linear Programming Solver
Linear Programming Solver deals with the optimization (maximization or minimization) of linear functions
subject to linear constraints.This technique has found its applications to important areas of product mix,
blending problems and diet problems.Oil refineries, chemical industries, steel industries and food processing
industry are also using linear programming with considerable success.Linear programming problems involving
only two variables can be effectively solved by a graphical technique which prrovides a pictorial representation
of the solution.
Step 1: formulate the given problem as a Linear Programming Solver problem
Step 2 : plot the given constraints as equalities on x1-x2 cordinate plane and determine the convex region
formed by them
Step 3 : determine the vertices of the convex region and find the value of objective function at each vertex.The
vertex which gives the optimal value of the objective function gives the desired optimal solution to the
problem.
General Linear Programming Problem
Any linear programming problem involving more than two variables may be expressed as follows
find the values of the variable x1,x2,............,xn which maximize (or minimize) the objective function
Z=c1x1+c2x2+..............+cnxn
subject to the constraints
a11x1+a12x2+.............+a1nxn<=b1
a21x1+a22x2+............. +a2nxn<=b2
.............................................................
am1x1+am2x2+..............+amnxn<=bm


and meet the non negative restrictions
x1,x2,...........xn>=0

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3. Any feasible solution which maximizes(or minimizes) the objective function of the
Linear Programming Solver called
its optimal solution .
Forms of linear programming problem
There are two forms of linear programming problem.They are:
Canonical form:
The general linear programming problem can be expressed as Maximize z=c1x1+c2x2+........cnxn subject to
the constraints
ai1x1+ai2x2+.........ainxn<=bi; x1,x2........xn>=0. This form is called its canonical form and has the fol owing
characteristics:
1. objective function is of maximization type
2. all constraints are of(<=)type
3 .all variables xi are non-negative
Standard form:
standard form has its following characteristics:
1. objective function is of maximization type
2. all constraints are expressed as equations
3. right hand side of each constraint is non negative
4. all variables are non negative
Working rules for solving linear programming problem (LPP)
Step 1. Identify the unknowns in the given LPP. Denote then by x and y.
Step 2. Formulate the objective function in terms of x and y. be sure whether it is to be maximized or
minimized.
Step 3. Translate all the constraints in the form of linear inequations.
Step 4. Solve these inequations simultaneously. Mark the common area by shaded region. This is the
feasible region .
Step 5. Find the coordinates of al the vertices of the feasible region
Step 6. Find the value of the objective function at each vertex of the feasible region.
Step 7. Find the values of x and y for which the objective function
z = ax + by has maximum or minimum value (as the case may be)
Linear Programming Solver Examples
Below are the example problems on linear programming-


Problem 1:
Solve the fol owing linear program

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