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operations research an introduction
Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions
Determine the optimal inventory policy for the situation in Example 16.2-2, assuming that the setup cost is $5.
Solve Example 16.2-1, assuming that the demand is continuous and uniform during the period and that the pdf of demand is uniform between 0 and 100. (Hint: Use the results of Problem 16-14.)
Colony Shop is stocking heavy coats for next winter. Colony pays $50 for a coat and sells it for $110. At the end of the winter season, Colony offers the coats at $55 each.The demand for coats during the winter season is more than 20 but less than or equal to 30, all with equal probabilities.
QuickStop provides its customers with coffee and donuts at 6:00 a.m. each day. The convenience store buys the donuts for 7 cents apiece and sells them for 25 cents apiece until 8:00 a.m. After 8:00 a.m., the donuts sell for 5 cents apiece. The number of customers buying donuts between 6:00 and 8:00
The U of A Bookstore offers a program of reproducing class notes for participating professors. Professor Yataha teaches a freshmen-level class with an enrollment of between 100 and 150 students, uniformly distributed. A copy costs $10 to produce, and it sells for $25. The students purchase their
The unit holding cost in a single-period inventory situation is $1. If the order quantity is 4 units, find the permissible range of the unit penalty cost implied by the optimal conditions. Assume that the demand occurs instantaneously at the start of the period and that the pdf of demand is as
The demand for an item during a single period occurs instantaneously at the start of the period. The associated pdf is uniform between 15 and 20 units. Because of the difficulty in estimating the cost parameters, the order quantity is determined such that the probability of either surplus or
Find the optimal solution for Example 16.1-2, assuming that the demand during lead time is N(100, 2). Assume that D = 10,000 gallons per month, h = $2 per gallon per month, p = $4 per gallon, and K = $20.
In Example 16.1-2, suppose that the demand during lead time is uniform between 40 and 60 gallons. Compare the solution with that obtained in Example 16.1-2, and interpret the results. (Hint: In both problems, E5x6 is the same, but the variance in the present problem is smaller.)
Solve Example 16.1-2, assuming that the demand during lead time is uniform between 0 and 50 gallons.
For the data given in Example 16.1-2, determine the following:(a) The approximate number of orders per month.(b) The expected monthly setup cost.(c) The expected holding cost per month.(d) The expected shortage cost per month.(e) The probability of running out of stock during lead time.
The daily demand for camera films at a gift shop is N(300, 5). The cost of holding a roll in the shop is $.02 per day, and the fixed cost of placing a replenishment order is $30.The shop’s inventory policy is to order 150 rolls whenever the inventory level drops to 80 units. It simultaneously
The daily demand for a popular CD in a music store is approximately N(200, 20). The cost of keeping the CD on the shelves is $.04 per disc per day. It costs the store $100 to place a new order. There is a 7-day lead time for delivery. Determine the store’s optimal inventory policy given that the
In Example 16.1-1, determine the optimal inventory policy for each of the following cases:*(a) Lead time = 15 days.(b) Lead time = 25 days.(c) Lead time = 10 days.(d) Lead time = 12 days.
In the two-player, two-finger Morra game, each player shows one or two fingers, and simultaneously guesses the number of fingers the opponent will show. The player making the correct guess wins an amount equal to the total number of fingers shown.Otherwise, the game is a draw. Set up the problem as
Colonel Blotto’s army is fighting for the control of two strategic locations. Blotto has two regiments and the enemy has three. A location will fall to the army with more regiments. Otherwise, the result of the battle is a draw.*(a) Formulate the problem as a two-person zero-sum game, and solve
UA and DU are devising their strategies for the 1994 national championship men’s college basketball game. Assessing the strengths of their respective “benches,” each coach comes up with four strategies for rotating the players during the game. The ability of each team to score 2-pointers,
On a picnic outing, 2 two-person teams are playing hide-and-seek. There are four hiding locations (A, B, C, and D), and the two members of the hiding team can hide separately in any two of the four locations. The other team can then search any two locations. The searching team gets a bonus point if
Consider the following two-person, zero-sum game:(a) Verify that the strategies 116 , 0, 56 2 for A and 149 54, 5 54, 02 for B are optimal, and determine the value of the game.(b) Show that the optimal value of the game equals A2 B B B3 5 50 50 1 1 1 A3 10 1 10
Solve the following games graphically. The payoff is for Player A. (a) B1 B2 B2 B3 (b) B B A 2 -3 8 5 00 8 A2 3 3 -6 A2 9 5 A3 5 7
