Chalmers university of technology econ 121 smda chapter 05 – network

MULTIPLE CHOICE

1.   Almost all network problems can be viewed as special cases of the

 a. transshipment problem. b. shortest path problem. c. maximal flow problem. d. minimal spanning tree problem.

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2.   The arcs in a network indicate all of the following except?

 a. routes b. paths c. constraints d. connections

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3.   A factory which ships items through the network would be represented by which type of node?

 a. demand b. supply c. random d. decision

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4.   A node which can both send to and receive from other nodes is a

 a. demand node. b. supply node. c. random node. d. transshipment node.

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5.   Demand quantities for demand nodes in a transshipment problem are customarily indicated by

 a. positive numbers. b. negative numbers. c. imaginary numbers. d. either positive or negative numbers.

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6.   Decision variables in network flow problems are represented by

 a. nodes. b. arcs. c. demands. d. supplies.

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7.   The number of constraints in network flow problems is determined by the number of

 a. nodes. b. arcs. c. demands. d. supplies.

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8.   How many constraints are there in a transshipment problem which has n nodes and m arcs?

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9.   In a transshipment problem, which of the following statements is a correct representation of the balance-of-flow rule if Total Supply < Total Demand?

 a. Inflow – Outflow ³ Supply or Demand b. Inflow + Outflow ³ Supply or Demand c. Inflow – Outflow £ Supply or Demand d. Inflow + Outflow £ Supply or Demand

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10.   Supply quantities for supply nodes in a transshipment problem are customarily indicated by

 a. positive numbers. b. negative numbers. c. imaginary numbers. d. either positive or negative numbers.

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11.   What is the correct constraint for node 2 in the following diagram?

 a. X12 + X23 = 100 b. X12 – X23 £ 100 c. –X12 + X23 ³–100 d. X12 – X23 ³ 100

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12.   The constraint X13 + X23 X34 ³ 50 indicates that

 a. 50 units are required at node 3. b. 50 units will be shipped from node 3. c. 50 units will be shipped in from node 1. d. 50 units must pass through node 3.

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13.   Which balance of flow rule should be applied at each node in a network flow problem when Total Supply > Total Demand?

 a. Inflow – Outflow £ Supply or Demand b. Inflow – Outflow ³ Supply or Demand c. Inflow – Outflow = Supply or Demand d. Inflow – Supply ³ Outflow or Demand

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14.   What formula would be entered in cell G18 in this Excel model?

 A B C D E F G H I J K L 1 2 3 4 Supply/ 5 Ship From To Unit Cost Nodes Net Flow Demand 6 55 1 LAV 2 PHO 60 1 LAV –100 –100 7 45 1 LAV 4 REN 120 2 PHO 50 50 8 5 2 PHO 3 LAX 160 3 LAX 30 30 9 0 3 LAX 5 SAN 70 4 REN 45 45 10 25 5 SAN 3 LAX 90 5 SAN 90 90 11 0 5 SAN 4 REN 70 6 DEN 35 35 12 0 5 SAN 6 DEN 90 7 SLC –150 –150 13 0 6 DEN 5 SAN 50 14 0 7 SLC 4 REN 190 15 115 7 SLC 5 SAN 90 16 35 7 SLC 6 DEN 100 17 18 Total 25600

 a. SUMPRODUCT(K6:K12,L6:L12) b. SUMPRODUCT(B6:B16,G6:G16) c. SUMPRODUCT(G6:G16,K6:K12) d. SUMPRODUCT(B6:G16,L6:L12)

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15.   How could a network be modified if demand exceeds supply?

 a. add extra supply arcs b. remove the extra demand arcs c. add a dummy supply d. add a dummy demand

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16.   What is the interpretation of units “shipped” along arcs from dummy supply nodes to demand nodes?

 a. Indicates unmet demand at demand nodes b. Indicates unmet supply at demand nodes c. Indicates unmet demand at supply nodes d. Indicates unmet supply at supply nodes

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17.   Consider the equipment replacement problem presented in the chapter. Recall that in the network model formulation of this problem a node represents a year when the equipment was purchased. An arc from node i to node j indicates that the equipment purchased in year i can be replaced at the beginning of year j. How could the network model below be modified to depict an equipment purchase in year 4 and operating costs only through the remainder of the planning window?

 a. Modify the cost on arc 4-5 to account for only operating costs. b. Add a second arc 4-5 to represent just the operating costs. c. Add a dummy node, 6, so that arc 4-6 represents just the operating costs. d. Add a dummy node, 6, so that arc 4-5 represents operating costs and 5-6 represents new equipment purchase.

