Chapter 6 study Dec anal

48. Show both the network and the linear programming formulation for this assignment problem. ​ ​ Task Person A B C D 1 9 5 4 2 2 12 6 3 5 3 11 6 5 7

Let X ij = 1 if person i is assigned to job j = 0 otherwise Min 9X 1A + 5X 1B + 4X 1C + 2X 1D + 12X 2A + 6X 2B + 3X 2C + 5X 2D + 11X 3A + 6X 3B + 5X 3C + 7X 3D s.t. X 1A + X 1B + X 1C + X 1D ≤ 1 X 2A + X 2B + X 2C + X 2D ≤ 1 X 3A + X 3B + X 3C + X 3D ≤ 1 X 4A + X 4B + X 4C + X 4D ≤ 1 X 1A + X 2A + X 3A + X 4A = 1 X 1B + X 2B + X 3B + X 4B = 1 X 1C + X 2C + X 3C + X 4C = 1 X 1D + X 2D + X 3D + X 4D = 1

54. Consider the network below. Formulate the LP for finding the shortest-route path from node 1 to node 7.

Min 10X 12 + 12X 13 + 4X 24 + 8X 25 + 7X 35 + 9X 36 + 4X 42 + 3X 45 + 6X 47 + 8X 52 + 7X 53 + 3X 54 + 4X 57 + 9X 63 + 3X 67 s.t. X 12 + X 13 = 1 −X 12 + X 24 + X 25 − X 42 − X 52 = 0 −X 13 + X 35 + X 36 − X 53 − X 63 = 0 −X 24 + X 42 + X 45 + X 47 − X 54 = 0 −X 25 − X 35 − X 45 + X 52 + X 53 + X 57 = 0 −X 36 + X 63 + X 67 = 0 X 47 + X 57 + X 67 = 1

55. Consider the following shortest-route problem involving six cities with the distances given. Draw the network for this problem and formulate the LP for finding the shortest distance from City 1 to City 6. Path Distance 1 to 2 3 1 to 3 2 2 to 4 4 2 to 5 5 3 to 4 3 3 to 5 7 4 to 6 6 5 to 6 2

Min 3X 12 + 2X 13 + 4X 24 + 5X 25 + 3X 34 + 7X 35 + 4X 42 + 3X 43 + 6X 46 + 5X 52 + 7X 53 + 2X 56 s.t. X 12 + X 13 = 1 −X 12 + X 24 + X 25 − X 42 − X 52 = 0 −X 13 + X 34 + X 35 − X 43 − X 53 = 0 −X 24 − X 34 + X 42 + X 43 + X 46 = 0 −X 25 − X 35 + X 52 + X 53 + X 56 = 0 X 46 + X 56 = 1 X ij ≥ 0 for all i and j

56. A beer distributor needs to plan how to make deliveries from its warehouse (node 1) to a supermarket (node 7), as shown in the network below. Develop the LP formulation for finding the shortest route from the warehouse to the supermarket. ​

Min 3X 12 + 3X 15 + 12X 16 + 5X 23 + 5X 32 + 6X 34 + 6X 43 + 4X 46 + 5X 47 + 8X 56 + 4X 64 + 8X 65 + 3X 67 s.t. X 12 + X 15 + X 16 = 1 −X 12 + X 23 = 0 −X 23 + X 32 + X 34 = 0 −X 34 + X 43 + X 46 + X 47 − 4X 64 = 0 −X 15 + X 56 = 0 −X 16 − X 46 − X 56 + X 64 + X 65 + X 67 = 0 X 47 + X 67 = 1 X ij ≥ 0 for all i and j

46. Canning Transport is to move goods from three factories to three distribution centers. Information about the move is given below. Give the network model and the linear programming model for this problem. Source Supply Destination Demand A 200 X 50 B 100 Y 125 C 150 Z 125 ​ Shipping costs are: ​ Destination Source X Y Z A 3 2 5 B 9 10 — C 5 6 4 ​ (Source B cannot ship to destination Z)

Min 3X AX + 2X AY + 5X AZ + 9X BX + 10X BY + 5X CX + 6X CY + 4X CZ s.t. X AX + X AY + X AZ ≤ 200 X BX + X BY ≤ 100 X CX + X CY + X CZ ≤ 150 X DX + X DY + X DZ ≤ 50 X AX + X BX + X CX + X DX = 250 X AY + X BY + X CY + X DY = 125 X AZ + X BZ + X CZ + X DZ = 125 X ij ≥ 0

