Transportation, Assignment, and Transshipment

1. Transportation, Assignment, and Transshipment Professor Ahmadi

2. Chapter 7Transportation, Assignment, and Transshipment Problems • The Transportation Problem: The Network Model and a Linear Programming Formulation • The Assignment Problem: The Network Model and a Linear Programming Formulation • The Transshipment Problem: The Network Model and a Linear Programming 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, and transshipment problems are all examples of network problems.

4. Transportation, Assignment, and Transshipment Problems • Each of the three models of this chapter (transportation, assignment, and transshipment models) can be formulated as linear programs. • For each of the three models, 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. • These three models can also be solved using a standard computer spreadsheet package.

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 • LP Formulation The linear programming formulation in terms of the amounts shipped from the origins to the destinations, xij, can be written as: Min SScijxij i j s.t. Sxij<si for each origin i j Sxij = dj for each destination j i xij> 0 for all i and j

8. Transportation Problem • LP Formulation Special Cases The following special-case modifications to the linear programming formulation can be made: • Minimum shipping guarantees from i to j: xij>Lij • Maximum route capacity from i to j: xij<Lij • Unacceptable routes: delete the variable

9. Example: BBC-1 Building Brick Company (BBC) has orders for 80 tons of bricks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. BBC has two plants. Plant 1 produces 50 and plant 2 produces 30 tons per week. How should end of week shipments be made to fill the above orders given the following delivery cost per ton: NorthwoodWestwoodEastwood Plant 1 24 30 40 Plant 2 30 40 42

10. Example: BBC-1 • LP Formulation • Decision Variables Defined xij = amount shipped from plant i to suburb j where i = 1 (Plant 1) and 2 (Plant 2) j = 1 (Northwood), 2 (Westwood), and 3 (Eastwood)

11. Transportation Problem • Network Representation of BBC-1 Northwood 1 25 24 Plant 1 50 30 Westwood 2 45 40 30 Plant 2 40 30 42 Eastwood 3 10 DESTINATIONS SOURCES

12. Example: BBC-1 • LP Formulation • Objective Function Minimize total shipping cost per week: Min 24x11 + 30x12 + 40x13 + 30x21 + 40x22 + 42x23 • Constraints s.t. x11 + x12 + x13< 50 (Plant 1 capacity) x21 + x22 + x23 < 30 (Plant 2 capacity) x11 + x21 = 25 (Northwood demand) x12 + x22 = 45 (Westwood demand) x13 + x23 = 10 (Eastwood demand) all xij> 0 (Non-negativity)

13. Example: BBC-1 • Optimal Solution FromToAmountCost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Total Cost = $2,490

14. 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.

15. Assignment Problem • Network Representation c11 1 1 c12 c13 c21 c22 2 2 c23 c31 c32 3 3 c33 WORKERS JOBS

16. Assignment Problem • Linear Programming Formulation Min SScijxij i j s.t. Sxij = 1 for each worker i j Sxij = 1 for each job j i xij = 0 or 1 for all i and j. • Note: A modification to the right-hand side of the first constraint set can be made if a worker is permitted to work more than one job.

17. Example: Assignment A 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. Project ABC Westside 50 36 16 Subcontractors Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 How should the contractors be assigned to minimize total distance (and total cost)?

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

19. Example: Assignment • LP Formulation • Decision Variables Defined xij = 1 if subcontractor i is assigned to project j = 0 otherwise where: i = 1 (Westside), 2 (Federated), 3 (Goliath), and 4 (Universal) j = 1 (A), 2 (B), and 3 (C)

20. Example: Assignment • LP Formulation • Objective Function Minimize total distance: Min 50x11 + 36x12 + 16x13 + 28x21 + 30x22 + 18x23 + 35x31 + 32x32 + 20x33 + 25x41 + 25x42 + 14x43

21. Example: Assignment • LP Formulation • Constraints x11 + x12 + x13< 1 (no more than one x21 + x22 + x23< 1 project assigned x31 + x32 + x33< 1 to any one x41 + x42 + x43< 1 subcontractor) x11 + x21 + x31 + x41 = 1 (each project must x12 + x22 + x32 + x42 = 1 be assigned to just x13 + x23 + x33 + x43 = 1 one subcontractor) all xij> 0 (non-negativity)

22. Example: Assignment • Optimal Assignment SubcontractorProjectDistance Westside C 16 Federated A 28 Universal B 25 Goliath (unassigned) Total Distance = 69 miles

23. Variations of Assignment Problem • Total number of agents not equal to total number of tasks • Maximization objective function • Unacceptable assignments

24. Transshipment Problem • Transshipment problemsare 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 as linear programs. • The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

25. Transshipment Problem • Network Representation c36 3 c13 c37 1 6 s1 d1 c14 c46 c15 4 c47 c23 c24 7 c56 2 d2 s2 c25 5 c57 SOURCES INTERMEDIATE NODES DESTINATIONS

26. Transshipment Problem • Linear Programming Formulation xij represents the shipment from node i to node j Min S cijxij all arcs s.t. S xij - S xij<si for each origin node i arcs outarcs in S xij - S xij= 0 for each intermediate arcs outarcs in node S xij - S xij = -djfor each destination arcs outarcs in node j (Note the order) xij> 0 for all i and j

27. Example: Transshipping Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) 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 Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers. Additional data is shown on the next slide.

28. Example: Transshipping Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: ThomasWashburn Arnold 5 8 Supershelf 7 4 The cost to install the shelving at the various locations are: ZroxHewesRockwright Thomas 1 5 8 Washburn 3 4 4

29. Example: Transshipping • Network Representation Zrox 5 ZROX 50 1 Arnold 1 Thomas 3 5 75 ARNOLD 5 8 8 Hewes 6 60 HEWES 3 7 Supershelf 2 Washburn 4 4 75 WASH BURN 4 4 Rockwright 7 40

30. Example: Transshipping • LP 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 (Thomas), 4 (Washburn) k = 5 (Zrox), 6 (Hewes), 7 (Rockwright) • Objective Function Defined Minimize Overall Shipping Costs: Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47

31. Example: Transshipping • Constraints Defined Amount out of Arnold: x13 + x14< 75 Amount out of Supershelf: x23 + x24< 75 Amount through Thomas: x13 + x23 - x35 - x36 - x37 = 0 Amount through Washburn: x14 + x24 - x45 - x46 - x47 = 0 Amount into Zrox: x35 + x45 = 50 Amount into Hewes: x36 + x46 = 60 Amount into Rockwright: x37 + x47 = 40 Non-negativity of variables: xij> 0, for all i and j.

32. Variations of Transshipment Problem • Total supply not equal to total demand • Maximization objective function • Route capacities or route minimums • Unacceptable routes