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operations research an introduction

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

6 In Problem 6 of Section 5.2, what is the most that Steelco should be willing to pay for an extra ton of coal?
5 In Problem 5 of Section 5.2, suppose that Mondo could purchase an additional unit of raw material at a cost of $6.Should the company do it? Explain.
4 In Problem 4 of Section 5.2, how much should Gepbab be willing to pay for another unit of capacity at plant 1?
3 In Problem 3 of Section 5.2, what is the most that one should be willing to pay for an additional ounce of chocolate?
2 In Problem 2 of Section 5.2, what is the most that Carco should be willing to pay to rent an additional Type 1 machine for one day?
1 In Problem 2 of Section 5.2, what is the most that Carco should be willing to pay for an extra ton of steel?
The following questions refer to Example 2.a What is the most that Tucker should pay for an extra hour of labor?b What is the most that Tucker should pay for an extra unit of raw material?c A new customer is willing to purchase 20 cars at a price of $25,000 per vehicle.Should Tucker fill her order?
Consider the Farmer Leary problem (Problem 1 in Section 5.2).a What is the most that Leary should pay for an additional hour of labor?b What is the most that Leary should pay for an additional acre of land?
Let’s reconsider Example 1 with the following changes. Suppose as many as 4,600 units of raw material are available, but they must be purchased at a cost of $4 per unit. Also, as many as 5,000 hours of labor are available, but they must be purchased at a cost of $6 per hour. The per-unit sales
In Example 1, what is the most that Winco should be willing to pay for an additional unit of raw material? How about an extra hour of labor?
7 Shoeco must meet (on time) the following demands for pairs of shoes: month 1—300; month 2—500; month 3—100; and month 4—100. At the beginning of month 1, 50 pairs of shoes are on hand, and Shoeco has three workers.A worker is paid $1,500 per month. Each worker can work up to 160 hours per
a In Example 1, suppose that a total of 980 units must be produced. Determine the new optimal z-value.b In Example 1, suppose that 4,500 units of raw material are available. What is the new optimal z-value? What if only 4,400 units of raw material are available?c In Example 2, suppose that 4,100
a Suppose Winco raises the price of product 2 by 50¢ per unit. What is the new optimal solution to the LP?b Suppose the sales price of product 1 is increased by 60¢ per unit. What is the new optimal solution to the LP?c Suppose the sales price of product 3 is decreased by 60¢. What is the new
Winco sells four types of products. The resources needed to produce one unit of each and the sales prices are given in Table 2. Currently, 4,600 units of raw material and 5,000 labor hours are available. To meet customer demands, exactly 950 total units must be produced.Customers also demand that
4 For the Dorian Auto problem (Example 2 in Chapter 3), a Find the range of values on the cost of a comedy ad for which the current basis remains optimal.b Find the range of values on the cost of a football ad for which the current basis remains optimal.c Find the range of values for required HIW
3 Show that if the weekly demand for soldiers is at least 20, then the current basis remains optimal, and Giapetto should still produce 20 soldiers and 60 trains.
2 Show that if available carpentry hours remain between 60 and 100, the current basis remains optimal. If between 60 and 100 carpentry hours are available, would Giapetto still produce 20 soldiers and 60 trains?
1 Show that if the contribution to profit for trains is between $1.50 and $3, the current basis remains optimal. If the contribution to profit for trains is $2.50, then what would be the new optimal solution?