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18.   The street intersections in a city road network represent

 a. nodes. b. arcs. c. resources. d. expenses.

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19.   The right hand side value for the starting node in a shortest path problem has a value of

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20.   The right hand side value for the ending node in a shortest path problem has a value of

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21.   What is the constraint for node 2 in the following shortest path problem?

 a. –X12 – X13 = 0 b. –X12 – X24 = 1 c. X12 + X13 = 0 d. –X12 + X24 = 0

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22.   An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is \$11 and the yield is 85 percent. The following network representation depicts this problem. What is the balance of flow constraint for node 3 (Refinery 1)?

 a. X13 + X23 – .95 X35 – .90 X36 – .90 X37 = 0 b. .80 X13 + .95 X23 – X35 – X36 – X37 = 0 c. .80 X13 + .95 X23 – .90 X36 – .90 X37 ³ 0 d. X13 + X23 – X35 – X36 – X37 ³ 0

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23.   An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is \$11 and the yield is 85 percent. The following flowchart depicts this problem. What is the balance of flow constraint for node 7 (Diesel)?

 a. X35 + X36 + X37 = 75 b. X37 + X47 ³ 75 c. .90 X37 + .95 X47 = 75 d. X37 + X47 –X36 – X35 – X45 – X46 ³ 75

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24.   A network flow problem that allows gains or losses along the arcs is called a

 a. non-constant network flow model. b. non-directional, shortest path model. c. generalized network flow model. d. transshipment model with linear side constraints.

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25.   What is the objective function for the following shortest path problem?

 a. –X12 – X13 = 0 b. MIN –50 X12 – 200 X13 + 100 X24 + 35 X34 c. MIN 50 X12 + 200 X13 + 100 X24 + 35 X34 d. MAX –50 X12 – 200 X13 + 100 X24 + 35 X34

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26.   Which formula should be used to determine the Net Flow values in cell K6 in the following spreadsheet model?

 A B C D E F G H I J K L 1 2 3 4 Supply/ 5 Ship From To Unit Cost Nodes Net Flow Demand 6 55 1 LAV 2 PHO 60 1 LAV –100 –100 7 45 1 LAV 4 REN 120 2 PHO 50 50 8 5 2 PHO 3 LAX 160 3 LAX 30 30 9 0 3 LAX 5 SAN 70 4 REN 45 45 10 25 5 SAN 3 LAX 90 5 SAN 90 90 11 0 5 SAN 4 REN 70 6 DEN 35 35 12 0 5 SAN 6 DEN 90 7 SLC –150 –150 13 0 6 DEN 5 SAN 50 14 0 7 SLC 4 REN 190 15 115 7 SLC 5 SAN 90 16 35 7 SLC 6 DEN 100 17 18 Total 25600

 a. SUMIF(\$C\$6:\$C\$16,I6,\$B\$6:\$B\$16)–SUMIF(\$E\$6:\$E\$16,I6,\$B\$6:\$B\$16) b. SUMIF(\$I\$6:\$I\$12,B6,\$B\$6:\$B\$16)–SUMIF(\$I\$6:\$I\$12,I6,\$B\$6:\$B\$16) c. SUMIF(\$E\$6:\$E\$16,I6,\$B\$6:\$B\$16)–SUMIF(\$C\$6:\$C\$16,I6,\$B\$6:\$B\$16) d. SUMPRODUCT(B6:B16,G6:G16)

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27.   Which property of network flow models guarantees integer solutions?

 a. linear constraints and balance of flow equation format b. linear objective function coefficients c. integer objective function coefficients d. integer constraint RHS values and balance of flow equation format

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28.   In generalized network flow problems

 a. solutions may not be integer values. b. flows along arcs may increase or decrease. c. it can be difficult to tell if total supply is adequate to meet total demand. d. all of these.

