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Chapter 6: Transportation, Assignment, and Transshipment Problems

Chapter 6: 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.

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Chapter 6: Transportation, Assignment, and Transshipment Problems

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  1. Chapter 6: 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. Examples include transportation, assignment, transshipment as well as shortest-route, maximal flow problems, minimal spanning tree and PERT/CPM problems. All network problems can be formulated as linear programs. However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure. If the right-hand side of the linear programming formulations are all integers, then optimal solution of the decision variables will also be integers.

  2. 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.

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

  4. 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

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

  6. Example: Transportation Problem The Navy has depots in Albany, BenSalem, and Winchester. Each of these three depots has 3,000 pounds of materials which the Navy wishes to ship to three installations, namely, San Diego, Norfolk, and Pensacola. These installations require 4,000, 2,500, and 2,500 pounds, respectively. The shipping costs per pound for are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements that will minimize the total shipping cost.

  7. Example: Transportation Problem (Continued) Destination Source San Diego Norfolk Pensacola Albany $12 $ 6 $ 5 BenSalem 20 11 9 Winchester 30 26 28

  8. Transportation Problem: Network Representation Source Destination c11 1 1 Albany 3000 San Diego 4000 c12 c13 c21 c22 BenSalem 3000 2 2 Norfolk 2500 c23 c31 c32 Winchester 3000 3 3 Pensacola 2500 c33

  9. Example: Transportation Problem (Continued) 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 Albanyx11x12x13 BenSalem x21x22x23 Winchester x31x32x33

  10. Example: Transportation Problem (Continued) 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

  11. Transportation Problem: Example #2 Define the Constraints Source availability: (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

  12. Example: Transportation Problem (Continued) Computer Output OBJECTIVE FUNCTION VALUE = 142000.000 VariableValueReduced Cost x11 1000.000 0.000 x12 2000.000 0.000 x13 0.000 1.000 x21 0.000 3.000 x22 500.000 0.000 x23 2500.000 0.000 x31 3000.000 0.000 x32 0.000 2.000 x33 0.000 6.000

  13. Transportation Problem: Example #2 Solution Summary • San Diego will receive 1000 lbs. from Albany • and 3000 lbs. from Winchester. • Norfolk will receive 2000 lbs. from Albany • and 500 lbs. from BenSalem. • Pensacola will receive 2500 lbs. from BenSalem. • The total shipping cost will be $142,000.

  14. 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. • Assign a zero shipping cost per unit • Maximum route capacity from i to j: • xij<Li • Remove the corresponding decision variable. No modification of LP formulation is necessary.

  15. 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. Transportation Problem Solve as a maximization problem.

  16. 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.

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

  18. 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

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

  20. Example: Assignment Problem 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?

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

  22. 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

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

  24. 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.

  25. 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:

  26. 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.

  27. 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

  28. 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

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

  30. 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.

  31. 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.

  32. 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 Thomas 1 5 8 Washburn 3 4 4

  33. 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

  34. 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

  35. 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.

  36. Transshipment Problem: Example Computer Output Objective Function Value = 1150.000 VariableValueReduced Costs X13 75.000 0.000 X14 0.000 2.000 X23 0.000 4.000 X24 75.000 0.000 X35 50.000 0.000 X36 25.000 0.000 X37 0.000 3.000 X45 0.000 3.000 X46 35.000 0.000 X47 40.000 0.000

  37. 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

  38. 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

  39. Transshipment Problem: Example Computer Output (continued) OBJECTIVE COEFFICIENT RANGES VariableLower LimitCurrent ValueUpper Limit X13 3.000 5.000 7.000 X14 6.000 8.000 No Limit X23 3.000 7.000 No Limit X24 No Limit 4.000 6.000 X35 No Limit 1.000 4.000 X36 3.000 5.000 7.000 X37 5.000 8.000 No Limit X45 0.000 3.000 No Limit X46 2.000 4.000 6.000 X47 No Limit 4.000 7.000

  40. Transshipment Problem: Example Computer Output (continued) RIGHT HAND SIDE RANGES ConstraintLower LimitCurrent ValueUpper Limit 1 75.000 75.000 No Limit 2 75.000 75.000 100.000 3 -75.000 0.000 0.000 4 -25.000 0.000 0.000 5 0.000 50.000 50.000 6 35.000 60.000 60.000 7 15.000 40.000 40.000

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