51. Write the linear program for this transshipment problem.

Min 3x 16 + 2x 14 + 3x 15 + 5x 24 + 6x 25 + 2x 32 + 8x 34 + 10x 35 + 5x 46 + 9x 47 + 12x 56 + 15x 57 s.t. x 16 + x 14 + x 35 ≤ 500 x 24 + x 25 − x 23 ≤ 400 x 32 + x 34 + x 35 ≤ 300 x 46 + x 47 − (x 14 + x 24 + x 34 ) = 0 x 56 + x 57 − (x 15 + x 25 + x 35 ) = 0 x 16 + x 46 + x 56 = 600 x 56 + x 57 = 600

53. RVW (Restored Volkswagens) buys 15 used VW's at each of two car auctions each week held at different locations. It then transports the cars to repair shops it contracts with. When they are restored to RVW's specifications, RVW sells 10 each to three different used car lots. Various costs are associated with the average purchase and transportation prices from each auction to each repair shop. There are also transportation costs from the repair shops to the used car lots. RVW is concerned with minimizing its total cost given the costs in the table below. a. Draw a network representation for this problem. ​ Repair Shops Used Car Lots S1 S2 L1 L2 L3 Auction 1 550 500 S1 250 300 500 Auction 2 600 450 S2 350 650 450 b. Formulate this problem as a transshipment linear programming model.

Min 50X 13 + 500X 14 + 600X 23 + 450X 24 + 250X 35 + 300X 36 + 500X 37 + 350X 45 + 650X 46 + 450X 47 s.t. X 13 + X 14 ≤ 15 X 23 + X 24 ≤ 15 X 13 + X 23 − X 35 − X 36 − X 37 = 0 X 14 + X 24 − X 45 − X 46 − X 47 = 0 X 35 + X 45 = 10 X 36 + X 46 = 10 X 37 + X 4 = 10 X ij ≥ 0 for all i and j

44. Write the LP formulation for this transportation problem.

Min 5X 1A + 6X 1B + 4X 2A + 2X 2B + 3X 3A + 6X 3B + 9X 4A + 7X 4B s.t. X 1A + X 1B ≤ 100 X 2A + X 2B ≤ 200 X 3A + X 3B ≤ 150 X 4A + X 4B ≤ 50 X 1A + X 2A + X 3A + X 4A = 250 X 1B + X 2B + X 3B + X 4B = 250 all X ij ≥ 0

57. Consider the following shortest-route problem involving seven cities. The distances between the cities are given below. Draw the network model for this problem and formulate the LP for finding the shortest route from City 1 to City 7. Path Distance 1 to 2 6 1 to 3 10 1 to 4 7 2 to 3 42 to 5 5 3 to 4 5 3 to 5 2 3 to 6 4 4 to 6 8 5 to 7 7 6 to 7 5

Min 6X 12 + 10X 13 + 7X 14 + 4X 23 + 5X 25 + 4X 32 + 5X 34 + 2X 35 + 4X 36 + 5X 43 + 8X 46 + 5X 52 + 2X 53 + 7X 57 + 4X 63 + 8X 64 + 5X 67 s.t. X 12 + X 13 + X 14 = 1 −X 12 + X 23 + X 25 − X 32 − X 52 = 0 −X 13 − X 23 + X 32 + X 34 + X 35 + X 36 − X 43 − X 53 − X 63 = 0 −X 14 − X 34 + X 43 + X 46 − X 64 = 0 −X 25 − X 35 + X 52 + X 53 + X 57 = 0 −X 36 − X 46 + X 63 + X 64 + X 67 = 0 X 57 + X 67 = 1 X ij ≥ 0 for all i and j

58. The network below shows the flows possible between pairs of six locations. Formulate an LP to find the maximal flow possible from node 1 to node 6.

Min X 61 s.t. X 12 + X 13 + X 15 − X 61 = 0 X 23 + X 24 − X 12 − X 32 = 0 X 32 + X 34 + X 35 − X 13 − X 23 − X 53 = 0 X 43 + X 46 − X 24 − X 34 = 0 X 53 + X 56 − X 15 − X 35 = 0 X 61 − X 36 − X 46 − X 56 = 0 X 12 ≤ 18 X 13 ≤ 20 X 15 ≤ 10 X 23 ≤ 9 X 24 ≤ 15 X 32 ≤ 4 X 34 ≤ 10 X 35 ≤ 8 X 43 ≤ 10 X 46 ≤ 14 X 53 ≤ 8 X 56 ≤ 10 X ij ≥ 0 for all i and j

59. A network of railway lines connects the main lines entering and leaving a city. Speed limits, track reconstruction, and train length restrictions lead to the flow diagram below, where the numbers represent how many cars can pass per hour. Formulate an LP to find the maximal flow in cars per hour from node 1 to node F.

