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operations research an introduction

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

33 In treating a brain tumor with radiation, physicians want the maximum amount of radiation possible to bombard the tissue containing the tumors. The constraint is, however, that there is a maximum amount of radiation that normal tissue can handle without suffering tissue damage. Physicians must
32 Alcoa produces 100-, 200-, and 300-foot-long aluminum ingots for customers. This week’s demand for ingots is shown in Table 115.Alcoa has 4 furnaces in which ingots can be produced.During a week, each furnace can be operated for 50 hours.Because ingots are produced by cutting long strips of
31 You are the sales manager for Eli Lilly. You want to have sales headquarters located in four of the cities in Table 113. The number of sales calls (in thousands) that must be made in each city are given in Table 113. For example, San Antonio requires 2,000 calls and is 602 miles from Phoenix.The
30 New York City has 10 trash districts and is trying todetermine which of the districts should be a site for dumpingtrash. It costs $1,000 to haul one ton of trash one mile. The location of each district, the number of tons of trash produced per year by the district, the annual fixed cost (in
29 A consulting company has 10 employees, each of whom can work on at most two team projects. Six projects are under consideration. Each project requires 4 of our 10 workers. The required workers and the revenue earned from each project are shown in Table 109.Each worker who is used on any project
28 The State of Texas frequently does tax audits of companies doing business in Texas. These companies often have headquarters located outside the state, so auditors must be sent to out-of-state locations. Each year, auditors must make 500 trips to cities in the Northeast, 400 trips to cities in
27 Four trucks are available to deliver milk to five groceries. The capacity and daily operating cost of each truck are shown in Table 107. The demand of each grocery store can be supplied by only one truck, but a truck may deliver to more than one grocery. The daily demands of each grocery are as
26 Houseco Developers is considering erecting three office buildings. The time required to complete each and the number of workers required to be on the job at all times are shown in Table 106. Once a building is completed, it brings in the following amount of rent per year: building 1,
25 Reconsider Problem 24. Suppose that at the beginningof year 1, power plants 1–4 have been constructed and are in operation. At the beginning of each year, PSI may shut down a plant that is operating or reopen a shut-down plant The costs associated with reopening or shutting down a plant are
24 PSI believes it will need the amounts of generating capacity shown in Table 103 during the next five years. The company has a choice of building (and then operating)power plants with the specifications shown in Table 104.Formulate an IP to minimize the total costs of meeting the generating
23 Arthur Ross, Inc., must complete many corporate tax returns during the period February 15–April 15. This year the company must begin and complete the five jobs shown in Table 102 during this eight-week period. Arthur Ross employs four full-time accountants who normally work 40 hours per week.
22 A company must complete three jobs. The amounts of processing time (in minutes) required are shown in Table 101. A job cannot be processed on machine j unless for all i< j the job has completed its processing on machine i.Once a job begins its processing on machine j, the job cannot be preempted
5 The Cubs are trying to determine which of the followingfree agent pitchers should be signed: Rick Sutcliffe (RS), Bruce Sutter (BS), Dennis Eckersley (DE), Steve Trout(ST), Tim Stoddard (TS). The cost of signing each pitcher and the number of victories each pitcher will add to the Cubs are shown
Use the branch-and-bound method to solve the following IPs: min z=3x1 + x2 s.t. x1+5x28 x + 2x2 4 x1, x20; x1 integer
8 Although the Hungarian method is an efficient method for solving an assignment problem, the branch-and-bound method can also be used to solve an assignment problem.Suppose a company has five factories and five warehouses.Each factory’s requirements must be met by a single warehouse, and each
7 Use branch-and-bound to determine a way (if any exists to place four queens on a 4 4 chessboard so that no queen can capture another queen. (Hint: Let xij 1 if a queen is placed in row i and column j of the chessboard and xij 0 otherwise. Then branch as in the machine-delay problem.Many nodes
6 LL Pea stores clothes at five different locations. Several times a day it sends an “order picker” out to each location to pick up orders. Then the order picker must return to the packaging area. Describe a TSP that could be used to minimize the time needed to pick up orders and return to the
5 a Use the NNH to find a solution to the TSP in Problem 2. Begin with LFR.b Use the CIH to find a solution to the TSP in Problem 2. Begin with the subtour LFR–LFP–LFR.
