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business
operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
Use Excel Solver to find the optimal solution to the following problems:Example 14 of Section 3.12
Use Excel Solver to find the optimal solution to the following problems:4 Problem 3 of Section 3.10
Use Excel Solver to find the optimal solution to the following problems:3 Example 11 of Chapter 3
Use Excel Solver to find the optimal solution to the following problems:2 Example 7 of Chapter 3
Use Excel Solver to find the optimal solution to the following problems: 1 Problem 2 of Section 3.4
13 The new president has just been elected and has set the following economic goals (listed from highest to lowest priority):Goal 1 Balance the budget (this means revenues are at least as large as costs).Goal 2 Cut spending by at most $150 billion.Goal 3 Raise at most $550 billion in taxes from the
8 Faber College is admitting students for the class of 2008. It has set four goals for this class, listed in order of priority:Goal 1 Entering class should be at least 5,000 students.Goal 2 Entering class should have an average SAT score of at least 640.Goal 3 Entering class should consist of at
7 There are four teachers in the Faber College Business School. Each semester, 200 students take each of the following courses: marketing, finance, production, and statistics. The “effectiveness” of each teacher in teaching each class is given in Table 61. Each teacher can teach a total of 200
5 Deancorp produces sausage by blending together beef head, pork chuck, mutton, and water. The cost per pound, fat per pound, and protein per pound for these ingredients is given in Table 59. Deancorp needs to produce 100 lb of sausage and has set the following goals, listed in order of
3 Highland Appliance must determine how many color TVs and VCRs should be stocked. It costs Highland $300 to purchase a color TV and $200 to purchase a VCR. A color TV requires 3 sq yd of storage space, and a VCR requires 1 sq yd of storage space. The sale of a color TV earns Highland a profit of
2 Fruit Computer Company is ready to make its annual purchase of computer chips. Fruit can purchase chips (in lots of 100) from three suppliers. Each chip is rated as being of excellent, good, or mediocre quality. During the coming year, Fruit will need 5,000 excellent chips, 3,000 good chips, and
1 Graphically determine the preemptive goal progamming solution to the Priceler example for the following priorities:a LIP is highest-priority goal, followed by HIW and then HIM.b HIM is highest-priority goal, followed by LIP and then HIW.c HIM is highest-priority goal, followed by HIW and then
The Leon Burnit Advertising Agency is trying to determine a TV advertising schedule for Priceler Auto Company. Priceler has three goals:Goal 1 Its ads should be seen by at least 40 million high-income men (HIM).Goal 2 Its ads should be seen by at least 60 million low-income people (LIP).Goal 3 Its
7 Clothco manufactures pants. During each of the next six months they can sell up to the numbers of pants given in Table 51.Demand that is not met during a month is lost. Thus, for example, Clothco can sell up to 500 pants during month 1.A pair of pants sells for $40, requires 2 hours of labor, and
6 Show that after any number of pivots the coefficient of xi in each row of the simplex tableau will equal the negative of the coefficient of xi in the same row.
5 Steelco’s main plant currently has a steel manufacturing area and shipping area located as shown in Figure 13(distances are in feet). The company must determine where to locate a casting facility and an assembly and storage facility to minimize the daily cost of moving material through the
3 During the next three months, Steelco faces the following demands for steel: 100 tons (month 1); 200 tons(month 2); 50 tons (month 3). During any month, a worker can produce up to 15 tons of steel. Each worker is paid$5,000 per month. Workers can be hired or fired at a cost of$3,000 per worker
1 Suppose that Mondo no longer must meet demands on time. For each quarter that demand for a motorcycle is unmet, a penalty or shortage cost of $110 per motorcycle short is assessed. Thus, demand can now be backlogged. All demands must be met, however, by the end of quarter 4.Modify the formulation
Mondo Motorcycles is determining its production schedule for the next four quarters. Demand for motorcycles will be as follows: quarter 1—40; quarter 2—70; quarter 3—50;quarter 4—20. Mondo incurs four types of costs.1 It costs Mondo $400 to manufacture each motorcycle.2 At the end of each
A baker has 30 oz of flour and 5 packages of yeast. Baking a loaf of bread requires 5 oz of flour and 1 package of yeast. Each loaf of bread can be sold for 30¢. The baker may purchase additional flour at 4¢/oz or sell leftover flour at the same price. Formulate and solve an LP to help the baker
