previous				contents				next

4. Sensitivity Analysis

4.5 Alternative Optimal Solutions

Let us now return to our original problem and examine the effect of lowering the cost coefficient in the objective function of the non basic variable x₁ by its reduced cost of 2.0. The objective function has now changed from:

Minimize - 5 x₁ - 7 x₂

to

Minimize - 7 x₁ - 7 x₂

The geometric representation of the problem now appears as in Figure 5.

Figure 5: Alternative Optimal Solutions

The line PQ in Figure 5 represents the original objective function and the line P'Q' the new objective. Note that there is now a choice of solutions at either vertex C or B. The solution at vertex B is:

x1  =  1.0
x2  =  3.0

OBJ  =   - 28.0

Note that the variable x₁ has now become basic in the optimal solution. Therefore, lowering the cost of the non basic variable x₁ by its reduced cost has made it basic. This is a general rule. The reduced cost of a non basic variable therefore also represents the required change in its cost coefficient for it to become basic.

From this last property of reduced costs, note that the reduced cost of a basic variable is zero. This is logical since no change in its cost coefficient is required for it to become basic since it already is. It is possible to have a non basic variable with a reduced cost of zero. This indicates that we can make this variable basic (and another suitable variable non basic) with no change in the objective function value. Thus, a non basic variable with zero reduced cost indicates that an alternative optimal solution to the problem exists.

previous				contents				next