4 There are four pins on a printed circuit. The distance between each pair of pins (in inches) is given in Table 76.a Suppose we want to place three wires between the pins in a way that connects all the wires and uses the minimum amount of wire. Solve this problem by using one of the techniques
3 A Hamiltonian path in a network is a closed path that passes exactly once through each node in the network before returning to its starting point. Taking a four-city TSP as an example, explain why solving a TSP is equivalent to finding the shortest Hamiltonian path in a network.
2 Each day, Sunco manufactures four types of gasoline:lead-free premium (LFP), lead-free regular (LFR), leaded premium (LP), and leaded regular (LR). Because of cleaning and resetting of machinery, the time required to produce a batch of gasoline depends on the type of gasoline last produced. For
1 Four jobs must be processed on a single machine. The time required to perform each job and the due date for each job are shown in Table 74. Use the branch-and-bound method to determine the order of performing the jobs that minimizes the total time the jobs are delayed. TABLE 74 Job Time to
Joe State lives in Gary, Indiana. He owns insurance agencies in Gary, Fort Wayne, Evansville, Terre Haute, and South Bend. Each December, he visits each of his insurance agencies.The distance between each agency (in miles) is shown in Table 65. What order of visiting his agencies will minimize the
Four jobs must be processed on a single machine. The time required to process each job and the date the job is due are shown in Table 63. The delay of a job is the number of days after the due date that a job is completed (if a job is completed on time or early, the job’s delay is zero). In what
3 Four projects are available for investment. The projects require the cash flows and yield the net present values (NPV)(in millions) shown in Table 62. If $6 million is available for investment at time 0, find the investment plan that maximizes NPV. TABLE 62 Cash Outflow Project at Time 0 ($) NPV
2 I am moving from New Jersey to Indiana and have rented a truck that can haul up to 1,100 cu ft of furniture. The volume and value of each item I am considering moving on the truck are given in Table 61. Which items should I bring to Indiana? To solve this problem as a knapsack problem, what
1 Show how the following problem can be expressed as a knapsack problem in which all variables must equal 0 or 1.NASA is determining how many of three types of objects should be brought on board the space shuttle. The weight and benefit of each of the items are given in Table 60. If the space
Use the branch-and-bound method to solve the following IPs: max z=4x+3x+x3 s.t. 3x1 + 2x2 + x3 7 2x1 + x2 + 2x3 11 x2, x3 integer, x1, x2 x3 (
9 Consider a long roll of wallpaper that repeats its pattern every yard. Four sheets of wallpaper must be cut from the roll. With reference to the beginning (point 0) of the wallpaper, the beginning and end of each sheet are located as shown in Table 78. Thus, sheet 1 begins 0.3 yd from the
10 A manufacturer of printed circuit boards uses programmable drill machines to drill six holes in each board.The x and y coordinates of each hole are given in Table 79.The time (in seconds) it takes the drill machine to move from one hole to the next is equal to the distance between the points.
4 A court decision has stated that the enrollment of each high school in Metropolis must be at least 20 percent black.The numbers of black and white high school students in each of the city’s five school districts are shown in Table 90.The distance (in miles) that a student in each district must
3 The Transylvania Olympic Gymnastics Team consists of six people. Transylvania must choose three people to enter both the balance beam and floor exercises. They must also enter a total of four people in each event. The score that each individual gymnast can attain in each event is shown in Table
2 Explain how you would use integer programming and piecewise linear functions to solve the following optimization problem. (Hint: Approximate x2 and y2 by piecewise linear functions.) max z = 3x + y s.t. x + y 1 x, y 0
1 In the Sailco problem of Section 3.10, suppose that a fixed cost of $200 is incurred during each quarter that production takes place. Formulate an IP to minimize Sailco’s total cost of meeting the demands for the four quarters.
