1. BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were graded. Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions.Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions. Review

2. BA 452 Lesson B.1 Transportation 2 2ReadingsReadings Chapter 6 Distribution and Network Models

3. BA 452 Lesson B.1 Transportation 3 3OverviewOverview

4. 4 4Overview Network Models are nodes, arcs, and functions (costs, supplies, demands, etc.) associated with the arcs and nodes, as in transportation, assignment, transshipment, and shortest-route problems. Transportation Problems are Resource Allocation Problems when outputs are fixed, and when outputs and inputs occur at different locations, so goods must be transported from origins to destinations. Transportation Problems with Modes of Transport re-interpret some of the different “origins” in a basic transportation problem to include not only location but modes of transportation (truck, rail, …). Assignment Problems are Transportation Problems when the “goods” are workers that are transported to jobs, and each worker either does all of a job or none of it, so the fraction completed is binary.

5. BA 452 Lesson B.1 Transportation 5 5 Tool Summary n Write the objective of maximizing a minimum as a linear program. For example, maximize min {2x, 3y} as maximize M subject to 2x > M and 3y > M. For example, maximize min {2x, 3y} as maximize M subject to 2x > M and 3y > M. n Define decision variable x ij = units moving from origin i to destination j. n Write origin constraints (with < or =): n Write destination constraints (with < or =): Overview

6. BA 452 Lesson B.1 Transportation 6 6 Tool Summary n Identify implicit assumptions needed to complete a formulation, such as all agents having an equal value of time. Overview

7. BA 452 Lesson B.1 Transportation 7 7 Network Models

8. BA 452 Lesson B.1 Transportation 8 8 Network Models Network Models are nodes, arcs, and functions (costs, supplies, demands, etc.) associated with the arcs and nodes. Transportation, assignment, transshipment, and shortest-route problems are examples.

9. BA 452 Lesson B.1 Transportation 9 9 n Each of the four network models (transportation, assignment, transshipment, and shortest-route problems) can be formulated as linear programs and solved by general-purpose linear programming codes. n For each of the four models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be integer values for the decision variables. n There are many computer packages (including The Management Scientist) that contain convenient separate computer codes for these models, which take advantage of their network structure. But do not use such codes on exams because they lack the flexibility of the general- purpose linear programming codes. Network Models

10. BA 452 Lesson B.1 Transportation 10 TransportationTransportation

11. BA 452 Lesson B.1 Transportation 11 TransportationOverview Transportation Problems are Resource Allocation Problems when outputs are fixed, and when outputs and inputs occur at different locations. Transportation Problems thus help determine the transportation of goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ) to minimize cost.

12. BA 452 Lesson B.1 Transportation 12 2 2 c 11 c 12 c 13 c 21 c 22 c 23 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s2s2s2s2 SourcesDestinations 3 3 2 2 1 1 1 1Transportation Here is the network representation for a transportation problem with two sources and three destinations.

13. BA 452 Lesson B.1 Transportation 13 x ij = number of units shipped from origin i to destination j x ij = number of units shipped from origin i to destination j c ij = cost per unit of shipping from origin i to destination j c ij = cost per unit of shipping from origin i to destination j s i = supply or capacity in units at origin i s i = supply or capacity in units at origin i d j = demand in units at destination j d j = demand in units at destination j n Notation: x ij > 0 for all i and j = n Linear programming formulation (supply inequality, demand equality). Transportation

14. BA 452 Lesson B.1 Transportation 14 Possible variations: Minimum shipping guarantee from i to j: Minimum shipping guarantee from i to j: x ij > L ij x ij > L ij Maximum route capacity from i to j: Maximum route capacity from i to j: x ij < L ij x ij < L ij Unacceptable route: Unacceptable route: Remove the corresponding decision variable. Remove the corresponding decision variable.Transportation

