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operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
4 Explain why the reward for a saddle point must be the smallest number in its row and the largest number in its column. Suppose a reward is the smallest in its row and the largest in its column. Must that reward yield a saddle point?(Hint: Think about the idea of weak duality discussed in Chapter
During the 8 to 9 P.M. time slot, two networks are vying for an audience of 100 million viewers. The networks must simultaneously announce the type of show they will air in that time slot. The possible choices for each network and the number of network 1 viewers(in millions) for each choice are
12 At present, littering is punished by a $50 fine, and there is a 10% chance that a litterer will be brought to justice. To cut down on littering, Gotham City is considering two alternatives:Alternative 1 Raise the littering fine by 20% (to $60).Alternative 2 Hire more police and increase by 20%
11 In Problem 5, suppose Rollo cannot hire the consulting firm, and his utility function for ending cash position is u(x) ln x. How much money should he invest in stocks and bonds?
7 Let x1 = undergraduate grade point average (GPA) of a student applying to State U’s MBA program x2 = GMAT score of the same student Suppose that preference between applicants is based on the following value function:v(x1, x2) = 200x1 +x2 - 0.1x2(x1)2 a Would the MBA program prefer a student
6 Willy Mutton has three potential bank robberies lined up. His chance of success and the size of the take are given in Table 18: These robberies must be attempted in order; if you “pass” on a robbery you may not go on to the next robbery. If Willie is caught, he loses all his money. What
4 Jay Boyville Corporation is being sued by Lark Dent.Lark can settle out of court and win $40,000, or go to court.If Lark goes to court, there is a 30% chance that she will win the case. If she wins, a small and a large settlement are equally likely (a small settlement nets $50,000, and a large
2 In Problem 1, suppose that the utility function for the value of the investment (x) one year from now is given by u(x) ln x. Determine which investment we should choose.Could we have predicted this answer without a table of logarithms?
5 Show that for a perfectly consistent decision maker, the ith entry in AwT = n (ith entry of wT).
3 Show that the method for determining k2 described in the text is valid.
Fruit Computer Company is certain that its market share during 2005 will be between 10% and 50% of the microcomputer market. Fruit is also sure that its profits during 2005 will be between $5 million and $30 million. Assess Fruit’s multiattribute utility function where u(x1, x2), where x1
10 Many colleges face the problem of whether athletes should be tested for drug use. Define c1 = Cost if athlete is falsely accused of drug use c2 = Cost if a drug user is not identified c3 = Cost due to invasion of privacy if a nonuser is tested Suppose that 5% of all athletes are drug users,
9 The government is attempting to determine whether immigrants should be tested for a contagious disease. Let’s assume that the decision will be made on a financial basis.Assume that each immigrant who is allowed into the country and has the disease costs the United States $100,000, and each
8 Joe owns a coin that is either a fair coin or a two-headed coin. Imelda believes that there is a 1 2 chance that the coin is two-headed. She must guess what kind of coin Joe has. If she guesses correctly, she pays Joe nothing, but if she guesses incorrectly, she must pay Joe $2. Before
7 Pat Sajork has two drawers. One drawer contains three gold coins, and the other contains one gold coin and two silver coins. We are allowed to choose one drawer, and we will be paid $500 for each gold coin and $100 for each silver coin in that drawer. Before choosing, we may pay Pat$200, and he
6 Abdul has one die in his left hand and one in his right hand. One die has six dots painted on each face, and the other has one dot painted on two of the faces and six dots painted on each of the other four faces. Greta is to pick one die (either “left” or “right”) and will receive $10 for
5 We are thinking of filming the Don Harnett story. We know that if the film is a flop, we will lose $4 million, and if the film is a success, we will earn $15 million. Beforehand, we believe that there is a 10% chance that the Don Harnett story will be a hit. Before filming, we have the option of
4 The NBS television network earns an average of$400,000 from a hit show and loses an average of $100,000 on a flop. Of all shows reviewed by the network, 25% turn out to be hits and 75% turn out to be flops. For $40,000, a market research firm will have an audience view a pilot of a prospective
