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operations research an introduction

Questions and Answers of Operations Research An Introduction

8 The Hatfields, Montagues, McCoys, and Capulets are going on their annual family picnic. Four cars are available to transport the families to the picnic. The cars can carry the following number of
7 Four workers are available to perform jobs 1–4.Unfortunately, three workers can do only certain jobs:worker 1, only job 1; worker 2, only jobs 1 and 2; worker 3, only job 2; worker 4, any job.
6 Seven types of packages are to be delivered by five trucks. There are three packages of each type, and the capacities of the five trucks are 6, 4, 5, 4, and 3 packages, respectively. Set up a
Five male and five female entertainers are at a dance. The goal of the matchmaker is to match each woman with a man in a way that maximizes the number of people who are matched with compatible mates.
Fly-by-Night Airlines must determine how many connecting flights daily can be arranged between Juneau, Alaska, and Dallas, Texas. Connecting flights must stop in Seattle and then stop in Los Angeles
Sunco Oil wants to ship the maximum possible amount of oil (per hour) via pipeline from node so to node si in Figure 6. On its way from node so to node si, oil must pass through some or all of
9 A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box is given in Table 7. The variable cost (in dollars) of producing each box is equal
7 At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine i years old is given in Table 5.The cost of purchasing a machine at the beginning of each year is
6 It costs $40 to buy a telephone from the department store. Assume that I can keep a telephone for at most five years and that the estimated maintenance cost each year of operation is as follows:
3 Formulate Problem 2 as a transshipment problem.
2 Find the shortest path from node 1 to node 5 in Figure 4.
1 Find the shortest path from node 1 to node 6 in Figure 3.
I have just purchased (at time 0) a new car for $12,000. The cost of maintaining a car during a year depends on its age at the beginning of the year, as given in Table 1. To avoid the high
28 Three professors must be assigned to teach six sections of finance. Each professor must teach two sections of finance, and each has ranked the six time periods during which finance is taught, as
27 During the month of July, Pittsburgh resident B. Fly must make four round-trip flights between Pittsburgh and Chicago. The dates of the trips are as shown in Table 87.B. Fly must purchase four
24 A company can produce as many as 35 units/month.The demands of its primary customers must be met on time each month; if it wishes, the company may also sell units to secondary customers each
23 A firm producing a single product has three plants and four customers. The three plants will produce 3,000, 5,000, and 5,000 units, respectively, during the next time period.The firm has made a
21 The Carter Caterer Company must have the following number of clean napkins available at the beginning of each of the next four days: day 1—15; day 2—12; day 3—18;day 4—6. After being used,
20 During each of the next two months you can produce as many as 50 units/month of a product at a cost of $12/unit during month 1 and $15/unit during month 2. The customer is willing to buy as many
18 During the next three quarters, Airco faces the following demands for air conditioner compressors: quarter 1—200;quarter 2—300; quarter 3—100. As many as 240 air compressors can be produced
16 For the Powerco problem, find the range of values of c23 for which the current basis remains optimal.
15 For the Powerco problem, find the range of values of c24 for which the current basis remains optimal.
14 Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce 500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per day.Oil is sent from the
12 Use the northwest corner method, the minimum-cost method, and Vogel’s method to find basic feasible solutions to the transportation problem in Table 73.
11 In Problem 10, suppose we increase si to 16 and d3 to 11. The problem is still balanced, and because 31 units(instead of 30 units) must be shipped, one would think that the total shipping costs
10 Find the optimal solution to the balanced transportation problem in Table 72 (minimization).
8 Using the northwest corner method to find a bfs, find(via the transportation simplex) an optimal solution to the transportation (minimization) problem shown in Table 71.
6 The Gotham City police have just received three calls for police. Five cars are available. The distance (in city blocks)of each car from each call is given in Table 69. Gotham City wants to
5 Currently, State University can store 200 files on hard disk, 100 files in computer memory, and 300 files on tape.Users want to store 300 word-processing files, 100 packaged-program files, and 100
3 A company must meet the following demands for a product: January, 30 units; February, 30 units; March, 20 units. Demand may be backlogged at a cost of$5/unit/month. All demand must be met by the
1 Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped to three customers. The
5 General Ford has two plants, two warehouses, and three customers. The locations of these are as follows:Plants: Detroit and Atlanta Warehouses: Denver and New York Customers: Los Angeles, Chicago,
4 Rework Problem 3 under the assumption that Galveston has a refinery capacity of 150,000 barrels per day and Mobile has one of 180,000 barrels per day. (Hint: Modify the method used to determine the
2 Sunco Oil produces oil at two wells. Well 1 can produce as many as 150,000 barrels per day, and well 2 can produce as many as 200,000 barrels per day. It is possible to ship oil directly from the
1 General Ford produces cars at L.A. and Detroit and has a warehouse in Atlanta; the company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in
10 Suppose cij is the smallest cost in row i and column j of an assignment problem. Must xij 1 in any optimal assignment?