Min X F1 s.t. X 12 + X 15 + X 16 − X F1 = 0 X 23 + X 24 − X 12 = 0 X 34 − X 23 = 0 X 48 + X 4F − X 24 − X 34 − X 84 = 0 X 57 − X 15 = 0 X 67 + X 69 − X 16 − X 76 = 0 X 76 + X 79 − X 57 − X 67 = 0 X 84 + X 89 + X 8F − X 48 − X 98 = 0 X 98 + X 9F − X 69 − X 79 − X 89 = 0 X F1 − X 4F − X 8F − X 9F = 0 X 12 ≤ 500 X 15 ≤ 300 X 16 ≤ 600 X 23 ≤ 300 X 24 ≤ 400 X 34 ≤ 150 X 48 ≤ 400 X 4F ≤ 600 X 57 ≤ 400 X 67 ≤ 300 X 69 ≤ 500 X 76 ≤ 200 X 79 ≤ 350 X 84 ≤ 200 X 89 ≤ 300 X 8F ≤ 450 X 98 ≤ 300 X 9F ≤ 500 X ij ≥ 0 for all i and j

60. A foreman is trying to assign crews to produce the maximum number of parts per hour of a certain product. He has three crews and four possible work centers. The estimated number of parts per hour for each crew at each work center is summarized below. Solve for the optimal assignment of crews to work centers. Work Center WC1 WC2 WC3 WC4 Crew A 15 20 18 30 Crew B 20 22 26 30 Crew C 25 26 27 30

OBJECTIVE FUNCTION VALUE = 82.000 VARIABLE VALUE REDUCED COST A A1 0.000 12.000 A A2 0.000 0.000 A A3 0.000 0.000 A A4 1.000 1.000 A B1 0.000 2.000 A B2 0.000 4.000 A B3 1.000 0.000 A B4 0.000 0.000 A C1 0.000 12.000 A C2 1.000 0.000 A C3 0.000 0.000 A C4 0.000 1.000 A D1 1.000 2.000 A D2 0.000 4.000 A D3 0.000 0.000 A D4 0.000 0.000 ​ CONSTRAINT SLACK/SURPLUS DUAL PRICE 1 0.000 18.000 2 0.000 23.000 3 0.000 24.000 4 0.000 −1.000 5 0.000 1.000 6 0.000 2.000 7 0.000 3.000 8 0.000 12.000 An optimal solution is: Crew Work Center Parts/Hour Crew A WC4 30 Crew B WC3 26 Crew C WC2 26 ---------- WC1 --- Total Parts Per Hour 82

50. A professor has been contacted by four not-for-profit agencies that are willing to work with student consulting teams. The agencies need help with such things as budgeting, information systems, coordinating volunteers, and forecasting.Although each of the four student teams could work with any of the agencies, the professor feels that there is a difference in the amount of time it would take each group to solve each problem. The professors estimate of the time, in days, is given in the table below. Use the computer solution to see which team works with which project. ​ Projects Team Budgeting Information Volunteers Forecasting A 32 35 15 27 B 38 40 18 35 C 41 42 25 38 D 45 45 30 42 ASSIGNMENT PROBLEM ************************ OBJECTIVE: MINIMIZATION SUMMARY OF UNIT COST OR REVENUE DATA ********************************************* TASK AGENT 1 2 3 4 ---------- ----- ----- ----- ----- 1 32 35 15 27 2 38 40 18 35 3 41 42 25 38 4 45 45 30 42 OPTIMAL ASSIGNMENTS COST/REVENUE ************************ *************** ASSIGN AGENT 3 TO TASK 1 41 ASSIGN AGENT 4 TO TASK 2 45 ASSIGN AGENT 2 TO TASK 3 18 ASSIGN AGENT 1 TO TASK 4 27 ------------------------------------------- ----- TOTAL COST/REVENUE 131

Team A works with the forecast, Team B works with volunteers, Team C works with budgeting, and Team D works with information. The total time is 131.

