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operations research an introduction

Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions

Consider the following system of equations:£1 23 x1 + £0 21 x2 + £1 42 x3 + £2 00 x4 = £3 42Determine if any of the following combinations forms a basis:(a) (p1, p2, p3)(b) (p1, p3, p4)(c) (p2, p3, p4)(d) (p1, p2, p3, p4)
Use vectors to determine graphically the type of solution for each of the sets of equations below: unique solution, an infinite number of solutions, or no solution. For the cases of unique solutions, indicate from the vector representation (and without solving the equations algebraically) whether
In the following sets of equations, (a) and (b) have unique (basic) solutions, (c) has an infinite number of solutions, and (d) has no solution. Show how these results can be verified using graphical vector representation. From this exercise, state the general conditions for vector
In the solution space in Figure 7.3 (drawn to scale), express the interior point (3, 1) as a convex combination of the extreme points A, B, C, and D by determining the weights associated with the extreme points.
Determine graphically the extreme points of the following convex set:Q = 5x1, x2 x1 + x2 … 3, x1 Ú 0, x2 Ú 06 Show that the entire feasible solution space can be determined as a convex combination of its extreme points. Hence, conclude that any convex (bounded) solution space is totally
Show that the set Q = 5x1, x2 x1 Ú 1 or x2 Ú 26 is not convex.
Show that the set Q = 5x1, x2 x1 + x2 … 3, x1 Ú 0, x2 Ú 06 is convex. Is the nonnegativity condition essential for the proof?
The estimates (a, m,b) are listed in the following table:Project (a) Project (b)Activity (a, m,b) Activity (a, m,b) Activity (a, m,b) Activity (a, m, b)1-2 (5, 6, 8) 3-6 (3, 4, 5) 1-2 (1, 3, 4) 3-7 (12, 13, 14)1-4 (1, 3, 4) 4-6 (4, 8, 10) 1-3 (5, 7, 8) 4-5 (10, 12, 15)1-5 (2, 4, 5) 4-7 (5, 6, 8)
Consider Problem
Use LP to determine the critical path for the project networks in Figure 6.45.
Use LP to determine the critical path for the project network in Figure 6.44.
(Job shop scheduling) Three jobs, J1, J2, and J3, are processed on 3 machines, M1, M2, and M3, according to the following sequences (processing times are shown in parentheses):J1: M3132 - M1142 - M2162 J2: M2112 - M3152 - M1192 J3: M3122 - M2182 - M1172 The order in which the jobs are processed on
Compute the floats and identify the red-flagged activities for the projects (a) and (b) in Figure 6.30, then develop the time schedules under the following conditions:Project (a)(i) Activity (1, 5) cannot start any earlier than time 14.(ii) Activities (5, 6) and (5, 7) use the same equipment, of
In the project of Example 6.5-2 (Figure 6.28), assume that the durations of activities B and F are changed from 6 and 11 days to 20 and 25 days, respectively.(a) Determine the critical path.(b) Determine the total and free floats for the network, and identify the red-flagged activities.(c) If
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
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 = 20, FF = 20, D = 8(b) TF = 8, FF
What are the total and free floats of a critical activity? Explain.
Given an activity (i, j) with duration Dij and its earliest start time i and its latest completion time j, determine the earliest completion and the latest start times of (i, j).
Determine the critical path for the project in Problem 6-51.
Determine the critical path for the project in Problem 6-50.
Determine the critical path for the project in Problem 6-49.
Determine the critical path for the project in Problem 6-47.
Determine the critical path for the project networks in Figure 6.45.
Determine the critical path for the project network in Figure 6.44.