15. BA 452 Lesson B.1 Transportation 15 Northwood Westwood Eastwood Northwood Westwood Eastwood Plant 1 24 30 40 Plant 1 24 30 40 Plant 2 30 40 42 Plant 2 30 40 42 Question: Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme has two plants, each of which can produce 50 tons per week. Delivery costs per ton from each plant to each suburban location are thus: Formulate then solve the linear program that determines how shipments should be made to fill the orders above. Transportation

16. BA 452 Lesson B.1 Transportation 16 Answer: Linear programming formulation (supply inequality, demand equality). n Variables: Xij = Tons shipped from Plant i to Destination j n Objective: Min 24 X11 + 30 X12 + 40 X13 + 30 X21 + 40 X22 + 42 X23 n Supply Constraints: X11 + X12 + X13 < 50 X21 + X22 + X23 < 50 n Demand Constraints: X11 + X21 = 25 X12 + X22 = 45 X13 + X23 = 10 Transportation

17. BA 452 Lesson B.1 Transportation 17 Transportation

18. BA 452 Lesson B.1 Transportation 18 n Define sources: Source 1 = Plant 1, Source 2 = Plant 2. n Define destinations: 1 = Northwood, 2 = Westwood, 3 = Eastwood. n Define costs: n Define 2 supplies: s 1 = 50, s 2 = 50. n Define 3 demands: d 1 = 25, d 2 = 45, d 3 = 10. n Define variables: Xij = number of units shipped from Source i to Destination j. shipped from Source i to Destination j. 2+3=5 2x3 = 6 Supply s 1 = 50 Demand d 2 = 45 Cost c 13 = 40 c 11 = 24 c 12 = 30 c 13 = 40 c 21 = 30 c 22 = 40 c 23 = 42 Transportation

19. BA 452 Lesson B.1 Transportation 19 Optimal shipments: From To Amount Cost From To Amount Cost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Plant 2 Eastwood 10 420 Total Cost = $2,490 Total Cost = $2,490 Variable names: Xij = number of units shipped from Plant i to Destination j. Destination 1 = Northwood; 2 = Westwood; 3 = Eastwood Transportation

20. BA 452 Lesson B.1 Transportation 20 Transportation

21. BA 452 Lesson B.1 Transportation 21 Optimal shipments: From To Amount Cost From To Amount Cost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Plant 2 Eastwood 10 420 Total Cost = $2,490 Total Cost = $2,490 Variable names: Origin i = Plant i Origin i = Plant i Destination 1 = Northwood Destination 1 = Northwood Destination 2 = Westwood Destination 2 = Westwood Destination 3 = Eastwood Destination 3 = Eastwood Cost from Plant 1 to Northwood Transportation

22. BA 452 Lesson B.1 Transportation 22 Transportation with Modes of Transport

23. BA 452 Lesson B.1 Transportation 23 Overview Transportation Problems with Modes of Transport re- interpret some of the different “origins” in a basic transportation problem to include not only location but modes of transportation. For example, instead of “San Diego” as an origin specifying location, we have “San Diego by Truck” as an origin specifying both location and mode of transportation. Transportation with Modes of Transport

24. BA 452 Lesson B.1 Transportation 24 Question: The Navy has 9,000 pounds of material in Albany, Georgia that it wishes to ship to three installations: San Diego, Norfolk, and Pensacola. They require 4,000, 2,500, and 2,500 pounds, respectively. Government regulations require equal distribution of shipping among the three carriers. n The shipping costs per pound by truck, railroad, and airplane are: n Formulate then solve the linear program that determines shipping arrangements (mode, destination, and quantity) that minimize the total shipping cost. Destination Destination Mode San Diego Norfolk Pensacola Truck $12 $ 6 $ 5 Railroad $20 $11 $ 9 Airplane $30 $26 $28 Transportation with Modes of Transport

