1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 6, Part A Distribution and Network Models n Transportation Problem Network Representation Network Representation General LP Formulation General LP Formulation n Assignment Problem Network Representation Network Representation General LP Formulation General LP Formulation n Transshipment Problem Network Representation Network Representation General LP Formulation General LP Formulation

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation, Assignment, and Transshipment Problems n 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. n 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 4 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation, Assignment, and Transshipment Problems n Each of the five models of this chapter can be formulated as linear programs and solved by general purpose linear programming codes. n For each of the five 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. n However, there are many computer packages (including The Management Scientist ) that contain separate computer codes for these models which take advantage of their network structure.

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation Problem n The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ), when the unit shipping cost from an origin, i, to a destination, j, is c ij. n The network representation for a transportation problem with two sources and three destinations is given on the next slide.

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation Problem n Network Representation 2 2 c 11 c 12 c 13 c 21 c 22 c 23 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s2s2 SourcesDestinations

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation Problem n Linear Programming Formulation Using the notation: Using the notation: x ij = number of units shipped from x ij = number of units shipped from origin i to destination j origin i to destination j c ij = cost per unit of shipping from c ij = cost per unit of shipping from origin i to destination j 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 continued

Modeling source supply capacity xi1xi1xi1xi1 ii Total supply from source i cannot be more than the capacity at source i: x i 1 + x i 2 + x i 3 + x i 4 xi2xi2xi2xi2 xi3xi3xi3xi3 xi4xi4xi4xi4

Modeling destination demand x1jx1jx1jx1j jj Total arrivals to destination j have to be at least equal to the demand at destination j: x 1j + x 2j + x 3j + x 4j x2jx2jx2jx2j x3jx3jx3jx3j x4jx4jx4jx4j

10 Slide © 2008 Thomson South-Western. All Rights Reserved x ij > 0 for all i and j Transportation Problem n Linear Programming Formulation (continued)

11 Slide © 2008 Thomson South-Western. All Rights Reserved n LP Formulation Special Cases The objective is maximizing profit or revenue: The objective is maximizing profit or revenue: 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 Problem Solve as a maximization problem.

12 Slide © 2008 Thomson South-Western. All Rights Reserved Transportation Problem: Example #1 Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood tons, Westwood tons, and Eastwood 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? AcmeAcme

13 Slide © 2008 Thomson South-Western. All Rights Reserved n Delivery Cost Per Ton Northwood Westwood Eastwood Northwood Westwood Eastwood Plant Plant Plant Plant Transportation Problem: Example #1 n Total Delivery Costs D 1 (Nw) D2 (Ww) ED3 (Ew) D 1 (Nw) D2 (Ww) ED3 (Ew) Plant 1 24x 11 30x 12 40x 13 Plant 1 24x 11 30x 12 40x 13 Plant 2 30x 21 40x 22 42x 23 Plant 2 30x 21 40x 22 42x 23

Network diagram for Acme 2 2 c 11 x 11 c 12 x 12 c 13 x 13 c 21 x 21 c 22 x 22 c 23 x 23 Plants (sources) Suburban locations (destinations) Total arrivals: Total departures: x x x 1 3 x x x 2 3 x x 2 1 x x 2 2 x x 2 3 Total costs:

15 Slide © 2008 Thomson South-Western. All Rights Reserved n Partial Spreadsheet Showing Problem Data Transportation Problem: Example #1

16 Slide © 2008 Thomson South-Western. All Rights Reserved n Partial Spreadsheet Showing Optimal Solution Transportation Problem: Example #1

17 Slide © 2008 Thomson South-Western. All Rights Reserved n Optimal Solution From To Amount Cost From To Amount Cost Plant 1 Northwood Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood Plant 2 Northwood Plant 2 Eastwood Plant 2 Eastwood Total Cost = $2,490 Total Cost = $2,490 Transportation Problem: Example #1

18 Slide © 2008 Thomson South-Western. All Rights Reserved n Partial Sensitivity Report (first half) Transportation Problem: Example #1 If transportation cost c13 is reduced by $4, from $40 to $36, there will be some non-zero shipments from source 1 to destination 3