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29.   What happens to the solution of a network flow model if side constraints are added that do not obey the balance of flow rules?

 a. The model solution is not guaranteed to be integer. b. The model solution will more accurately reflect reality. c. The model solution will be integer but more accurate. d. The model solution is not guaranteed to be feasible.

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30.   Consider modeling a warehouse with three in-flow arcs and three outflow arcs. The warehouse node is a transshipment node but has a capacity of 100. How would one modify the network model to avoid adding a side constraint that limits either the sum of in-flows or the sum of the out-flows to 100?

 a. Place a limit of 34 on each in-flow arc. b. Add a side constraint limiting the out-flow arcs sum to 100. c. Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100. d. It cannot be accomplished, a side constraint must be added.

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31.   The equipment replacement problem is an example of which network problem?

 a. transportation problem. b. shortest path problem. c. maximal flow problem. d. minimal spanning tree problem.

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32.   If a side constraint for a network flow model cannot be avoided, and non-integer solutions result, how can the solution be expressed as an integer solution?

 a. Force all the arc flow decision variables to be integer. b. Round off all the non-integer arc flow decision variables. c. Increase the supply until the solutions are all integer using a dummy supply node. d. Increase the demand until the solutions are all integer using a dummy demand node.

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33.   A maximal flow problem differs from other network models in which way?

 a. arcs are two directional b. multiple supply nodes are used c. arcs have limited capacity d. arcs have unlimited capacity

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34.   Maximal flow problems are converted to transshipment problems by

 a. connecting the supply and demand nodes with a return arc b. adding extra supply nodes c. adding supply limits on the supply nodes d. requiring integer solutions

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35.   What is the objective function in the following maximal flow problem?

 a. MIN X41 b. MAX X12 + X13 c. MAX X14 d. MAX X41

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36.   What is the constraint for node 2 in the following maximal flow problem?

 a. X12 – X23 – X24 = 0 b. X12 + X23 + X24 = 0 c. X12 £ 4 d. X12 + X13 – X23 = 0

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37.   What is missing from transportation problems compared to transshipment problems?

 a. arcs b. demand nodes c. transshipment nodes d. supply nodes

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38.   Which method is preferred for solving fully connected transportation problems?

 a. linear programming b. network flow methods c. trial and error d. simulation

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39.   When might a network flow model for a transportation/assignment problem be preferable to a matrix form for the problem?

 a. When an integer solution is required. b. When the problem is large and not fully connected. c. When the problem is large and fully connected. d. When supply exceeds demand.

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40.   Which method is preferred for solving minimal spanning tree problems?

 a. linear programming b. transshipment models c. simulation d. manual algorithms

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41.   How many arcs are required to make a spanning tree in a network with n nodes and m arcs?

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42.   The minimal spanning tree solution algorithm works by defining a subnetwork and

 a. adding the least expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork. b. adding the most expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork. c. adding the least expensive arc which connects unconnected nodes in the current subnetwork. d. adding the least expensive arc which connects the most recently added node in the current subnetwork to the closest node not in the current subnetwork.

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PROBLEM

43.   Draw the network representation of the following network flow problem.

 MIN: 5 X12 + 3 X13 + 2 X14 + 3 X24 + 2 X34 Subject to: –X12 – X13 – X14 = –10 X12 – X24 = 2 X13 – X34 = 3 X14 + X24 + X34 = 5 Xij ³ 0 for all i and j

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44.   A company wants to determine the optimal replacement policy for its delivery truck. New trucks cost \$30,000. The company does not keep trucks longer than 2 years and has estimated the annual operating costs and trade-in values for trucks during each of the 2 years as:

 Age in years 0-1 1-2 Operating Cost \$15,000 \$16,500 Trade-in Value \$20,000 \$16,000

Draw the network representation of this problem.

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45.   A company wants to determine the optimal replacement policy for its photocopier. The company does not keep photocopiers longer than 4 years. The company has estimated the annual costs for photocopiers during each of the 4 years and developed the following network representation of the problem.

