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operations research an introduction

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

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 experience, she believes that the number of papers she can sell each day is governed by the probability
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 distribution shown in Table 4. Each car is sold for $15,000. If the demand for 2004 cars exceeds the number
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 plants have different types of furnaces, so the resources needed to manufacture a ton of steel
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 must be processed through two assembly lines: 1 and 2. The number of hours required by 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 up to 1,000 bushels of alfalfa can be sold, but demand for beef is unlimited. If an acre of land is
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 $200 per thousand tools; customer 2 pays $150 per thousand tools. To produce 1,000 tools at plant 1,
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 workers, who work 40 hours per week. (Their salaries are considered a fixed cost.)Workers are paid $6 per
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 material are available. Up to 90 hours of labor can be purchased at $1 per hour. To maximize Gepbab
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 of hops, and 2 lb of malt. A barrel of beer sells for $50 and requires 2 lb of corn, 1 lb of hops,
4 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; type 2 sells for $500. All
3 Wivco produces product 1 and product 2 by processing raw material. Up to 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 1
8 Consider the LP:a Solve this LP with LINDO and use your output to show that the optimal solution is degenerate.b Use your LINDO output to find an example of Oddities 1–3. max 9x + 8x2 + 5x3 + 4x4 s.t. x1 + x4 200 x2+x3 150 350 2x1 + x2 + x3 + x4 = 550 X1, X2, X3, X40
6 Steelco uses coal, iron, and labor to produce three types of steel. The inputs (and sales price) for one ton of each type of steel are shown in Table 8. Up to 200 tons of coal can be purchased at a price of $10 per ton. Up to 60 tons of iron can be purchased at $8 per ton, and up to 100 labor
5 Mondo produces motorcycles at three plants. At each plant, the labor, raw material, and production costs(excluding labor cost) required to build a motorcycle are as shown in Table 7. Each plant has sufficient machine capacity to produce up to 750 motorcycles per week. Each of Mondo’s workers
4 Gepbab Corporation produces three products at two different plants. The cost of producing a unit at each plant is shown in Table 6. Each plant can produce a total of 10,000 units. At least 6,000 units of product 1, at least 8,000 units of product 2, and at least 5,000 units of product 3 must be
3 Consider the diet problem discussed in Section 3.4. Use the LINDO output in Figure 8 to answer the following questions.a If a Brownie costs 30¢, then what would be the new optimal solution to the problem?b If a bottle of cola cost 35¢, then what would be the new optimal solution to the
2 Carco manufactures cars and trucks. Each car contributes $300 to profit, and each truck contributes $400.The resources required to manufacture a car and a truck are shown in Table 5. Each day, Carco can rent up to 98 Type 1 machines at a cost of $50 per machine. The company has 73 Type 2 machines
1 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. Each acre planted with wheat yields 5 bushels, and each acre planted with corn yields 4 bushels.Wheat sells for $30 per bushel, and corn sells for $50 per bushel. To harvest an
Tucker Inc. must produce 1,000 Tucker automobiles. The company has four production plants. The cost of producing a Tucker at each plant, along with the raw material and labor needed, is shown in Table 3.The autoworkers’ labor union requires that at least 400 cars be produced at plant 3;3,300
5 Radioco manufactures two types of radios. The only scarce resource that is needed to produce radios is labor. At present, the company has two laborers. Laborer 1 is willing to work up to 40 hours per week and is paid $5 per hour.Laborer 2 will work up to 50 hours per week for $6 per hour. The
29 You are the mayor of Gotham City, and you must determine a tax policy for the city. Five types of taxes are used to raise money:a Property taxes. Let p property tax percentage rate.b A sales tax on all items except food, drugs, and durable goods. Let s sales tax percentage rate.c A sales tax
23 During the 1972 football season, the games shown in Table 76 were played by the Miami Dolphins, the Buffalo Bills, and the New York Jets. Suppose that on the basis of these games, we want to rate these three teams. Let M=Miami rating, J = Jets rating, and B = Bills rating. Given values of M, J,
18 Suppose we have obtained the tableau in Table 75 for a maximization problem. State conditions on a1, a2, a3, b, c1, and c2 that are required to make the following statements true:a The current solution is optimal, and there are alternative optimal solutions.b The current basic solution is not a
16 A camper is considering taking two types of items on a camping trip. Item 1 weighs a1 lb, and item 2 weighs a2 lb. Each type 1 item earns the camper a benefit of c1 units, and each type 2 item earns the camper c2 units. The knapsack can hold items weighing at most b lb.a Assuming that the camper
14 A hospital outpatient clinic performs four types of operations. The profit per operation, as well as the minutes of X-ray time and laboratory time used are given in Table 72. The clinic has 500 private rooms and 500 intensive care rooms. Type 1 and Type 2 operations require a patient to stay in
13 Jobs at Indiana University are rated on three factors:Factor 1 Complexity of duties Factor 2 Education required Factor 3 Mental and or visual demands For each job at IU, the requirement for each factor has been rated on a scale of 1–4, with a 4 in factor 1 representing high complexity of duty,
11 Consider the following LP:a Find all the basic feasible solutions for this LP.b Show that when the simplex is used to solve this LP, every basic feasible solution must be examined before the optimal solution is found.By generalizing this example, Klee and Minty (1972)constructed (for n 2, 3, .