25. BA 452 Lesson B.1 Transportation 25 Transportation with Modes of Transport

26. BA 452 Lesson B.1 Transportation 26 n Define the variables. We want to determine the pounds of material, x ij, to be shipped by mode i to destination j. n Variable names: n Define the objective. Minimize the total shipping cost. Min: (shipping cost per pound for each mode-destination pairing) x (number of pounds shipped by mode-destination pairing). Min: (shipping cost per pound for each mode-destination pairing) x (number of pounds shipped by mode-destination pairing). Min: 12x 11 + 6x 12 + 5x 13 + 20x 21 + 11x 22 + 9x 23 Min: 12x 11 + 6x 12 + 5x 13 + 20x 21 + 11x 22 + 9x 23 + 30x 31 + 26x 32 + 28x 33 + 30x 31 + 26x 32 + 28x 33 San Diego Norfolk Pensacola San Diego Norfolk Pensacola Truck x 11 x 12 x 13 Truck x 11 x 12 x 13 Railroad x 21 x 22 x 23 Railroad x 21 x 22 x 23 Airplane x 31 x 32 x 33 Airplane x 31 x 32 x 33 Transportation with Modes of Transport

27. BA 452 Lesson B.1 Transportation 27 n Define the constraints of equal use of transportation modes: (1) x 11 + x 12 + x 13 = 3000 (1) x 11 + x 12 + x 13 = 3000 (2) x 21 + x 22 + x 23 = 3000 (2) x 21 + x 22 + x 23 = 3000 (3) x 31 + x 32 + x 33 = 3000 (3) x 31 + x 32 + x 33 = 3000 n Define the destination material constraints: (4) x 11 + x 21 + x 31 = 4000 (4) x 11 + x 21 + x 31 = 4000 (5) x 12 + x 22 + x 32 = 2500 (5) x 12 + x 22 + x 32 = 2500 (6) x 13 + x 23 + x 33 = 2500 (6) x 13 + x 23 + x 33 = 2500 Transportation with Modes of Transport

28. BA 452 Lesson B.1 Transportation 28 Linear programming summary. n Variables: Xij = Pounds shipped by Mode i to Destination j n Objective: Min 12 X11 + 6 X12 + 5 X13 + 20 X21 + 11 X22 + 9 X23 + 30 X31 + 26 X32 + 28 X33 + 20 X21 + 11 X22 + 9 X23 + 30 X31 + 26 X32 + 28 X33 n Mode (Supply equality) Constraints: X11 + X12 + X13 = 3000 X21 + X22 + X23 = 3000 X31 + X32 + X33 = 3000 n Destination Constraints: X11 + X21 + X31 = 4000 X12 + X22 + X32 = 2500 X13 + X23 + X33 = 2500 Transportation with Modes of Transport

29. BA 452 Lesson B.1 Transportation 29 San Diego Norfolk Pensacola San Diego Norfolk Pensacola Truck X11 X12 X13 Truck X11 X12 X13 Railroad X21 X22 X23 Railroad X21 X22 X23 Airplane X31 X32 X33 Airplane X31 X32 X33 Solution Summary: San Diego receives 1000 lbs. by truck San Diego receives 1000 lbs. by truck and 3000 lbs. by airplane. Norfolk receives 2000 lbs. by truck Norfolk receives 2000 lbs. by truck and 500 lbs. by railroad. and 500 lbs. by railroad. Pensacola receives 2500 lbs. by railroad. Pensacola receives 2500 lbs. by railroad. The total shipping cost is $142,000. The total shipping cost is $142,000. Variable names: Units to San Diego by truck Transportation with Modes of Transport

30. BA 452 Lesson B.1 Transportation 30 The Management Science Transportation module is not available. Remember, in that formulation, the supply constraints are inequalities. x ij > 0 for all i and j But in Example 2, the “origins” are the modes of shipping, and the supply constraint on each mode is an equality. = Transportation with Modes of Transport

31. BA 452 Lesson B.1 Transportation 31 AssignmentAssignment

32. BA 452 Lesson B.1 Transportation 32 Overview Assignment Problems are Transportation Problems when the “goods” are workers that are transported to jobs, and each worker either does all of a job or none of it. Assignment Problems thus minimize the total cost of assigning of m workers (or agents) to m jobs (or tasks). The simplest way to model all-or-nothing in any linear program is to restrict the fraction of the job completed to be a binary (0 or 1) decision variable. Assignment

