Presentation on theme: "Transportation, Assignment, and Transshipment Problems"— Presentation transcript:
1Transportation, Assignment, and Transshipment Problems Transportation ProblemNetwork RepresentationGeneral LP FormulationAssignment ProblemTransshipment Problem
2Transportation, 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 of this chapter as well as the PERT/CPM problems (in another chapter) are all examples of network problems.
3Transportation, Assignment, and Transshipment Problems Each of the three models of this chapter can be formulated as linear programs and solved by general purpose linear programming codes.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.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.
4Transportation 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.
5Transportation Problem Network Representation1d1c111c12s1c132d2c212c22s2c233d3SourcesDestinations
6Transportation Problem LP FormulationThe LP formulation in terms of the amounts shipped from the origins to the destinations, xij , can be written as:Min cijxiji js.t. xij < si for each origin ijxij = dj for each destination jixij > 0 for all i and j
7Transportation Problem LP Formulation Special CasesThe following special-case modifications to the linear programming formulation can be made:Minimum shipping guarantee from i to j:xij > LijMaximum route capacity from i to j:xij < LijUnacceptable route:Remove the corresponding decision variable.
8Acme Example: Acme Block Co. Acme Block Company has orders for 80 tons ofconcrete blocks at three suburban locationsas follows: Northwood tons,Westwood tons, andEastwood tons. Acmehas two plants, each of whichcan produce 50 tons per week.Delivery cost per ton from each plantto each suburban location is shown on the next slide.How should end of week shipments be made to fillthe above orders?Acme
9Example: Acme Block Co. Delivery Cost Per Ton Northwood Westwood EastwoodPlantPlant
10Example: Acme Block Co.Partial Spreadsheet Showing Problem Data
15Assignment ProblemAn 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.
16Assignment Problem Network Representation c11 c12 c13 Agents Tasks c21
17Assignment Problem LP Formulation Min cijxij i j s.t. xij = for each agent ijxij = for each task jixij = 0 or 1 for all i and jNote: A modification to the right-hand side of the first constraint set can be made if a worker is permitted to work more than 1 job.
18Assignment Problem LP Formulation Special Cases Number of agents exceeds the number of tasks:xij < for each agent ijNumber of tasks exceeds the number of agents:Add enough dummy agents to equalize thenumber of agents and the number of tasks.The objective function coefficients for thesenew variable would be zero.
19Assignment 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 a tasks:xij < a for each agent ij
20Example: Who Does What?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.ProjectsSubcontractor A B CWestsideFederatedGoliathUniversalHow should the contractors be assigned to minimize total mileage costs?
21Example: Who Does What? Network Representation A Subcontractors 50West.A3616SubcontractorsProjects28Fed.30B183532Gol.C202525Univ.14
22Example: Who Does What? Linear Programming Formulation Min 50x11+36x12+16x13+28x21+30x22+18x23+35x31+32x32+20x33+25x41+25x42+14x43s.t. x11+x12+x13 < 1x21+x22+x23 < 1x31+x32+x33 < 1x41+x42+x43 < 1x11+x21+x31+x41 = 1x12+x22+x32+x42 = 1x13+x23+x33+x43 = 1xij = 0 or 1 for all i and jAgentsTasks
23Example: Who Does What? The optimal assignment is: Subcontractor Project DistanceWestside CFederated AGoliath (unassigned)Universal BTotal Distance = 69 miles
24Transshipment 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.
25Transshipment Problem Network Representationc363c131c376s1d1c14c15c464c47DemandSupplyc23c24c5627d2s2c255c57SourcesDestinationsIntermediate Nodes
26Transshipment Problem Linear Programming Formulationxij represents the shipment from node i to node jMin cijxiji js.t. xij < si for each origin ijxik - xkj = for each intermediatei j node kxij = dj for each destination jixij > for all i and j
27Example: Zeron Shelving 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.
28Example: Zeron Shelving Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:Zeron N Zeron SArnoldSupershelfThe costs to install the shelving at the various locations are:Zrox Hewes RockriteThomasWashburn
35Example: Zeron Shelving Optimal Solution (continued)OBJECTIVE COEFFICIENT RANGESVariable Lower Limit Current Value Upper LimitXX No LimitX No LimitX No LimitX No LimitXX No LimitX No LimitXX No Limit
36Example: Zeron Shelving Optimal Solution (continued)RIGHT HAND SIDE RANGESConstraint Lower Limit Current Value Upper LimitNo Limit