maximum number of items he can store (Here it is ). Stated mathematically, x + y ≤ 50000 (investment constraint) or x + y ≤ and x + y ≤ (storage constraint) The dealer wants to invest in such a way so as to maximise his profit, say, Z which stated as a function of x and y is given by Z = x + y (called objective function ) ... ( ) Mathematically, the given problems now reduces to: Maximise Z = x + y x + y ≤ x + y ≤ x ≥ , y ≥ So, we have to maximise the linear function Z subject to certain conditions determined by a set of linear inequalities with variables as non-negative. There are also some other problems where we have to minimise a linear function subject to certain conditions determined by a set of linear inequalities with variables as non-negative.
Such problems are called Linear Programming Problems. Thus, a Linear Programming Problem is one that is concerned with finding the optimal value (maximum or minimum value) of a linear function (called objective function ) of several variables (say x and y ), subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). The term linear implies that all the mathematical relations used in the problem are linear relations while the term programming refers to the method of determining a particular programme or plan of action. Before we proceed further, we now formally define some terms (which have been used above) which we shall be using in the linear programming problems: Objective function Linear function Z = ax + by , where a , b are constants, which has to be maximised or minimized is called a linear objective function.
In the above example, Z = x + y is a linear objective function. Variables x and y are called decision variables . Constraints The linear inequalities or equations or restrictions on the variables of