Write out the LP formulation for this problem.

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46.   A company needs to ship 100 units from Roanoke to Washington at the lowest possible cost. The costs associated with shipping between the cities are:

 To From Lexington Washington Charlottesville Roanoke 50 – 80 Lexington – 50 40 Charlottesville – 30

Draw the network representation of this problem.

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47.   A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are listed below. Also, the decision variable associated with each pair of cities is shown next to the cost.

 To From Portland Spokane Salt Lake City Denver Seattle 100 (X12) 500 (X13) 600 (X14) – Portland – 350 (X23) 300 (X24) – Spokane – – 250 (X34) 200 (X35) Salt Lake City – – – 200 (X45)

Write out the LP formulation for this problem.

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48.   A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are:

 To From Portland Spokane Salt Lake City Denver Seattle 100 500 600 – Portland – 350 300 – Spokane – – 250 200 Salt Lake City – – – 200

What values should go into cells G6:L13 in the following Excel spreadsheet?

 A B C D E F G H I J K L 1 2 3 4 Supply/ 5 Ship From To Unit Cost Nodes Net Flow Demand 6 1 SEA 2 POR 1 SEA 7 1 SEA 3 SPO 2 POR 8 1 SEA 4 SLC 3 SPO 9 2 POR 3 SPO 4 SLC 10 2 POR 4 SLC 5 DEN 11 3 SPO 4 SLC 12 3 SPO 5 DEN 13 4 SLC 5 DEN 14 15 Total cost

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49.   A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are:

 To From Portland Spokane Salt Lake City Denver Seattle 100 500 600 – Portland – 350 300 – Spokane – – 250 200 Salt Lake City – – – 200

What values would you enter in the Risk Solver Platform (RSP) task pane for the following Excel spreadsheet?

Objective Cell:

Variables Cells:

Constraints Cells:

 A B C D E F G H I J K L 1 2 3 4 Supply/ 5 Ship From To Unit Cost Nodes Net Flow Demand 6 1 SEA 2 POR 100 1 SEA –100 –100 7 1 SEA 3 SPO 500 2 POR 0 0 8 1 SEA 4 SLC 600 3 SPO 0 0 9 2 POR 3 SPO 350 4 SLC 0 0 10 2 POR 4 SLC 300 5 DEN 100 100 11 3 SPO 4 SLC 250 12 3 SPO 5 DEN 200 13 4 SLC 5 DEN 200 14 15 Total cost

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50.   A trucking company wants to find the quickest route from Seattle to Denver. What values should be placed in cells L6:L10 of the following Excel spreadsheet?

 A B C D E F G H I J K L 1 2 3 4 Select Driving Supply/ 5 Route From To Time Nodes Net Flow Demand 6 0 1 SEA 2 POR 3 1 SEA –1 7 0 1 SEA 3 SPO 4 2 POR 0 8 1 1 SEA 4 SLC 12 3 SPO 0 9 0 1 SEA 5 DEN 18 4 SLC 0 10 0 2 POR 3 SPO 9 5 DEN 1 11 0 2 POR 4 SLC 12 12 0 2 POR 5 DEN 16 13 0 3 SPO 4 SLC 10 14 0 3 SPO 5 DEN 15 15 1 4 SLC 5 DEN 5 16 17 Total Driving Time 17

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51.   An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following network representation depicts this problem.

Write out the LP formulation for this problem.

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52.   An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. What formula should be entered in cell E6 (and copied to cells E7:E15) in this spreadsheet?

 A B C D E F G H I J K L M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Total cost

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53.   An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem.

What values would you enter in the Risk Solver Platform (RSP) task pane for the following Excel spreadsheet?