Use the branch-and-bound method to solve the following IPs: max z=3x1 + x2 s.t. 5x + 2x2 10 4x1 + x2 7 x1, x20; x2 integer
39 Huntco produces tomato sauce at five different plants.The capacity (in tons) of each plant is given in Table 40. The tomato sauce is stored at one of three warehouses. The perton cost (in hundreds of dollars) of producing tomato sauce at each plant and shipping it to each warehouse is given in
38 A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box are given in Table 39. The variable cost (in dollars)of producing each box is equal to the box’s volume. A fixed cost of $1,000 is incurred to produce any of a particular box. If
37 The Indiana University Business School has two rooms that each seat 50 students, one room that seats 100 students, and one room that seats 150 students. Classes are held five hours a day. The four types of requests for rooms are listed in Table 37. The business school must decide how many
36 An Ohio company, Clevcinn, consists of three subsidiaries. Each has the respective average payroll, unemployment reserve fund, and estimated payroll given in Table 36. (All figures are in millions of dollars.) Any employer in the state of Ohio whose reserve/average payroll ratio is less than 1
35 A power plant has three boilers. If a given boiler is operated, it can be used to produce a quantity of steam (in tons) between the minimum and maximum given in Table 34. The cost of producing a ton of steam on each boiler is also given. Steam from the boilers is used to produce power on three
34 The Smalltown Fire Department currently has seven conventional ladder companies and seven alarm boxes. The two closest ladder companies to each alarm box are given in Table 33. The city fathers want to maximize the number of conventional ladder companies that can be replaced with tower ladder
33 Boris Milkem’s financial firm owns six assets. The expected sales price (in millions of dollars) for each asset is given in Table 32. If asset 1 is sold in year 2, the firm receives $20 million. To maintain a regular cash flow, Milkem must sell at least $20 million of assets during year 1, at
32 Simon’s Mall has 10,000 sq ft of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and maximum number of each type of store (along with the square footage of each type) is given in Table 30. The annual profit made by each type of store
31 A Sunco oil delivery truck contains five compartments, holding up to 2,700, 2,800, 1,100, 1,800, and 3,400 gallons of fuel, respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands, penalty per gallon short, and the maximum allowed
30 Cousin Bruzie of radio station WABC schedules radio commercials in 60-second blocks. This hour, the station has sold commercial time for commercials of 15, 16, 20, 25, 30 35, 40, and 50 seconds. Formulate an integer programming model that can be used to determine the minimum number of 60-second
29 You have been assigned to arrange the songs on the cassette version of Madonna’s latest album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette must total between 14 and 16 minutes in length. The length and type of each song are given in Table 28. The assignment
28 Thailand inducts naval draftees at three drafting centers. Then the draftees must each be sent to one of three naval bases for training. The cost of transporting a draftee from a drafting center to a base is given in Table 26. Each year, 1,000 men are inducted at center 1; 600 at center 2;and
27 The Father Domino Company sells copying machines.A major factor in making a sale is Domino’s quick service.Domino sells copiers in six cities: Boston, New York,Philadelphia, Washington, Providence, and Atlantic City.The annual sales of copiers projected depend on whether a service
26 Governor Blue of the state of Berry is attempting to get the state legislature to gerrymander Berry’s congressional districts. The state consists of 10 cities, and the numbers of registered Republicans and Democrats (in thousands) in each city are shown in Table 23. Berry has five
40 To satisfy telecommunication needs for the next 20 years, Telstar Corporation estimates that the number of circuits required between the United States and Germany, France, Switzerland, and the United Kingdom will be as given in Table 45.Two types of circuits may be created: cable and
41 A large drug company must determine how many sales representatives to assign to each of four sales districts. The cost of having n representatives in a district is ($88,000 $80,000n) per year. If a rep is based in a given district, the time it takes to complete a call on a doctor is given in
9 During the next five periods, the demands in Table 58 must be met on time. At the beginning of period 1, theinventory level is 0. Each period that production occurs a setup cost of $250 and a per-unit production cost of $2 are incurred. At the end of each period a per-unit holding cost of $1 is
8 Suppose we have branched on a subproblem (call it subproblem 0, having optimal solution SOL0) and have obtained the following two subproblems:Subproblem 1 Subproblem 0 Constraint x1 i.Subproblem 2 Subproblem 0 Constraint x1 i 1 (i is some integer).Prove that there will exist at least
7 Use the branch-and-bound method to find the optimal solution to the following IP: max z = =7x1 + 3x2 s.t. 2x1 + x2 9 3x1 + 2x2 13 x1, x20; x1, x2 integer
Use branch-and-bound to solve the following IPs: max z = 4x1 + 5x2 s.t. 3x + 2x2 10 x1 + 4x2 11 3x + 3x = 13 x1, x20; x1, x2 integer
Use branch-and-bound to solve the following IPs: max z = 4x + 5x2 s.t. x1 + 4x2 5 3x + 2x2 7 x1, x20; x1, x2 integer
Use branch-and-bound to solve the following IPs: max z = 4x1 + 3x2 s.t. 4x1 + 9x2 26 8x1 + 5x2 17 x1, x20; x1, x2 integer
Use branch-and-bound to solve the following IPs: max z = s.t. 2x + 3x x1 + 2x2 10 3x1 + 4x2 25 x1, x20; x1, x2 integer
Use branch-and-bound to solve the following IPs: The Dorian Auto example of Section 3.2.
