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operations research an introduction

Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions

*4. In Table 5.5 of Example 5.1-2, where a dummy destination is added, suppose that the Detroit plant must ship out all its production. How can this restriction be implemented in the model?
5. In Example 5.1-2, suppose that for the case where the demand exceeds the supply(Table 5.4), a penalty is levied at the rate of $200 and $300 for each undelivered car at Denver and Miami, respectively. Additionally, no deliveries are made from the Los Angeles plant to the Miami distribution
*6. Three electric power plants with capacities of 25, 40, and 30 million kWh supply electricity to three cities. The maximum demands at the three cities are estimated at 30, 35, and 25 million kWh. The price per million kWh at the three cities is given in Table 5.6.During the month of August,
(b) Determine an optimal distribution plan for the utility company.
(c) Determine the cost of the additional power purchased by each of the three cities.
7. Solve Problem 6, assuming that there is a 10% power transmission loss through the network.
8. Three refineries with daily capacities of 6, 5, and 8 million gallons, respectively, supply three distribution areas with daily demands of 4, 8, and 7 million gallons, respectively.Gasoline is transported to the three distribution areas through a network of pipelines.The transportation cost is
(b) Determine the optimum shipping schedule in the network.
5.1 Definition of the Transportation Model 199 TABLE 5.7 Mileage Chart for Problem 8 Distribution area 1 2 3 1Refinery 2 3120 300 200 180 100 250 80 120
*9. In Problem 8, suppose that the capacity of refinery 3 is 6 million gallons only and that distribution area 1 must receive all its demand. Additionally, any shortages at areas 2 and 3 will incur a penalty of 5 cents per gallon.(a) Formula,te the problem as a transportation model.
(b) Determine the optimum shipping schedule.
10. In Problem 8, suppose that the daily demand at area 3 drops to 4 million gallons. Surplus production at refineries 1 and 2 is diverted to other distribution areas by truck. The transportation cost per 100 gallons is $1.50 from refinery 1 and $2.20 from refinery 2. Refinery 3 can divert its
(b) Determine the optimum shipping schedule.
11. Three orchards supply crates of oranges to four retailers. The daily demand amounts at the four retailers are 150,150,400, and 100 crates, respectively. Supplies at the three orchards are dictated by available regular labor and are estimated at 150,200, and 250 crates daily. However, both
(c) How many crates should orchards 1 and 2 supply using overtime labor?
12. Cars are shipped from three distribution centers to five dealers. The shipping cost is based on the mileage between the sources and the destinations, and is independent of whether the truck makes the trip with partial or full loads. Table 5.9 summarizes the mileage between the distribution
(b) Determine the optimal shipping schedule.TABLE 5.8 Transportation Cost/Crate for Problem 11 II 1Retailer 2 3 4 1Orchard 2 3$1$2$1$2$4$3$3$1$5$2$2$3 200 Chapter 5 Transportation Model and Its Variants TABLE 5.9 Mileage Chart and Supply and Demand for Problem 12 5.2 Dealer 1 2 3 4 5 Supply 1 100
13. MG Auto, of Example 5.1-1, produces four car models: MI, M2, M3, and M4. The Detroit.plant produces models Ml, M2, and M4. Models MI and M2 are also produced in New Orleans. The Los Angeles plant manufactures models M3 and M4. The capacities of the various plants and the demands at the
(b) Determine the optimum shipping schedule.(Hint: Add four new destinations corresponding to the new combinations [MI, M2], [M3, M4], [Ml, M2], and [M2, M4]. The demands at the new destinations are determined from the given percentages.)TABLE 5.10 Capacities and Demands for Problem 13 Model M1 M2
*2. In Example 5.2-2, suppose that the sharpening service offers 3-day service for $1 a blade on Monday and Tuesday (days 1 and 2). Reformulate the problem, and interpret the optimum solution.
3. In Example 5.2-2, if a blade is not used the day it is sharpened, a holding cost of 50 cents per blade per day is incurred. Reformulate the model, and interpret the optimum solution.
4. JoShop wants to assign four different categories of machines to five types of tasks. The numbers of machines available in the four categories are 25, 30,20, and 30. The numbers of jobs in the five tasks are 20, 20, 30,10, and 25. Machine category 4 cannot be assigned to task type 4. Table 5.14
*5. The demand for a perishable item over the next four months is 400,300,420, and 380 tons, respectively. The supply capacities for the same months are 500, 600, 200, and 300 tons. The purchase price per ton varies from month to month and is estimated at $100,$140, $120, and $150, respectively.