3 Farmer Jones must determine whether to plant corn or wheat. If he plants corn and the weather is warm, he earns$8,000; if he plants corn and the weather is cold, he earns$5,000. If he plants wheat and the weather is warm, he earns $7,000; if he plants wheat and the weather is cold, he earns
2 A nuclear power company is deciding whether or not to build a nuclear power plant at Diablo Canyon or at Roy Rogers City. The cost of building the power plant is $10 million at Diablo and $20 million at Roy Rogers City. If the company builds at Diablo, however, and an earthquake occurs at Diablo
1 A customer has approached a bank for a $50,000 oneyear loan at 12% interest. If the bank does not approve the loan, the $50,000 will be invested in bonds that earn a 6%annual return. Without further information, the bank feels that there is a 4% chance that the customer will totally default on
Fruit Computer Company manufactures memory chips in lots of ten chips. From past experience, Fruit knows that 80% of all lots contain 10% (1 out of 10) defective chips, and 20%of all lots contain 50% (5 out of 10) defective chips. If a good (that is, 10% defective) batch of chips is sent on to the
16 You have just been chosen to appear on Hoosier Millionaire! The rules are as follows: There are four hidden cards. One says “STOP” and the other three have dollar amounts of $150,000, $200,000, and $1,000,000. You get to choose a card. If the card says “STOP,” you win no money.At any
14 A patient enters the hospital with severe abdominal pains. Based on past experience, Doctor Craig believes there is a 28% chance that the patient has appendicitis and a 72%chance that the patient has nonspecific abdominal pains. Dr.Craig may operate on the patient now or wait 12 hours to gain a
13 We are going to see the movie Fatal Repulsion. There are three parking lots we may park in. One is one block east of the theater (call this lot 1); one lot is directly behind the theater (lot 0); and one lot is one block west of the theater (lot 1). We are approaching the theater from the
11 The Indiana Hoosiers trail the Purdue Boilermakers by a 14–0 score late in the fourth quarter of a football game.Indiana’s guardian angel has informed Indiana that before the game ends they will have the ball two more times, and they will score a touchdown each time. The Indiana coach is
10 Yvonne Delaney is playing Chris Becker a single point for the women’s world tennis championship. She has won the coin toss and elected to serve. If she tries a hard serve, her probability of getting the serve into play is .60. Given that the hard serve is in play, she has a .60 chance of
9 The American chess master Jonathan Meller is playing the Soviet expert Yuri Gasparov in a two-game exhibition match. Each win earns a player one point, and each draw earns a half point. The player who has the most points after two games wins the match. If the players are tied after two games,
8 I am a contestant on the TV show Remote Jeopardy, which works as follows. I am first asked a question about Stupid Videos. If I answer correctly, I earn $100. I believe that I have an 80% chance of answering such a question correctly. If I answer incorrectly, the game is over, and I win nothing.
7 Erica is going to fly to London on August 5 and return home on August 20. It is now July 1. On July 1, she may buy a one-way ticket (for $350) or a round-trip ticket (for$660). She may also wait until August 1 to buy a ticket. On August 1, a one-way ticket will cost $370, and a round-trip ticket
5 During the summer, Olympic swimmer Adam Johnson swims every day. On sunny summer days, he goes to an outdoor pool, where he may swim for no charge. On rainy days, he must go to a domed pool. At the beginning of the summer, he has the option of purchasing a $15 season pass to the domed pool, which
3 I am managing the Chicago Cubs. Suppose there is a runner on first base with nobody out and we want to determine whether we should bunt. Assume that a bunt will yield one of two results: (1) With probability .80, the bunt will be successful, in which case the batter is out and the runner on first
2. If the repairer believes that machine 2 is satisfactory, there is a 60% chance that its annual maintenance cost will be $0 and a 40% chance it will be $100. If the repairer believes that machine 2 is unsatisfactory, there is a 20%chance that the annual maintenance cost will be $0, a 40%chance it
2 The decision sciences department is trying to determine which of two copying machines to purchase. Both machines will satisfy the department’s needs for the next ten years.Machine 1 costs $2,000 and has a maintenance agreement which, for an annual fee of $150, covers all repairs. Machine 2