Robin travels between two cities and may use one of two routes: Route A is a fast four-lane highway, and route B is a long winding road. She has the habit of driving“superfast.” The highway patrol has a limited police force. If the full force is allocated to the route driven by Robin, she is
Solve the coin-tossing game of Example 15.4-2 graphically
Let aij be the (i, j)th element of a payoff matrix with m strategies for player A and n strategies for player B. The payoff is for player A. Prove that max imin jaij … min imax iaij
Two companies promote two competing products. Currently, each product controls 50%of the market. Because of recent improvements in the two products, each company plans to launch an advertising campaign. If neither company advertises, equal market shares will continue. If either company launches a
In the games below, the payoff is for player A. Specify the range for the value of the game in each case. *(a) B1 B2 B3 B4 (b) B B2 B3 BA A 1 9 6 0 A1 -1 9 6 8 A2 2 3 8 4 A2 -2 10 4 6 A3 -5 -2 10 -3 A3 5 3 0 7 A4 7 4 -2 -5 A 7 -2 8 4 (c) A B B2 B3 (d) B B B2 B3 BA 3 9 1 A1 3 7 1 3 A2 5 2 3 A2 4 8
In games (a) and (b) below, the payoff is for player A. Determine the values of p and q that will make (A2, B2) a saddle point: (a) (b) B B B2 B3 B B2 B3 A 1 6 A 2 4 5 A2 P 5 10 A2 10 7 q A3 6 2 3 A3 4 P 6
In games (a) and (b) given below, the payoff is for player A. Each game has a pure strategy solution. In each case, determine the strategies that define the saddle point and the value of the game. *(a) B B B3 B4 (b) A 6 9 2 8 A 5 A2. 8 9 4 5 A2 -3 A3 7 5 2 10 5 A3 9 B1 B2 B3 r- -4 -5 B4 9 -4 -8 -2
One of N machines must be selected for manufacturing Q units of a specific product.The minimum and maximum demands for the product are Q* and Q**, respectively.The total production cost for Q items on machine i involves a fixed cost Ki and a variable cost per unit ci, and it is given as TCi = Ki +
For the upcoming planting season, Farmer McCoy can plant corn (a1), wheat (a2), or soybeans (a3) or use the land for grazing (a4). The payoffs associated with the different actions are influenced by the amount of rain: heavy rainfall (s1), moderate rainfall (s2), light rainfall (s3), or drought
Hank is an intelligent student and usually makes good grades, provided that he can review the course material the night before the test. For tomorrow’s test, Hank is faced with a small problem: His fraternity brothers are having an all-night party in which he would like to participate. Hank has
An investment of $10,000 in a high-risk venture has a 50-50 chance over the next year of increasing to $14,000 or decreasing to $8000. Thus the net return can be either $4000 or - $2000.(a) Assuming a risk-neutral investor and a utility scale from 0 to 100, determine the utility of $0 net return on
The Golden family has just moved to a location where earthquakes are not uncommon.They must decide whether to build their house according to the high-standard earthquake code. The construction cost using the earthquake code is $850,000;otherwise, a comparable house can be constructed for only
You are a student at the University of Arkansas and desperately want to attend the next Razorbacks basketball game. The problem is that the admission ticket costs $10, and you have only $5. You can bet your $5 in a poker game, with a 50-50 chance of either doubling your money or losing all of
Historical data at Acme Manufacturing estimate a 5% chance that a batch of widgets will be unacceptable (bad). A bad batch has 15% defective items, and a good batch includes only 4% defective items. Letting a = u1 and a = u2 represent a good and a bad batch, respectively, the associated prior
In Problem 15-13, suppose that AFC management has decided to test-market its Wings ’N Things in selective locations. The outcome of the test is either “good” (a1) or“bad” (a2). The test yields the following conditional probabilities with and without the advertising campaign:The symbols v
The farmer has the additional option of using the land as a grazing range, in which case a payoff of $7500 is guaranteed. The farmer has also secured additional information from a broker regarding the degree of stability of future commodity prices. The broker’s assessment of“favorable” and
Consider Farmer McCoy’s decision situation in Problem
You are the author of what promises to be a successful novel. You have the option to either publish the novel yourself or through a publisher. The publisher is offering you $20,000 for signing the contract. If the novel is successful, it will sell 200,000 copies. Else, it will sell 10,000 copies
In Example 15.2-2, suppose that you have the additional option of investing the original$10,000 in a safe certificate of deposit that yields 4% interest. The friend’s advice applies to investing in the stock market only.(a) Develop the associated decision tree.(b) What is the optimal decision in
Elektra receives 75% of its electronic components from vendor A and the remaining 25% from vendor B. The percentage of defectives from vendors A and B are 1% and 2%, respectively. When a random sample of size 5 from a received lot is inspected, only one defective unit is found. Determine the
