LINEAR PROGRAMMING L. Kantorovich LINEAR PROGRAMMING . Linear Programming Problem and its Mathematical Formulation We begin our discussion with the above example of furniture dealer which will further lead to a mathematical formulation of the problem in two variables. In this example, we observe (i) The dealer can invest his money in buying tables or chairs or combination thereof.
Further he would earn different profits by following different investment strategies. (ii) There are certain overriding conditions or constraints viz., his investment is limited to a maximum of Rs , and so is his storage space which is for a maximum of pieces. Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ , i.e., tables. His profit in this case will be Rs ( × ), i.e., Rs .
Suppose he chooses to buy chairs only and no tables. With his capital of Rs , , he can buy 50000 ÷ , i.e. chairs. But he can store only pieces.
Therefore, he is forced to buy only chairs which will give him a total profit of Rs ( × ), i.e., Rs . There are many other possibilities, for instance, he may choose to buy tables and chairs, as he can store only pieces. Total profit in this case would be Rs ( × + × ), i.e., Rs and so on. We, thus, find that the dealer can invest his money in different ways and he would earn different profits by following different investment strategies.
Now the problem is : How should he invest his money in order to get maximum profit? To answer this question, let us try to formulate the problem mathematically. . .
Mathematical formulation of the problem Let x be the number of tables and y be the number of chairs that the dealer buys. Obviously, x and y must be non-negative, i.e., (Non-negative constraints) x y ≥ ≥ The dealer is constrained by the maximum amount he can invest (Here it is Rs , ) and by the