1 Oilco must determine whether or not to drill for oil in the South China Sea. It costs $100,000, and if oil is found, the value is estimated to be $600,000. At present, Oilco believes there is a 45% chance that the field contains oil.Before drilling, Oilco can hire (for $10,000) a geologist to
An art dealer’s client is willing to buy the painting Sunplant at $50,000. The dealer can buy the painting today for $40,000 or can wait a day and buy the painting tomorrow (if it has not been sold) for $30,000. The dealer may also wait another day and buy the painting(if it is still available)
Alternative 1 Test market Chocola locally, then utilize the results of the market study to determine whether or not to market Chocola nationally.Alternative 2 Immediately (without test marketing) market Chocola nationally.Alternative 3 Immediately (without test marketing) decide not to market
Colaco currently has assets of $150,000 and wants to decide whether to market a new chocolate-flavored soda, Chocola. Colaco has three alternatives:
Tversky and Kahneman asked 72 respondents to choose between lottery 1 and lottery 2 and lottery 3 and lottery 4.Lottery 1: A .001 chance at winning $5,000 and a.999 chance of winning $0 Lottery 2: A sure gain of $5 Lottery 3: A .001 chance of losing $5,000 and a.999 chance of losing $0 Lottery 4: A
3 You are given a choice between lottery 1 and lottery 2.You are also given a choice between lottery 3 and lottery 4.Lottery 1: A sure gain of $240 Lottery 2: 25% chance to gain $1,000 and 75%chance to gain nothing Lottery 3: A sure loss of $750 Lottery 4: A 75% chance to lose $1,000 and a 25%
2.662549p3, use prospect theory to determine the certainty equivalent of the lottery.c Intuively explain why your answer in part (b) is smaller than your answer in part (a).d What implications does this problem have for the method used in Section 13.2 to estimate a person’s utility function?
1 Explain how prospect theory and/or framing explains the Allais Paradox. (See Problem 11 of Section 13.2.)2 Suppose a decision maker has a utility function u(x) x1/3.We flip a fair coin and receive $10 for heads and $0 for tails.a Using expected utility theory, determine the certainty equivalent
16 Suppose my utility function for my asset position is u(x) x1/2. I have $10,000 at present. Consider the following lottery:L: With probability 1 2, L yields a payoff of $1,025.L: With probability 1 2, L yields a payoff of $199.a If I don’t have the right to play L, find an equation that
14† (The Ellsberg Paradox) An urn contains 90 balls. It is known that 30 are red and that each of the other 60 is either yellow or black. One ball will be drawn at random from the urn. Consider the following four options:Option 1 We receive $1,000 if a red ball is drawn.Option 2 We receive $1,000
13 Joe is a risk-averse decision maker. Which of the following lotteries will he prefer?L1: With probability .10, Joe loses $100.L2: With probability .90, Joe receives $0.L2: With probability .10, Joe loses $190.L2: With probability .90, Joe receives $10.
12 (The St. Petersburg Paradox) Let L represent the following lottery. I toss a coin until it comes up heads. If the first heads is obtained on the nth toss of the coin, I receive a payoff of $2n.a If I were a risk-neutral decision maker, what would be the certainty equivalent of L? Is this
11 (The Allais Paradox) Suppose we are offered a choice between the following two lotteries:L1: With probability 1, we receive $1 million.L2: With probability .10, we receive $5 million.L2: With probability .89, we receive $1 million.L2: With probability .01, we receive $0.Which lottery do we
10 My current income is $40,000. I believe that I owe$8,000 in taxes. For $500, I can hire a CPA to review my tax return; there is a 20% chance that she will save me $4,000 in taxes. My utility function for (disposable income) (current income) (taxes) (payment to accountant) is given by x
9 We now have $5,000 in assets and are given a choice between investment 1 and investment 2. With investment 1, 80% of the time we increase our asset position by $295,000, and 20% of the time we increase our asset position by$95,000. With investment 2, 50% of the time we increase our asset position
8 We are going to invest $1,000 for a period of 6 months.Two potential investments are available: T-bills and gold. If the $1,000 is invested in T-bills, we are certain to end the 6-month period with $1,296. If we invest in gold, there is a3 4 chance that we will end the 6-month period with $400
6 A decision maker has a utility function for monetary gains x given by u(x) (x 10,000)1/2.a Show that the person is indifferent between the status quo and L: With probability 1 3, he or she gains $80,000 L: With probability 2 3, he or she loses $10,000 b If there is a 10% chance that a
5 Show that a decision maker who has a linear utility function will rank two lotteries according to their expected value.