63. A computer manufacturing company wants to develop a monthly plan for shipping finished products from three of its manufacturing facilities to three regional warehouses. It is thinking about using a transportation LP formulation to exactly match capacities and requirements. Data on transportation costs (in dollars per unit), capacities, and requirements are given below. ​ Warehouse Plant 1 2 3 Capacities A 2.41 1.63 2.09 4,000 B 3.18 5.62 1.74 6,000 C 4.12 3.16 3.09 3,000 Requirement 8,000 2,000 3,000 ​ a. How many variables are involved in the LP formulation? b. How many constraints are there in this problem? c. What is the constraint corresponding to Plant B? d. What is the constraint corresponding to Warehouse 3?

The problem formulation is shown below. Use it to answer questions a through d. ​ X ij = each combination of plant i and warehouse j Min 2.41X A1 + 1.63X A2 + 2.09X A3 + 3.18X B1 + 5.62X B2 + 1.74X B3 + 4.12X C1 + 3.16X C2 + 3.09X C3 s.t. X A1 + X A2 + X A3 = 4,000 (capacities) X A1 + X B1 + X C1 = 8,000 (requirements) X B1 + X B2 + X B3 = 6,000 X A2 + X B2 + X C2 = 2,000 X C1 +X C2 + X C3 = 3,000 X A3 + X B3 + X C3 = 3,000 ​ a. 9 variables b. 6 constraints c. X B1 + X B2 + X B3 = 6,000 d. X A3 + X B3 +X C3 = 3,000

1. Whenever total supply is less than total demand in a transportation problem, the LP model does not determine how the unsatisfied demand is handled. a. True b. False

True

62. A clothing distributor has four warehouses that serve four large cities. Each warehouse has a monthly capacity of 5,000 blue jeans. It is considering using a transportation LP approach to match demand and capacity. The following table provides data on shipping cost, capacity, and demand constraints on a per-month basis. Develop a linear programming model for this problem. ​ Warehouse City E City F City G City H A 0.53 0.21 0.52 0.41 B 0.31 0.38 0.41 0.29 C 0.56 0.32 0.54 0.33D 0.42 0.55 0.34 0.52 City Demand 2,000 3,000 3,500 5,500

X ij = each combination of warehouse i and city j Min 0.53X AE + 0.21X AF + 0.52X AG + 0.41X AH + 0.31X BE + 0.38X BF + 0.41X BG + 0.29X BH + 0.56X CE + 0.32X CF + 0.54X CG + 0.33X CH + 0.42X DE + 0.55X DF + 0.34X DG + 0.52X DH s.t. X AE + X AF + X AG + X AH 5,000 X AE + X BE + X CE + X DE = 2,000 X BE + X BF + X BG + X BH 5,000 X AF + X BF + X CF + X DF = 3,000 X CE + X CF + X CG + X CH 5,000 X AG + X BG + X CG + X DG = 3,500 X DE + X DF + X DG + X DH 5,000 X AH + X BH + X CH + X DH = 5,500

27. We represent the number of units shipped from origin i to destination j by a. x ij . b. x ji . c. o ij . d. o ji .

a

31. The assignment problem is a special case of the a. transportation problem. b. transshipment problem. c. maximal flow problem. d. shortest-route problem.

a

33. The assignment problem constraint x 31 + x 32 + x 33 + x 34 ≤ 2 means a. agent 3 can be assigned to two tasks. b. agent 2 can be assigned to three tasks. c. a mixture of agents 1, 2, 3, and 4 will be assigned to tasks. d. there is no feasible solution.

a

47. After some special presentations, the employees of AV Center have to move projectors back to classrooms. The table below indicates the buildings where the projectors are now (the sources), where they need to go (the destinations), and a measure of the distance between sites. Destination Source Business Education Parsons Hall Holmstedt Hall Supply Baker Hall 10 9 5 2 35 Tirey Hall 12 11 1 6 10 Arena 15 14 7 6 20 Demand 12 20 10 10 ​ a. If you were going to write this as a linear programming model, how many decision variables would there be, and how many constraints would there be? The solution to this problem is shown below. Use it to answer questions b through e. TRANSPORTATION PROBLEM ***************************** OPTIMAL TRANSPORTATION SCHEDULE **************************************** FROM TO DESTINATION FROM ORIGIN 1 2 3 4 ---------------- --- ------ ------ ------ ------ 1 12 20 0 3 2 0 0 10 0 3 0 0 0 7 TOTAL TRANSPORTATION COST OR REVENUE IS 358 NOTE: THE TOTAL SUPPLY EXCEEDS THE TOTAL DEMAND BY 13 ORIGIN EXCESS SUPPLY ---------- ----------------------- 3 13 b. How many projectors are moved from Baker to Business? c. How many projectors are moved from Tirey to Parsons? d. How many projectors are moved from Arena to Education? e. Which site(s) has (have) projectors left?