6. The demand for a special small engine over the next five quarters is 200, 150, 300,250, and 400 units. The manufacturer supplying the engine has different production capacities estimated at 180,230,430,300, and 300 for the five quarters. Back-ordering is not allowed, but the manufacturer may use
7. Periodic preventive maintenance is carried out on aircraft engines, where an important component must be replaced. The numbers of aircraft scheduled for such maintenance over the next six months are estimated at 200, 180,300,198,230, and 290, respectively. All maintenance work is done during the
8. The National Parks Service is receiving four bids for logging at three pine forests in Arkansas. The three locations include 10,000,20,000, and 30,000 acres.A single bidder can bid for at most 50% of the total acreage available. The bids per acre at the three locations are given in Table 5.15.
(b) Determine the acreage that should be assigned to each of the four bidders.
1. Compare the starting solutions obtained by the northwest-comer, least-cost, and Vogel methods for each of the following models:*(a) (b) (c)0 2 1 6 1 2 6 7 5 1 8 12 2 1 5 7 0 4 2 12 2 4 0 14 t 2 4 3 7 3 1 5 11 3 6 7 4 5 S 10 10 10 10 9 10 11(1 d 5.3.2 jd yis Iterative Computations of the
1. Consider the transportation models in Table 5.26.(a) Use the northwest-corner method to find the starting solution.(b) Develop the iterations that lead to the optimum solution.(c) TORA Experiment. Use TORA's Iterations module to compare the effect of using the northwest-corner rule, least-cost
2. In the transportation problem in Table 5.27, the total demand exceeds the total supply.Suppose that the penalty costs per unit of unsatisfied demand are $5, $3, and $2 for destinations 1,2, and 3, respectively. Use the least-cost starting solution and compute the iterations leading to the
*5. In a 3 X 3 transportation problem, let Xij be the amount shipped from source i to destination j and let Cij be the corresponding transportation cost per unit. The amounts of supply at sources 1,2, and 3 are 15,30, and 85 units, respectively, and the demands at destinations 1,2, and 3 are 20,30,
6. The transportation problem in Table 5.29 gives the indicated degenerate basic solution(i.e., at least one of the basic variables is zero). Suppose that the multipliers associated with this solution are UI = 1, Uz = -1, VI = 2, V2 = 2, and V3 = 5 and that the unit cost for all (basic and
(b) Determine the value of () that will guarantee the optimality of the given solution.(Hint: Locate the zero basic variable.)7. Consider the problem m /I Minimize z = 2: 2:CijXij
1. Write the dual problem for the LP of the transportation problem in Example 5.3-5(Table 5.21). Compute the associated optimum dual objective value using the optimal dual values given in Table 5.25, and show that it equals the optimal cost given in the example.
2. In the transportation model, one of the dual variables assumes an arbitrary value. This means that for the same basic solution, the values of the associated dual variables are not unique. The result appears to contradict the theory of linear programming, where the dual values are determined as
2. JoShop needs to assign 4 jobs to 4 workers. The cost of performing a job is a function of the skills of the workers. Table 5.41 summarizes the cost of the assignments. Worker 1 cannot do job 3 and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method.
3. In the JoShop model of Problem 2, suppose that an additional (fifth) worker becomes available for performing the four jobs at the respective costs of $60, $45, $30, and $80. Is it economical to replace one of the current four workers with the new one?
4. In the model of Problem 2, suppose that JoShop has just received a fifth job and that the respective costs of performing it by the four current workers are $20, $10, $20, and $80.Should the new job take priority over any of the four jobs JoShop already has?