4 Show that a decision maker who has a strictly convex utility function will exhibit risk-seeking behavior.
3 Answer Problem 1 for a utility function u(x) 2x 1.
2 Answer Problem 1 for a utility function u(x) x2.
1 Suppose my utility function for asset position x is given by u(x) ln x.a Am I risk-averse, risk-neutral, or risk-seeking?b I now have $20,000 and am considering the following two lotteries:L1: With probability 1, I lose $1,000.L2: With probability .9, I gain $0.L2: With probability .1, I lose
Joan’s utility function for her asset position x is given by u(x) = x1/2. Currently, Joan’s assets consist of $10,000 in cash and a $90,000 home. During a given year, there is a .001 chance that Joan’s home will be destroyed by fire or other causes. How much would Joan be willing to pay for
5 Alden Construction is bidding against Forbes Construction for a project. Alden believes that Forbes’s bid is a random variable B with the following mass function: P(B $6,000) .40, P(B $8,000) .30, P(B $11,000) .30.It will cost Alden $6,000 to complete the project. Use each of the
4 Suppose that Pizza King and Noble Greek stop advertising but must determine the price they will charge for each pizza sold. Pizza King believes that Noble Greek’s price is a random variable D having the following mass function: P(D $6) .25, P(D $8) .50, P(D $10) .25. If Pizza King
3 For Example 1, show that ordering 11 or more papers is dominated by ordering 10 papers.
2 Sodaco is considering producing a new product:Chocovan soda. Sodaco estimates that the annual demand for Chocovan, D (in thousands of cases), has the following mass function: P(D 30) .30, P(D 50) .40, P(D 80) .30. Each case of Chocovan sells for $5 and incurs a variable cost of $3. It
News vendor Phyllis Pauley sells newspapers at the corner of Kirkwood Avenue and Indiana Street, and each day she must determine how many newspapers to order. Phyllis pays the company 20¢ for each paper and sells the papers for 25¢ each. Newspapers that are unsold at the end of the day are
Suppose that at time t (measured in hours, and the present t 0), the rate a(t) at which customers enter a bank is a(t) 100t. During the next 2 hours, how many customers will enter the bank? 50 100 150 50 200 200 a(t) a(t) = 100t A = 0.1 0=10 11 12 13 14 15 16 17 18 to 10 11 12 13 14 15 16 17 18
1 The present is t 0. At a time t years from now, I earn income at a rate e2t. How much money do I earn during the next 5 years?
Forfind F'( y). F(y) = y dx x
For each of the following functions, use Leibniz’s rule to find F'( y): 1 F(y) = f (2y + x) dx
For each of the following functions, use Leibniz’s rule to find F'( y): 2 F(y) = f yx dx
For each of the following functions, use Leibniz’s rule to find F'( y): 3 F(y) = 6(5x)f(x) dx + 4(x-5)f(x) dx
Let X be the number of dots that show when a die is tossed. Then for i=1, 2, 3, 4, 5, 6, P(X= i) 1/6. The cumulative distribution function (cdf) for X is shown in Figure 4. T 9 2 3 4 5 T F(x) T 33 17 T 5to 2/3 97
Consider a continuous random variable X having a density function f (x) given by [2x if 0 x 1 f(x) [0 otherwise Find the cdf for X. Also find P( X 1).
Find the mean and variance for the continuous random variable X having the following density function f(x) = (2x if 0 x1 lo otherwise
3 Consider a continuous random variable X with the density function (called the exponential densitya Find and sketch the cdf for X.b Find the mean and variance of X. (Hint: Use integration by parts.)c Find -x F(x)= {* if x 0 otherwise
4 I have 100 units of a product in stock. The demand D for the item is a continuous random variable with the following density function:a Find the probability that supply is insufficient to meet demand.b What is the expected number of items sold? What is the variance of the number of items sold?