3 Consider the following IP:The optimal tableau for this IP’s linear programming relaxation is given in Table 88. Use the cutting plane algorithm to find the optimal solution max z s.t. 2x14x2 2x1 + x2 5 -4x1 + 4x2 5 x1, x20; x1, x2 integer
2 Consider the following IP:The optimal tableau for this IP’s linear programming relaxation is given in Table 87. Use the cutting plane algorithm to find the optimal solution. min z=6x+8x2 s.t. 3x1 + x2 4 x1 + 2x2 4 x1, x20; x1, x2 integer
1 Consider the following IP:The optimal tableau for this IP’s linear programming relaxation is given in Table 86. Use the cutting plane algorithm to solve this IP. max z=14x + 18x2 s.t. -x1 + 3x2 6 7x1 + x2 35 x1, x20; x1, x2 integer
6 Why are the values of u0, u1, . . . , un in (44) unique?
5 Use implicit enumeration to solve Problem 1 of Section 9.2.
11 Four jobs must be processed on a single machine. The time required to perform each job, the due date, and the penalty(in dollars) per day the job is late are given in Table 80.Use branch-and-bound to determine the order of performing the jobs that will minimize the total penalty costs due to
Use implicit enumeration to solve the following 0–1 IP: max z = -7x - 3x22x3x42x5 s.t. -4x12x2 + x3 2x4 - 2x4x5-3 -4x12x24x3 + x4 + 2x5 -7 == x=0 or 1 (i = 1, 2, 3, 4, 5)
Use implicit enumeration to solve the following max z= 3x1 + x2 + 2x3 - x4 + x5. s.t. 2x1 + x2 x12x2 - 3x4 3x3x4 + 2x5 = 2 x = 0 or 1
Use implicit enumeration to solve the following max z = 2x1 - x2 + x3 s.t. x+2x2x31 x1 + x2 + x3 2 x; = 0 or 1
3 Finco is considering investing in five projects. Each requires a cash outflow at time 0 and yields an NPV as described in Table 82 (all dollars in millions). At time 0, $10 million is available for investment. Projects 1 and 2 are mutually exclusive (that is, Finco cannot undertake
4 Use implicit enumeration to find the optimal solution to Example 5 (the set-covering problem).
29 Three fires have just broken out in New York. Fires 1 and 2 each require two fire engines, and fire 3 requires three fire engines. The “cost” of responding to each fire depends on the time at which the fire engines arrive. Let tij be the time (in minutes) when the jth engine arrives at fire
26 Powerhouse produces capacitors at three locations: Los Angeles, Chicago, and New York. Capacitors are shipped from these locations to public utilities in five regions of the country: northeast (NE), northwest (NW), midwest (MW), southeast (SE), and southwest (SW). The cost of producing and
25 My home has four valuable paintings that are up for sale. Four customers are bidding for the paintings. Customer 1 is willing to buy two paintings, but each other customer is willing to purchase at most one painting. The prices that each customer is willing to pay are given in Table 84. Usethe
22 Braneast Airlines must staff the daily flights between New York and Chicago shown in Table 81. Each of Braneast’s crews lives in either New York or Chicago. Each day a crew must fly one New York–Chicago and one Chicago–NewYork flight with at least 1 hour of downtime between
19 A company is considering hiring people for four types of jobs. It would like to hire the number of people in Table 79 for each type of job.Four types of people can be hired by the company. Each type is qualified to perform two types of jobs according toTable 80. A total of 20 Type 1, 30 Type 2,
17 A company produces cars in Atlanta, Boston, Chicago, and Los Angeles. The cars are then shipped to warehouses in Memphis, Milwaukee, New York City, Denver, and San Francisco. The number of cars available at each plant is given in Table 75.Each warehouse needs to have available the number of cars
13 Find the optimal solution to Problem 12. TABLE 71 12 14 16 60 60 14 13 19 17 15 18 40 70 10 50 50 40 40
9 Solve the following LP: min z = 2x + 3x2 + 4x3 + 3x4 s.t. x1 + x2 +x2 4 x3 + x4 5 x1 +x3 3 x2 + x = 6 6 min x 0 (j = 1, 2, 3, 4) x;
7 There are three school districts in the town of Busville.The number of black and white students in each district are shown in Table 70. The Supreme Court requires the schools in Busville to be racially balanced. Thus, each school must have exactly 300 students, and each school must have the same
4 Appletree Cleaning has five maids. To complete cleaning my house, they must vacuum, clean the kitchen, clean the bathroom, and do general straightening up. The time it takes each maid to do each job is shown in Table 67. Each maidis assigned one job. Use the Hungarian method to determine