9 Show that step 3 of the Hungarian method is equivalent to performing the following operations: (1) Add k to each cost that lies in a covered row. (2) Subtract k from each cost that lies in an
7 Any transportation problem can be formulated as an assignment problem. To illustrate the idea, determine an assignment problem that could be used to find the optimal solution to the transportation
5 Greydog Bus Company operates buses between Boston and Washington, D.C. A bus trip between these two cities takes 6 hours. Federal law requires that a driver rest for four or more hours between
3 Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert island with Jennifer Aniston, Courteney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility
time it takes each person to perform each job is given in Table 50. Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs.
Machineco has four machines and four jobs to be completed. Each machine must be assigned to complete one job. The time required to set up each machine for completing each job is shown in Table 43.
4 If s3 and d3 are both decreased by 2, what is the new optimal solution?
3 If s2 and d3 are both increased by 3, what is the new optimal solution?
2 Determine the range of values of c34 for which the current basis remains optimal.
1 Determine the range of values of c14 for which the current basis remains optimal.
Use the transportation simplex to solve Problems 1–8 in Section 7.1. Begin with the bfs found in Section 7.2
4 How should Vogel’s method be modified to solve a maximization problem?
3 Use Vogel’s method to find a bfs for Problems 5 and 6 of Section 7.1.
2 Use the minimum-cost method to find a bfs for Problems 4, 7, and 8 of Section 7.1. (Hint: For a maximization problem, call the minimum-cost method the maximumprofit method or the maximum-revenue
1 Use the northwest corner method to find a bfs for Problems 1, 2, and 3 of Section 7.1.
12 Explain how each of the following would modify the formulation of the Sailco problem as a balanced transportation problem:a Suppose demand could be backlogged at a cost of$30/sailboat/month.
10 Touche Young has three auditors. Each can work as many as 160 hours during the next month, during which time three projects must be completed. Project 1 will take 130 hours; project 2, 140 hours;
9 For the examples and problems of this section, discuss whether it is reasonable to assume that the proportionality assumption holds for the objective function.
8 The Ayatola Oil Company controls two oil fields. Field 1 can produce up to 40 million barrels of oil per day, and field 2 can produce up to 50 million barrels of oil per day.At field 1, it costs $3
6 A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per day, and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary,
4 Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel(regardless of type) and the costs at each plant are shown in Table 8. Each week, 100
2 Referring to Problem 1, suppose that extra units could be purchased and shipped to either warehouse for a total cost of $100 per unit and that all customer demand must be met. Formulate a balanced
1 A company supplies goods to three customers, who each require 30 units. The company has two warehouses.Warehouse 1 has 40 units available, and warehouse 2 has 30 units available. The costs of
Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter is three months). Demand is as follows: first quarter, 40 sailboats; second
Two reservoirs are available to supply the water needs of three cities. Each reservoir can supply up to 50 million gallons of water per day. Each city would like to receive 40 million gallons per
Powerco has three electric power plants that supply the needs of four cities.† Each power plant can supply the following numbers of kilowatt-hours (kwh) of electricity: plant 1—35 million; plant
1 Farmer Jones must determine how many acres of corn and wheat to plant this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week.An acre of corn yields 10
2 Answer these questions about Problem 1.a Is (x1 = 2, x2 = 3) in the feasible region?b Is (x1= 4, x2 = 3) in the feasible region?c Is (x1 = 2, x2 = -1) in the feasible region?d Is (x1 = 3,
3 Using the variables x1 = number of bushels of corn produced and x2 = number of bushels of wheat produced, reformulate Farmer Jones’s LP.
