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operations research an introduction

Questions and Answers of Operations Research An Introduction

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
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
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%
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,
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
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);
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
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
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
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
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
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
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
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
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
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%
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
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)
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
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
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
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
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
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
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,
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
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
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
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
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
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
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
3 A customer has approached a bank for a loan. Without further information, the bank believes there is a 4% chance that the customer will default on the loan. The bank can run a credit check on the
4 Of all 40-year-old women, 1% have breast cancer. If a woman has breast cancer, a mammogram will give a positive indication for cancer 90% of the time. If a woman does not have breast cancer, a
5 Three out of every 1,000 low-risk 50-year-old males have colon cancer. If a man has colon cancer, a test for hidden blood in the stool will indicate hidden blood half the time. If he does not have
6 You have made it to the final round of “Let’s Make a Deal.” You know there is $1 million behind either door 1, door 2, or door 3. It is equally likely that the prize is behind any of the
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
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).
Consider the discrete random variable X having P(X = i) 1/6 for i = 1, 2, 3, 4, 5, 6.Find E(X) and var X.
Find the mean and variance for the continuous random variable X having the following density function f(x) = (2x if 0 x1 lo otherwise
Each summer in Gotham City is classified as being either a rainy summer or a sunny summer.The profits earned by Gotham City’s two leading industries (the Gotham City Hotel and the Gotham City
I pay $1 to play the following game: I toss a die and receive $3 for each dot that shows.Determine the mean and variance of my profit.
In Example 9, suppose I owned both the hotel and the umbrella store. Find the mean and the variance of the total profit I would earn during a summer
1 I have 100 items of a product in stock. The probability mass function for the product’s demand D is P(D = 90) =P(D = 100) = P(D = 110) =1/3 .a Find the mass function, mean, and variance of
2 I draw 5 cards from a deck (replacing each card immediately after it is drawn). I receive $4 for each heart that is drawn. Find the mean and variance of my total payoff.
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
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
5 An urn contains 10 red balls and 30 blue balls.a Suppose you draw 4 balls from the urn. Let Xi be the number of red balls drawn on the ith ball (Xi = 0 or 1). After each ball is drawn, it is put
6 Let X be the following discrete random variable: P(X =-1) = P(X = 0) = P(X = 1)=1/3. Let Y= X2. Show that cov(X, Y) = 0, but X and Y are not independent random variables.
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
Family income in Bloomington is normally distributed, with m = $30,000 and s =$8,000. The poorest 10% of all families in Bloomington are eligible for federal aid. What should the aid cutoff be?

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