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operations research an introduction

Questions and Answers of Operations Research An Introduction

John must work at least 20 hours a week to supplement his income while attending school.He has the opportunity to work in two retail stores. In store 1, he can work between 4.5 and 12 hours a week,
For the diet model, determine the optimum solution given the feed mix does not exceed 500 lb a day? Does the solution make sense?
For the diet model, suppose that the daily availability of corn is limited to 400 lb. Identify the new solution space, and determine the new optimum solution.
Identify the direction of decrease in z in each of the following cases:*(a) Minimize z = 4x1 - 2x2.(b) Minimize z = -6x1 + 2x2.(c) Minimize z = -3x1 - 6x2.
TORA Experiment. In the Reddy Mikks model, suppose that the following constraint is added to the problem:x2 Ú 3 Use TORA to show that the resulting model has conflicting constraints that cannot be
TORA Experiment. In the Reddy Mikks model, use TORA to show that the removal of the raw material constraints (constraints 1 and 2) would result in an unbounded solution space. What can be said in
TORA Experiment. Consider the following LP model:Maximize z = 5x1 + 4x2 subject to 6x1 + 4x2 … 24 6x1 + 3x2 … 22.5 x1 + x2 … 5 x1 + 2x2 … 6-x1 + x2 … 1 x2 … 2 x1, x2 Ú 0 In LP, a
TORA Experiment. Enter the following LP into TORA, and select the graphic solution mode to reveal the LP graphic screen.Minimize z = 3x1 + 8x2 subject to x1 + x2 Ú 8 2x1 - 3x2 … 0 x1 + 2x2 … 30
Determination of the Optimum LP Solution by Enumerating All Feasible Corner Points.The remarkable observation gleaned from the graphical LP solution is that the optimum, when finite, is always
An assembly line consisting of three consecutive stations produces two radio models: HiFi-1 and HiFi-2. The following table provides the assembly times for the three workstations.Minutes per unit
A furniture company manufactures desks and chairs. The sawing department cuts the lumber for both products, which is then sent to separate assembly departments. Assembled items are sent to the
The Burroughs Garment Company manufactures men’s shirts and women’s blouses for Walmark Discount Stores. Walmark will accept all the production supplied by Burroughs.The production process
Top Toys is planning a new radio and TV advertising campaign. A radio commercial costs$300 and a TV ad costs $2000. A total budget of $20,000 is allocated to the campaign.However, to ensure that each
Show & Sell can advertise its products on local radio and television (TV). The advertising budget is limited to $10,000 a month. Each minute of radio advertising costs $15, and each minute of TV
Wild West produces two types of cowboy hats. A Type 1 hat requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a total
Jack is an aspiring freshman at Ulern University. He realizes that “all work and no play make Jack a dull boy.” Jack wants to apportion his available time of about 10 hrs a day between work and
In the Ma-and-Pa grocery store, shelf space is limited and must be used effectively to increase profit. Two cereal items, Grano and Wheatie, compete for a total shelf space of 60 ft2. A box of Grano
ChemLabs uses raw materials I and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively.One unit of solution A
The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical and humanistic. To satisfy the
An individual wishes to invest $5000 over the next year in two types of investment:Investment A yields 5%, and investment B yields 8%. Market research recommends an allocation of at least 25% in A
Alumco manufactures aluminum sheets and aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 560 bars. The
A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 110 units of A per day. Both products
A company that operates 10 hrs a day manufactures two products on three sequential processes. The following table summarizes the data of the problem:Minutes per unit Product Process 1 Process 2
Determine the solution space and the optimum solution of the Reddy Mikks model for each of the following independent changes:(a) The maximum daily demand for interior paint is 1.9 tons and that for
Identify the direction of increase in z in each of the following cases:*(a) Maximize z = x1 - x2.(b) Maximize z = -8x1 - 3x2.(c) Maximize z = -x1 + 3x2.*(d) Maximize z = -3x1 + x2.
Suppose that Reddy Mikks sells its exterior paint to a single wholesaler at a quantity discount.The profit per ton is $5000 if the contractor buys no more than 5 tons daily and $4300
For the feasible solution x1 = 1, x2 = 2 of the Reddy Mikks model, determine the unused amounts of raw materials M1 and M2.