Objective Cell:

Variables Cells:

Constraints Cells:

 A B C D E F G H I J K L M 1 2 3 4 Unit Net Supply/ 5 Flow from Node Yield Flow into Node Cost Nodes Flow Demand 6 1 Crude A 0.90 3 Refinery 1 15 1 Crude A –120 7 1 Crude A 0.85 4 Refinery 2 13 2 Crude B –60 8 2 Crude B 0.80 3 Refinery 1 9 3 Refinery 1 0 9 2 Crude B 0.85 4 Refinery 2 11 4 Refinery 2 0 10 3 Refinery 1 0.95 5 Lube Oil 4 5 Lube Oil 75 11 3 Refinery 1 0.90 6 Gasoline 7 6 Gasoline 50 12 3 Refinery 1 0.90 7 Diesel 8 7 Diesel 25 13 4 Refinery 2 0.90 5 Lube Oil 3 14 4 Refinery 2 0.95 6 Gasoline 9 15 4 Refinery 2 0.95 7 Diesel 6 16 17 Total cost

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54.   Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:

 Distance Center 1 Center 2 Center 3 Center 4 Supply Plant A 45 60 53 75 500 Plant B 81 27 49 62 700 Plant C 55 40 35 60 650 Demand 350 325 400 375

Draw the transportation network for Clifton’s distribution problem.

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55.   The following network depicts a transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.

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56.   Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:

 Distance Center 1 Center 2 Center 3 Center 4 Supply Plant A 45 60 53 75 500 Plant B 81 27 49 62 700 Plant C 55 40 35 60 650 Demand 350 325 400 375

Draw the balanced transportation network for Clifton’s distribution problem.

ANS:

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57.   The following network depicts a balanced transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.

PTS:   1

58.   Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:

 Plane 1 Plane 2 Plane 3 Repair Person 1 11 9 21 Repair Person 2 17 7 13 Repair Person 3 9 12 17 Repair Person 4 14 8 28 Repair Person 5 12 5 12

Draw the network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

ANS:

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59.   The following network depicts an assignment/transportation problem for Joe Fix’s repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.

PTS:   1

60.   Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:

 Plane 1 Plane 2 Plane 3 Repair Person 1 11 9 21 Repair Person 2 17 7 13 Repair Person 3 9 12 17 Repair Person 4 14 8 28 Repair Person 5 12 5 12

Draw the balanced network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

ANS:

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61.   The following network depicts a balanced assignment/transportation problem for Joe Fix’s repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.

PTS:   1

62.   A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs \$25 per hour while contracted labor costs \$45 per hour. Draw the network flow model that captures this problem.

PTS:   1

63.   A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs \$25 per hour while contracted labor costs \$45 per hour. The following network flow model captures this problem.

Write out the LP formulation for this problem.

PTS:   1

64.   A company wants to manage its distribution network which is depicted below. Identify the supply, demand and transshipment nodes in this problem.

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65.   Draw the network and indicate how many units are flowing along each arc based on the following Risk Solver Platform (RSP) solution.

 Units Unit Net Supply/ of Flow From To Cost Nodes Flow Demand 5 1 A 2 B 20 1 A –40 –40 35 1 A 3 C 15 2 B 5 5 0 2 B 4 D 30 3 C 5 5 25 3 C 4 D 10 4 D 10 10 5 3 C 5 E 25 5 E 5 5 15 4 D 6 F 10 6 F 15 15 0 5 E 6 F 30 Total 1150

ANS:

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66.   A railroad needs to move the maximum amount of material through its rail network. Formulate the LP model to determine this maximum amount based on the following network diagram.

PTS:   1

67.   Draw the network and solution for the maximal flow problem represented by the following Excel spreadsheet.

 Units Upper Net Supply/ of Flow From To Bound Nodes Flow Demand 4 1 A 2 B 4 1 A 0 0 8 1 A 3 C 8 2 B 0 0 4 2 B 4 D 6 3 C 0 0 0 2 B 5 E 2 4 D 0 0 4 3 C 4 D 4 5 E 0 0 4 3 C 5 E 5 8 4 D 5 E 9 12 5 E 1 A 999 12 Maximal flow

ANS:

PTS:   1

68.   Draw the network representation of this LP model. What type of problem is it?

 MAX X41 Subject to: X41 – X12 – X13 = 0 X12 – X24 = 0 X13 – X34 = 0 X24 + X34 – X41 = 0 0 £ X12 £ 5, 0 £ X13 £ 4, 0 £ X24 £ 3, 0 £ X34 £ 2, 0 £ X41 £¥

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69.   Solve the following minimal spanning tree problem starting at node 1.