9 Use the Big M method and the two-phase method to find the optimal solution to the following LP: min z = -3x1 + x2 s.t. x12x22 -x1 + x2 3 x1, x20
8 Use the simplex method to find the optimal solution to the following LP: max z = 5x1 + x2 s.t. 2x1 + x2 6 x1x20 XX
7 Use the simplex algorithm to find two optimal solutions to the following LP. How many optimal solutions does this LP have? Find a third optimal solution. max z = 4x1 + x2 s.t. 2x + 3x2 4 x1 + x2 1 4x1 + x2 2 X1, X20
6 Use the Big M method and the two-phase method to find the optimal solution to the following LP: max z=x1 + x2 s.t. 2x1 + x2 3 3x1 + x2 = 3.5 x + x 1 X1, X20
5 Use the simplex algorithm to find the optimal solution to the following LP: min z=-x1-2x2 s.t. 2x1 + x2 5 x + x 3 x1, x20
4 Use the simplex algorithm to find the optimal solution to the following LP: max z 5x-x x2 s.t. x13x21 x1 - 4x2 3 x1, x2 0
3 Use the Big M method and the two-phase method to find the optimal solution to the following LP: max z=5x-x2 s.t. 2x1 + x2 = 6 x1 + x2 4 x + 2x2 5 x1 x1, x20
2 Use the simplex algorithm to find the optimal solution to the following LP: min z = -4x1 + x2 s.t. s.t. 3x1 + x2 6 -x+2x=0 x1,x20
1 Use the simplex algorithm to find two optimal solutions to the following LP: max z=5x+3x2 + x3 s.t. x1+2+3x3 6 5x + 3x2+6x3 15 x3, x1, x20
14 HAL computer must determine which of seven research and development (R&D) projects to undertake. For each project four quantities are of interest:a the net present value (NPV in millions of dollars) of the project b the annual growth rate in sales generated by the project c the probability
12 A small aerospace company is considering eight projects:Project 1 Develop an automated test facility.Project 2 Barcode all company inventory and machinery.Project 3 Introduce a CAD/CAM system.Project 4 Buy a new lathe and deburring system.Project 5 Institute FMS (flexible manufacturing
11 Gotham City is trying to determine the type and location of recreational facilities to be built during the next decade. Four types of facilities are under consideration: golf courses, swimming pools, gymnasiams, and tennis courts.Six sites are under consideration. If a golf course is built, it
10 Ricky’s Record Store now employs five full-time employees and three part-time employees. The normal workload is 40 hours per week for full-time and 20 hours per week for part-time employees. Each full-time employee is paid $6 per hour for work up to 40 hours per week and can sell 5 records per
9 During the next four quarters, Wivco faces the following demands for globots: quarter 1—13 globots; quarter 2—14 globots; quarter 3—12 globots; quarter 4—15 globots.Globots may be produced by regular-time labor or byovertime labor. Production capacity (number of globots)and production
6 The Touche Young accounting firm must complete three jobs during the next month. Job 1 will require 500 hours of work, job 2 will require 300 hours of work, and job 3 will require 100 hours of work. Currently, the firm consists of 5 partners, 5 senior employees, and 5 junior employees, each of
4 A company produces two products. Relevant information for each product is shown in Table 58. The company has a goal of $48 in profits and incurs a $1 penalty for each dollar it falls short of this goal. A total of 32 hours of labor are available. A $2 penalty is incurred for each hour of
4 Show how you could use linear programming to solve the following problem: s.t. max z = 12x1 - 3x2| 4x1 + x2 4 2x1 - x2 = 0.5 x1, x20
