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Linear Programming Problem

The mathematical statement of the general form of a linear programming problem (abbreviated LLP) may be written as follows.

Optimize (maximize or minimize)

Z = c1x1 + c2x2 +.....+ cnxn

Subject to

a11x1 + a12x2 +......+ a1nxn ( ≤, =, ≥ ) b1
a21x1 + a22x2 +......+ a2nxn ( ≤, =, ≥ ) b2

          .     .                     .
          .     .                     .
          .     .                     .
am1x1 + am2x2 +.....+ amnxn ( ≤, =, ≥ ) bm

             x1,x2,......,xn ≥ 0


(i)    x1, x2,..., xn are the varibles whose values we wish to determine and are called the decision or structural variables.

(ii)    the linear function Z which is to be maximized or minimized is called the objective function of the general LPP.

(iii)    the inequalities (1) are called the constraints of the general LPP,

(iv)    the set of inequalities (2) is known as the set of non-negative restriction of the general LPP,

(v)    the constant cj ( j = 1,2,..., n ) represents the contribution (profit or cost) to the objective function of the jth variable,

(vi)    the coefficients  aij ( i = 1,2,...,m;j = 1, 2,...,n ) are referred to as the technological or substitution coefficients.

(vii) bi  ( i = 1,2,....,m ) is the constant representing the requirement or availability of the jth constraint, and

(viii)    the expression ( ≤, =, ≥ ) means that only one of the relationship in the set( ≤, =, ≥ ) would hold for a particular constraint.

It is appropriate at this stage to give some definitions which pertain to general linear programming problems.

Definition. A set of values of the decision varibles which satisfy the constraints of general LPP is called a solution to the general LPP.

Definition. Any solution to the general LPP which also satisfies the non-negative restrictions of the problem is called a feasible solution. The set of all feasible solutions constitutes what is called the feasible region.

Definition. Any feasible solution which optimizes the objective function of a general LPP is called an optimum solution to the general LPP.

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