Sensitivity Analysis Slack Variables and Shadow Prices

For any optimization technique such as linear programming, we would be interested in the sensitivity of the obtained solution at or near the optimal point to small changes in the constraints and other estimated values. In other words, a sensitivity analysis is of importance. It should be noted that estimates regarding the requirements of resources, the availability of resources, the costs associated with resources, or the profit margins associated with products, etc are subject to some error by virtue of being only estimates. By means of a sensitivity analysis we can check or find out the effect of small changes in the estimated figures on the optimal solution to the problem. For instance, we would be interested in knowing that by relaxing our product mix constraints slightly in our earlier problem, how would the profit margin change? We can make the marginal change in the profit margin with marginal relaxation of the product mix constraints.

Expressing the machine hour constraints (200) as an equality by incorporating the idle capacity available within the constraint (denoting idle capacity as M), we have

2Y + 4X + M = 200

Similarly, we can incorporate the idle capacity of materials and express the materials constraints as an equality as given below:

6Y + 3X + N = 240

The above two equations give us the following relations:

X = 40 – M/3 + N/9

Y= 20 + M/6 + 2N/9

And we have, our earlier relation for the profit margin:

P = 7X + 9Y

Incorporating the preceding two relations and substituting for X and Y in the above relation for the profit margin, we get

P = 460 – 5/6M – 11/9 N

or P = 460 – 0.83M – 1.22 N

This is an important relation in terms of the meaning of the coefficients of M and N. It states that if there was one unit of machine capacity unused, the profit margin would have been affected by 0.83 rupees, this 0.83 being the coefficient of the idle capacity variable or slack variable M. Similarly, if the materials capacity was unused by one unit then the profit margin would have been reduced by 1.22 rupees, where 1.22 is the coefficient of the slack variable for materials N. Putting it in reverse, if we had one unit of extra machine capacity, we would have made 0.83 rupees in addition to the maximum (optimal value) profit of Rs 460. Similarly if we had one unit of extra materials capacity, we would have made Rs 1.22 in addition to the earlier obtained maximum profit of Rs 460. Therefore, these coefficients of the slack variables provide an insight as to how a slight relaxation of the constraints might prove useful or profitable. The coefficients of the slack variables are many a time referred to as the shadow prices of the constraints. Shadow price of a constraint is the marginal increase/decrease in the objective achieved by a marginal change in the constraint. It is very important for decision making, because in practice there is always some amount of freedom available with the constraints. Although the budget may be of Rs 1 lakh, still there is the flexibility of allowing for a few thousand more. Although the raw material availability may be one ton, still the possibility of procuring a few more kilograms of the materials exists, albeit with difficulty. The availability of a machine capacity is limited, there is still a possibility that a few more hours can be squeezed in. The management would like to always know as to whether these little flexibilities available to them might make a significant difference with respect to the achievement of the desired objectives such as profits or costs. Therefore, a constraint which has a considerable positive shadow price can be relaxed within the available flexibility whereas a constraint which has a very small shadow price need not be relaxed.