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operations research an introduction

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

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 customer. The check will yield either a favorable or an unfavorable report. From past experience, 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 mammogram will give a positive indication for cancer 9% of the time. If a 40-year-old woman’s
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 colon cancer, a test for hidden blood in the stool will indicate hidden blood 3% of the time.If the
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 three. The two doors without a prize have nothing behind them. You randomly choose door 2, but before
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).
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 Umbrella Store) depend on the summer’s weather, as shown in Table 1. Of all summers, 20% are rainy, and
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 the number of items sold.b Find the mass function, mean, and variance of the amount of demand that
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 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?
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 back into the urn.Are the random variables X1, X2, X3, and X4 independent random variables?b Repeat
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 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
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?
Daily demand for chocolate bars at the Gillis Grocery has a mean of 100 and a variance of 3,000 (chocolate bars)2. At present, the store has 3,500 chocolate bars in stock. What is the probability that the store will run out of chocolate bars during the next 30 days?Also, how many should Gillis have
1 The daily demand for milk (in gallons) at Gillis Grocery is N(1,000, 100). How many gallons must be in stock at the beginning of the day if Gillis is to have only a 5% chance of running out of milk by the end of the day?
2 Before burning out, a light bulb gives X hours of light, where X is N(500, 400). If we have 3 bulbs, what is the probability that they will give a total of at least 1,460 hours of light?
3 The number of traffic accidents occurring in Bloomington in a single day has a mean and a variance of 3. What is the probability that during a given year (365-day period), there will be at least 1,000 traffic accidents in Bloomington?
4 Suppose that the number of ounces of soda put into a Pepsi can is normally distributed, with m = 12.05 oz and s = .03 oz.a Legally, a can must contain at least 12 oz of soda.What fraction of cans will contain at least 12 oz of soda?b What fraction of cans will contain under 11.9 oz of soda?c
5 Suppose the annual return on Disney stock follows a normal distribution, with mean .12 and standard deviation .30.a What is the probability that Disney’s value will decrease during a year?b What is the probability that the return on Disney during a year will be at least 20%?c What is the
6 The daily demand for six-packs of Coke at Mr. D’s follows a normal distribution, with a mean of 120 and a standard deviation of 30. Every Monday, the delivery driver delivers Coke to Mr. D’s. If the store wants to have only a 1% chance of running out of Coke by the end of the week, how many
7 The Coke factory fills bottles of soda by setting a timer on a filling machine. It has been observed that the number of ounces the machine puts in a bottle has a standard deviation of .05 oz. If 99.9% of all bottles are to have at least 16 oz of soda, to what amount should the average amount be
8 We assemble a large part by joining two smaller parts together. In the past, the smaller parts we have produced have had a mean length of 1 and a standard deviation of.01. Assume that the lengths of the smaller parts are normally distributed and are independent.a What fraction of the larger
9 Weekly Ford sales follow a normal distribution, with a mean of 50,000 cars and a standard deviation of 14,000 cars.a There is a 1% chance that Ford will sell more than cars during the next year.b The chance that Ford will sell between 2.4 and 2.7 million cars during the next year is .
10 Warren Dinner has invested in nine different investments. The profits earned on the different investments are independent. The return on each investment follows a normal distribution, with a mean of $500 and a standard deviation of $100.a There is a 1% chance that the total return on the nine
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!
2 Prove Equation (26).
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
1 Let f (x)=xe-x a Find f '(x) and f''(x).b For what values of x is f (x) increasing? Decreasing?c Find the first-order Taylor series expansion for f (x)about x 1.
2 Let f (x1, x2) = x1 ln(x2 - x1). Determine all first-order and second-order partial derivatives.
3 Some t years from now, air conditioners are sold at a rate of t per year. How many air conditioners will be sold during the next five years?
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
5 Let Xi be the price (in dollars) of stock i one year from now. X1 is N(15, 100) and X2 is N(20, 2025). Today I buy three shares of stock 1 for $12/share and two shares of stock 2 for $17/share. Assume that X1 and X2 are independent random variables.a Find the mean and variance of the value of my
6 An airplane has four engines. On a flight from New York to Paris, each engine has a 0.001 chance of failing. The plane will crash if at any time two or fewer engines are working properly. Assume that the failures of different engines are independent.a What is the probability that the plane will
7 Suppose that each engine can be inspected before the flight. After inspection, each engine is labeled as being in either good or bad condition. You are given that P(inspection says engine is in good condition | engine will fail) .001 P(inspection says engine is in bad condition | engine will
7 Suppose that each engine can be inspected before the flight. After inspection, each engine is labeled as being in either good or bad condition. You are given that P(inspection says engine is in good condition | engine will fail) .001 P(inspection says engine is in bad condition | engine will
6 An airplane has four engines. On a flight from New York to Paris, each engine has a 0.001 chance of failing. The plane will crash if at any time two or fewer engines are working properly. Assume that the failures of different engines are independent.a What is the probability that the plane will
5 Let Xi be the price (in dollars) of stock i one year from now. X1 is N(15, 100) and X2 is N(20, 2025). Today I buy three shares of stock 1 for $12/share and two shares of stock 2 for $17/share. Assume that X1 and X2 are independent random variables.a Find the mean and variance of the value of my
3 Some t years from now, air conditioners are sold at a rate of t per year. How many air conditioners will be sold during the next five years?
2 Let f (x1, x2) = x1 ln(x2 - x1). Determine all first-order and second-order partial derivatives.
1 Let f (x)=xe-x a Find f '(x) and f''(x).b For what values of x is f (x) increasing? Decreasing?c Find the first-order Taylor series expansion for f (x)about x 1.
