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operations research an introduction

Questions and Answers of Operations Research An Introduction

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
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
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
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
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.
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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 =
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 =
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 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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
A retailer receives 70% of its batteries from Factory A and 30% from Factory B.The percentages of defectives produced by A and B are known to be 3% and 5%, respectively. A customer has just bought a
Bayes’ theorem.2 Given the two events A and B, show that P5A0B6 =P5B0A6P5A6 P5B6, P5B6 7 0
Prove that if the probability P5A0B6 = P5A6, then A and B must be independent.
Graduating high school seniors with an ACT score of at least 26 can seek admission in two universities, A and B. The probability of being accepted in A is .4 and in B .25.The chance of being accepted
The stock of WalMark Stores, Inc. trades on the New York Stock Exchange under the symbol WMS. Historically, the price of WMS goes upward with the Dow 65% of the time and goes downward with the Dow
In Example 14.1-2, suppose that you are told that the outcome is less than 6.(a) Determine the probability of getting an even number.(b) Determine the probability of getting an odd number larger than
Ann, Jim, John, and Nancy are scheduled to compete in a racquetball tournament. Ann is twice as likely to beat Jim, and Jim is at the same level as John. Nancy’s past winning record against John is
You can toss a fair coin up to 7 times. You will win $100 if three tails appear before a head is encountered. What are your chances of winning?
Two dice are rolled independently and the two numbers that turn up are recorded.Determine the following:(a) The probability that the two numbers are odd with values less than 5.(b) The probability
A fair 6-faced die is tossed twice. Letting E and F represent the outcomes of the two tosses, compute the following probabilities:(a) The sum of E and F is 10.(b) The sum of E and F is even.(c) The
Answer Problem 14-2 assuming that in a room full of n persons at least one person shares your birthday.
Consider a random gathering of n persons. Determine the smallest n that will make it more likely that two persons or more have the same birthday. (Hint: Assume no leap years and that all days of the
In a survey conducted in the State of Arkansas high schools to study the correlation between senior year scores in mathematics and enrollment in engineering colleges, 400 out of 1000 surveyed seniors
A small publisher reprints a novel to satisfy the demand over the next 12 months. The demand estimates for the successive months are 100, 120, 50, 70, 90, 105, 115, 95, 80, 85, 100, and 110. The
The demand for fishing poles is at its minimum during the month of December and reaches its maximum during the month of April. Fishing Hole, Inc., estimates the December demand at 50 poles. It
Find the optimal inventory policy for the following six-period inventory situation:The unit production cost is $2 for all the periods.Period i Di (units) Ki ($) hi ($)1 10 20 1 2 15 17 1 3 7 10 1 4
Find the optimal inventory policy for the following five-period model. The unit production cost is $10 for all periods. The unit holding cost is $1 per period.Period i Demand Di (units) Setup cost K1
Solve the following 10-period deterministic inventory model. Assume an initial inventory of 50 units.Period i Demand Di (units)Unit production cost ($)Unit holding cost ($)Setup cost ($)1 150 6 1 100
Solve Example 13.4-3, assuming that the initial inventory is 80 units. You may use excelWagnerWhitin.xls to check your calculations.
Develop the backward recursive equation for the model, assuming that the inventoryholding cost is based on the average inventory in the period.

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