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 = 40x_{1} + 35x_{2}, Subject to

2x_{1} + 3X_{2 }< or = 60, Raw materials constraint per week.

4x_{1} + 3X_{2 } < or = 96, Capacity constraint per week.

x_{1},x_{2} > or = 0

The optimal solution to this problem gives production of 18 units of x_{1 }and 8 units of x_{2 }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 y_{l} represent the rent per kg of raw material and y_{2} the rent per capacity hour, the firm would receive a total rent equal to 6Oy_{1} + 96y_{2}. 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 x_{1} and x_{2}. We know that production of one unit of x_{1} 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

2y_{1}+4y_{2 } > 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.,

3y_{1} + 3Y_{2} > or = 35

Besides, the rent cannot be negative. Therefore, y_{1} and y_{2} should both be non negative. In complete form, the problem can be expressed as:

Minimise: Z* = 60y_{1} + 96y_{2}, Subject to:

2y_{1} + 4y_{2} > or = 40

3y_{1} + 3y_{2} > or = 35

y_{1}, y_{2} > or = 0

This problem is absolutely the same as the dual to the given problem. These rates y_{1} and y_{2} are obtainable from the solution of the dual as y_{1} = 10/3 and y_{2} = 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 y_{1} and y_{2}, 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 x_{1} 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 x_{1} 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 MP_{R} 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 MP_{R} + 96 MPc, Subject to:

2MP_{R} + 4MPc > or = 40

3MP_{R} + 3MPc > or = 35

MP_{R}, MPc > or = 0

Now let us consider the economic significance of the surplus variables S_{1} and S_{2} 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 S_{1} in the optimal solution represents the opportunity cost of the product x_{1} while the value of S_{2} represents the opportunity cost of product x_{2.} Production of an additional unit of x_{1} 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 x_{2 }the opportunity cost equals zero.

Further if x_{1}, x_{2} is a feasible solution to the primal and y_{1}, y_{2} is the feasible solution to its dual then c_{1}x_{1} + c_{2}x_{2} < or = b_{1}y_{1} + b_{2}y_{2} 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.