1 Department of Business Administration SPRING Management Science by Asst. Prof. Sami Fethi © 2007 Pearson Education

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 2 Chapter Topics The Transportation Model Solution of a Transportation Problem The Assignment Model Solution of an Assignment Problem

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 3 Overview Part of a larger class of linear programming problems known as network flow models. Possess special mathematical features that enabled development of very efficient, unique solution methods. Methods are variations of traditional simplex procedure. Detailed description of methods is contained in CD-ROM Module B, Transportation and Assignment Solution Methods. Text focuses on model formulation and solution with Excel and QM for windows.

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 4 The Transportation Model Characteristics A product is transported from a number of sources to a number of destinations at the minimum possible cost. Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. The linear programming model has constraints for supply at each source and demand at each destination. All constraints are equalities in a balanced transportation model where supply equals demand. Constraints contain inequalities in unbalanced models where supply does not equal demand.

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 5 The Transportation Model Characteristics Transportation modelling is an iterative procedure for solving problems that involve minimizing the cost of shipping products from a series of sources to a series of destinations.  Origin Points (or sources) can be factories, warehouses, car rental agencies like AVIS or any other points from which goods are shipped.  Destinations are any points that receive goods. Transportation models are useful when considering alternative facility locations. The choice of a new location depends on which will yield the minimum cost for the entire system

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 6 The Transportation Model Characteristics To use the transportation model we need to know the following:  The origin points and the capacity or supply per period at each  The destination points and the demand per period at each  The cost of shipping one unit from each origin to each destination  The transportation problem can be regarded as a generalization of the assignment problem. In an assignment problem we have a number of origins each processing one item, and the same number of destinations each requiring one item and are asked to empty the origins and fill the destinations in such a way the Total Effectiveness is optimized. In a transportation problem we have “m” origins, with origin “i” processing “ai” items and “n” destinations (possibly different number from “m”) with destination “j” requiring “bj” items, and with  ai =  bj.

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 7  We are given the m.n costs associated with shipping one item from any origin to any destination, and are asked to empty the origins and fill the destinations in such a way that the total cost is minimized.  The assignment problem is seen to be a special case of transportation problem in which m = n, all the ai and bj are unity.  The iterative routine to be presented as the mode of the solution of the transportation problem involves a succession of “feasible solutions” of gradually diminishing cost; eventually the minimum achievable cost is reached, and the corresponding feasible solution is optimal. The Transportation Model Characteristics

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 8 Transportation Model Example1 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 9 Transportation Model Example (2 )

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 10 Transportation Model Example (3)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 11 Transportation Model Example (4)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 12 Transportation Model Example (5)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 13 Transportation Model Example (6)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 14 Transportation Model Example 2 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 15 Transportation Model Example (2)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 16 Transportation Model Example (3)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 17 Transportation Model Example (4)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 18 Special form of linear programming model similar to the transportation model. Supply at each source and demand at each destination limited to one unit. In a balanced model supply equals demand. In an unbalanced model supply does not equal demand. The Assignment Model Characteristics

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 19 The Assignment Model Characteristics

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 20 Assignment Model Example 1 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 21 Assignment Model Example (2)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 22 Assignment Model Example 2 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 23 Assignment Model Example 3 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 24 Assignment Model Example 4 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 25 Assignment Model Example 4 (2)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 26 Assignment Model Example (3)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 27 Assignment Model Example (4)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 28 Assignment Model Example 5 (1)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 29 Assignment Model Example 5 (2)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 30 Assignment Model Example 6 (1) (minimize)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 31 Assignment Model Example 6 (2) (minimize)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 32 Assignment Model Example 6 (3) (minimize)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 33 Assignment Model Example 6 (4) (minimize)

© 2007/08, Sami Fethi, EMU, All Right Reserved. Operations Research Ch 6: Transportation and Assignment 34 End of chapter