a. 12 decision variables, 7 constraints b. 12 c. 10 d. 0 e. Arena

24. The problem that deals with the distribution of goods from several sources to several destinations is a(n) a. maximal flow problem. b. transportation problem. c. assignment problem. d. shortest-route problem.

b

32. Which of the following is NOT true regarding an LP model of the assignment problem? a. Costs appear in the objective function only. b. All constraints are of the ≥ form. c. All constraint left-hand-side coefficient values are 1. d. All decision variable values are either 0 or 1.

b

35. Constraints in a transshipment problem a. correspond to arcs. b. include a variable for every arc. c. require the sum of the shipments out of an origin node to equal supply. d. All of these are correct.

b

38. The shortest-route problem finds the shortest route a. from the source to any other node. b. from any node to any other node. c. from any node to the destination. d. None of these are correct.

b

39. Consider a maximal flow problem in which vehicle traffic entering a city is routed among several routes before eventually leaving the city. When represented with a network, a. the nodes represent stoplights. b. the arcs represent one-way streets. c. the nodes represent locations where speed limits change. d. None of these are correct.

b

41. If a transportation problem has four origins and five destinations, the LP formulation of the problem will have a. 5 constraints. b. 9 constraints. c. 18 constraints. d. 20 constraints.

b

In a maximal flow problem, a. the flow out of a node is less than the flow into the node. b. the objective is to determine the maximum amount of flow that can enter and exit a network system in a given period of time. c. the number of arcs entering a node is equal to the number of arcs exiting the node. d. None of these are correct.

b

25. The parts of a network that represent the origins are called a. capacities. b. flows. c. nodes. d. arcs.

c

28. Which of the following is NOT true regarding the linear programming formulation of a transportation problem? a. Costs appear only in the objective function. b. The number of variables is calculated as number of origins times number of destinations. c. The number of constraints is calculated as number of origins times number of destinations. d. The constraints; left-hand-side coefficients are either 0 or 1.

c

29. The difference between the transportation and assignment problems is that a. total supply must equal total demand in the transportation problem. b. the number of origins must equal the number of destinations in the transportation problem. c. each supply and demand value is 1 in the assignment problem. d. there are many differences between the transportation and assignment problems.

c

34. Arcs in a transshipment problem a. must connect every node to a transshipment node. b. represent the cost of shipments. c. indicate the direction of the flow. d. All of these are correct.

c

37. Consider a shortest-route problem in which a bank courier must travel between branches and the main operations center. When represented with a network, a. the branches are the arcs and the operations center is the node. b. the branches are the nodes and the operations center is the source. c. the branches and the operations center are all nodes and the streets are the arcs. d. the branches are the network and the operations center is the node.

c

26. The objective of the transportation problem is to a. identify one origin that can satisfy total demand at the destinations and at the same time minimize totalshipping cost. b. minimize the number of origins used to satisfy total demand at the destinations. c. minimize the number of shipments necessary to satisfy total demand at the destinations. d. minimize the cost of shipping products from several origins to several destinations.

d

30. In the general linear programming model of the assignment problem, a. one agent can do parts of several tasks. b. one task can be done by several agents. c. each agent is assigned to its own best task. d. one agent is assigned to one and only one task.

d

36. In a transshipment problem, shipments a. cannot occur between two origin nodes. b. cannot occur between an origin node and a destination node. c. cannot occur between a transshipment node and a destination node. d. can occur between any two nodes.

d

42. Which of the following is NOT a characteristic of assignment problems? a. Costs appear in the objective function only. b. The RHS of all constraints is 1. c. The value of all decision variables is either 0 or 1. d. The signs of constraints are always less than or equal to

d

43. The network flows into and out of demand nodes are what makes the production and inventory application modeled in the textbook a a. shortest-route model. b. maximal flow model. c. transportation model. d. transshipment model.