Data in a community college show that 80% of new students who took calculus in high school do well, compared with 50% of those who did not take calculus. Admissions for the current academic year show that only 40% of the new students have completed a course in calculus. What is the probability that
Aspiration Level Criterion. Acme Manufacturing uses a chemical in one of its processes.The shelf life is 1 month, and any amount left is destroyed. The amount, x, in gallons of the chemical used by Acme is represented by the following distribution:f1x2 = •200 x2 , 100 … x … 200 0, otherwise
Rapport (1967). A manufacturer has used linear programming to determine the optimum production mix of the various TV models it produces. Recent information received by the manufacturer indicates that there is a 40% chance that the supplier of a component used in one of the models may raise the
Cohan and Associates (1984). Modern forest management uses controlled fires to reduce fire hazards and to stimulate new forest growth. Management has the option to postpone or plan a burning. In a specific forest tract, if burning is postponed, a general administrative cost of $300 is incurred. If
The outer diameter,d, of a cylinder is processed on an automatic machine with upper and lower tolerance limits of d + tU and d - tL. The production process follows a normal distribution with mean m and standard deviation s. Oversized cylinders are reworked at the cost of c1 dollars each. Undersized
An automatic machine produces a (thousands of) units of a product per day. As a increases, the proportion of defectives, p, goes up according to the following probability density function:f1p2 = eapa-1, 0 … p … 1 0, otherwise Each defective item incurs a loss of $50. A good item yields $5
In Problem 15-21, suppose that the store wishes to extend the decision problem to a 2-day horizon. The alternatives for the second day depend on the demand in the first day. If demand on day 1 equals the amount stocked, the store will continue to order the same quantity for day 2; if it exceeds the
Daily demands for loaves of bread at a grocery store are specified by the following probability distribution:n 100 150 200 250 300 pn .20 .25 .30 .15 .10 The store buys a loaf for 55 cents and sells it for $1.20 each. Any unsold loaves at the end of the day are disposed of at 25 cents each. Assume
Sunray Electric Coop uses a fleet of 20 trucks to service its electric network. The company wants to develop a preventive maintenance schedule for the fleet. The probability of a breakdown in year 1 is zero. For year 2, the breakdown probability is .03, increasing annually by .01 for years 3
Rework Problem 15-17, assuming that the demand can be high, medium, and low with probabilities .7, .2, and .1, respectively. Expansion of a small plant will occur only if demand in the first 2 years is high. The following table provides estimates of the annual income. Ignore the time value of
Rework Problem 15-17, assuming that decisions are made taking into account the time value of money at an annual interest rate of 10%. [Note: You need compound interest tables to solve this problem. You can use Excel function NPV(i, R) to compute the present value of cash flows stored in range R,
TriStar plans to open a new plant in Arkansas. The company can open a full-size plant now or a small-size plant that can be expanded 2 years later if warranted by high demand. The time horizon for the decision problem is 10 years. TriStar estimates that the probabilities for high and low demands
Acme Manufacturing produces lots of widget with .8%, 1%, 1.2%, and 1.4% defectives according to the respective probabilities .4, .3, .25, and .05. Three customers, A, B, and C, are contracted to receive batches with no more than .8%, 1.2%, and 1.4% defectives, respectively. If the defectives are
You have the chance to play the following game in a gambling casino. A fair die is rolled twice, leading to four outcomes: (1) both rolls show the same even number,(2) both rolls show the same odd number, (3) the two rolls show either even followed by odd or odd followed by even, and (4) all other
A fair coin is flipped three successive times. You receive $1.00 for each head (H) that turns up and an additional $.25 for each two successive heads that appear (remember that HHH includes two sets of HH). However, you give back $1.10 for each tail that shows up. You have the option to either play
AFC is about to launch its new Wings ‘N Things fast food nationally. The research department is convinced that Wings ‘N Things will be a great success and wants to introduce it immediately in all AFC outlets without advertising. The marketing department sees “things” differently and wants
You have the chance to invest your money in either a 7.5% bond that sells at face value or an aggressive growth stock that pays only 1% dividend. If inflation occurs, the interest rate will go up to 8%, in which case the principal value of the bond will go down by 10%, and the stock value will go