2 Explain why the Phase I LP will usually have alternative optimal solutions.
1 Use the two-phase simplex method to solve the Section 4.12 problems.
Bevco manufactures an orange-flavored soft drink called Oranj by combining orange soda and orange juice. Each ounce of orange soda contains 0.5 oz of sugar and 1 mg of vitamin C. Each ounce of orange juice contains 0.25 oz of sugar and 3 mg of vitamin C. It costs Bevco 2¢ to produce an ounce of
6 Consider an LP (maximization problem) in which each basic feasible solution is nondegenerate. Suppose that xi is the only variable in our current tableau having a negative coefficient in row 0. Show that any optimal solution to the LP must have xi > 0.
5 Show that if Bland’s Rule to prevent cycling is applied to Problem 4, then cycling does not occur.
4 Use the simplex method to solve Problem 10 of Section 3.3.
3 Suppose that in solving an LP, we obtain the tableau in Table 22. Although x1 can enter the basis, this LP is unbounded.Why?
7 Explain why the set of optimal solutions to an LP is a convex set.
6 Suppose you have found this optimal tableau (Table 17) for a maximization problem. Use the fact that each nonbasic variable has a strictly positive coefficient in row 0 to show that x1 = 4, x2 = 3, s1 = s2 = 0 is the unique optimal solution to this LP. (Hint: Can any extreme point having s1
1 Show that if a toy soldier sold for $28, then the Giapetto problem would have alternative optimal solutions.
7 It has been suggested that at each iteration of the simplex algorithm, the entering variable should be (in a maximization problem) the variable that would bring about the greatest increase in the objective function. Although this usually results in fewer pivots than the rule of entering the most
4 Suppose you want to solve the Dorian problem (Example 2 in Chapter 3) by the simplex algorithm. What difficulty would occur?
1 Use the simplex algorithm to solve the Giapetto problem(Example 1 in Chapter 3).
7 Recall that Example 5 of Chapter 3 is an unbounded LP. Find a direction of unboundedness along which we can move for which the objective function becomes arbitrarily large.
6 For an LP in standard form with constraints Ax = b and x >= 0, show that d is a direction of unboundedness if and only if Ad = 0 and d >= 0.
2 For the Dorian problem (Example 2 in Chapter 3), show how the basic feasible solutions to the LP in standard form correspond to the extreme points of the feasible region.
1 For the Giapetto problem (Example 1 in Chapter 3), show how the basic feasible solutions to the LP in standard form correspond to the extreme points of the feasible region.
Why Does an LP Have an Optimal bfs?
2 Convert the Dorian problem (Example 2 in Chapter 3)to standard form.
1 Convert the Giapetto problem (Example 1 in Chapter 3)to standard form.
How to Convert an LP to Standard Form
8. TORA Experiment. Consider the Diet Model and let the objective function be given as Minimize z = .8x1 + .8X2 Use TORA to show that the optimum solution is associated with two distinct corner points and that both points yield the same objective value. In this case, the problem is said to have
*7. An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon, and 3% carbon.The costs per ton for scraps A and Bare $100 and $80, respectively. TIle specifications
6. Day Trader wants to invest a sum of money that would generate an annual yield of at least $10,000. Two stock groups are available: blue chips and high tech, with average annual yields of 10% and 25%, respectively. TIlOugh high-tech stocks provide higher yield, they are more risky, and Trader
*5. OilCo is building a refinery to produce four products: diesel, gasoline, lubricants, and jet fuel. The minimum demand (in bblJday) for each of these products is 14,000,30,000, 10,000, and 8,000, respectively. Iran and Dubai are under contract to ship crude to OilCo.Because of the production
4. 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 5 and 12 hours a week, and in store 2 he is allowed between 6 and 10 hours. Both stores pay the same hourly wage. In
3. For the diet model, what type of optimum solution would the model yield if the feed mix should not exceed 800 Ib a day? Does the solution make sense?
2. For the diet model, suppose that the daily availability of corn is limited to 450 lb. Identify the new solution space, and determine the new optimum solution.
1. Identify the direction of decrease in z in each of the following cases:*(a) Minimize z = 4xI - 2x2'(b) Minimize z = -3xl + X2'(c) Minimize z = -Xl - 2X2'
22. TORA Experiment. In the Reddy Mikks model, suppose that the following constraint is added to the problem.X2 ~ 3 Use TORA to show that the resulting model has conflicting constraints that cannot be satisfied simultaneously and hence it has no feasible solution.
