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operations research an introduction

Questions and Answers of Operations Research An Introduction

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
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
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
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
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
An opinion survey involves designing and printing questionnaires, hiring and training personnel, selecting participants, mailing questionnaires, and analyzing the data.Construct the project network,
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
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
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
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
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
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
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
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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%
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
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
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,
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
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
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,
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
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
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,
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
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
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,
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.
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%
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
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
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
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
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
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
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
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
In Problem 5-8, 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
In Problem 5-8, 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
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
Solve Problem 5-6, assuming that there is a 10% power transmission loss through the network.
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
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 $300 and $190 for each undelivered car at Denver and Miami,
In Table 5.4 of Example 5.1-2, where a dummy plant is added, what does the solution mean when the dummy plant “ships” 150 cars to Denver and 50 cars to Miami?*5-4. In Table 5.5 of Example 5.1-2,
In each of the following cases, determine whether a dummy source or a dummy destination must be added to balance the model.(a) Supply: a1 = 100, a2 = 50, a3 = 40, a4 = 60 Demand: b1 = 100, b2 = 50,
True or False?(a) To balance a transportation model, it is necessary to add a dummy source or a dummy destination bur never both.(b) The amounts shipped to a dummy destination represent surplus at
In the Reddy Mikks model, the company is considering the production of a cheaper brand of exterior paint whose input requirements per ton include .75 ton of each of raw materials M1 and M2. Market
In the TOYCO model, suppose that a new toy (fire engine) requires 3, 2, and 4 minutes, respectively, on operations 1, 2, and 3. Determine the optimal solution when the revenue per unit is given
In the TOYCO model, suppose that the company can reduce the unit times on operations 1, 2, and 3 for toy trains from the current levels of 1, 3, and 1 minutes to .5, 1, and .5 minutes, respectively.
In the original TOYCO model, toy trains are not part of the optimal product mix. The company recognizes that market competition will not allow raising the unit price of the toy. Instead, the company
Show that the 100% optimality rule (Problem 3-88, Chapter 3) is derived from 1reduced costs2 Ú 0 for maximization problems and 1reduced costs2 … 0 for minimization problems.
Investigate the optimality of the Reddy Mikks solution (Example 4.3-1) for each of the following objective functions. If necessary, use post-optimal analysis to determine the new optimum. (The
Investigate the optimality of the TOYCO solution for each of the following objective functions. Where necessary, use post-optimal analysis to determine the new optimum.(The optimum tableau of TOYCO
Secondary Constraints. Instead of solving a problem using all of its constraints, we can start by identifying the so-called secondary constraints. These are the constraints that we suspect are least
In the TOYCO model, suppose the fourth operation has the following specifications: The maximum production rate based on 480 minutes a day is 120 units of product 1, 480 units of product 2, or 240

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