*5. A business executive must make the four round trips listed in Table 5.42 between the head office in Dallas and a branch office in Atlanta.The price of a round-trip ticket from Dallas is $400. A discount of 25% is granted if the dates of arrival and departure of a ticket span a weekend (Saturday
*6. Figure 5.6 gives a schematic layout of a machine shop with its existing work centers designated by squares 1,2,3, and 4. Four new work centers, I, II, III, and IV, are to be added to the shop at the locations designated by circlesa, b,c, andd. The objective is to assign the new centers to the
7. In the Industrial Engineering Department at the University of Arkansas, INEG 4904 is a capstone design course intended to allow teams of students to apply the knowledge and skills learned .in the undergraduate curriculum to a practical problem. The members of each team select a project manager,
1. The network in Figure 5.9 gives the shipping routes from nodes 1 and 2 to nodes 5 and 6 by way of nodes 3 and 4. The unit shipping costs are shown on the respective arcs.(a) Develop the corresponding transshipment model.(b) Solve the problem, and show how the shipments are routed from the
2. In Problem 1, suppose that source node 1 can be linked to source node 2 with a unit shipping cost of $1. The unit shipping cost from node 1 to node 3 is increased to $5. Formulate the problem as a transshipment model, and find the optimum shipping schedule.
3. The network in Figure 5.10 shows the routes for shipping cars from three plants (nodes 1, 2, and 3) to three dealers (nodes 6 to 8) by way of two distribution centers (nodes 4 and 5). The shipping costs per car (in $100) are shown on the arcs.(a) Solve the problem as a transshipment model.(b)
*4. Consider the transportation problem in which two factories supply three stores with a commodity. The numbers of supply units available at sources 1 and 2 are 200 and 300;those demanded at stores 1,2, and 3 are 100,200, and 50, respectively. Units may be transshipped among the factories and the
S. Consider the oil pipeline network shown in Figure 5.11. The different nodes represent pumping and receiving stations. Distances in miles between the stations are shown on the network. The transportation cost per gallon between two nodes is directly proportional to the length of the pipeline.
6. Shortest-Route Problem. Find the shortest route between nodes 1 and 7 of the network in Figure 5.12 by formulating the problem as a transshipment model. The distances between the different nodes are shown on the network. (Hint: Assume that node 1 has a net supply of 1 unit, and node 7 has a net
7. In the transshipment model of Example 5.5-1, define Xij as the amount shipped from node i to node j. The problem can be formulated as a linear program in which each node produces a constraint equation. Develop the linear program, and show that the resulting formulation has the characteristic
8. An employment agency must provide the following laborers over the next 5 months:Month No. of laborers 1100 2120 380 4170 550 234 Chapter 5 Transportation Model and Its Variants Because the cost of labor depends on the length of employment, it may be more economical to keep more laborers than
1. Design of an offshore natural-gas pipeline network connecting well heads in the Gulf of Mexico to an inshore delivery point. The objective of the model is to minimize the cost of constructing the pipeline.
2. Determination of the shortest route between two cities in an existing network of roads.
3. Determination of the maximum capacity (in tons per year) of a coal slurry pipeline network joining coal mines in Wyoming with power plants in Houston. (Slurry pipelines transport coal by pumping water through specially designed pipes.)
4. Determination of the time schedule (start and completion dates) for the activities of a construction project.
S. Determination of the minimum-cost flow schedule from oil fields to refineries through a pipeline network.
*1. For each network in Figure 6.5 determine (a) a path, (b) a cycle, (c) a tree, and (d) a spanning tree.
2. Determine the sets N and A for the networks in Figure 6.5.
*4. Consider eight equal squares arranged in three rows, with two squares in the first row, fOUT in the second, and two in the third.The squares of each row are arranged symmetrically about the vertical axis. It is desired to fill the squares with distinct numbers in the range 1 through 8 so that
5. Three inmates escorted by 3 guards must be transported by boat from the mainland to a penitentiary island to serve their sentences. The boat cannot transfer more than two persons in either direction.The inmates are certain to overpower the guards if they outnumber them at any time. Develop a