19 Slide © 2008 Thomson South-Western. All Rights Reserved n Partial Sensitivity Report (second half) Transportation Problem: Example #1 If Westwood demand is increased by 1 ton, from 45 tons to 46 tons, the total cost will increase by $36

20 Slide © 2008 Thomson South-Western. All Rights Reserved Assignment Problem n 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 c ij. n It assumes all workers are assigned and each job is performed. n 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. n The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

21 Slide © 2008 Thomson South-Western. All Rights Reserved Assignment Problem n Network Representation c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 AgentsTasks

22 Slide © 2008 Thomson South-Western. All Rights Reserved n Linear Programming Formulation Using the notation: Using the 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 Assignment Problem continued

23 Slide © 2008 Thomson South-Western. All Rights Reserved n Linear Programming Formulation (continued) Assignment Problem x ij > 0 for all i and j

24 Slide © 2008 Thomson South-Western. All Rights Reserved n LP Formulation Special Cases Number of agents exceeds the number of tasks: Number of agents exceeds the number of tasks: Number of tasks exceeds the number of agents: Number of tasks exceeds the number of agents: Add enough dummy agents to equalize the Add enough dummy agents to equalize the number of agents and the number of tasks. number of agents and the number of tasks. The objective function coefficients for these The objective function coefficients for these new variable would be zero. new variable would be zero. Assignment Problem Extra agents simply remain unassigned.

25 Slide © 2008 Thomson South-Western. All Rights Reserved Assignment Problem n LP Formulation Special Cases (continued) The assignment alternatives are evaluated in terms of revenue or profit: The assignment alternatives are evaluated in terms of revenue or profit: Solve as a maximization problem. Solve as a maximization problem. 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:

26 Slide © 2008 Thomson South-Western. All Rights Reserved 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 Projects Subcontractor A B C Westside Federated Federated Goliath Universal Universal How should the contractors be assigned to minimize total mileage costs? Assignment Problem: Example

27 Slide © 2008 Thomson South-Western. All Rights Reserved n Network Representation West. CC BB AA Univ.Univ. Gol.Gol. Fed. Fed. Projects Subcontractors Assignment Problem: Example

28 Slide © 2008 Thomson South-Western. All Rights Reserved n Linear Programming Formulation Min 50 x x x x x x 23 Min 50 x x x x x x x x x x x x x x x x x x 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 Assignment Problem: Example

29 Slide © 2008 Thomson South-Western. All Rights Reserved n The optimal assignment is: Subcontractor Project Distance Subcontractor Project Distance Westside C 16 Westside C 16 Federated A 28 Federated A 28 Goliath (unassigned) Universal B 25 Total Distance = 69 miles Total Distance = 69 miles Assignment Problem: Example

30 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 6 Essentials – Part A

31 Slide © 2008 Thomson South-Western. All Rights Reserved Transshipment Problem n Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes)before reaching a particular destination node. n Transshipment problems can be converted to larger transportation problems and solved by a special transportation program. n Transshipment problems can also be solved by general purpose linear programming codes. n The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

32 Slide © 2008 Thomson South-Western. All Rights Reserved Transshipment Problem n Network Representation c 13 c 14 c 23 c 24 c 25 c 15 s1s1s1s1 c 36 c 37 c 46 c 47 c 56 c 57 d1d1d1d1 d2d2d2d2 Intermediate Nodes Sources Destinations s2s2s2s2 Demand Supply

33 Slide © 2008 Thomson South-Western. All Rights Reserved Transshipment Problem n Linear Programming Formulation Using the notation: Using the notation: x ij = number of units shipped from node i to node j x ij = number of units shipped from node i to node j c ij = cost per unit of shipping from node i to node j c ij = cost per unit of shipping from node i to node j s i = supply at origin node i s i = supply at origin node i d j = demand at destination node j d j = demand at destination node j continued