2 Five workers are available to perform four jobs. The time it takes each worker to perform each job is given in Table 65. The goal is to assign workers to jobs so as to minimize the total time required to perform the four jobs.Use the Hungarian method to solve the problem TABLE 64 To ($) From
6 A company must meet the following demands for cash at the beginning of each of the next six months: month 1,$200; month 2, $100; month 3, $50; month 4, $80; month 5, $160; month 6, $140. At the beginning of month 1, the company has $150 in cash and $200 worth of bond 1, $100 worth of bond 2, and
3 In Problem 2, assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at either Galveston or Mobile. To refine 1,000 barrels of oil costs $12 at Mobile and $10 at Galveston. Assuming that both Mobile and Galveston have infinite refinery
Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce as many as 150 widgets per day, and the Denver factory can produce as many as 200 widgets per day. Widgets are shipped by air to customers in Los Angeles and Boston. The customers in
8 The Chicago board of education is taking bids on the city’s four school bus routes. Four companies have made the bids in Table 57.a Suppose each bidder can be assigned only one route.Use the assignment method to minimize Chicago’s cost of running the four bus routesb Suppose that each company
6 Five male characters (Billie, John, Fish, Glen, and Larry)and five female characters (Ally, Georgia, Jane, Rene, and Nell) from Ally McBeal are marooned on a desert island.The problem is to determine what percentage of time each woman on the island should spend with each man. For example, Ally
4 A company is taking bids on four construction jobs. Three people have placed bids on the jobs. Their bids (in thousands of dollars) are given in Table 53 (a * indicates that the persondid not bid on the given job). Person 1 can do only one job, but persons 2 and 3 can each do as many as two
2 Doc Councillman is putting together a relay team for the 400-meter relay. Each swimmer must swim 100 meters of breaststroke, backstroke, butterfly, or freestyle. Doc believes that each swimmer will attain the times given inTable 51. To minimize the team’s time for the race, which swimmer should
5 Two plants supply three customers with medical supplies. The unit costs of shipping from the plants to the customers, along with the supplies and demands, are given in Table 42.a The company’s goal is to minimize the cost of meeting customers’ demands. Find two optimal bfs for this
11 Paperco recycles newsprint, uncoated paper, and coated paper into recycled newsprint, recycled uncoated paper, and recycled coated paper. Recycled newsprint can be produced by processing newsprint or uncoated paper.Recycled coated paper can be produced by recycling any type of paper. Recycled
7 The U.S. government is auctioning off oil leases at two sites: 1 and 2. At each site, 100,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules state that no bidder can receive more than 40% of the land being auctioned.
5 A hospital needs to purchase 3 gallons of a perishable medicine for use during the current month and 4 gallons for use during the next month. Because the medicine isperishable, it can only be used during the month of purchase.Two companies (Daisy and Laroach) sell the medicine. The medicine is in
3 A shoe company forecasts the following demands during the next six months: month 1—200; month 2—260; month 3—240; month 4—340; month 5—190; month 6—150. It costs $7 to produce a pair of shoes with regular-time labor(RT) and $11 with overtime labor (OT). During each month, regular
37 Use LINDO to solve the Sailco problem of Section 3.10. Then use the output to answer the following questions:a If month 1 demand decreased to 35 sailboats, what would be the total cost of satisfying the demands during the next four months?b If the cost of producing a sailboat with regular-time
36 Consider an LP with three constraints. The righthand sides are 10, 15, and 20, respectively. In the optimal tableau, s2 is a basic variable in the second constraint, which has a right-hand side of 12. Determine the range of values of b2 for which the current basis remains optimal. (Hint: If
35 Consider the following LP and its optimal tableau(Table 68): = max z c + Cx s.t. C2X2 911x1 +12x2 b a21x1 + a22x2 b Determine c1, c2, b1, b2, a11, a12, a21, and a22.