d

10. A dummy origin in a transportation problem is used when supply exceeds demand. a. True b. False

false

12. In the LP formulation of a maximal flow problem, a conservation of flow constraint ensures that an arcs flow capacity is not exceeded. a. True b. False

false

17. A transportation problem with three sources and four destinations will have seven variables in the objective function. a. True b. False

false

19. In a transportation problem with total supply equal to total demand, if there are four origins and seven destinations, and there is a unique optimal solution, the optimal solution will utilize 11 shipping routes. t or f

false

20. In an assignment problem, one agent can be assigned to several tasks. a. True b. False

false

21. In a capacitated transshipment problem, some or all of the transfer points are subject to capacity restrictions. a. True b. False

false

3. A transportation problem with three sources and four destinations will have seven decision variables. a. True b. False

false

6. When the number of agents exceeds the number of tasks in an assignment problem, one or more dummy tasks must be introduced in the LP formulation or else the LP will not have a feasible solution. a. True b. False

false

11. When a route in a transportation problem is unacceptable, the corresponding variable can be removed from the LP formulation. a. True b. False

true

13. The maximal flow problem can be formulated as a capacitated transshipment problem and determines the maximum amount of flow (such as messages, vehicles, fluid, etc.) that can enter and exit a network system in a given period of time. a. True b. False

true

14. The direction of flow in the shortest-route problem is always out of the origin node and into the destination node. a. True b. False

true

15. A transshipment problem is a generalization of the transportation problem in which certain nodes are neither supply nodes nor destination nodes. a. True b. False

true

16. The assignment problem is a special case of the transportation problem in which one agent is assigned to one, and only one, task. a. True b. False

true

18. Flow in a transportation network is limited to one direction. a. True b. False

true

2. Converting a transportation problem LP from cost minimization to profit maximization requires only changing the objective function; the conversion does not affect the constraints. a. True b. False

true

22. In a transportation problem, excess supply will appear as slack in the linear programming solution. a. True b. False

true

23. There are two specific types of problems common in supply chain models that can be solved using linear programing: transportation problems and transshipment problems. a. True b. False

true

4. If a transportation problem has four origins and five destinations, the LP formulation of the problem will have nine constraints. a. True b. False

true

5. When a transportation problem has capacity limitations on one or more of its routes, it is known as a capacitated transportation problem. a. True b. False

true

7. A transshipment constraint must contain a variable for every arc entering or leaving the node. a. True b. False

true

8. The shortest-route problem is a special case of the transshipment problem. true false

true

9. Transshipment problems allow shipments both in and out of some nodes, while transportation problems do not. a. True b. False

true

61. A plant manager for a sporting goods manufacturer is in charge of assigning the manufacture of four new aluminum products to four different departments. Because of varying expertise and workloads, the different departments canproduce the new products at various rates. If only one product is to be produced by each department and the daily output rates are given in the table below, which department should manufacture which product to maximize total daily product output? (Note: Department 1 does not have the facilities to produce golf clubs.) ​ ​ Department Baseball Bats Tennis Rackets Golf Clubs Racquetball Rackets 1 100 60 X 80 2 100 80 140 100 3 110 75 150 120 4 85 50 100 75 ​ Formulate this assignment problem as a linear program.

xi represent all possible combinations of departments and products. For example: x 1 = 1 if Department 1 is assigned baseball bats; = 0 otherwise x2 = 1 if Department 1 is assigned tennis rackets; = 0 otherwise x5 = 1 if Department 2 is assigned baseball bats; = 0 otherwise x15 = 1 if Department 4 is assigned golf clubs; = 0 otherwise (Note: x 3 is not used because Dept.1/golf clubs is not a feasible assignment.) ​ Min Z = 100x 1 +60x 2 +80x 4 +100x 5 +80x 6 +140x 7 +100x 8 +110x 9 +75x 10 +150x 11 +120x 12 +85x 13 +50x 14 +100x 15 +75x 16 s.t. x 1 + x 2 + x 4 = 1 x 5 + x 6 + x 7 + x 8 = 1 x 9 + x 10 + x 11 + x 12 = 1 x 13 + x 14 + x 15 + x 16 = 1 x 1 + x 5 + x 9 + x 13 = 1 x 2 + x 6 + x 10 + x 14 = 1 x 7 + x 11 + x 15 = 1 x 4 + x 8 + x 12 + x 16 = 1