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operations research an introduction

Questions and Answers of Operations Research An Introduction

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,
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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,
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 -
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
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
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
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
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
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
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
For the problems of Section 10.1, use the product form of the inverse to perform the revised simplex method.
Use the product form of the inverse to compute B1-1 and B2-1 for the Dakota problem that was solved by the revised simplex in Section 10.1
4 The university library has one copying machine for the students to use. Students arrive at the machine with the distribution of interarrival times shown in Table 18. The time to make a copy is
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 50 20 30 5 27 10 3 15 2 14 19
5 A salesperson in a large bicycle shop is paid a bonus ifhe sells more than 4 bicycles a day. The probability of selling more than 4 bicycles a day is only .40. If the number of bicycles sold is
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 3 6 1 3, 2 2 8 4 8 2
A local accounting firm in Smalltown orders boxes of floppy disks (10 disks to a box)from a store in Megalopolis. The per-box price charged by the store depends on the number of boxes purchased (see
4 The number of crimes in each of a city’s three police precincts depends on the number of patrol cars assigned to each precinct (see Table 11). Five patrol cars are available.Use dynamic
6 A heart specialist schedules 16 patients each day, 1 every 30 minutes, starting at 9 A.M. Patients are expected to arrive for their appointments at the scheduled times.However, past experience
7 Suppose we are considering the selection of the reorder point, R, of a (Q, R) inventory policy. With this policy, we order up to Q when the inventory level falls to R or less.The probability
2 Use either of the approaches outlined in this section to solve the following knapsack problem: max z = 5x + 4x2 + 2x3 s.t. 4x + 3x2 + 2x3 8 x1, x2, x30; x1, x2, x3 integer
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the
3 An airport hotel has 100 rooms. On any given night, it takes up to 105 reservations, because of the possibility of no-shows. Past records indicate that the number of daily reservations is uniformly
1 J. R. Carrington has $4 million to invest in three oil well sites. The amount of revenue earned from site i(i 1, 2, 3)depends on the amount of money invested in site i (see Table 10). Assuming
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the
2 A news vendor sells newspapers and tries to maximize profits. The number of papers sold each day is a random variable. However, analysis of the past month’s data shows the distribution of daily
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the
Finco has $6,000 to invest, and three investments are available. If dj dollars (in thousands)are invested in investment j, then a net present value (in thousands) of rj(dj) is obtained, where the
5 A machine operator processes two types of jobs, A and B, during the course of the day. Analyses of past data show that 40% of all jobs are type A jobs, and 60% are type B jobs. Type A jobs have

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