33. BA 452 Lesson B.1 Transportation 33 An assignment problem is thus a special case of a transportation problem in which all supplies and all demands equal to 1; hence assignment problems may be solved as linear programs. And although the only sensible solution quantities are binary (0 or 1), the special form of the problem and of The Management Scientist guarantees all solutions are binary (0 or 1). Assignment

34. BA 452 Lesson B.1 Transportation 34 22 33 11 22 33 11 c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 AgentsTasks Here is the network representation of an assignment problem with three workers (agents) and three jobs (tasks): Assignment

35. BA 452 Lesson B.1 Transportation 35 Notation: Notation: x ij = 1 if agent i is assigned to task j x ij = 1 if agent i is assigned to task j 0 otherwise 0 otherwise c ij = cost of assigning agent i to task j c ij = cost of assigning agent i to task j x ij > 0 for all i and j s.t.Assignment

36. BA 452 Lesson B.1 Transportation 36 Possible variations: Number of agents exceeds the number of tasks: Number of agents exceeds the number of tasks: Extra agents simply remain unassigned. An assignment is unacceptable: An assignment is unacceptable: Remove the corresponding decision variable. Remove the corresponding decision variable. An agent is permitted to work t tasks: An agent is permitted to work t tasks:Assignment

37. BA 452 Lesson B.1 Transportation 37 Question: Russell electrical contractors pay their subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. Projects Projects Subcontractor A B C Westside 50 36 16 Federated 28 30 18 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 Universal 25 25 14 Assume each subcontractor can perform at most one project. Formulate then solve the linear program that assigns contractors to minimize total mileage costs. Assignment

38. BA 452 Lesson B.1 Transportation 38 50 36 16 28 30 18 35 32 20 25 25 14 West.West. CC BB AA Univ.Univ. Gol.Gol. Fed. Fed. Projects SubcontractorsAssignmentAnswer:

39. BA 452 Lesson B.1 Transportation 39 Project A Project B Project C Project A Project B Project C Westside x 11 x 12 x 13 Westside x 11 x 12 x 13 Federated x 21 x 22 x 23 Federated x 21 x 22 x 23 Goliath x 31 x 32 x 33 Goliath x 31 x 32 x 33 Universal x 41 x 42 x 43 Universal x 41 x 42 x 43 Variable names: n There will be 1 variable for each agent-task pair, so 12 variables all together. n There will be 1 constraint for each agent and for each task, so 7 constraints all together. Assignment

40. BA 452 Lesson B.1 Transportation 40 Min 50x 11 +36x 12 +16x 13 +28x 21 +30x 22 +18x 23 Min 50x 11 +36x 12 +16x 13 +28x 21 +30x 22 +18x 23 +35x 31 +32x 32 +20x 33 +25x 41 +25x 42 +14x 43 +35x 31 +32x 32 +20x 33 +25x 41 +25x 42 +14x 43 s.t. x 11 +x 12 +x 13 < 1 s.t. x 11 +x 12 +x 13 < 1 x 21 +x 22 +x 23 < 1 x 21 +x 22 +x 23 < 1 x 31 +x 32 +x 33 < 1 x 31 +x 32 +x 33 < 1 x 41 +x 42 +x 43 < 1 x 41 +x 42 +x 43 < 1 x 11 +x 21 +x 31 +x 41 = 1 x 11 +x 21 +x 31 +x 41 = 1 x 12 +x 22 +x 32 +x 42 = 1 x 12 +x 22 +x 32 +x 42 = 1 x 13 +x 23 +x 33 +x 43 = 1 x 13 +x 23 +x 33 +x 43 = 1 x ij = 0 or 1 for all i and j x ij = 0 or 1 for all i and j Agents Tasks Project A Project B Project C Project A Project B Project C Westside x 11 x 12 x 13 Westside x 11 x 12 x 13 Federated x 21 x 22 x 23 Federated x 21 x 22 x 23 Goliath x 31 x 32 x 33 Goliath x 31 x 32 x 33 Universal x 41 x 42 x 43 Universal x 41 x 42 x 43 Variable names: Assignment