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70.   Solve the following minimal spanning tree problem starting at node 1.

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71.   Solve the following minimal spanning tree problem starting at node 1.

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PROJECT

72.   Project 5.1 Recruit Training

You are a military training analyst in charge of initial training for the XXX career field and must decide how to best train the new recruits to satisfy the requirements for skilled recruits. There are six different courses (A, B, C, D, E, F) used for training in the XXX career field and four different sequences of courses that can be taken to achieve the required skill level. These sequences are A-E, B, C-F, and A-D-F. The table below provides information on the six courses.

 Course Cost Per Student Min. Num. of Trainees Max. Num. of Trainees A 25 15 40 B 55 10 50 C 30 15 50 D 10 15 50 E 20 10 50 F 15 10 50

There are 100 recruits available for training and a demand for 100 skilled recruits. Assume all recruits pass each course and that you are trying to put students in classes in order to minimize the total cost of training. Assume non-integer solutions are acceptable. Further, assume each course will be held.

 a. Draw a network flow diagram describing the problem. b. Formulate the associated network flow linear program. c. Implement a spreadsheet model and use Risk Solver Platform (RSP) to obtain a solution to the problem. Use your model to answer the following questions. What is the expected student load for each course? Should any course be expanded? Should any course or sequence be considered for elimination?

Next, assume that not all students pass each course. In fact only 90% of the students pass courses A, E, and F and only 95% of the students pass courses B, C, and D. Each course is considered independent. The requirement for 100 skilled recruits remains. Your job is now to determine the number of recruits to place into the training program to obtain the 100 trained recruits while continuing to minimize the total cost of training.

 d. Re-draw the network flow diagram describing the problem to accommodate the above changes. e. Formulate the associated generalized network flow linear program. f. Implement a spreadsheet model of this changed model and use Risk Solver Platform (RSP) to obtain a solution to the expanded problem. How many recruits are needed and what is the change in total training cost?

PTS:   1

73.   Project 5.2 Small Production Planning Project

(Fixed Charge Problem via Network Flow with Side Constraints)

Jack Small Enterprises runs two factories in Ohio, one in Toledo and one in Centerville. His factories produce a variety of products. Two of his product lines are polished wood clocks which he adorns with a regional theme. Naturally, clocks popular in the southwest are not as popular in the northeast, and vice versa. Each plant makes both of the clocks. These clocks are shipped to St Louis for distribution to the southeast and western states and to Pittsburg for distribution to the south and northeast.

Jack is considering streamlining his plants by removing certain production lines from certain plants. Among his options is potentially eliminating the clock production line at either the Toledo or the Centerville plant. Each plant carries a fixed operating cost for setting up the line and a unit production cost, both in terms of money and factory worker hours. This information is summarized in the table below.

 Production Cost per Clock Clocks Produced per Hour Available Plant Fixed Cost for Line Southwest Clock Northeast Clock Southwest Clock Northwest Clock Hours per Month Toledo \$20,000 \$10 \$12 5 6 500 Centerville \$24,000 \$  9 \$13 5.5 6.2 675

The Southwest clocks are sold for \$23 each and the Northwest clocks are sold for \$25 each. Demand rates used for production planning are 1875 Southwest clocks for sale out of the St Louis distribution center and 2000 Northeast clocks for sale out of the Pittsburg distribution center. Assume all these units are sold. The per clock transportation costs from plant to distribution center is given in the following table.

 (cost per clock shipped) Cost to Ship to Distribution Center Plant St Louis Pittsburg Toledo \$2 \$4 Centerville \$3 \$2

Develop a generalized network flow model for this problem and implement this model in solver. Use the model to answer the following questions.

 a. Should any of the production lines be shut down? b. How should worker hours be allocated to produce the clocks to meet the demand forecasts? Are there any excess hours, and if so how many? c. What is the expected monthly profit? d. If a plant is closed, what are the estimated monthly savings?

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