You have the chance to invest in three mutual funds: utility, aggressive growth, and global. The value of your investment will change depending on the market conditions.There is a 20% chance the market will go down, 45% chance it will remain moderate, and 35% chance it will perform well. The
Farmer McCoy can plant either corn or soybeans. The probabilities that the next harvest prices will go up, stay the same, or go down are .25, .30, and .45, respectively.If the prices go up, the corn crop will net $30,000 and the soybeans will net $10,000.If the prices remain unchanged, McCoy will
You have been invited to play the Fortune Wheel game on television. The wheel operates electronically with two buttons that produce hard (H) or soft (S) spin. The wheel itself is divided into white (W) and red (R) half-circle regions. You have been told that the wheel is designed to stop on white
An individual is in the process of buying a car and has narrowed the choices to three models: M1, M2, and M3. The deciding factors include purchase price (PP), maintenance cost (MC), cost of city driving (CD), and cost of rural driving (RD). The following table provides the relevant data for 3
A school district is in dire need to reduce expenses to meet new budgetary restrictions at its elementary schools. Two options are available: delete the physical education program (E), or delete the music program (M). The superintendent has formed a committee with equal-vote representation from the
A professor of political science wants to predict the outcome of a school board election.Three candidates, Ivy (I), Bahrn (B), and Smith (S), are running for one position.There are three categories of voters: left (L), center (C), and right (R). The candidates are judged based on three factors:
A new author sets three criteria for selecting a publisher for an OR textbook: royalty percentage (R), marketing (M), and advance payment (A). Two publishers, H and P, have expressed interest in the book. Using the following comparison matrices, rank the two publishers and assess the consistency of
Kevin and June Park (K and J) are in the process of buying a new house. Three houses, A, B, and C, are available. The Parks have agreed on two criteria for the selection of the house—amount of yard work (Y) and proximity to place of work (W)—and have developed the following comparison matrices.
The personnel department at C&H has narrowed the search for a new hire to three candidates: Steve (S), Jane (J), and Maisa (M). The final selection is based on three criteria: personal interview (I), experience (E), and references (R). The department uses matrix a (given below) to establish the
Copy the weights in a logical order into the solution summary section of the spreadsheet excelAHP.xls, then develop the formula for evaluating the first alternative, UA, and copy it to evaluate the remaining two alternatives.7
Consider the two-hierarchal data of Problem
Suppose that the following weights are specified for the situation of Martin and Jane(Figure 15.2):p = .5, q = .5 p1 = .4, p2 = .6 p11 = .129, p12 = .277, p13 = .594 p21 = .545, p22 = .273, p23 = .182 q1 = .6, q2 = .4 q11 = .2, q12 = .3, q13 = .5 q21 = .5, q22 = .2, q23 = .3 Based on this
An automatic device is used to count the volume of traffic at a busy intersection. The arrival time is recorded and translated into an absolute time starting from zero. The following table provides the arrival times (in minutes) for the first 60 cars. Use Excel to construct a suitable histogram.
The following data represent the period (in seconds) needed to transmit a message.25.8 67.3 35.2 36.4 58.7 47.9 94.8 61.3 59.3 93.4 17.8 34.7 56.4 22.1 48.1 48.2 35.8 65.3 30.1 72.5 5.8 70.9 88.9 76.4 17.3 77.4 66.1 23.9 23.8 36.8 5.6 36.4 93.5 36.4 76.7 89.3 39.2 78.7 51.9 63.6 89.5 58.6 12.8 28.6
The following data represent the interarrival time (in minutes) at a service facility:4.3 3.4 .9 .7 5.8 3.4 2.7 7.8 4.4 .8 4.4 1.9 3.4 3.1 5.1 1.4.1 4.1 4.9 4.8 15.9 6.7 2.1 2.3 2.5 3.3 3.8 6.1 2.8 5.9 2.1 2.8 3.4 3.1 .4 2.7 .9 2.9 4.5 3.8 6.1 3.4 1.1 4.2 2.9 4.6 7.2 5.1 2.6 .9 4.9 2.4 4.1 5.1 11.5
The inside diameter of a cylinder is normally distributed with a mean of 1 cm and a standard deviation of .01 cm. A solid rod is assembled inside each cylinder. The diameter of the rod is also normally distributed with a mean of .99 cm and a standard deviation of.01 cm. Determine the percentage of
The weights of individuals who seek a helicopter ride in an amusement park have a mean of 180 lb and a standard deviation of 15 lb. The helicopter can carry five persons but has a maximum weight capacity of 1000 lb. What is the probability that the helicopter will not take off with five persons
The college of engineering at U of A requires a minimum ACT score of 27. The test scores among high school seniors in a given school district are normally distributed with mean 23 and standard deviation 4.(a) Determine the percentage of high school seniors who are potential engineering recruits.(b)
Prove the formulas for the mean and variance of the exponential distribution.