2 Use the simplex algorithm to solve the following LP: max z = 2x1 + x2 s.t. 3x1 + x2 6 x1 + x2 4 x 0, x urs
Use the two-phase simplex method to solve the following LP: min z = 40x1 + 10x2+7x5 + 14x6 s.t. x- x2 -2x1 + x2 x1 + X3 +2x5 = 0 -2x5 = 0 +x5x6 = 3 2x2 + x3 + x4 + 2x3 + x6 = 4 All x, 0
To illustrate Case 1, we now modify Bevco’s problem so that 36 mg of vitamin C are required.From Section 4.12, we know that this problem is infeasible. This means that the optimal Phase I solution should have w 0 (Case 1). To show that this is true, we begin with the original problem: min z = 2x1
First we use the two-phase simplex to solve the Bevco problem of Section 4.12. Recall that the Bevco problem was min z = 2x1 + 3x2 s.t. 2x1 +24 x1 + 3x2 20 x1 + x2 = 10 x1,x20
Use the Big M method to solve the following LP 6 min z = x + x s.t. x1 + x2 = 2 2x1 + 2x2 = 4 x1, x2 0
Use the Big M method to solve the following LP 5 min z=x+x s.t. x2 2x1 + x2 + x3 = 420 x1 + x2 + 2x3 = 2 X1, X2, X3 IV
Use the Big M method to solve the following LP 4 min z = 3x1 s.t. 2x1 + x2 6 3x + 2x = 4 x1, x20
Use the Big M method to solve the following LP 3 max z = 3x1 + x2 s.t. x1 + x2 3 2x + x2 4 x1 + x2 = 3 x1, x20
Use the Big M method to solve the following LP 2 min z s.t. 2x + 3x2 2x1 + x2 = 4 x1 x2-1 x1,x20
Use the Big M method to solve the following LP 1 min z=4x+4x2 + x3 s.t. 2x1 + x2 3 2x1 + x2+3x3 3 X1, X2, X30
4 Show that if ties are broken in favor of lower-numbered rows, then cycling occurs when the simplex method is used to solve the following LP: max z = -3x1 + x2 - 6x3 9x1 + x29x3 - 2x4 0 x + -2x3-0 -9x1 x2 +9x3 + 2x4 1 x; 0 (i=1, 2, 3, 4)
3 Show that if ties in the ratio test are broken by favoring row 1 over row 2, then cycling occurs when the following LP is solved by the simplex max z = 2x + 3x2 x3 - 12x4 s.t -2x19x2 + x3 + 9x4 0 + x2--2xg 0 x; 0 (i=1, 2, 3, 4)
2 Find the optimal solution to the following LP: min z=-x - x2 s.t. x1 + x2 1 x1,x20
1 Even if an LP’s initial tableau is nondegenerate, later tableaus may exhibit degeneracy. Degenerate tableaus often occur in the tableau following a tie in the ratio test. To illustrate this, solve the following LP:Also graph the feasible region and show which extreme points correspond to more
6 Show that the following LP is unbounded: min z=-x1-3x2 s.t. x1 - 2x24 -x1 + x2 3 X1, X20
5 Show that the following LP is unbounded max z = x + 2x2 s.t. -x1 + x22 -2x1 + x2 1 X1, X2
2 State a rule that can be used to determine if a min problem has an unbounded optimal solution(that is, z can be made arbitrarily small). Use the rule to show thatis an unbounded LP. min z = -2x1 - 3x2 s.t. x1 x21 x - 2x2 2 x1 x1, x2 0
1 Show that the following LP is unboundedFind a point in the feasible region with z >= 10,000. max z 2x2 s.t. x1-x24 -x1 + x2 1 X120
Recall from Section 3.3 that for some LPs, there exist points in the feasible region for which z assumes arbitrarily large (in max problems) or arbitrarily small (in min problems)values. When this situation occurs, we say that LP is unbounded. In this section, we showhow the simplex algorithm can
9 Characterize all optimal solutions to the following LP: max z=-8x5 s.t. x1 + X3 + 3x4 + 2x5 = 2 x2 + 2x3 + 4x4 + 5.xs = All x, 0
8 Consider an LP with the optimal tableau shown in Table 18.a Does this LP have more than one bfs that is optimal?b How many optimal solutions does this LP have? (Hint: If the value of x3 is increased, then how does this change the values of the basic variables and the z-value?) TABLE 18 X X2 X X
