Slides by John Loucks St. Edward’s University

1. Slides by John Loucks St. Edward’s University Modifications by A. Asef-Vaziri

2. Chapter 6, Part ADistribution and Network Models • Transportation Problem • Network Representation • General LP Formulation • Assignment Problem • Network Representation • General LP Formulation • Transshipment Problem • Network Representation • General LP Formulation

3. Transportation, Assignment, and Transshipment Problems • A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes. • Transportation, assignment, transshipment, shortest-route, and maximal flow problems of this chapter as well as the minimal spanning tree and PERT/CPM problems (in others chapter) are all examples of network problems.

4. Transportation, Assignment, and Transshipment Problems • Each of the five problems of this chapter can be formulated as linear programs and solved by general purpose linear programming codes. • For each of the five problems, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables. • However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.

5. Transportation Problem • The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij. • The network representation for a transportation problem with two sources and three destinations is given on the next slide.

6. Transportation Problem • Network Representation 1 d1 c11 1 c12 s1 c13 2 d2 c21 c22 2 s2 c23 3 d3 Sources Destinations

7. Transportation Problem: Example #1 Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown on the next slide. How should end of week shipments be made to fill the above orders?

8. Transportation Problem: Example #1 • Delivery Cost Per Ton NorthwoodWestwoodEastwood Plant 1 24 30 40 Plant 2 30 40 42 Decision Variables. the tons of concrete blocks, xij, to be shipped from source i to destination j. NorthwoodWestwoodEastwood Plant 1 x11x12x13 Plant 1 x21x22x23

9. Transportation Problem: Example #2 • Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per ton for each origin to destination) × (number of pounds shipped from each origin to each destination). Min: 24x11+ 30x12+ 40x13+ 30x21+ 40x22+ 42x23

10. Transportation Problem: Example #2 = Constraints • Define the Constraints Supply constraints: (1) x11 + x12 + x13= 50 (2) x21 + x22 + x23= 50 Demand constraints: (4) x11 + x21 = 25 (5) x12 + x22 = 45 (6) x13 + x23 = 10 Non-negativity of variables: xij> 0, i = 1, 2 and j = 1, 2, 3

11. Transportation Problem: Example #2 ≤ and ≥ Constraints • Define the Constraints Supply constraints: (1) x11 + x12 + x13≤ 50 (2) x21 + x22 + x23 ≤ 50 Demand constraints: (4) x11 + x21 ≥ 25 (5) x12 + x22 ≥ 45 (6) x13 + x23 ≥ 10 Non-negativity of variables: xij> 0, i = 1, 2 and j = 1, 2, 3

12. Transportation Problem: Example #1 • Partial Spreadsheet Showing Problem Data

13. Transportation Problem: Example #1

14. Transportation Problem: Example #1 Sensitivity Report

15. Transportation Problem: Example #2 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. The shipping costs per pound for truck, railroad, and airplane transit are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and quantity) that will minimize the total shipping cost.

16. Transportation Problem: Example #2 Destination Mode San Diego Norfolk Pensacola Truck $12 $ 6 $ 5 Railroad 20 11 9 Airplane 30 26 28

17. Transportation Problem: Example #2 • Define the Decision Variables We want to determine the pounds of material, xij, to be shipped by mode i to destination j. The following table summarizes the decision variables: San Diego Norfolk Pensacola Truckx11x12x13 Railroad x21x22x23 Airplane x31x32x33

18. Transportation Problem: Example #2 • Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing). Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23 + 30x31 + 26x32 + 28x33

19. Transportation Problem: Example #2 • Define the Constraints Equal use of transportation modes: (1) x11 + x12 + x13 = 3000 (2) x21 + x22 + x23 = 3000 (3) x31 + x32 + x33 = 3000 Destination material requirements: (4) x11 + x21 + x31 = 4000 (5) x12 + x22 + x32 = 2500 (6) x13 + x23 + x33 = 2500 Non-negativity of variables: xij> 0, i = 1, 2, 3 and j = 1, 2, 3

20. Transportation Problem: Example #2

21. Transportation Problem • Linear Programming Formulation Using the notation: xij = number of units shipped from origin i to destination j cij= cost per unit of shipping from origin i to destination j si = supply or capacity in units at origin i dj = demand in units at destination j continued

22. Transportation Problem • Linear Programming Formulation (continued) > xij> 0 for all i and j

23. Transportation Problem • LP Formulation Special Cases • Total supply exceeds total demand: • Total demand exceeds total supply: Add a dummy origin with supply equal to the shortage amount. Assign a zero shipping cost per unit. The amount “shipped” from the dummy origin (in the solution) will not actually be shipped. No modification of LP formulation is necessary.

24. Transportation Problem • LP Formulation Special Cases (continued) • The objective is maximizing profit or revenue: • Minimum shipping guarantee from i to j: xij>Lij • Maximum route capacity from i to j: xij<Lij • Unacceptable route: Remove the corresponding decision variable. Solve as a maximization problem.