Determine the best feasible solution among the following (feasible and infeasible) solutions of the Reddy Mikks model:(a) x1 = 1, x2 = 2.(b) x1 = 3, x2 = 1.(c) x1 = 3, x2 = 1.5.(d) x1 = 2, x2 = 1.(e)
For the Reddy Mikks model, construct each of the following constraints, and express it with a linear left-hand side and a constant right-hand side:*(a) The daily demand for interior paint exceeds
You are given two identical balls made of a tough alloy. The hardness test fails if a ball dropped from a floor of a 120-storey building is dented upon impact. A ball can be reused in fresh drops
You have 10 identical cartons each holding 10 water bottles. All bottles weigh 10 oz.each, except for one defective carton in which each of the 10 bottles weighs on 9 oz. only.A scale is available
The squares of a rectangular board of 11 rows and 9 columns are numbered sequentially 1 through 99 with a hidden monetary reward between 0 and 50 dollars assigned to each square. A game using the
You have five chains, each consisting of four solid links. You need to make a bracelet by connecting all five chains. It costs 2 cents to break a link and 3 cents to re-solder it.(a) Identify two
An upright symmetrical triangle is divided into four layers: The bottom layer consists of four (equally-spaced) dots, designated as A, B, C, and D. The next layer includes dots E, F, and G, and the
During the construction of a house, six joists of 24 ft each must be trimmed to the correct length of 23 ft. The operations for cutting a joist involve the following sequence:Operation Time
In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can throw either a fast or a curve ball at random. If Joe correctly predicts a curve ball, he can maintain a .400
Amy, Jim, John, and Kelly are standing on the east bank of a river and wish to cross to the west side using a canoe. The canoe can hold at most two people at a time. Amy, being the most athletic, can
Determine the optimal solution of the garden problem. (Hint: Use the constraint to express the objective function in terms of one variable, then use differential calculus.)
In the garden problem, identify three feasible solutions, and determine which one is better.
In the tickets example,(a) Provide an infeasible alternative.(b) Identify a fourth feasible alternative and determine its cost.
2. UA and DU are setting up their strategies for the 1994 national championship college basketball game. Assessing the strengths of their respective "benches," each coach comes up with four
4. Consider the following two-person, zero-sum game: A B B By S 50 50 A2 1 .1 A 10 110 (a) Verify that the strategies (1.0.) for A and (0) for B are optimal, and deter- mine the value of the game.
3. Solve the following games graphically. The payoff is for Player A. B B 8 00 (a) B B By (b) A 1 -3 7 A A 2 4 -6 Az 6 A> S 7
(b) For 1000 ~ Q ~ 4000, solve the problem for the following set of data: Machine i K; ($) C($) 1 100 5 234 2 40 12 150 3 90 8
2. For the upcoming planting season, Farmer McCoy can plant corn (a1)' plant wheat (a2), plant soybeans (a3)' or use the land for grazing (a4)' The payoffs associated with the different actions are
*1. 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_
The symbols at and a2 represent the "favorable" and "unfavorable" assessment by the broker, and 5J, 52, and 53 represent, respectively, the up, same, and down change in future prices.(a) Draw the
5. Consider Farmer McCoy's decision situation in Problem 2, Set 13.2a. The farmer has the additional option of using the land as a grazing range, in which case he is guaranteed a payoff of $7500. The
13. Daily demands for loaves of bread at a grocery store are specified by the following probability distribution:The store buys a loaf for 55 cents and sells it for $1.20 each. Any unsold loaves at
11. Rework Problem 9, 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
9. TriStar plans to open a new plant in Arkansas. The company can open a full-sized plant now or a small-sized plant that can be expanded 2 years later if high. demand conditions prevail. The time
6. 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
5. A professor of political science wants to predict the outcome of a school board election.Three candidates, Ivy (I), Bahrn (8), and Smith (5) are running for one position. The professor places the
