 92 days, 21 days, and 60 days. What would be the optimal power-of-two ordering policy?
5 Suppose we order three types of appliances for the appliance store Ohm City. The optimal reorder intervals are 9.2 days, 21.2 days, and 38.1 days. What would be the optimal power-of-two ordering policy?
4 The efficiency of an inventory system is often measured by the turnover ratio. The turnover ratio (TR) is defined by TR a Does a high turnover ratio indicate an efficient inventory system?b If the EOQ model is being used, determine TR in terms of K, D, h, and q.c Suppose D is increased. Show
3 Father Dominic’s Pizza Parlor receives 30 calls per hour for delivery of pizza. It costs Father Dominic’s $10 to send out a truck to deliver pizzas. It is estimated that each minute a customer spends waiting for a pizza costs the pizza parlor 20¢ in lost future business.a How often should
2 Money in my savings account earns interest at 10%annual rate. Each time I go to the bank, I waste 15 minutes in line. My time is worth $10 per hour. During each year, I need to withdraw $10,000 to pay my bills.a How often should I go to the bank?b Each time I go to the bank, how much money should
1 Each month, a gas station sells 4,000 gallons of gasoline.Each time the parent company refills the station’s tanks, it charges the station $50 plus 70¢, per gallon. The annual cost of holding a gallon of gasoline is 30¢.a How large should the station’s orders be?b How many orders per year
Each hour, D students want to ride a bus from the student union to Fraternity Row. The administration places a value of h dollars on each hour that a student is forced to wait for a bus. It costs the university K dollars to send a bus from the student union to Fraternity Row. Assuming that demand
A department store sells 10,000 cameras per year. The store orders cameras from a regional warehouse. Each time an order is placed, an ordering cost of $5 is incurred. The store pays $100 for each camera, and the cost of holding $1 worth of inventory for a year is estimated to be the annual capital
Braneast Airlines uses 500 taillights per year. Each time an order for taillights is placed, an ordering cost of $5 is incurred. Each light costs 40¢, and the holding cost is 8¢/light/year.Assume that demand occurs at a constant rate and shortages are not allowed. What is the EOQ? How many orders
12 Edwina, a commodities broker, has acquired an option to buy 1,000 oz of gold at $50/oz. If she takes the option and if Congress relaxes import quotas, she can sell the gold for$80/oz. If she takes the option and Congress does not relax the import quotas, however, the company will lose
15 a Suppose you are given a choice between the following options:A1: Win $30 for sure A2: 80% chance of winning $45 and 20% chance of A2: winning nothingB1: 25% chance of winning $30 B2: 20% chance of winning $45 Most people prefer A1 to A2 and B2 to B1. Explain why this behavior violates the
1 National Express Carriers is interested in two attributes:Attribute 1 The average cost of delivering a letter (known to be between $1 and $5)Attribute 2 Percentage of all letters reaching their destination on time (known to be between 70% and 100%)a Would National’s multiattribute utility
2 Keeney and Raiffa (1976) discuss the assessment of a blood bank’s multiattribute utility function. For simplicity, we assume that the blood bank must determine at the beginning of each week how many pints of blood should be ordered. Any blood left over at the end of the week spoils(it is
4 Gotham City is trying to determine how many ambulances it should have and how to staff them. Each ambulance may be staffed with paramedics or emergency medical technicians. Paramedics are considered to provide better service and are paid higher salaries. Budgetary limitations have forced the city
5 Public service Indiana (PSI) is considering two sites for a nuclear power plant. The following two attributes will influence its determination about where to build the plant:Attribute 1 Cost of the plant (in millions of dollars)Attribute 2 Acres of land damaged by building the plant Assume that
6 Consider the four points A, B, C, and D in Figure 16.Assume that more of each attribute is desirable and that a decision maker’s utility function exhibits mui. Consider the following two lotteriespoorly. Suppose that Germany will suffer defeat if it performs poorly on either front. If these
1 Each professor’s annual salary increase is determined by performance in three areas: teaching, research, and service to the university. The administration has come up with the following pairwise comparison matrix for these objectivesThe administration has compared two professors with regard to
2 A business is about to purchase a new personal computer.Three objectives are important in determining which computer should be purchased: cost, user-friendliness, and software availability. The pairwise comparison matrix for these objectives is as follows:Three computers are being considered for
3 Woody is ready to select his mate for life and has determined that beauty, intelligence, and personality are the key factors in selecting a satisfactory mate. His pairwise comparison matrix for these objectives is as follows:Three women (Jennifer Lopez, Britney Spears, and Mandy Moore) are