21. 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 this case about the optimal solution of the model?
20. TORA Experiment. Consider the following LP model:Maximize z = 5xI + 4X2 subject to 6xI + 4X2 ::; 24 6x1 + 3X2 ::; 22.5 XI + X2::; 5 XI + 2X2::; 6- XI + X2::; 1 xz::; 2 XJ, X2 ~ 0 In LP, a constraint is said to be redundant if its removal from the model leaves the feasible solution space
*18. An assembly line consisting of three consecutive stations produces two radio models: HiFi1 and HiFi-2. The following table provides the assembly times for the three workstations.Minutes per unit Workstation 12 3HiFi-1 65 4HiFi-2 45 62.The daily maintenance for stations 1,2, and 3 consumes
17. 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 for finishing to the painting department. The daily capacity of the sawing department is 200 chairs or 80
16. 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 includes cutting, sewing, and packaging. Burroughs employs 25 workers in the cutting department, 35 in the
15. Top Toys is planning a new radio and TV advertising campaign. A radio commercial costs$300 and a TV ad cosls $2000.A total budget of $20,000 is allocated to the campaign.However, to ensure that each medium will have at least one radio commercial and one TV ad, the most that can be allocated to
*14. Wyoming Electric Coop owns a steam-turbine power-generating plant. Because Wyoming is rich in coal deposits, the plant generates its steam from coal. 111is, however, may result in emission that does not meet the Environmental Protection Agency standards.EPA regulations limit sulfur dioxide
13. 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 ofTY commercials $300. Show & Sell likes to advertise on radio at least twice as much as on TV. In the
12. Wild West produces two types of cowboy hats. A type 1 hat requires twice as much labor time as a type 2. If the all available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for the two types are 150 and 200 hats
11. Jack is an aspiring freshman at Diem University. He realizes that "all work and no play make Jack a dull boy." As a result, Jack wants to apportion his available time of about 10 hours a day between work and play. He estimates that play is twice as much fun as work. He also wants to study at
10. 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 occupies.2 ft2 and a box ofWheatie needs .4 ft2. The maximum daily demands of Grano and Wheatie
9. 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 consumes .5 unit of raw materiall and .6 unit of raw material II, and one unit of solution Buses
8. 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, such as woodworking, word processing, and car maintenance; and humanistic, such as history, music, and fine arts. To satisfy the
*7. 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 and at most 50% in B. Moreover, investment in A should be at least half the investment in B. How
6. 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 580 bars. The profit per ton is $40 per sheet and $35 per bar. Determine the optimal daily production mix.
*5. 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 100 units of A per day. Both products use one raw material, of which the maximum daily availability is 240 lb. The usage rates of the
4. A company that operates 10 hours a day manufactures two products on three sequential processes. TIle following table summarizes the data of the problem:Minutes per unit Product 12 Process 1 10 5Process 2 620 Process 3 810 Unit profit$2$3 Determine the optimal mix of the two products.
3. 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 exterior paint is at most 2.5 tons.(b) The daily demand for interior paint is at least 2 tons.(c) The daily demand for interior paint is
2. Identify the direction of increase in z in each of the following cases:*(a) Maximize z = Xl - X2'(b) Maximize z = - 5xI - 6X2'(c) Maximize z = -Xl + 2X2'*(d) Maximize z = -3XI + X2'
1. Determine the feasible space for each of the following independent constraints, given that Xl, X2 :::: O.*(a) - 3XI + X2 5; 6.(b) Xl - 2X2 :::: 5.(c) 2Xl - 3X2 5; 12.*(d) XI - X2 5; O.(e) -Xl + X2 :::: O.
4. Suppose that Reddy Mikks sells its exterior paint to a single wholesaler at a quantity discount.1l1e profit per ton is $5000 if the contractor buys no more than 2 tons daily and $4500 otherwise. Express the objective function mathematically. Is the resulting function linear?
*3. For the feasible solution XI = 2, x2 = 2 of the Reddy Mikks model, determine the unused amounts of raw materials Ml and M2.
2. Determine the best feasible solution among the following (feasible and infeasible) solutions of the Reddy Mikks model:(a) XI = 1, X2 = 4.(b) Xl = 2, X2 = 2.(c) XI = 3, x2 = 1.5.(d) X I = 2, X2 = 1.(e) XI = 2, X2 = -l.