The following table gives the activities for buying a new car. Construct the project network:Activity Predecessor(s) Duration (days)A: Conduct feasibility study — 3 B: Find potential buyer for present car A 14 C: List possible models A 1 D: Research all possible models C 3 E: Conduct interview
The widening of a road section requires relocating (“reconductoring”) 1700 ft of 13.8-kV overhead primary line. The following table summarizes the activities of the project.Construct the associated project network.Activity Predecessor(s) Duration (days)A: Job review — 1 B: Advise customers of
The activities involved in a candlelight choir service are listed in the following table.Construct the project network.Activity Predecessor(s) Duration (days)A: Select music — 2 B: Learn music A 14 C: Make copies and buy books A 14 D: Tryouts B, C 3 E: Rehearsals D 70 F: Rent candelabra D 14 G:
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.Activity Predecessor(s) Duration (days)A: Forecast sales volume — 10 B: Study competitive market — 7 C: Design
The activities in the following table describe the construction of a new house. Construct the associated project network.Activity Predecessor(s) Duration (days)A: Clear site — 1 B: Bring utilities to site — 2 C: Excavate A 1 D: Pour foundation C 2 E: Outside plumbing B, C 6 F: Frame house D 10
An opinion survey involves designing and printing questionnaires, hiring and training personnel, selecting participants, mailing questionnaires, and analyzing the data.Construct the project network, stating all assumptions.
In Problem 6-44, 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
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 concrete. The digging of one section cannot start until that of the preceding section has been completed.The same restriction applies to
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 B and D.(d) H follows both C and G.(e) K and L
Construct the project network comprised of activities A to M 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 and M.(e) E and H precede I and J.(f) C,
Guéret and Associate (2002),Section 12.1. A military telecommunication system connecting 9 sites is given in Figure 6.43. 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 requirement?
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
Model each of the following problems as a linear program, then solve using Solver or AMPL.(a) Problem 6-32.(b) Problem 6-35.(c) Problem 6-39.
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 case
The 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
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 Toys productions mix 1 1, 2, 3 2 2, 3 3 1, 3, 4 4 1, 3, 4 All toys require approximately the same per-unit labor and material. The daily capacities of
A parent has five (teenage) children and five household chores to assign to them.Past experience has shown that forcing chores on a child is counterproductive. With this in mind, the children are asked to list their preferences among the five chores, as the following table shows:Child Preferred
In Problem 6-33, 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
Chicken feed is transported by trucks from three silos to four farms. Some of the silos cannot ship directly to some of the farms. The capacities of the other routes are limited by the number of trucks available and the number of trips made daily. The following table shows the daily amounts of
Suppose that the maximum daily capacity of pump 6 in the network of Figure 6.41 is limited to 50 million bbl per day. Remodel the network to include this restriction. Then determine the maximum capacity of the network.
Three 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. The pipeline network is served by three pumping stations, as shown in Figure 6.41. The product flows in the
Determine the maximal flow and the optimum flow in each arc for the network in Figure 6.40.
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 3S 5 and 4S 5?
For the network in Figure 6.20, determine two additional cuts, and find their capacities.
Adapt amplEx6.3-6b.txt for Problem 6-14, to find the shortest route between node 1 and node 6. The input data must be the raw probabilities. Use AMPL programming facilities to print/display the optimum transmission route and its success probability.
Modify solverEx6.3-6.xls to find the shortest route between the following pairs of nodes:(a) Node 1 to node 5.(b) Node 1 to node 4.
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.
Six kids, Joe, Kay, Jim, Bob, Rae, and Kim, play a 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
The Tell-All mobile-phone company services six geographical areas. The satellite distances(in miles) among the six areas are given in Figure 6.39. Tell-All needs to determine the most efficient message routes that should be established between each two areas in the network.
Apply Floyd’s algorithm to the network in Figure 6.38. 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.
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 1 to node 4.(d) From node 3 to node 2.
Use Dijkstra’s algorithm to determine the optimal solution of each of the following problems:(a) Problem 6-13.(b) Problem 6-14.(c) Problem 6-16.
Use Dijkstra’s algorithm to find the shortest route between node 1 and every other node in the network of Figure 6.37.
The network in Figure 6.36 gives the distances in miles between pairs of cities 1, 2, …, and 8. Use Dijkstra’s algorithm to find the shortest route between the following cities:(a) Cities 1 and 7.(b) Cities 1 and 6.*(c) Cities 4 and 8.(d) Cities 2 and 7.
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. After
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 0 to 100 as 30, 50, and 70,
Production Planning. DirectCo 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 consecutive months. In the latter case, a
Figure 6.35 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 maximizes the probability of a
Reconstruct the equipment replacement model of Example 6.3-1, assuming that a car must be kept in service for 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
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 dij
In Figure 6.34 of Problem 6-10, 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 7, and a low-pressure group that includes wells 5, 6, 8, and 9. Because of pressure difference, it is not possible to link the
Figure 6.34 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 delivery
In intermodal transportation, loaded truck trailers are shipped between railroad terminals on special flatbed carts. Figure 6.33 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
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
Solve Example 6.2-1 starting at node 6 (instead of node 1), and show that the algorithm produces the same solution.