5 How many optimal basic feasible solutions does the following LP have? max z = 2x + 2x2 s.t. x1 + x2 6 2x1 + x2 13 All x, 0 xi
4 Find all optimal solutions to the following LP: max z = 3x + 3x2 s.t. x1 + x2 1 All x; 0
3 Find alternative optimal solutions to the following LP: max zx1 + x2 s.t. x1 + 2x3 1 All x; 0
2 Show that the following LP has alternative optimal solutions; find three of them. max z= s.t. -3x+6x2 5x1 + 7x2 35 -x1 + 2x2 2
4 Use the simplex algorithm to find the optimal solution to the following LP: min z = -3x1 + 8x2 s.t. 4x + 2x 12 2x + 3x2 6
3 Use the simplex algorithm to find the optimal solution to the following LP: min z = 2x1 - 5x2 s.t. 3x1 + 8x2 12 2x + 3x 6 x1, x20
2 Use the simplex algorithm to find the optimal solution to the following LP: min z=-x1 - x2 -x1-x2 s.t. x-x21 x + x2 2
1 Use the simplex algorithm to find the optimal solution to the following LP: min z = 4x1 - x2 s.t. 2x1 + x28 x25 x1-x24 X1X20
6 Use the simplex algorithm to solve the following LP: max z = x1 + x2 + x3 p.t. x1 + 2x2 + 2x3 20
5 Use the simplex algorithm to solve the following LP: max zx1 + x2 s.t. = 4x1 + x2 100 x+x280 x140 X1, X20
3 Use the simplex algorithm to solve the following problem: max z=2x-x+x3 s.t. Z 3x1 + x2 x3 60 2x1 + x2 + 2x3 20 2x1 + 2x2 + x3 20 x1, x2, x3 0
2 Use the simplex algorithm to solve the following LP: max z = 2x + 3x2 s.t. x + 2x2 6 2x1 + x2 8 8 x1, x20
We call this format the row 0 version of the objective function (row 0 for short).The Dakota Furniture Company manufactures desks, tables, and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry.The amount of each resource needed
5 For the Dorian problem, represent the point (10,40) in the form i=k cd+bi
4 For the Leather Limited problem, represent the point(10, 20) in the form =k cd + ,bi.
3 Widgetco produces two products: 1 and 2. Each requires the amounts of raw material and labor, and sells for the price given in Table 3.Up to 350 units of raw material can be purchased at $2 per unit, while up to 400 hours of labor can be purchased at$1.50 per hour. To maximize profit, Widgetco
3 Convert the following LP to standard form: min z = 3x1 + x2 s.t. x1 3 x1 + x2 4 2x1 - x2 = 3 X1, X20
Leather Limited manufactures two types of belts: the deluxe model and the regular model.Each type requires 1 sq yd of leather. A regular belt requires 1 hour of skilled labor, and a deluxe belt requires 2 hours. Each week, 40 sq yd of leather and 60 hours of skilled labor are available. Each
7 Given a point yk in Karmarkar’s method, express the LP’s original objective function as a function of yk. Use the answer to this question to give a reason why [Diag(xk)]cT is projected, rather than cT.
6 Show that the point xk in Karmarkar’s method is feasible for the original LP.
5 Prove Lemma 1.
4 Suppose an LP contained lower-bound constraints of the following form: xj Lj. Suggest an algorithm that could be used to solve such a problem efficiently.
5 Give an example to show why Theorem 1 does not hold for an LP with an unbounded feasible region.
4 Give an economic interpretation to explain why 3 priced out favorably in the plant 2 tableau 2 subproblem.
Woodco sells 3-ft, 5-ft, and 9-ft pieces of lumber. Woodco’s customers demand 25 3-ft boards, 20 5-ft boards, and 15 9-ft boards. Woodco, who must meet its demands by cutting up 17-ft boards, wants to minimize the waste incurred. Formulate an LP to help Woodco accomplish its goal, and solve the

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