1. 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 that of exterior paint by at least 1 ton.(b) The daily usage of raw material M2 in tons is at most 6
What Is Operations Research?
6. During the construction of a house, six joists of 24 feet each must be trimmed to the correct length of 23 feet. The operations for cutting a joist involve the following sequence:1.~Operation 1. Place joist on saw horses 2. Measure correct length (23 feet)3. Mark cutting line for circular saw 4.
*5. 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 .500 batting average, else if Jim throws a curve ball and Joe prepares for a fast ball, his batting average
*(C)1 What is the smallest time for moving all four people to the other side of the river?
(b) Define the criterion for evaluating the alternatives.
(a) Identify at least two feasible plans for crossing the river (remember, the canoe is the only mode of transportation and it cannot be shuttled empty).
5, and 10 minutes, respectively. If two people are in the canoe, the slower person dictates the crossing time. The objective is for all four people to be on the other side of the river in the shortest time possible.
4. 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 row across the river in 1 minute. Jim, John, and Kelly would take 2,
3. Determine the optimal solution of the rectangle problem. (Hint: Use the constraint to express the objective function in terms of one variable, then use differential
2. In the rectangle problem, identify two feasible solutions and determine which one is better.
L In the tickets example, identify a fourth feasible alternative.
1 Use the linear congruential generator to obtain a sequence of 10 random numbers, given that a 17, c 43, m 100, and x0 31.
7 We all hate to bring small change to the store. Using random numbers, we can eliminate the need for change and give the store and the customer a fair shake.a Suppose you buy something that costs $.20. How could you use random numbers (built into the cash register system) to decide whether you
6 Show that on any iteration of the acceptance–rejection method, there is a probability 1/M(b-a) that a value of the random variable is generated.
4 For Problem 2, develop a computer program for the process generator. Generate 100 random variates and compare the mean and variance of this sample against the theoretical mean and variance of this distribution. Now repeat the experiment for the following numbers of random variates: 250; 500;
2 Perform the simulation for Pierre’s Bakery for 25 more days (days 16 through 40) for policy C in Table 11. Compare the answer with the results in the table. Use the random numbers in Table 5 to solve the problem
1 Simulate the single-server queuing system described in Section 21.2 for the first 25 departures from the system to develop an estimate for the expected time in the waiting line. Is this a reasonable estimate? Explain. Use the random numbers in Table 5 to solve the problem
Pierre’s Bakery bakes and sells french bread. Each morning, the bakery satisfies the demand for the day using freshly baked bread. Pierre’s can bake the bread only in batches of a dozen loaves each. Each loaf costs 25¢ to make. For simplicity, we assume that the total daily demand for bread
1 For the years 1961–1970, the annual return on General Motors stock and the return on the Standard and Poor’s market index were as given in Table 20 (file Beta.xls).a Let Y return on General Motors stock during a year and X return on Standard and Poor’s index during a year. Financial
4 To determine how price influences sales, a company changed the price of a product over a 20-week period. The price charged each week and the number of units sold are given in Table 23 (file Price.xls). Develop a model to relate sales to price.
5 Confederate Express Service is attempting to determine how its shipping costs for a month depend on the number of units shipped during a month. For the last 15 months, the number of units shipped and total shipping cost are given in Table 24 (file Ship.xls).a Determine a relationship between
6 In Example 1, we ran a regression with only x1 (miles driven) as an independent variable. We found thecoefficient of x1 in this regression to be 51.68. This appears to indicate (contrary to what we would expect) that increasing the miles driven will lead to decreased maintenance costs. Explain
2 We are to predict sales for a motel chain based on the information in Table 26 (file Motel.xls).a Use this data and multiple regression to make predictions for the motel chain’s sales during the next four quarters. Assume that advertising during each of the next four quarters is $50,000.b Use
7 Suppose we are trying to fit a curve to data, and part (i)of Figure 17 is relevant. Explain why the points of the form (x 1i , ln yi) should, when plotted, indicate a straight-line relationship.
8 Consider the regression in which we estimated cost of running an insurance company as a function of the number of home and car insurance policies. If there were a 1%increase in the number of car insurance policies, by what percentage would we predict that total costs would increase?
9 In the example in which we predicted the number of customers to enter the credit union, suppose that we had used five (instead of four) dummy variables to represent the days of the week. What problem would have arisen?
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