1 Cengage Learning. All Rights Reserved. May not be scanned, copied duplicated, or posted to a publicly accessible website, in whole or in part. 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
40
Embed
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. - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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.
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.
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
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.
Define the Decision VariablesWe want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:
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.
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.
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 C Westside 50 36 16
Federated 28 30 18 Goliath 35 32 20
Universal 25 25 14How should the contractors be assigned so that totalmileage is minimized?
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.
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.
LP Formulation Special Cases• Total supply not equal to total demand• Maximization objective function• Route capacities or route minimums• Unacceptable routesThe LP model modifications required here areidentical to those required for the special
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.