*4. 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, Hand P, have expressed interest in
3. 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: yard work
*2. TIle personnel department at C&H has narrowed the search for a prospective employee to three candidates: Steve (S), Jane (J), and Maisa (M). The final selection is based on three criteria:
5 Use dynamic programming to solve a knapsack problem in which the knapsack can hold up to 13 lb (see Table 12). TABLE 12 Item Weight (lb) Benefit 1 23 2 3 357 3 12 25 7 50
17 Seas Beginning sells clothing by mail order. An important question is: When should the company strike a customer from its mailing list? At present, the company does so if a customer fails to order
16 A telephone sales force can model its contact with customers as a Markov chain. The six states of the chain are as follows:State 1 Sale completed during most recent call State 2 Sale lost during
13 Define bij probability of ending up in absorbing state aj given that we begin in transient state ti rij ijth entry of R qik ikth entry of Q B (s m) m matrix whose ijth entry is bij Suppose
10 In caring for elderly patients at a mental hospital, a major goal of the hospital is successful placement of the patients in boarding homes or nursing homes. The movement of patients between the
5 Each week, the number of acceptable-quality units of a drug that are processed by a machine is observed:100, 50–100, 1–50, 0 (indicating that the machine was broken during the week). Given last
1† The State College admissions office has modeled the path of a student through State College as a Markov chain:Each student’s state is observed at the beginning of each fall semester. For
12 Consider the following nonergodic chain: 010 P=001 100 a Why is this chain nonergodic? b Explain why Theorem 1 fails for this chain. (Hint: Show that lim P11(n) does not exist by listing the
11 Consider the following nonergodic chain: 004-11 004-26 P 12120 12120 0 0 00 0 0 a Why is the chain nonergodic? 3 b Explain why Theorem 1 fails for this chain. Hint: Find out if the following
9 Two types of squirrels—gray and black—have been seen in Pine Valley. At the beginning of each year, we determine which of the following is true:There are only gray squirrels in Pine
3 For each of the following Markov chains, determine the long-run fraction of the time that each state will be occupied a 1312 2312 [.8 2 0] b 0.2.8 .8 20 c Find all mean first passage times for part
4 For each of the following chains, determine whether the Markov chain is ergodic. Also, for each chain, determine the recurrent, transient, and absorbing states 0.8 .8.2 P 3 .7 = 201 .4 .5.1 .1 P=
3 Consider the following transition matrix:a Which states are transient?b Which states are recurrent?P2 is not ergodic because there are two closed classes of states (class 1 = {1, 2} and class 2 =
4 A business believes that its needs for cash during the next month are described by the random variable shown in Table 23. At the beginning of the month, the business has$10,000 available, and the
1 The Chocochip Cookie Store bakes its cookies every morning before opening. It costs the store 15¢ to bake each cookie, and each cookie is sold for 35¢. At the end of the day, leftover cookies may
1 Consider a two-item inventory system with the attributes in Table 21.a Draw an exchange curve for these products (use AOC and AII as the x- and y-coordinates).b Currently, management is ordering
1 Develop an ABC graph for the data in Table 16. Which items should be classified A, B, and C? TABLE 16 Unit Cost Item Annual Usage (in dollars) -234567890 1 20,000 20 23,000 10 20,000 3 30,000 2
3 Suppose that the EOQ is 100, average annual demand is 1,000 units, and the lead time demand is a random variable having the distribution shown in Table 15.a What value of SLM1 corresponds to a
6 In this problem, use marginal analysis to determine the optimal reorder point for the lost sales case.a Show that the average inventory level for the lost sales case may be written as [ E(X)+
4 The lead time demand for bathing suits is governed by the discrete random variable shown in Table 11. The company sells an average of 10,400 suits per year. The cost of placing an order for bathing