Use branch-and-bound to solve the following IPs: max z=5x+2x2 s.t. 3x1 + x2 12 x1 + x 5 +325 x1, x20; x1, x2 integer
The Telfa Corporation manufactures tables and chairs. A table requires 1 hour of labor and 9 square board feet of wood, and a chair requires 1 hour of labor and 5 square board feet of wood. Currently, 6 hours of labor and 45 square board feet of wood are available.Each table contributes $8 to
45 Write a LINGO program that can solve a fixed-charge problem of the type described in Example 3. Assume there is a limited demand for each product. Then use your program to solve a four-product, three-resource fixed-charge problem with the parameters shown in Tables 53, 54, and 55. TABLE 53
44 Venture capital firm JD is trying to determine in which of 10 projects it should invest. It knows how much money is available for investment each of the next N years, the NPV of each project, and the cash required by each project during each of the next N years (see Table 52).a Write a LINGO
43 Ford has four automobile plants. Each is capable of producing the Taurus, Lincoln, or Escort, but it can only produce one of these cars. The fixed cost of operating each plant for a year and the variable cost of producing a car of each type at each plant are given in Table 51.Ford faces the
42 In this assignment, we will use integer programming and the concept of bond duration to show how Wall Street firms can select an optimal bond portfolio. The duration of a bond (or any stream of payments) is defined as follows: Let C(t) be the payment of the bond at time t (t 1, 2, . . . ,
25 Speaker’s Clearinghouse must disburse sweepstakes checks to winners in four different regions of the country:Southeast (SE), Northeast (NE), Far West (FW), and Midwest (MW). The average daily amount of the checks written to winners in each region of the country is as follows:SE, $40,000; NE,
24 Breadco Bakeries is a new bakery chain that sells bread to customers throughout the state of Indiana. Breadco is considering building bakeries in three locations:Evansville, Indianapolis, and South Bend. Each bakery can bake as many as 900,000 loaves of bread each year. The cost of building a
7 In Example 7 (Euing Gas), suppose that x 300. What would be the values of y1, y2, y3, z1, z2, z3, and z4? How about if x 1,200?
6 To graduate from Basketweavers University with a major in operations research, a student must complete at least two math courses, at least two OR courses, and at least two computer courses. Some courses can be used to fulfill more than one requirement: Calculus can fulfill the math requirement;
5 How would the following restrictions modify the formulation of Example 6 (Dorian car sizes)? (Do each part separately.)a If midsize cars are produced, then compacts must also be produced.b Either compacts or large cars must be manufactured.
4 Suppose we add the following restriction to Example 1(Stockco): If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text?
3 A manufacturer can sell product 1 at a profit of $2/unit and product 2 at a profit of $5/unit. Three units of raw material are needed to manufacture 1 unit of product 1, and6 units of raw material are needed to manufacture 1 unit of product 2. A total of 120 units of raw material are available.If
1 Coach Night is trying to choose the starting lineup for the basketball team. The team consists of seven players who have been rated (on a scale of 1 poor to 3 excellent)according to their ball-handling, shooting, rebounding, and defensive abilities. The positions that each player is allowed
Dorian Auto has a $20,000 advertising budget. Dorian can purchase full-page ads in two magazines: Inside Jocks (IJ) and Family Square (FS). An exposure occurs when a person reads a Dorian Auto ad for the first time. The number of exposures generated by each ad in IJ is as follows: ads 1–6, 10,000
Euing Gas produces two types of gasoline (gas 1 and gas 2) from two types of oil (oil 1 and oil 2). Each gallon of gas 1 must contain at least 50 percent oil 1, and each gallon of gas 2 must contain at least 60 percent oil 1. Each gallon of gas 1 can be sold for 12¢, and each gallon of gas 2 can
Dorian Auto is considering manufacturing three types of autos: compact, midsize, and large. The resources required for, and the profits yielded by, each type of car are shown in Table 8. Currently, 6,000 tons of steel and 60,000 hours of labor are available. For production of a type of car to be
There are six cities (cities 1–6) in Kilroy County. The county must determine where to build fire stations. The county wants to build the minimum number of fire stations needed to ensure that at least one fire station is within 15 minutes (driving time) of each city. The times (in minutes)
J. C. Nickles receives credit card payments from four regions of the country (West, Midwest, East, and South). The average daily value of payments mailed by customers from each region is as follows: the West, $70,000; the Midwest, $50,000; the East, $60,000;the South, $40,000. Nickles must decide
Gandhi Cloth Company is capable of manufacturing three types of clothing: shirts, shorts, and pants. The manufacture of each type of clothing requires that Gandhi have the appropriate type of machinery available. The machinery needed to manufacture each type of clothing must be rented at the
Modify the Stockco formulation to account for each of the following requirements:1 Stockco can invest in at most two investments.2 If Stockco invests in investment 2, they must also invest in investment 1.3 If Stockco invests in investment 2, they cannot invest in investment 4.