1. Solve Example 6.2-1 starting at node 5 (instead of node 1), and show that the algorithm produces the same solution.
Determine the minimal spanning tree of the network of Example 6.2-1 under each of the following separate conditions:*(a) Nodes 5 and 6 are linked by a 2-mile cable.(b) Nodes 2 and 5 cannot be linked.(c) Nodes 2 and 6 are linked by a 4-mile cable.(d) The cable between nodes 1 and 2 is 8 miles
3. In intermodal transportation, loaded truck trailers are Shipped between railroad terminals on special flatbed carts. Figure 6.8 shows the location of the main railroad terminals in the United States and the existing railroad tracks. The objective is to decide which tracks should be "revitalized"
4. Figure 6.9 gives the mileage of the feasible links connecting nine offshore natural gas wellheads with an inshore delivery point. Because wellhead 1 is the closest to shore, it is equipped with sufficient pumping and storage capacity to pump the output of the remaining eight wells to the
*5. In Figure 6.9 of Problem 4, suppose that the wellheads can be divided into two groups depending on gas pressure: a high-pressure group that includes wells 2,3,4, and 6, and a low-pressure group that includes wells 5, 7, 8, and 9. Because of pressure difference, it is not possible to link the
6. Electro produces 15 electronic parts on 10 machines.The company wants to group the machines into cells designed to minimize the "dissimilarities" among the parts processed in each cell. A measure of "dissimilarity," dij , among the parts processed on machines i and j can be expressed as where
*1. Reconstruct the equipment replacement model of Example 6.3-1, assuming that a car must be kept in service at least 2 years, with a maximum service life of 4 years. The planning horizon is from the start of year 1 to the end of year 5. The following table provides the necessary data.Replacement
2. Figure 6.14 provides the communication network between two stations, 1 and 7. The probability that a link in the network will operate without failure is shown on each arc.Messages are sent from station 1 to station 7, and the objective is to determine the route. that will maximize the
3. Production Planning. Directeo sells an item whose demands over the next 4 months are 100,140,210, and 180 units, respectively. The company can stock just enough supply to meet each month's demand, or it can overstock to meet the demand for two or more successive and consecutive months. In the
*4. Knapsack Problem. A hiker has a 5-ft3 backpack and needs to decide on the most valuable items to take on the hiking trip. There are three items from which to choose. Their volumes are 2, 3, and 4 ft3, and the hiker estimates their associated values on a scale from oto 100 as 30,50, and 70,
5. An old-fashioned electric toaster has two spring-loaded base-hinged doors. The two doors open outward in opposite directions away from the heating element. A slice of bread is toasted one side at a time by pushing open one of the doors with one hand and placing the slice with the other hand.
1. In Example 6.3-5, use Floyd's algorithm to determine the shortest routes between each of the following pairs of nodes:*(a) From node 5 to node 1.(b) From node 3 to node 5.(c) From node 5 to node 3.(d) From node 5 to node 2.
2. Apply Floyd's algorithm to the network in Figure 6.22. Arcs (7,6) and (6,4) are unidirectional, and all the distances are in miles. Determine the shortest route between the following pairs of nodes:(a) From node 1 to node 7.(b) From node 7 to node 1.(c) From node 6 to node 7.
3. TIle Tell-All mobile-phone company services six geographical areas. The satellite distances(in miles) among the six areas are given in Figure 6.23. Tell-All needs to determine the most efficient message routes that should be established between each two areas in the network.
*4. Six kids, Joe, Kay, Jim, Bob, Rae, and Kim, playa variation of hide and seek. The hiding place of a child is known only to a select few of the other children. A child is then paired with another with the objective of finding the partner's hiding place. This may be achieved through a chain of
1. In Example 6.3-6, use LP to determine the shortest routes between the following pairs of nodes:*(a) Node 1 to node 5.(b) Node 2 to node 5.
1. Modify solverEx6.3-6.xls to find the shortest route between the following pairs of nodes:(a) Node 1 to node 5.(b) Node 4 to node 3.2. Adapt amplEx6.3-6b.txt for Problem 2, Set 6.3a, to find the shortest route between node 1 and node 7. The input data must be the raw probabilities. Use AMPL
*1. In Example 6.4-2,(a) Determine the surplus capacities for all the arcs.(b) Determine the amount of flow through nodes 2,3, and 4.(c) Can the network flow be increased by increasing the capacities in the directions 3-5 and4-5?
2. Determine the maximal flow and the optimum flow in each arc for the network in Figure 6.32.
3. ll1fee refineries send a gasoline product to two distribution terminals through a pipeline network. Any demand that cannot be satisfied through the network is acquired from other sources. TIle pipeline network is served by three pumping stations, as shown in Figure 6.33. The product flows in the
4. Suppose that the maximum daily capacity of pump 6 in the network of Figure 6.33 is limited to 50 million bbl per day. Remodel the network to include this restriction. Then determine the maximum capacity of the network.