1 The City of Rulertown consists of the unit interval [0, 1] (see Figure 2). Rulertown needs to determine where to build the city’s only fire station. It knows that for small x, the probability
7 Suppose demand is a continuous random variable having a probability density function f (d), and c(d, q) is given by Equation (2). Show that if q units are ordered, the expected cost E(q) may be
8 Vivian and Wayne are planning on going to Lamaze natural childbirth classes. Lamaze classes meet once a week for five weeks. Each class gives 20% of the knowledge needed for “natural”
7 Every four years, Blockbuster Publishers revises its textbooks. It has been three years since the best-selling book, The Joy of OR, has been revised. At present, 2,000 copies of the book are in
5 The power at Ice Station Lion is supplied via solar cells.Once a year, a plane flies in and sells solar cells to the ice station at a price of $20 per cell. Because of uncertainty about future
2 Each day, a news vendor must determine how many New York Herald Wonderfuls to order. She pays 15¢ for each paper and sells each for 30¢. Any leftover papers are a total loss. From past
1 In August 2003, a car dealer is trying to determine how many 2004 models should be ordered. Each car costs the dealer $10,000. The demand for the dealer’s 2004 models has the probability
Use the upper-bounded simplex algorithm to solve the following LPs max z=4x+3x2 x + 6x2 6 x25 X1, X20 s.t. 2x1 x21
Use the upper-bounded simplex algorithm to solve the following LPs min z -4x1 - 9x2 5x + 6x2 10 2x-3x2 4 x1 2 321 X1, X20 s.t. 3x1 + 5x2 6
Use the upper-bounded simplex algorithm to solve the following LPs max z 4x + 3x2+5x31 s.t. 2x1 + 2x2 + x3 + x + x 9 S.L. 4x1 x2-x3 + x 2x+ X3 + x 6 5 2 x1 3 32 4 X3 5 X4 x 7 X1, X2, X3, X4, X5 0
Solve the following LP: Z max z=4x+2x2 + 3x3 s.t. 2x1 + x2 x3 10 9 = x + x + x 2x + 2x2 + 4x3 20 4 x x2 3 X3 1
Use the Dantzig–Wolfe decomposition algorithm to solve the following problem max z= = 3x1 + 6x2 + 5x3 s.t. x1 + 2x2 + x3 4 4 2x + 3x2 + 2x3 6 x1 + x2 2 2x1 + x2 3 x1, x2, x30
Use the Dantzig–Wolfe decomposition algorithm to solve the following problem max z= x + 2x2 + x3 8 8 4x1 + 2x2 + 3x3 + 4x4 + 2x5 x1 + 2x2 + 2x3 s.t. x4 + x 3 X1, X2, X3, X4,0
Use the Dantzig–Wolfe decomposition algorithm to solve the following problem max z = 7x + 5x2 + 3x3 x1 x2+x3 5 3 2x2 + x3 8 x1, x2, x3 0 X3 s.t. x+2x2+x3 10
Steelco manufactures two types of steel (steel 1 and steel 2) at two locations (plants 1 and 2). Three resources are needed to manufacture a ton of steel: iron, coal, and blast furnace time. The two
Remember that B-1 is always found under the columns corresponding to the starting basis.) s.t. x1 min z=3x1 + x2 - 3x3 x2+x34 x1 +x3 6 2x2 - x3 = 5 x1, x2, x30
Use the revised simplex method to solve the following LPs max z = 4x1 + x2 s.t. x1 + x2 4 2x1 + x2 6 3x2 6 xxxz
Use the revised simplex method to solve the following LPs max z=3x1 + x2 + x3 s.t. x1 + x2 + x 6 2x1 -x34 x2 + x3 2 x1, x2, x30
16 Cornco produces two products: PS and QT. The sales price for each product and the maximum quantity of each that can be sold during each of the next three months are given in Table 16.Each product
15 Old Macdonald’s 200-acre farm sells wheat, alfalfa, and beef. Wheat sells for $30 per bushel, alfalfa sells for$200 per bushel, and beef sells for $300 per ton. Up to 1,000 bushels of wheat and
13 A company produces tools at two plants and sells them to three customers. The cost of producing 1,000 tools at aplant and shipping them to a customer is given in Table 14.Customers 1 and 3 pay
8 Wivco produces two products: 1 and 2. The relevant data are shown in Table 12. Each week, up to 400 units of raw material can be purchased at a cost of $1.50 per unit.The company employs four
6 Gepbab Production Company uses labor and raw material to produce three products. The resource requirements and sales price for the three products are as shown in Table 10.Currently, 60 units of raw
5 Beerco manufactures ale and beer from corn, hops, and malt. Currently, 40 lb of corn, 30 lb of hops, and 40 lb of malt are available. A barrel of ale sells for $40 and requires 1 lb of corn, 1 lb

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