previous				contents				next

4. Sensitivity Analysis

4.4 Reduced Cost

Variable x₁ is therefore non basic at its lower bound of zero in the optimal solution. Suppose we force variable x₁ off its lower bound by 0.5 by adding to the problem a bound:

x₁ ³ 0.5

The geometric representation of the modified problem is shown in Figure 4.

Figure 4: 2-variable Problem with Bound Added

The new solution to the problem now lies at vertex C where:

x1  =  0.5
x2  =  3.5

OBJ  =   - 27.0

Note that forcing the non-basic variable off its lower bound has worsened the value of the objective function (it is greater and we are minimising). We have seen that:

OBJ with variable x1 at its lower bound           	=  - 28.0
OBJ with x1 forced off its lower bound by 0.5	=  - 27.0

We can therefore define the unit cost of forcing the non basic variable x₁ off its lower bound as:

Change in Objective function value	=     1 	=   2.0
	Change in x1		     0.5

Therefore the unit cost of forcing the non basic variable x₁ off its bound is +2.0 This value is called the reduced cost of variable x₁ and implies that for every unit we force the variable x₁ off its bound by the objective function value will change by +2.0.

It is important to note that all sensitivity analysis information addresses individual changes to the problem only. That is for instance, if we force one non basic variable off its bound to form a new problem the solution will be worsened by its reduced cost / unit. We cannot then proceed to force another non basic variable off its bound and measure the detriment in the objective function using the reduced cost information from the original problem. This is true of all the sensitivity analysis information presented. Therefore, although sensitivity analysis is extremely useful, it does have its limitations.

previous				contents				next