4 In determining where to invest my money, two objectives—expected rate of return and degree of risk—are considered equally important. Two investments (1 and 2)have the following pairwise comparison matrices: Expected return:a How should I rank these investments?b Now suppose another investment
6 A consumer is trying to determine which type of frozen dinner to eat. He considers three attributes to be important:taste, nutritional value, and price. Nutritional value is considered to be determined by cholesterol and sodium levels. Three types of dinners are under consideration. The pairwise
7 You are trying to determine which MBA program to attend. You have been accepted at two programs: Indiana and Northwestern. You have chosen three attributes to use in helping you make your decision:Attribute 1 Cost Attribute 1 Starting salary Attribute 1 Ambience of school (can we party
8 You have been hired by Arthur Ross to determine which of the following accounts receivable procedures should be used in an audit of the Keating Five and Dime Store:a Analytic review b Confirmations c Test of subsequent collections (receipts)The three criteria used to distinguish between the
9 You are trying to determine which of two secretarial candidates (Jack and Jill) to hire. The three objectives that are important to your decision are personality, typing ability, and intelligence. You have assessed the following pairwise comparison matrix:If you follow the AHP method which
1 We have $1,000 to invest. All the money must be placed in one of three investments: gold, stock, or money market certificates. If $1,000 is placed in an investment, the value of the investment one year from now depends on the state of the economy (see Table 16). Assume that each state of the
3 Consider the following four lotteries:a Most people prefer L1 to L2 and L4 to L3. Explain why.b Suppose a decision maker subscribes to the Von Neumann–Morgenstern axioms and prefers L1 to L2.Show that he or she must also prefer L3 to L4. L .50 $1,000,000 $3,000,000 L .50 $0 .10 $1,000,000 L3
5 Rollo Megabux has $1 million to invest in stocks or bonds. The percentage yield on each investment during the coming year depends on whether the economy has a good or a bad year (see Table 17). It is equally likely that the economy will have a good or a bad year.a If Rollo is risk-neutral, how
8 The Pine Valley Board of Education is trying to determine its multiattribute utility function with respect to the following attributes:Attribute 1 Average score of students on an English achievement test Attribute 2 Average score of students on a mathematics achievement testThe board believes
9 BeatTrop Foods is trying to choose one of three companies to merge with. In making this decision seven factors are important:Factor 1 Contribution to profitability Factor 2 Growth potential Factor 3 Labor environment Factor 4 R&D ability of company Factor 5 Organizational fit Factor 6
10 You are trying to determine which city to live in. New York and Chicago are under consideration. Four objectives will determine your decision: affordability of housingcultural opportunities, quality of schools and universities, and crime level. The weight for each objective is in Table 19. For
13 In Section 13.2, we discussed the concept of the risk premium of a lottery and a risk-averse decision maker. In many situations, we would like to measure the degree of risk aversion associated with a utility function, and how a decision maker’s risk aversion depends on his or her wealth.In
1 Find the value and optimal strategy for the game in Table 6. TABLE 6 2 2 1 3
2 Find the value and the optimal strategies for the twoperson zero-sum game in Table 7. TABLE 7 577 4656 5655 8645 655
3 Mad Max wants to travel from New York to Dallas by the shortest possible route. He may travel over the routes shown in Table 8. Unfortunately, the Wicked Witch can block one road leading out of Atlanta and one road leading out of Nashville. Mad Max will not know which roads have been blocked
1 Find the value and the optimal strategies for the twoperson zero-sum game in Table 15. TABLE 15 1 2 2 20 0 3 3
3 Find the value and optimal strategies for the two-person zero-sum game in Table 16. TABLE 16 213 4 3 2
9 Consider a two-person zero-sum game with the reward matrix in Table 17. Suppose this game does not have a saddle point. Show that the optimal strategy for the row player is to play the first row a fraction (d c)/(a d b c) of the time and the optimal strategy for the column player is to play the
10 Consider the following simplified version of football.On each play the offense chooses to run or pass. At the same time, the defense chooses to play a run defense or pass defense. The number of yards gained on each play is determined by the reward matrix in Table 18. The offense’s goal is to
11 Use the idea of dominated strategies to determine optimal strategies for the reward matrix in Table 19. TABLE 19 -5 -10 -1 -10 47278 12787 -10 7 -5 -5 -10 -10 2 -1 785 20 20 -1 -1 -1 2 -10 7 -1 -10
Find the value and optimal strategies for the two-person zero-sum game in Table 26. TABLE 26 Reward Matrix for Example 5 Column Player Row Player 30 40 36 60 10 36 Column Maximum 60 40 36 Row Minimum 30 720 10
Two players in the game of Two-Finger Morra simultaneously put out either one or two fingers. Each player must also announce the number of fingers that he believes his opponent has put out. If neither or both players correctly guess the number of fingers put out by the opponent, the game is a draw.