34 Consider the following LP and its partial optimal tableau (Table 67):a Complete the optimal tableau.b Find the dual to this LP and its optimal solution. max z 20x + 10x2 s.t. x1 + x2 = 150 x1 40 X220 x1,x20
33 Consider the following LP: max z=c1x1 + Cx2 s.t. 3x1 + 4x2 6
32 Ballco manufactures large softballs, regular softballs, and hardballs. Each type of ball requires time in three departments: cutting, sewing, and packaging, as shown in Table 65 (in minutes). Because of marketing considerations, at least 1,000 regular softballs must be produced. Eachregular
31 In this problem, we discuss how shadow prices can be interpreted for blending problems (see Section 3.8). To illustrate the ideas, we discuss Problem 2 of Section 3.8. If we define x6J = pounds of grade 6 oranges in juice x9J = pounds of grade 9 oranges in juice x6B = pounds of grade 6 oranges
30 The following questions pertain to the Finco investment example of Section 3.11. The LINDO output for this problem is shown in Figure 17.a If Finco has $2,000 more on hand at time 0, by how much would their time 3 cash increase?b Observe that if Finco were given a dollar at time 1, the cash
29 The following questions pertain to the Star Oil capital budgeting example of Section 3.6. The LINDO output for this problem is shown in Figure 16.a Find and interpret the shadow price for each constraint.b If the NPV of investment 1 were $5 million, would the optimal solution to the problem
28 Consider the following two LPs:Suppose that BV {x1, x2} is an optimal basis for both LPs, and the optimal solution to LP 1 is x1 50, x2 500, z 550. Also suppose that for LP 1, the shadow price of both Constraint 1 and Constraint 2 10 3 0. Find the optimal solution to LP 2 and the optimal
27 The following question concerns the Rylon example discussed in Section 3.9. After defining RB = ounces of Regular Brute produced annually LB= ounces of Luxury Brute produced annually RC = ounces of Regular Chanelle produced annually LC =ounces of Luxury Chanelle produced annually RM =
26 Wivco produces two products: 1 and 2. The relevant data are shown in Table 64. Each week, as many as 400 units of raw material can be purchased at a cost of $1.50 per unit. The company employs four workers, who work 40 hours per week(their salaries are considered a fixed cost). Workers can be
25 Consider the following LP:It is given thata Show that the basic solution with basic variables x1, x2, and x3 is optimal. Find the optimal solution.b Write down the dual to this LP and find its optimal solution.c Show that if we multiply the right-hand side of each constraint by a non-negative
24 Consider the following LP:a Find the dual to this LP and show that it has the same feasible region as the original LP.b Use weak duality to show that the optimal objective function value for the LP (and its dual) must be 0. max z = -3x1 + x2 + 2x3 s.t. x2 + 2x3 3 -x1 + 3x3 = -1 -2x-3x2 -2 x1,
23 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 of hops, and 2 lb of malt. A barrel of beer sells for $50 and requires 2 lb of corn, 1 lb of hops,
22 Radioco manufactures two types of radios. The only scarce resource that is needed to produce radios is labor.The company now has two laborers. Laborer 1 is willing to work up to 40 hours per week and is paid $5 per hour.Laborer 2 is willing to work up to 50 hours per week and is paid $6 per
21 z = 8, x1 = 2, x2 = 0 is the optimal solution to the following LP:Use the graphical approach to answer the following questions:a Determine the range of values of c1 for which the current basis remains optimal.b Determine the range of values of c2 for which the current basis remains optimal.c
20 Use the Theorem of Complementary Slackness to find the optimal solution to the following LP and its dual: max z = 3x1 + 4x2 + x3 + 5x4 s.t. x1 + 2x2 + x3 + 2x4 5 2x + 3x2 x3 + 3x = 8 X1, X2, X3, X4 0