Three inmates escorted by three 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 anywhere at any time.
In Example 6.1-1,(a) Specify the smallest number and locations of additional bridges needed to construct(i) a round-trip starting from A, and (ii) a trip that starts from A and ends in C.Construct the resulting network, and determine the legs of the trip.(b) During World War II, two of the bridges
Draw the network defined by N = 51, 2, 3, 4, 56 A = 511, 22, 11, 52, 12, 32, 12, 42, 13, 42, 13, 52, 14, 32, 14, 52, 15, 226
Determine the sets N and A for the networks in Figure 6.32.
For each network in Figure 6.32, determine (a) a path, (b) a cycle, (c) a tree, and (d) a spanning tree.
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,
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
A business executive must make the four round-trips listed in Table 5.41 between the head office in Dallas and a branch office in Atlanta.The price of a round-trip ticket from Dallas is $400. A 25% discount is granted if the dates of arrival and departure of a ticket span a weekend (Saturday and
In the model of Problem 5-33, 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. Moreover, job 1 cannot be displaced by the newly arriving job. Should the new job take priority over any of the four
In the JoShop model of Problem 5-33, 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?
JoShop needs to assign four jobs to four workers. The cost of performing a job is a function of the skills of the workers. Table 5.40 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.
Consider the assignment models in Table 5.39.(a) Solve by the Hungarian method.(b) TORA Experiment. Express the problem as an LP, and solve it with TORA.(c) TORA Experiment. Use TORA to solve the problem as a transportation model.(d) Solver Experiment. Modify Excel file solverEx5.3-1.xls to solve
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 the
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.
Consider the problem Minimize z = a m i=1 a n j=1 cij xij subject to a n j=1 xij Ú ai, i = 1, 2,c, m a m i=1 xij Ú bj, j = 1, 2,c, n xij Ú 0, all i and j It may appear logical to assume that the optimum solution will require the first (second) set of inequalities to be replaced with equations if
The transportation problem in Table 5.37 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 u1 = 1, u2 = -1, v 1 = 2, v 2 = 2, and v 3 = 5 and that the unit cost for all (basic and
In a 3 * 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,
In the unbalanced transportation problem in Table 5.36, if a unit from a source is not shipped out (to any of the destinations), a storage cost is incurred at the rate of $5, $4, and $3 per unit for sources 1, 2, and 3, respectively. Additionally, all the supply at source 2 must be shipped out
Solve Problem 5-24, assuming that the demand at destination 1 must be satisfied completely.(a) Find the optimal solution.(b) Solver Experiment. Solve the problem by modifying file solverEx5.3-1.xls.(c) AMPL Experiment. Solve the problem by modifying file amplEx5.3-1b.txt.
In the transportation problem in Table 5.35, the total demand exceeds the total supply.Suppose that the penalty costs per unit of unsatisfied demand are $2, $5, and $3 for destinations 1, 2, and 3, respectively. Use the least-cost starting solution and compute the iterations leading to the optimum
Consider the transportation models in Table 5.34.(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
Compare the starting solutions obtained by the northwest-corner, least-cost, and Vogel methods for each of the models in Table 5.33.
The National Parks Service is receiving four bids for logging at three pine forests in Arkansas. The three locations include 20,000, 30,000, and 10,000 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 5.32.
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
The demand for a special small engine over the next five quarters is 200, 150, 300, 250, and 400 units, respectively. 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
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.
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 30, 10, 20, 25, and 20. Machine category 4 cannot be assigned to task type 4. Table 5.31
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.
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.
In Example 5.2-1, suppose that the holding cost per unit is period-dependent and is given by 20, 15, and 35 cents for periods 1, 2, and 3, respectively. The penalty cost is $1 per period and the production costs remain as given in the example. Determine the optimum solution and interpret the
MG Auto, of Example 5.1-1, produces four car models: M1, M2, M3, and M4. The Detroit plant produces models M1, M2, and M4. Models M1 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 distribution

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