2 Find each player’s optimal strategy and the value of the two-person zero-sum game in Table 31. TABLE 31 4 5 14 2 163 100 2
3 Find each player’s optimal strategy and the value of the two-person zero-sum game in Table 32. TABLE 32 2 3 4 6 1 5 1
8 Wivco has observed the daily production and the daily variable production costs of widgets at the New York City.0 plant. The data in Table 33 have been collected. Wivco believes that daily production and daily variable production costs are related as follows: For some numbers a and b, Daily
Two prisoners who escaped and participated in a robbery have been recaptured and are awaiting trial for their new crime. Although they are both guilty, the Gotham City district attorney is not sure he has enough evidence to convict them. To entice them to testify against each other, the district
1 Find an equilibrium point (if one exists in pure strategies)for the two-person nonconstant-sum game in Table 41. TABLE 41 (9,-1) (-2,-3) (8,7) (-9, 11)
2 Find an equilibrium point in pure strategies (if any exists)for the two-person nonconstant-sum game in Table 42. TABLE 42 (9,9) (-10, 10) (10, 10) (-1, 1)
4 Given that each player’s goal is to maximize her expected reward, show that for the game in Table 43 each player’s choice of the mixed strategy ( 1/2, 1/2) is an equilibrium point TABLE 43 Player 2 Player 1 Strategy 1 Strategy 2 Strategy 1 (2,-1) (-2, 1) Strategy 2 (-2,1) (2,-1)
Consider a three-person game with the following characteristic function v({ }) = v({1}) = v({2}) = v({3}) = 0 v({1, 2})=0.1, v({1, 3})=0.2, v({2, 3})=0.2, v({1, 2, 3}) = 11 x. Let x (0.05, 0.90, 0.05) and y = (0.10, 0.80, 0.10). Show that y> (13)x
1 Consider the four-player game with the following characteristic functionShow that this game has an empty core. v({1, 2, 3})=({1, 2, 4}) = v({1, 3, 4}) = v({2, 3, 4})=75 v({1, 2, 3, 4}) = 100 v({3, 4})=60 v(S) = 0 for all other coalitions
8 Consider the following three-person game:a Find the core of this game.b Find the Shapley value for this game.c Find an imputation dominating the imputation v({ }) = 0, v({1}) = 0.2, v({2}) = v({3}) = 0, v({1, 2}) = 1.5, v({1,3})=1.6, v({1, 2, 3})=2 v({2, 3}) = 1.8,
2 A total of 90,000 customers frequent the Ruby and the Swamp supermarkets. To induce customers to enter, each store gives away a free item. Each week, the giveaway item is announced in the Monday newspaper. Of course, neither store knows which item the other store will choose to give away this
4 Find optimal strategies for each player and the value of the two-person zero-sum game in Table 52. TABLE 52 222 20 1 2 12 10 4 24 8 -2
6 The stock in Alden Corporation is held by three people.Person 1 owns 1%, person 2 owns 49%, and person 3 owns 50%. To pass a resolution at the annual stockholders’meeting, 51% of the stock is needed. A coalition receives a reward of 1 if it can pass a resolution and a reward of 0 if it cannot
7 In addition to the core and the Shapley value, the stable set is an alternative solution concept for n-person games. A set I of imputations is called a stable set if each imputation in I is undominated and every imputation that is not in I is dominated by some member of I. Consider the
5 Airway (a Midwestern department store chain) and Corvett (an Eastern department store chain) are determining whether to expand their geographical bases. The only viable manner by which expansion might be carried out is for a chain to open stores in the other’s area. If neither chain expands,
3 Consider the two-person zero-sum game in Table 51.a Write down each player’s LP.b We are told that player 1’s optimal strategy has x1 > 0, x2 0, and x3 > 0. Find the value of the game and each player’s optimal strategies.c Suppose the column player plays the nonoptimal strategy ( 1/2,
1 Two competing firms are deciding whether to locate a new store at point A, B, or C. There are 52 prospective customers for the two stores. Twenty customers live in village A, 20 customers live in village B, and 12 customers live in village C (see Figure 6). Each customer will shop at the nearer
12 Consider an n-person game in which the only winning coalitions are those containing player 1 and at least one other player. If a winning coalition receives a reward of $1, find the Shapley value to each player.
