Economic interpretation of linear programming duality

We see that the primal and the dual of linear programming are related mathematically, we can now show that they are also related in economic sense. Consider the economic interpretation of the duality of linear programming — first for a maximisation problem and then for a minimisation problem.

The maximisation problem: Consider the following linear programming problem.

The optimal solution to this problem dives production of 18 units of Xi and 8 units of x2 per week. It yields the maximum prof of a Rs. 1000,

Maximise Z = 40x1 + 35x2, Subject to

2x1 + 3X2 < or = 60, Raw materials constraint per week.

4x1 + 3X2 < or = 96, Capacity constraint per week.

x1,x2 > or = 0

The optimal solution to this problem gives production of 18 units of x1 and 8 units of x2 per week. It yields the maximum profit of a Rs. 1000.

Now, to rent the facilities of the firm for one week, the firm has 60 kg of raw material and 96 capacity hours. If we let yl represent the rent per kg of raw material and y2 the rent per capacity hour, the firm would receive a total rent equal to 6Oy1 + 96y2. We shall compute the minimum value of the rent so that the firm will know what minimum offer shall be economically acceptable to it. The lower limit can be set up after keeping in mind that the alternative to renting must be at least as favourable as using the capacity itself. The rent of the resources should be at least equal to the earnings from producing products x1 and x2. We know that production of one unit of x1 requires 2 kg of raw material and 4 capacity hours. Thus, the total rent for these amounts of resources should be greater than, or equal to, the profit obtainable from one unit of the product, i.e. Rs. 40. Hence

2y1+4y2 > or = 40

similarly, the resources consumed in producing one unit of product x2 are 3 kg of raw material and 3 capacity hours. The total rent of these resources should equal to atleast Rs. 35, the unit profit of product x2. i.e.,

3y1 + 3Y2 > or = 35

Besides, the rent cannot be negative. Therefore, y1 and y2 should both be non negative. In complete form, the problem can be expressed as:

Minimise: Z* = 60y1 + 96y2, Subject to:

2y1 + 4y2 > or = 40

3y1 + 3y2 > or = 35

y1, y2 > or = 0

This problem is absolutely the same as the dual to the given problem. These rates y1 and y2 are obtainable from the solution of the dual as y1 = 10/3 and y2 = 25/3. Also, these values can be obtained from ∆j row of simplex tableau showing optimal solution to the primal problem. As we have seen, values of the objective function of the primal and the dual are identical. Naturally, the minimum total rent acceptable to the firm is equal to the maximum profit that it can earn by producing the output itself using the given resources.

The individual rents of y1 and y2, are called the shadow prices or imputed prices. They indicate the worth of the resources. These prices, of the two resources, materials and capacity hours, are imputed from the profit obtained front utilizing their services, and are not derived from the from the original cost of these resources.

Now we know that each unit of product x1 contributes Rs. 40 to the profit. The imputed price of material and capacity is respectively, Rs. 10/3 per kg and 25/3 per hour, we can find the total imputed cost of the resources used in making a unit of the product as

2kg at Rs. 10/3 per kg

i.e., 2 X 10/3 = 20/3 Rs.

4 hours at Rs. 25/3 per kg. = 4 X 25/3  = 100/3 Rs.

The total cost is 120/3 = 40 Rs.

Thus, the total imputed cost of producing one unit of product x1 equals the per unit profit obtainable from it.

Similarly, from each unit of product x2, the total imputed cost of resources employed would be:

3kg at Rs.10/3 per kg = 3 x lO/3 = 10 Rs.

3 hours at Rs. 25/3 per hour 3 x 25/3 = 25 Rs.

The total cost is 10 + 25 = 35 Rs.

This obviously equals the profit per unit of the product. This proves that the valuation of the resources is such that their total value equals the total profit obtained at the optimum level of production.

The shadow prices are also called the marginal value products or marginal profitability of the resources. Thus, if there were a market for renting resources, the firm would be willing to take some materials if the price of the material were less than Rs. 10/3 per kg, and capacity hours, if the price is less than Rs. 25/3 per hour.

If we denote marginal profitability of resources as MPR and the marginal profitability of capacity as MPc, respectively, the shadow prices of the two resources, we can write the dual as follows:

Minimise H = 50 MPR + 96 MPc, Subject to:

2MPR + 4MPc > or = 40

3MPR + 3MPc > or = 35

MPR, MPc > or = 0

Now let us consider the economic significance of the surplus variables S1 and S2 in the dual. The numerical values of these variables can be obtained from ∆j  row in the optimal solution of the primal. The value of S1 in the optimal solution represents the opportunity cost of the product x1 while the value of S2 represents the opportunity cost of product x2. Production of an additional unit of x1 will give the firm a profit of Rs. 40 and, at the same time, the firm would use up resources worth 2 x 10/3 + 4 x 25/3 = Rs. 40. Thus, the net effect of producing one unit of product would be 40 –  40 = 0. Similarly, for product x2 the opportunity cost equals zero.

Further if x1, x2 is a feasible solution to the primal and y1, y2 is the feasible solution to its dual then c1x1 + c2x2 < or = b1y1 + b2y2 i.e. [profit obtained is less than or equal to the rents to be paid. This would induce the producer to rent the resources rather than produce the goods himself.

The concept of dual and shadow prices help us in determining the upper and lower bounds for changes in requirement vectors and coefficients in the objective function. Such that the feasibility of the LPP is not disturbed.

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