8 Formulate an IP to solve the Dorian Auto problem for the advertising data that exhibit increasing returns as more ads are placed in a magazine (pages 495–496).
9 How can integer programming be used to ensure that the variable x can assume only the values 1, 2, 3, and 4?
23 At a machine tool plant, five jobs must be completed each day. The time it takes to do each job depends on the machine used to do the job. If a machine is used at all, there is a setup time required. The relevant times are given in Table 20. The company’s goal is to minimize the sum of the
22 Consider the following puzzle. You are to pick out 4 three-letter “words” from the following list:DBA DEG ADI FFD GHI BCD FDF BAI For each word, you earn a score equal to the position that the word’s third letter appears in the alphabet. For example, DBA earns a score of 1, DEG earns a
21 Eastinghouse sells air conditioners. The annual demand for air conditioners in each region of the country is as follows: East, 100,000; South, 150,000; Midwest, 110,000;West, 90,000. Eastinghouse is considering building the air conditioners in four different cities: New York, Atlanta, Chicago,
20 WSP Publishing sells textbooks to college students.WSP has two sales reps available to assign to the A–G state area. The number of college students (in thousands) in each state is given in Figure 9. Each sales rep must be assigned to two adjacent states. For example, a sales rep could be
19 Comquat owns four production plants at which personal computers are produced. Comquat can sell up to 20,000 computers per year at a price of $3,500 per computer. For each plant the production capacity, the production cost per computer, and the fixed cost of operating a plant for a year are given
18 Use LINDO, LINGO, or Excel Solver to find the optimal solution to the following IP:Bookco Publishers is considering publishing five textbooks. The maximum number of copies of each textbook that can be sold, the variable cost of producing each textbook, the sales price of each textbook, and the
17 A product can be produced on four different machines.Each machine has a fixed setup cost, variable production costs per-unit-processed, and a production capacity given in Table 15. A total of 2,000 units of the product must be produced. Formulate an IP whose solution will tell us how to minimize
16 The Lotus Point Condo Project will contain both homes and apartments. The site can accommodate up to 10,000 dwelling units. The project must contain a recreation project:either a swimming–tennis complex or a sailboat marina, but not both. If a marina is built, then the number of homes in the
15 Fruit Computer produces two types of computers: Pear computers and Apricot computers. Relevant data are given in Table 14. A total of 3,000 chips and 1,200 hours of labor are available. Formulate an IP to help Fruit maximize profits.
14 The manager of State University’s DED computer wants to be able to access five different files. These files are scattered on 10 disks as shown in Table 13. The amount of storage required by each disk is as follows: disk 1, 3K; disk 2, 5K; disk 3, 1K; disk 4, 2K; disk 5, 1K; disk 6, 4K; disk 7,
13 Glueco produces three types of glue on two different production lines. Each line can be utilized by up to seven workers at a time. Workers are paid $500 per week on production line 1, and $900 per week on production line 2.A week of production costs $1,000 to set up production line 1 and $2,000
12 A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago, and Atlanta. Each warehouse can ship 100 units per week. The weekly fixed cost of keeping each warehouse open is $400 for New York, $500 for Los Angeles, $300 for Chicago, and $150 for Atlanta.Region 1 of