11 Three doctors have banded together to form a joint practice: the Port Charles Trio. The overhead for the practice is $40,000 per year. Each doctor brings in annual revenues and incurs annual variable costs as follows: doctor 1—$155,000 in revenue, $40,000 in variable cost; doctor 2—$160,000
10 Indiana University leases WATS lines and is charged according to the following rules: $400 per month for each of the first five lines; $300 per month for each of the next five lines; $100 per month for each additional line. The College of Arts and Sciences makes 150 calls per hour, the School of
9 Howard Whose has left an estate of $200,000 to support his three ex-wives. Unfortunately, Howard’s attorney has determined that each ex-wife needs the following amount of money to take care of Howard’s children: wife 1—$100,000;wife 2—$200,000; wife 3—$300,000. Howard’s attorney must
7 The Gotham City airport runway is 5,000 ft long and costs $100,000 per year to maintain. Last year there were 2,000 landings at the airport. Four types of planes landed.The length of runway required by each type of plane and the number of landings of each type are shown in Table 49.Assuming that
6 For the four-player garbage game, find an imputation that dominates (-1, -1, -1, -1).
5 Show that for n > 2, the n-player garbage game has an empty core.
4 Show that for n 2, the core of the garbage game is the imputation (-1, -1).
3 The game of Odd Man Out is a three-player coin toss game in which each player must choose heads or tails. If all the players make the same choice, the house pays each player $1; otherwise, the odd man out pays each of the other players $1.a Find the characteristic function for this game.b Find
2 Show that if v({3, 4}) in Problem 1 were changed to 50, then the game’s core would consist of a single point.
Suppose three types of planes (Piper Cubs, DC-10s, and 707s) use an airport. A Piper Cub requires a 100-yd runway, a DC-10 requires a 150-yd runway, and a 707 requires a 400-yd runway. Suppose the cost (in dollars) of maintaining a runway for one year is equal to the length of the runway. Because
Find the Shapley value for the drug game.
Find the core of the land development game.
Determine the core of the garbage game
Find the core of the drug game.
For the land development game (Example 13), let x = ($19,000, $1,000, $10,000) and y = ($19,800, $100, $10,100). Show that y >{1,3}x.
Player 1 owns a piece of land and values the land at $10,000. Player 2 is a subdivider who can develop the land and increase its worth to $20,000. Player 3 is a subdivider who can develop the land and increase its worth to $30,000. There are no other prospective buyers.Find the characteristic
Each of four property owners has one bag of garbage and must dump it on somebody’s property. If b bags of garbage are dumped on the coalition of property owners, then the coalition receives a reward of b. Find the characteristic function for this game
Joe Willie has invented a new drug. Joe cannot manufacture the drug himself, but he can sell the drug’s formula to company 2 or company 3. The lucky company will split a $1 million profit with Joe Willie. Find the characteristic function for this game.
