Linear Programming

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LINEAR PROGRAMMING

INTRODUCTION:

The term ‛programming′ means planning and it refers to a particular plan of action amongst several alternatives for maximizing profit or minimizing cost etc. Programming problems deal with determining optimal allocation of limited resources to meet the given objectives, such as cost, maximum profit, highest margin or least time, when resources have alternative uses.

The term ‛linear’ means that all inequations or equations used and the function to be maximized or minimized are linear. That is why linear programming deals with that class of problems for which all relations among the variables involved are linear.

Formally, linear programming deals with the optimization (maximization or minimization) of a linear function of a number of variables subject to a ¹equations in variables involved.

The general form of a linear programming problem is

Optimize (Maximize or Minimize) Z = c1x1 + c2x2 + ……..+ cnxn Subject to

a11 x1 + a12x2 + ….. + a1n xn (≤ , = , ≥) b1

a21 x1+ a22x2+ ….. + a2nxn (≤ , = , ≥ ) b2

. . . .

am1 x1+ am2 x2 + … + amn xn {≤ , = , ≥ { bmC

x1, x2….., xn ≥ 0

OBJECTIVE FUNCTION:

IF C1, C2.., Cn are constants and X1,X2,…..,Xn are variables, then the linear function Z = C1X1 + C2X2 + … + CnXn which is to be maximized or minimized is called the objective function. The objective function describes the primary purpose of the formulation of a linear programming problem and it is always non- negative. In business applications, the profit function which is to be maximized or the cost function which is to be minimized is called objective function.

CONSTRAINTS:

The inequations or equations in the variable of a LPP which describe the conditions under which the...
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