Customers shopping at Walmark Store are both urban and suburban. Urban customers arrive at the rate of 5 per minute, and suburban customers arrive at the rate of 10 per minute. Arrivals are totally random. Determine the probability that the interarrival time for all customers is less than 8 seconds.
Prove the formulas for the mean and variance of the Poisson distribution.
Customers arrive randomly at a checkout counter at the average rate of 10 per hour.(a) Determine the probability that the counter is idle.(b) What is the probability that at least one person is in line awaiting service?
The Poisson distribution with parameter l approximates the binomial distribution with parameters (n, p) when nS , pS 0, and npS l. Demonstrate this result for the situation where a manufactured lot is known to contain 1% defective items. If a sample of 10 items is taken from the lot, compute the
Customers arrive at a service facility according to a Poisson distribution at the rate of three per minute. What is the probability that at least one customer will arrive in any given 45-second interval?
Prove the formulas for the mean and variance of the binomial distribution.
Suppose that you throw 2 fair dice simultaneously. If there is a match, you receive 50 cents.Otherwise, you pay 10 cents. Determine the expected payoff of the game.
In a gambling casino, you play the game of selecting a number from 1 to 6 before the operator rolls three fair dice simultaneously. The casino pays you as many dollars as the number of dice that match your selection. If there is no match, you pay the casino only$1. Determine your long-run expected
A fortune-teller claims to predict whether people will amass financial wealth in their lifetime by examining their handwriting. To verify this claim, 10 millionaires and 10 university professors were asked to provide samples of their handwriting. The samples are then paired, one millionaire and one
Suppose that four fair coins are tossed independently. What is the probability that exactly one of the coins will be different from the remaining three?
A fair die is rolled 10 times. What is the probability that the rolled die will not show an even number?
The joint pdf of x1 and x2 is*(a) Find the marginal pdfs p1(x1) and p2(x2).*(b) Are x1 and x2 independent?(c) Compute E5x1 + x26.(d) Compute cov5x1, x26.(e) Compute var55x1 - 6x26. x2 = 1 x = 2 x = 3 x = 1 .2 0 .2 P(x1,x2)=x = 2 0 .2 0 x = 3 .2 0 .2
Given the pdf f(x) and y = cx +d, where c and d are constants, prove that E5y6 = cE5x6 + d var5y6 = c2 var5x6
For the pdf f(x), prove that var5x6 = E5x26 - 1E5x622
Show that the mean and variance of a uniform random variable x, a … x …b, are E5x6 =b + a 2var 5x6 =1b - a22 12
Compute the mean and variance of the random variable in Problem 14-16.
Compute the mean and variance of the random variable defined in Problem 14-15.
The owner of a newspaper stand receives 50 copies of Al Ahram newspaper every morning. The number of copies sold, x, varies randomly according to the following probability distribution:p1x2 = c 145, x = 35, 36,c, 49 130, x = 50, 51,c, 59 133, x = 60, 61,c, 70(a) Determine the probability that the
The results of Example 14.3-1 and of Problem 14-18 show positive averages for both the surplus and shortage of stamps. Are these results inconsistent? Explain.
In Example 14.3-1, compute the average shortage of stamps per month. (Hint: Shortage can occur if I need more than 20 stamps.)
The daily demand for unleaded gasoline is uniformly distributed between 750 and 1250 gallons. The 1100-gallon gasoline tank is refilled daily at midnight. What is the probability that the tank will be empty just before a refill?
Consider the following function:f1x2 =k x2, 10 … x … 20*(a) Determine the value of the constant k that will render f(x) a pdf.(b) Determine the CDF, and find the probability that x is (i) larger than 12 and(ii) between 13 and 15.
The number of units, x, needed of an item is discrete from 1 to 6. The probability, p(x), is directly proportional to the number of units needed. The constant of proportionality is K.(a) Determine the pdf and CDF of x, and graph the resulting functions.(b) Find the probability that x is an even
Statistics show that 70% of all men have some form of prostate cancer. The PSA test will show positive 90% of the time for afflicted men and 10% of the time for healthy men. What is the probability that a man who tested positive does have prostate cancer?
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