19 Consider the following LP:This LP is unbounded. Use this fact to show that the following LP has no feasible solution: max z = -2x + 6x2 s.t. x1 + x22 -x1 + x2 1 X1, X22
18 Consider the following LP:After subtracting an excess variable e1 from the first constraint, adding a slack variable s2 to the second constraint, and adding artificial variables a1 and a3 to the first and third constraints, the optimal tableau for this LP is as shown in Table 60.a Find the dual
17 Use the dual simplex method to solve the following LP: max z = -2x-x2 s.t. x1 + x2 5 x12x8 x1,x20
16 Zales Jewelers uses rubies and sapphires to produce two types of rings. A Type 1 ring requires 2 rubies, 3 sapphires, and 1 hour of jeweler’s labor. A Type 2 ring requires 3 rubies, 2 sapphires, and 2 hours of jeweler’s labor.Each Type 1 ring sells for $400, and each Type 2 ring sells for
14 Consider the following LP and its optimal tableau(Table 59):a Find the dual to this LP and its optimal solution.b Find the range of values of the right-hand side of the third constraint for which the current basis remains optimal.Also find the new optimal solution if the righthand side of the
13 Consider the following LP and its optimal tableau(Table 58):a Find the dual to this LP and its optimal solution.b Find the range of values of the objective function coefficient of x3 for which the current basis remains optimal.c Find the range of values of the objective function coefficient of
11 Consider the following LP:The optimal solution to this LP is z = 9, x1 = 1, x2 = 6.Graphically find the range of values of b2 for which the current basis remains optimal.12 Farmer Leary grows wheat and corn on his 45-acre farm. He can sell at most 140 bushels of wheat and 120 bushels of corn.
10 Consider the following LP and its optimal tableau(Table 57):a Find the dual of this LP and its optimal solution.b Find the range of values of b2 for which the current basis remains optimal. Also find the new optimal solution if b2 5. TABLE 57 Z X X2 S $2 rhs 1 0 0 0 0 - 1 0 01373 0 10 1 2313 4
9 Consider the following LP and its optimal tableau(Table 56):a Find the dual of this LP and its optimal solution.b Find the range of values of the objective function coefficient of x1 for which the current basis remains optimal.c Find the range of values of the objective function coefficient for
8 Wivco produces product 1 and product 2 by processing raw material. As much as 90 lb of raw material may be purchased at a cost of $10/lb. One pound of raw material can be used to produce either 1 lb of product 1 or 0.33 lb of product 2. Using a pound of raw material to produce a pound of product
7 Consider the following LP: max z = 3x + 4x2 s.t. 2x1 + x2 8 4x1 + x2 10 x1, x20 The optimal solution to this LP is z = 32, x1 = 0, x2 = 8, s = 0, s = 2. Graphically find the range of values of c for which the current basis remains optimal.
6 Consider the following LP and its optimal tableau(Table 55):a Find the dual of this LP and its optimal solution.b Find the range of values of b2 for which the current basis remains optimal. If b2 12, what is the new optimal solution? s.t. x2-x3 max z = 3x1 + x 2x1 + x2 + x3 8 4x1 + x2 x3 10 |
5 The following LP has the optimal tableau shown in Table 54.a Find the dual of this LP and its optimal solution.b Find the range of values of the objective function coefficient of x2 for which the current basis remains optimal.c Find the range of values of the objective function coefficient of x1
4 Carco manufactures cars and trucks. Each car contributes $300 to profit and each truck, $400. Theresources required to manufacture a car and a truck are shown in Table 53. Each day, Carco can rent up to 98 Type 1 machines at a cost of $50 per machine. The company now has 73 Type 2 machines and

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