Eli Lilly believes that the year’s demand for Prozac will be normally distributed, with m = 60 million d.o.t. (days of therapy) and s = 5 million d.o.t. How many units should be produced this year if Lilly wants to have only a 1% chance of running out of Prozac? Density 0.08 0.07 0.06 0.05 0.04
Suppose we toss a coin n times, and the probability of obtaining heads each time is p. Let q = 1 - p. If successive coin tosses are independent events, then the mass function describing the random variable X = number of heads is the well-known binomial random variable defined byThe z-transform for
Let the random variable X be defined as the number of coin tosses needed to obtain the first heads, given that successive tosses are independent, the probability that each toss is heads is given by p, and the probability that each coin is tails is given by q 1 p. Then X follows a geometric random
For a given m, the Poisson random variable has the mass function Find the mean and variance of a Poisson random variable. P(X = k) = e L^ (k = 0, 1, 2, ). n!
3 Suppose we toss a coin. Successive coin tosses are independent and yield heads with probability p. The negative binomial random variable with parameter k assumes a value n if it takes n failures until the kth success occurs. Use z-transforms to determine the probability mass function for the
4 Let X be a continuous random variable with density functiona What is k?b Find the cdf for X.c Find E(X) and var X.d Find P(2 f(x) = 0 if 0 x 4 otherwise
Suppose that at time t (measured in hours, and the present t 0), the rate a(t) at which customers enter a bank is a(t) 100t. During the next 2 hours, how many customers will enter the bank? 50 100 150 50 200 200 a(t) a(t) = 100t A = 0.1 0=10 11 12 13 14 15 16 17 18 to 10 11 12 13 14 15 16 17 18
1 The present is t 0. At a time t years from now, I earn income at a rate e2t. How much money do I earn during the next 5 years?
2 If money is continuously discounted at a rate of r% per year, then $1 earned t years in the future is equivalent to ert dollars earned at the present time. Use this fact to determine the present value of the income earned in Problem 1.
3 At time 0, a company has I units of inventory in stock.Customers demand the product at a constant rate of d units per year (assume that I d). The cost of holding 1 unit of stock in inventory for a time is $h. Determine the total holding cost incurred during the next year.
Forfind F'( y). F(y) = y dx x
For each of the following functions, use Leibniz’s rule to find F'( y): 1 F(y) = f (2y + x) dx
For each of the following functions, use Leibniz’s rule to find F'( y): 2 F(y) = f yx dx
For each of the following functions, use Leibniz’s rule to find F'( y): 3 F(y) = 6(5x)f(x) dx + 4(x-5)f(x) dx
Suppose we draw a single card from a deck of 52 cards. What is the probability that a heart or spade is drawn?
What is the probability that the drawn card is not a 2? Suppose we draw a single card from a deck of 52 cards.
3 Given that a red card has been drawn, what is the probability that it is a diamond? Are the events E1 = red card is drawn E2 = diamond is drawn independent events? Suppose we draw a single card from a deck of 52 cards.
Show that the events E1 = spade is drawn E2 = 2 is drawn are independent events. Suppose we draw a single card from a deck of 52 cards.
Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).a What is the probability that the total of the two dice will add up to 7 or 11?
b What is the probability that the total of the two dice will add up to a number other than 2 or 12? Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).
c Are the events E1 = first die shows a 3 E2 = total of the two dice is 6 independent events? Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).
d Are the events 1 2 . 4 Bayes’ Rule 713 E1 = first die shows a 3 E2 = total of the two dice is 7 independent events? Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).
e Given that the total of the two dice is 5, what is the probability that the first die showed 2 dots? Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).
f Given that the first die shows 5, what is the probability that the total of the two dice is even? Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show).
Suppose that 1% of all children have tuberculosis (TB). When a child who has TB is given the Mantoux test, a positive test result occurs 95% of the time. When a child who does not have TB is given the Mantoux test, a positive test result occurs 1% of the time. Given that a child is tested and a
1 A desk contains three drawers. Drawer 1 contains two gold coins. Drawer 2 contains one gold coin and one silver coin. Drawer 3 contains two silver coins. I randomly choose a drawer and then randomly choose a coin. If a silver coin is chosen, what is the probability that I chose drawer 3?
2 Cliff Colby wants to determine whether his South Japan oil field will yield oil. He has hired geologist Digger Barnes to run tests on the field. If there is oil in the field, there is a 95% chance that Digger’s tests will indicate oil. If the field contains no oil, there is a 5% chance that
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