Discuss how you would use the material covered in this module for a future position in management.

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In this chapter, we examine three special types of linear programming model formulations- transportation, transshipment, and assignment problems.

They are part of a larger class of linear programming problems known as network flow problems. We are considering these problems in a separate chapter because they represent a popular group of linear programming applications. These problems have special mathematical characteristics that have enabled management scientists to develop very efficient, unique mathematical solution approaches to them.

These solution approaches are variations of the traditional simplex solution procedure. Like the simplex method, we have placed these detailed manual, mathematical solution procedures-called the transportation method and assignment method-on the companion Web site that accompanies this text. As in previous chapters, we will focus on model formulation and solution by using the computer, specifically by using Excel and QM for Windows.

The Transportation Model The transportation model is formulated for a class of problems with the following unique characteristics: (1) A product is transported from a number of sources to a number of destinations at the minimum possible cost; and (2) each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. Although the general transportation model can be applied to a wide variety of problems, it is this particular application to the transportation of goods that is most familiar and from which the problem draws its name.

Assumption was that solutions could be fractional or real numbers (i.e., non-integer). However, non-integer solutions are not always practical. When only integer solutions are practical or logical, it is sometimes assumed that non- integer solution values can be "rounded off" to the nearest feasible integer values.

This method would cause little concern if, for example, x1 = 8,000.4 nails were rounded off to 8,000 nails because nails cost only a few cents apiece. However, if we are considering the production of jet aircraft and x1 = 7.4 jet airliners, rounding off could affect profit (or cost) by millions of dollars. In this case we need to solve the problem so that an optimal integer solution is guaranteed. In this chapter the different forms of integer linear programming models are presented.

Integer Programming Models There are three basic types of integer linear programming models-a total integer model, a 0-1 integer model, and a mixed integer model.

In a total integer model, all the decision variables are required to have integer solution values. In a 0-1 integer model, all the decision variables have integer values of zero or one. Finally, in a mixed integer model, some of the decision variables (but not all) are required to have integer solutions.

The following three examples demonstrate these types of integer programming models.
A Total Integer Model Example The owner of a machine shop is planning to expand by purchasing some new machines-presses and lathes.

The owner has estimated that each press purchased will increase profit by $100 per day and each lathe will increase profit by $150 daily. The number of machines the owner can purchase is limited by the cost of the machines and the available floor space in the shop.