5 A Japanese electronics company and an American electronics company are both considering working on developing a superconductor. If both companies work on the superconductor, they will have to share the market, and each company will lose $10 billion. If only one company works on the
3 The New York City Council is ready to vote on two bills that authorize the construction of new roads in Manhattan and Brooklyn. If the two boroughs join forces, they can pass both bills, but neither borough by itself has enough power to pass a bill. If a bill is passed, then it will cost the
Angry Max drives toward James Bound on a deserted road. Each person has two strategies:swerve or don’t swerve. The reward matrix in Table 39 needs no explanation! Find the equilibrium point(s) for this game
The Vulcans and the Klingons are engaged in an arms race in which each nation is assumed to have two possible strategies: develop a new missile or maintain the status quo.The reward matrix is assumed to be as shown in Table 38. This reward matrix is based on the assumption that if only one nation
Competing restaurants Hot Dog King and Hot Dog Chef are attempting to determine their advertising budgets for next year. The two restaurants will have combined sales of $240 million and can spend either $6 million or $10 million on advertising. If one restaurant spends more money than the other,
9 Suppose we add a constant c to every element in a reward matrix A. Call the new game matrix A. Show that A and A have the same optimal strategies and that value of A (value of A) c.
7 Interpret the complementary slackness conditions for the row and the column players’ LP’s.
6 For a two-person zero-sum game with an m n reward matrix, let x (x1, x2, . . . , xm) be a solution to the row player’s LP and y ( y1, y2, . . . , yn) be a solution to the column player’s LP. Show that if the row player departs from his optimal strategy, he cannot increase his
5 A two-person zero-sum game with an n n reward matrix A is a symmetric game if A AT.a Explain why a game having A AT is called a symmetric game.b Show that a symmetric game must have a value of zero.c Show that if (x1, x2, . . . ,xn) is an optimal strategy for the row player, then (x1,
4 Two armies are advancing on two cities. The first army is commanded by General Custard and has four regiments the second army is commanded by General Peabody and has three regiments. At each city, the army that sends more regiments to the city captures both the city and the opposing army’s
1 A soldier can hide in one of five foxholes (1, 2, 3, 4, or 5) (see Figure 5). A gunner has a single shot and may fire at any of the four spots A, B, C, or D. A shot will kill a soldier if the soldier is in a foxhole adjacent to the spot where the shot was fired. For example, a shot fired at spot
Two players simultaneously utter one of the three words stone, paper, or scissors and show corresponding hand signs. If both players utter the same word, then the game is a draw.Otherwise, one player wins $1 from the other player according to the following: Scissors defeats (cuts) paper, paper
8 State University is about to play Ivy College for the state tennis championship. The State team has two players (A and B), and the Ivy team has three players (X, Y, and Z). The following facts are known about the players’ relative abilities X will always beat B; Y will always beat A; A will
7 Mo and Bo each have a quarter and a penny.Simultaneously, they each display a coin. If the coins match, then Mo wins both coins; if they don’t match, then Bo wins both coins. Determine optimal strategies for this game.
6 Two competing firms must simultaneously determine how much of a product to produce. The total profit earned by the two firms is always $1,000. If firm 1’s production level is low and firm 2’s is also low, then firm 1 earns a profit of $500; if firm 1’s level is low and 2’s is high, then
5 For Example 3, show that if player 2 deviates from her optimal strategy, then player 1 can ensure that she earns an expected reward that is more than the value ( 1/3) of the game.
4 For Example 3, show that if player 1 deviates from her optimal strategy, then player 2 can ensure that player 1 earns an expected reward that is less than the value ( 1/3) of the game.
2 Player 1 writes an integer between 1 and 20 on a slip of paper. Without showing this slip of paper to player 2, player 1 tells player 2 what he has written. Player 1 may lie or tell the truth. Player 2 must then guess whether or not player 1 has told the truth. If caught in a lie, player 1 must
A fair coin is tossed, and the result is shown to player 1. Player 1 must then decide whether to pass or bet. If player 1 passes, then he must pay player 2 $1. If player 1 bets, then player 2 (who does not know the result of the coin toss) may either fold or call the bet. If player 2 folds, then
Two players (called Odd and Even) simultaneously choose the number of fingers (1 or 2)to put out. If the sum of the fingers put out by both players is odd, then Odd wins $1 from Even. If the sum of the fingers is even, then Even wins $1 from Odd. We consider the row player to be Odd and the column

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