Intermediate node convention x1jx1jx1jx1j jj Subtract arrivals (sinks) to intermediate node j -x 1j - x 2j - x 3j - x 4j x2jx2jx2jx2j x3jx3jx3jx3j x4jx4jx4jx4j xj5xj5xj5xj5 xj6xj6xj6xj6 xj7xj7xj7xj7 xj8xj8xj8xj8 Add departures (sources) from intermediate node j + x j5 + x j6 + x j7 + x j8 = 0 Total quantity generated at node j is zero

35 Slide © 2008 Thomson South-Western. All Rights Reserved Transshipment Problem x ij > 0 for all i and j n Linear Programming Formulation (continued) continued

36 Slide © 2008 Thomson South-Western. All Rights Reserved Transshipment Problem n LP Formulation Special Cases Total supply not equal to total demand Total supply not equal to total demand Maximization objective function Maximization objective function Route capacities or route minimums Route capacities or route minimums Unacceptable routes Unacceptable routes The LP model modifications required here are identical to those required for the special cases in the transportation problem.

37 Slide © 2008 Thomson South-Western. All Rights Reserved 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. Transshipment Problem: Example

38 Slide © 2008 Thomson South-Western. All Rights Reserved Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: Zeron N Zeron S Zeron N Zeron S Arnold 5 8 Arnold 5 8 Supershelf 7 4 Supershelf 7 4 The costs to install the shelving at the various locations are: Zrox Hewes Rockrite Zrox Hewes Rockrite Thomas Thomas Washburn Washburn Transshipment Problem: Example

39 Slide © 2008 Thomson South-Western. All Rights Reserved n Network Representation ARNOLD WASH BURN ZROX HEWES Arnold SuperShelf Hewes Zrox ZeronN ZeronS Rock-Rite Transshipment Problem: Example

40 Slide © 2008 Thomson South-Western. All Rights Reserved n Linear Programming Formulation Decision Variables Defined Decision Variables Defined x ij = amount shipped from manufacturer i to supplier j x jk = amount shipped from supplier j to customer k x jk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) Objective Function Defined Objective Function Defined Minimize Overall Shipping Costs: Min 5 x x x x x x x 37 Min 5 x x x x x x x x x x x x x 47 Transshipment Problem: Example

41 Slide © 2008 Thomson South-Western. All Rights Reserved n Constraints Defined Amount Out of Arnold: x 13 + x 14 < 75 Amount Out of Supershelf: x 23 + x 24 < 75 Amount Through Zeron N: x 13 + x 23 - x 35 - x 36 - x 37 = 0 Amount Through Zeron S: x 14 + x 24 - x 45 - x 46 - x 47 = 0 Amount Into Zrox: x 35 + x 45 = 50 Amount Into Hewes: x 36 + x 46 = 60 Amount Into Rockrite: x 37 + x 47 = 40 Non-negativity of Variables: x ij > 0, for all i and j. Transshipment Problem: Example

42 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution Objective Function Value = Objective Function Value = Variable Value Reduced Costs Variable Value Reduced Costs X X X X X X X X X X X X X X X X X X X X Transshipment Problem: Example

43 Slide © 2008 Thomson South-Western. All Rights Reserved n Solution ARNOLD WASH BURN ZROX HEWES Arnold SuperShelf Hewes Zrox ZeronN ZeronS Rock-Rite Transshipment Problem: Example

44 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution (continued) Constraint Slack/Surplus Dual Prices Constraint Slack/Surplus Dual Prices Transshipment Problem: Example

45 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution (continued) OBJECTIVE COEFFICIENT RANGES OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit Variable Lower Limit Current Value Upper Limit X X X No Limit X No Limit X No Limit X No Limit X24 No Limit X24 No Limit X35 No Limit X35 No Limit X X X No Limit X No Limit X No Limit X No Limit X X X47 No Limit X47 No Limit Transshipment Problem: Example

46 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution (continued) RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit Constraint Lower Limit Current Value Upper Limit No Limit No Limit Transshipment Problem: Example

47 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 6, Part A