2 Prove Equation (26).
10 Warren Dinner has invested in nine different investments. The profits earned on the different investments are independent. The return on each investment follows a normal distribution, with a mean of $500 and a standard deviation of $100.a There is a 1% chance that the total return on the nine
9 Weekly Ford sales follow a normal distribution, with a mean of 50,000 cars and a standard deviation of 14,000 cars.a There is a 1% chance that Ford will sell more than cars during the next year.b The chance that Ford will sell between 2.4 and 2.7 million cars during the next year is .
8 We assemble a large part by joining two smaller parts together. In the past, the smaller parts we have produced have had a mean length of 1 and a standard deviation of.01. Assume that the lengths of the smaller parts are normally distributed and are independent.a What fraction of the larger
7 The Coke factory fills bottles of soda by setting a timer on a filling machine. It has been observed that the number of ounces the machine puts in a bottle has a standard deviation of .05 oz. If 99.9% of all bottles are to have at least 16 oz of soda, to what amount should the average amount be
6 The daily demand for six-packs of Coke at Mr. D’s follows a normal distribution, with a mean of 120 and a standard deviation of 30. Every Monday, the delivery driver delivers Coke to Mr. D’s. If the store wants to have only a 1% chance of running out of Coke by the end of the week, how many
5 Suppose the annual return on Disney stock follows a normal distribution, with mean .12 and standard deviation .30.a What is the probability that Disney’s value will decrease during a year?b What is the probability that the return on Disney during a year will be at least 20%?c What is the
4 Suppose that the number of ounces of soda put into a Pepsi can is normally distributed, with m = 12.05 oz and s = .03 oz.a Legally, a can must contain at least 12 oz of soda.What fraction of cans will contain at least 12 oz of soda?b What fraction of cans will contain under 11.9 oz of soda?c
3 The number of traffic accidents occurring in Bloomington in a single day has a mean and a variance of 3. What is the probability that during a given year (365-day period), there will be at least 1,000 traffic accidents in Bloomington?
2 Before burning out, a light bulb gives X hours of light, where X is N(500, 400). If we have 3 bulbs, what is the probability that they will give a total of at least 1,460 hours of light?
1 The daily demand for milk (in gallons) at Gillis Grocery is N(1,000, 100). How many gallons must be in stock at the beginning of the day if Gillis is to have only a 5% chance of running out of milk by the end of the day?
Daily demand for chocolate bars at the Gillis Grocery has a mean of 100 and a variance of 3,000 (chocolate bars)2. At present, the store has 3,500 chocolate bars in stock. What is the probability that the store will run out of chocolate bars during the next 30 days?Also, how many should Gillis have
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?
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.
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 back into the urn.Are the random variables X1, X2, X3, and X4 independent random variables?b Repeat
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.
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 the number of items sold.b Find the mass function, mean, and variance of the amount of demand that
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
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.
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 Umbrella Store) depend on the summer’s weather, as shown in Table 1. Of all summers, 20% are rainy, and
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.
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 three. The two doors without a prize have nothing behind them. You randomly choose door 2, but before
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 colon cancer, a test for hidden blood in the stool will indicate hidden blood 3% of the time.If 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 mammogram will give a positive indication for cancer 9% of the time. If a 40-year-old woman’s
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 customer. The check will yield either a favorable or an unfavorable report. From past experience, the
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
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?
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
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).
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).
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).
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).
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).
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?
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.
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.
What is the probability that the drawn card is not a 2? Suppose we draw a single card from a deck of 52 cards.
Suppose we draw a single card from a deck of 52 cards. What is the probability that a heart or spade is drawn?
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.
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.
On the given set S, determine whether each function is convex, concave, or neither 4 f(x) = x (0 a 1); S = (0, )
17 Suppose a function’s Hessian has both positive and negative entries on its diagonal. Show that the function is neither concave nor convex.
18 Show that if f (x) is a non-negative, increasing concave function, then ln [ f (x)] is also a concave function.
19 Show that if a function f (x1, x2, . . . , xn) is quasiconcave on a convex set S, then for any number a the set Sa all points satisfying f(x1, x2, . . . , xn) a is a convex set.
20 Show that Theorem 1 is untrue if f is a quasi-concave function.
21 Suppose the constraints of an NLP are of the form gi(x1, x2, . . . , xn) bi(i 1, 2, . . . m). Show that if each of the gi is a convex function, then the NLP’s feasible region is convex.Group C
22 If f (x1, x2) is a concave function on R2, show that for any numbera, the set of (x1, x2) satisfying f (x1, x2) a is a convex set.
23 Let Z be a N(0, 1) random variable, and let F(x) be the cumulative distribution function for Z. Show that on S (∞, 0], F(x) is an increasing convex function, and on S [0, ∞), F(x) is an increasing concave function.
24 Recall the Dakota LP discussed in Chapter 6. Let v(L, FH, CH) be the maximum revenue that can be earned when L sq board ft of lumber, FH finishing hours, and CH carpentry hours are available.a Show that v(L, FH, CH) is a concave function.b Explain why this result shows that the value of each
It costs a monopolist $5/unit to produce a product. If he produces x units of the product, then each can be sold for 10 -x dollars (0
LetFind for 0 x
16 From Problem 12, it follows that the sum of concave functions is concave. Is the sum of quasi-concave functions necessarily quasi-concave?

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