6. In Problem 5, suppose that transshipping is allowed between silos 1 and 2 and silos 2 and 3. Suppose also that transshipping is allowed between farms 1 and 2,2 and 3, and 3 and 4.The maximum two-way daily capacity on the proposed transshipping routes is 50 (thousand)lb. What is the effect of
8. Four factories are engaged in the production of four types of toys. The following table lists the toys that can be produced by each factory.Factory 12 34 Toy productions mix 1,2,3 2,3 1,4 3,4 All toys require approximately the same per-unit labor and material.The daily capacities of the four
9. TIle academic council at the U of A is seeking representation from among six students who are affiliated with four honor societies. The academic council representation includes three areas: mathematics, art, and engineering. At most two students in each area can be on the council. The following
10. Maximal/minimal flow in networks with lower bounds.The maximal flow algorithm given in this section assumes that all the arcs have zero lower bounds. In some models, the lower bounds may be strictly positive, and we may be interested in finding the maximal or minimal flow in the network (see
1. Model each of the following problems as a linear program, then solve using Solver and AMPL.(a) Problem 2, Set 6.4b.(b) Problem 5,Set 6.4b(c) Problem 9, Set 6.4b.
2. Jim lives in Denver, Colorado, and likes to spend his annual vacation in Yellowstone National Park in Wyoming. Being a nature lover, Jim tries to drive a different scenic route each year. After consulting the appropriate maps, Jim has represented his preferred routes between Denver (D) and
3. (Gueret and Associates, 2002, Section 12.1) A military telecommunication system connecting 9 sites is given in Figure 6.37. Sites 4 and 7 must continue to communicate even if as many as three other sites are destroyed by enemy actions. Does the present communication network meet this
1. Construct the project network comprised of activities A to L with the following precedence relationships:(a) A, B, and C, the first activities of the project, can be executed concurrently.(b) A and B precede D.(c) B precedes E, F, and H.(d) F and C precede G.(e) E and H precede I and J.(I) C, D,
2. Construct the project network comprised of activities A to P that satisfies the following precedence relationships:(a) A, B, and C, the first activities of the project, can be executed concurrently.(b) D, E, and F follow A.(c) I and G follow both Band D.(d) H follows both C and G.(e) K and L
*3. The footings of a building can be completed in four consecutive sections. The activities for each section include (1) digging, (2) placing steel, and (3) pouring cqncrete. The digging of one section cannot start until that of the preceding section has been completed. The same restriction
4. In Problem 3, suppose that 10% of the plumbing work can be started simultaneously with the digging of the first section but before any concrete is poured. After each section of the footings is completed, an additional 5% of the plumbing can be started provided that the preceding 5% portion is
6. The activities in the following table describe the construction of a new house. Construct the associated project network.A:B:C:D:£:F:G:H:l:J:K:L:M:N:0:P:Q:R:s:T:Activity Clear site Bring utilities to site Excavate Pour foundation Outside plumbing Frame house Do electric wiring Lay floor Lay
7. A company is in the process of preparing a budget for launching a new product. The following table provides the associated activities and their durations. Construct the project network.A:B:c:D:E:F:G:Activity Forecast sales volume Study competitive market Design item and facilities Prepare
8. The activities involved in a candlelight choir service are listed in the following table.Construct the project network.A:B:Activity Select music Learn music Predecessor(s)A Duration (days)2 14 C:D:E:F:G:H:I:J:K:L:M:N:0:Make copies and buy books Tryouts Rehearsals Rent candelabra Decorate
*1. Determine the critical path for the project network in Figure 6.43. 2. Determine the critical path for the project networks in Figure 6.44. 2 FIGURE 6.43 Project networks for Problem 1, Set 6.5b 7
1. Given an activity (i, j) with duration Dij and its earliest start time q and its latest completion time t:i. j , determine the earliest completion and the latest start times of (i,j).
2. What are the total and free floats of a critical activity? Explain.
*3. For each of the following activities, determine the maximum delay in the starting time relative to its earliest start time that will allow all the immediately succeeding activities to be scheduled anywhere between their earliest and latest completion times.(a) TF = 10, FF = 10, D = 4(b) TF =
4. In Example 6.5-4, use the floats to answer the following:(a) If activity B is started at time 1, and activity C is started at time 5, determine the earliest start times for E and F.
(b) If activity B is started at time 3, and activity C is started at time 7, determine the earliest start times for E and F.
(c) How is the scheduling of other activities impacted if activity B starts at time 6?

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