4 Truckco manufactures two types of trucks: 1 and 2.Each truck must go through the painting shop and assembly shop. If the painting shop were completely devoted to painting Type 1 trucks, then 800
5 Why don’t we allow an LP to have< or > constraints?
1 Graphically solve Problem 1 of Section 3.1.
2 Graphically solve Problem 4 of Section 3.1.
3 Leary Chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B,
4 For each of the following, determine the direction in which the objective function increases a z = 4x1 - x2 b z=-x+2x2 czx13x2
5 Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3. A desk contributes $40 to profit, and a chair contributes $25. Marketing restrictions require that the
6 Farmer Jane owns 45 acres of land. She is going to plant each with wheat or corn. Each acre planted with wheat yields $200 profit; each with corn yields $300 profit. The labor and fertilizer used
An auto company manufactures cars and trucks. Each vehicle must be processed in the paint shop and body assembly shop. If the paint shop were only painting trucks, then 40 per day could be painted.
Suppose that auto dealers require that the auto company in Example 3 produce at least 30 trucks and 20 cars. Find the optimal solution to the new LP.
Graphically solve the following LP: max z= = 2x1 - x2 s.t. x1-x21 2x1 + x2 = 6 X1, X2 0
5 True or false: For an LP to be unbounded, the LP’s feasible region must be unbounded.
6 True or false: Every LP with an unbounded feasible region has an unbounded optimal solution.
7 If an LP’s feasible region is not unbounded, we say the LP’s feasible region is bounded. Suppose an LP has a bounded feasible region. Explain why you can find the optimal solution to the LP
Identify which of Case apply to the following LP: max z=x1 + x2 s.t. x1 + x2 4 x1-x25 X1, X20
Identify which of Case apply to the following LP: max z = 4x1 + x2 s.t. 8x1 + 2x2 16 5x + 2x2 12
Identify which of Case apply to the following LP: max z = -x + 3x2 s.t. x1 x24 x + 2x = 4 x1 x1, x20
Identify which of Case apply to the following LP: max z s.t. 3x1 + x2 2x1 + x2 6 x + 3x2 9 x1, x20
8 Graphically find all optimal solutions to the following LP min z = x - x2 s.t. x1 + x2 6 x1 x2 3 x2-x = 3 X1, X20
9 Graphically determine two optimal solutions to the following LP: min z=3x+5x2 s.t. 3x+2x236 3x + 5x2 = 45
10 Money manager Boris Milkem deals with French currency (the franc) and American currency (the dollar). At 12 midnight, he can buy francs by paying .25 dollars per franc and dollars by paying 3
My diet requires that all the food I eat come from one of the four “basic food groups”(chocolate cake, ice cream, soda, and cheesecake). At present, the following four foods are available for
1 There are three factories on the Momiss River (1, 2, and 3). Each emits two types of pollutants (1 and 2) into the river. If the waste from each factory is processed, the pollution in the river can
2 U.S. Labs manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. U.S. Labs purchases pig valves from three different
3 Peg and Al Fundy have a limited food budget, so Peg is trying to feed the family as cheaply as possible. However, she still wants to make sure her family members meet their daily nutritional
4 Goldilocks needs to find at least 12 lb of gold and at least 18 lb of silver to pay the monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day that Goldilocks
6 A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in Table 4. Union rules state that
1 In the post office example, suppose that each full-time employee works 8 hours per day. Thus, Monday’s requirement of 17 workers may be viewed as a requirement of 8(17) 136 hours. The post
2 During each 4-hour period, the Smalltown police force requires the following number of on-duty police officers: 12 midnight to 4 A.M.—8; 4 to 8 A.M.—7; 8 A.M. to 12 noon—6; 12 noon to 4
3 Suppose that the post office can force employees to work one day of overtime each week. For example, an employee whose regular shift is Monday to Friday can also be required to work on Saturday.
4 Suppose the post office had 25 full-time employees and was not allowed to hire or fire any employees. Formulate an LP that could be used to schedule the employees in order to maximize the number of
5 Each day, workers at the Gotham City Police Department work two 6-hour shifts chosen from 12 A.M. to 6 A.M., 6 A.M.to 12 P.M., 12 P.M. to 6 P.M., and 6 P.M. to 12 A.M. The following number of
6 During each 6-hour period of the day, the Bloomington Police Department needs at least the number of policemen shown in Table 5. Policemen can be hired to work either 12 consecutive hours or 18
7 Each hour from 10 A.M. to 7 P.M., Bank One receives checks and must process them. Its goal is to process all the checks the same day they are received. The bank has 13 check-processing machines,
For a discount rate of 15%, consider a project with the cash flows shown in Figure 8.a Compute project NPV if cash flows are at the end of the year.b Compute project NPV if cash flows are at the

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