41. BA 452 Lesson B.1 Transportation 41 Project A Project B Project C Project A Project B Project C Westside x 11 x 12 x 13 Westside x 11 x 12 x 13 Federated x 21 x 22 x 23 Federated x 21 x 22 x 23 Goliath x 31 x 32 x 33 Goliath x 31 x 32 x 33 Universal x 41 x 42 x 43 Universal x 41 x 42 x 43 Agent 1 capacity: x 11 +x 12 +x 13 < 1 Task 3 done: x 13 +x 23 +x 33 +x 43 = 1 Assignment

42. BA 452 Lesson B.1 Transportation 42 Project A Project B Project C Project A Project B Project C Westside x 11 x 12 x 13 Westside x 11 x 12 x 13 Federated x 21 x 22 x 23 Federated x 21 x 22 x 23 Goliath x 31 x 32 x 33 Goliath x 31 x 32 x 33 Universal x 41 x 42 x 43 Universal x 41 x 42 x 43 Variable names: Optimal assignment: Subcontractor Project Distance Subcontractor Project Distance Westside C 16 Westside C 16 Federated A 28 Federated A 28 Goliath (unassigned) Goliath (unassigned) Universal B 25 Universal B 25 Total distance = 69 miles Total distance = 69 milesAssignment

43. BA 452 Lesson B.1 Transportation 43 Assignment

44. BA 452 Lesson B.1 Transportation 44 Projects Projects Subcontractor A B C Westside 50 36 16 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 Optimal assignment: Subcontractor Project Distance Subcontractor Project Distance Westside C 16 Westside C 16 Federated A 28 Federated A 28 Goliath (unassigned) Goliath (unassigned) Universal B 25 Universal B 25 Total distance = 69 miles Total distance = 69 milesAssignment

45. BA 452 Lesson B.1 Transportation 45 Question: Now change Example 3 to take into account the recent marriage of the Goliath subcontractor to your youngest daughter. That is, you have to assign Goliath one of the jobs. How should the contractors now be assigned to minimize total mileage costs? Assignment

46. BA 452 Lesson B.1 Transportation 46 Alternative notation: WA = 0 if Westside does not get task A 1 if Westside does get task A 1 if Westside does get task A and so on. Min 50WA+36WB+16WC+28FA+30FB+18FC Min 50WA+36WB+16WC+28FA+30FB+18FC +35GA+32GB+20GC+25UA+25UB+14UC +35GA+32GB+20GC+25UA+25UB+14UC s.t. WA+WB+WC < 1 s.t. WA+WB+WC < 1 FA+FB+FC < 1 FA+FB+FC < 1 GA+GB+GC = 1 GA+GB+GC = 1 UA+UB+UC < 1 UA+UB+UC < 1 WA+FA+GA+UA = 1 WA+FA+GA+UA = 1 WB+FB+GB+UB = 1 WB+FB+GB+UB = 1 WC+FC+GC+UC = 1 WC+FC+GC+UC = 1 Agents TasksAssignment

47. BA 452 Lesson B.1 Transportation 47 Goliath gets a task: GA+GB+GC = 1 Task A gets done: WA+FA+GA+UA=1 Assignment

48. BA 452 Lesson B.1 Transportation 48 Optimal assignment: Subcontractor Project Distance Subcontractor Project Distance Westside C 16 Westside C 16 Federated (unassigned) Federated (unassigned) Goliath B 32 Goliath B 32 Universal A 25 Universal A 25 Total distance = 73 miles Total distance = 73 milesAssignment

49. BA 452 Lesson B.1 Transportation 49 BA 452 Quantitative Analysis End of Lesson B.1