11 If x and y are both integers, how would you ensure that whenever x 2, then y 3?
10 If x and y are integers, how could you ensure that x y 3, 2x 5y 12, or both are satisfied by x and y?
Stockco is considering four investments. Investment 1 will yield a net present value (NPV)of $16,000; investment 2, an NPV of $22,000; investment 3, an NPV of $12,000; and investment 4, an NPV of $8,000. Each investment requires a certain cash outflow at the present time: investment 1, $5,000;
36 Cook County needs to build two hospitals. There arenine cities where the hospitals can be built. The number of hospital visits made annually by the inhabitants of each city and the x and y coordinates of each city are as shown in Table 123.To minimize the total distance patients must travel to
19 Hallco runs a day shift and a night shift. No matter how many units are produced, the only production cost during a shift is a setup cost. It costs $8,000 to run the day shift and $4,500 to run the night shift. Demand for the next two days is as follows: day 1, 2,000; night 1, 3,000; day 2,
18 Monsanto annually produces 359 million lb of the chemical maleic anhydride. A total of four reactors are available to produce maleic anhydride. Each reactor can be run on one of three settings. The cost (in thousands of dollars) and pounds produced (in millions) annually for each reactor and
17 Newcor’s steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 5% molybdenum by weight. The steel is manufactured by combining three types of metal: steel ingots, scrap steel, and alloys. Four steel ingots are available for purchase. The weight (in tons), cost
16 Eastinghouse ships 12,000 capacitors per month to their customers. The capacitors may be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in Table 96. The fixed cost for a plant is
15 At Blair General Hospital, six types of surgical operations are performed. The types of operations each surgeon is qualified to perform (indicated by an X) are given in Table 95. Suppose that surgeon 1 and surgeon 2 dislike each other and cannot be on duty at the same time. Formulatean IP whose
14 A soda delivery truck starts at location 1 and must deliver soda to locations 2, 3, 4, and 5 before returning to location 1. The distance between these locations is given in Table 94. The soda truck wants to minimize the total distance traveled. In what order should the delivery truck make its
13 Use the implicit enumeration method to find the optimal solution to the following 0–1 IP: max z = s.t. 5x17x2 +10x3 + 3x4x5 -x-3x2 + 3x3 x4 - 2x 0 2x15x2 + 3x3 2x4 - 2x5 3 - x2 x3 + + x4 x5 = 2 All variables 0 or 1
12 Use the implicit enumeration method to find the optimal solution to Problem 5.
11 Use the branch-and-bound approach to find the optimal solution to the traveling salesperson problem shown in Table 93. TABLE 93 City City 1 2 3 4 5 3 31 12345 1442 34 7 4 2 2 7 4 4 7 2227
10 Consider a country where there are 1¢, 5¢, 10¢, 20¢, 25¢, and 50¢ pieces. You work at the Two-Twelve Convenience Store and must give a customer 91¢ in change.Formulate an IP that can be used to minimize the number of coins needed to give the correct change. Use what you know about
9 Use the branch-and-bound method to solve the following IP: max z = x + 2x2. s.t. x1 + x2 10 2x + 5x2 30 x1, x20; x1, x2 integer
8 Use the branch-and-bound method to solve the following IP: min z = 3x1 + x s.t. 2x1 - x2 6 x1 + x2 4 x1, x20; x integer
7 Use the branch-and-bound method to solve the following IP: max z = 3x1 + x2 s.t. 5x1 + x2 12 2x1 + x2 8 x1,x20; x1, x2 integer
6 State University must purchase 1,100 computers from three vendors. Vendor 1 charges $500 per computer plus a delivery charge of $5,000. Vendor 2 charges $350 per computer plus a delivery charge of $4,000. Vendor 3 charges$250 per computer plus a delivery charge of $6,000. Vendor 1 will sell the
20 After listening to a seminar on the virtues of the Japanese theory of production, Hallco has cut its day shift setup cost to $1,000 per shift and its night shift setup cost to $3,500 per shift. Determine a production schedule that minimizes the sum of setup and inventory costs. All demand must
21 Gotham City has been divided into eight districts. The time (in minutes) it takes an ambulance to travel from one district to another is shown in Table 100. The population of each district (in thousands) is as follows: district 1, 40;district 2, 30; district 3, 35; district 4, 20; district 5,
35 When you lease 800-phone numbers from AT&T fortelemarketing, AT&T uses a Solver model to tell you where you should locate calling centers to minimize your operating costs over a 10-year horizon. To illustrate the model, suppose you are considering 7 calling center locations: Boston, New
34 It is currently the beginning of 2003. Gotham City is trying to sell municipal bonds to support improvements in recreational facilities and highways. The face value and due date at which principal comes due of the bonds are in Table 118.Gold and Silver (GS) wants to underwrite Gotham City’s

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