25. Assignment Problem • An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij. • It assumes all workers are assigned and each job is performed. • An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs. • The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

26. Assignment Problem • Network Representation c11 1 1 c12 c13 Agents Tasks c21 c22 2 2 c23 c31 c32 3 3 c33

27. Assignment Problem: Example An electrical contractor pays his 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 SubcontractorABC Westside 50 36 16 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 How should the contractors be assigned so that total mileage is minimized?

28. Assignment Problem: Example • Network Representation 50 West. A 36 16 Subcontractors Projects 28 30 Fed. B 18 32 35 Gol. C 20 25 25 Univ. 14

29. Assignment Problem: Example • Linear Programming Formulation Min 50x11+36x12+16x13+28x21+30x22+18x23 +35x31+32x32+20x33+25x41+25x42+14x43 s.t.x11+x12+x13 < 1 x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j Agents Tasks

30. Assignment Problem: Example • Linear Programming Formulation Min 50x11+36x12+16x13+28x21+30x22+18x23 +35x31+32x32+20x33+25x41+25x42+14x43 s.t.x11+x12+x13 < 1 x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 ≥ 1 x12+x22+x32+x42 ≥ 1 x13+x23+x33+x43 ≥ 1 xij = 0 or 1 for all i and j Agents Tasks

31. Transportation Problem: Example #2

32. Assignment Problem: Example • The optimal assignment is: SubcontractorProjectDistance Westside C 16 Federated A 28 Goliath (unassigned) Universal B 25 Total Distance = 69 miles

33. Assignment Problem • Linear Programming Formulation Using the notation: xij = 1 if agent i is assigned to task j 0 otherwise cij= cost of assigning agent i to task j continued

34. Assignment Problem • Linear Programming Formulation (continued) xij> 0 for all i and j

35. Assignment Problem • LP Formulation Special Cases • Number of agents exceeds the number of tasks: • Number of tasks exceeds the number of agents: • Add enough dummy agents to equalize the • number of agents and the number of tasks. • The objective function coefficients for these • new variable would be zero. Extra agents simply remain unassigned.

36. Assignment Problem • LP Formulation Special Cases (continued) • The assignment alternatives are evaluated in terms of revenue or profit: • Solve as a maximization problem. • An assignment is unacceptable: Remove the corresponding decision variable. • An agent is permitted to work t tasks:

37. Transshipment Problem • Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes)before reaching a particular destination node. • Transshipment problems can be converted to larger transportation problems and solved by a special transportation program. • Transshipment problems can also be solved by general purpose linear programming codes. • The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

38. Transshipment Problem • Network Representation c36 3 c13 c37 1 6 s1 d1 c14 c46 c15 4 Demand c47 Supply c23 c56 c24 7 2 d2 s2 c25 5 c57 Destinations Sources Intermediate Nodes

39. Transshipment Problem • Linear Programming Formulation Using the notation: xij = number of units shipped from node i to node j cij = cost per unit of shipping from node i to node j si= supply at origin node i dj= demand at destination node j continued

40. Transshipment Problem • Linear Programming Formulation (continued) xij> 0 for all i and j continued

41. Transshipment Problem • LP Formulation Special Cases • Total supply not equal to total demand • Maximization objective function • Route capacities or route minimums • Unacceptable routes • The LP model modifications required here are • identical to those required for the special cases in • the transportation problem.

42. Transshipment Problem: Example The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers. Additional data is shown on the next slide.

43. Transshipment Problem: Example Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: Zeron NZeron S Arnold 5 8 Supershelf 7 4 The costs to install the shelving at the various locations are: ZroxHewesRockrite Zeron N 1 5 8 Zeron S 3 4 4

44. Transshipment Problem: Example • Network Representation ZROX Zrox 50 1 5 Zeron N Arnold 75 ARNOLD 5 8 8 Hewes 60 HEWES 3 7 Super Shelf Zeron S 4 75 WASH BURN 4 4 Rock- Rite 40

45. Transshipment Problem: Example • Linear Programming Formulation • Decision Variables Defined xij = amount shipped from manufacturer i to supplier j xjk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) • Objective Function Defined Minimize Overall Shipping Costs: Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47

46. Transshipment Problem: Example • Constraints Defined Amount Out of Arnold: x13 + x14< 75 Amount Out of Supershelf: x23 + x24< 75 Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0 Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0 Amount Into Zrox: x35 + x45 = 50 Amount Into Hewes: x36 + x46 = 60 Amount Into Rockrite: x37 + x47 = 40 Non-negativity of Variables: xij> 0, for all i and j.

47. Transshipment Problem: Example

48. Transshipment Problem: Example • Solution Zrox ZROX 50 50 75 1 5 Zeron N Arnold 75 ARNOLD 5 25 8 8 Hewes 60 35 HEWES 3 4 7 Super Shelf Zeron S 40 75 WASH BURN 4 4 75 Rock- Rite 40

49. Transshipment Problem: Example

50. Transshipment Problem: Example • Computer Output (continued) ConstraintSlack/SurplusDual Values 1 0.000 0.000 2 0.000 2.000 3 0.000 -5.000 4 0.000 -6.000 5 0.000 -6.000 6 0.000 -10.000 7 0.000 -10.000