Graphical Method :

The graphic method of solving linear programming problems consists of the following steps :

Step 1 : Plot the Constraints :  To plot the constraints, treat each constraint as equalities so as it represents a straight line. Because of Non-negativity constraints the graph will be in the first quadrant. Depending upon inequalities mark the regions either below or above the line.

Step 2 : Identify the feasible region : The feasible region is the region in which all the constraints are satisfied. In other words, it is the common portion of all the regions represented by all the constraints of the problem.

Step 3 : Locate the solution points :  The feasible region contains an infinite number of points.  In this step, search those points which will make objective function optimum. Note that such points will be only from corner points of the solution space. This is because of extreme point theorem. “Let the solution space of an L.P.P. be a complex region bounded by lines in the plane. Then the objective function of L.P.P. attends its maximum                (or minimum) at the vertices (corners of the feasible space region).”

Step 4 : Select any of the following two methods :

(a)     ISO – Profit (or ISO – Cost) Method :

(i)      Choose a convenient profit (or cost) and draw as iso – profit (or iso – cost) line so that it falls within shaded region.

(ii)     Move this iso – profit (or iso – cost) line parallel to itself further (closer) from (to) the origin.

(iii)    Identify the optimum solution as the co – ordinate of that point on the feasible region touched by the highest possible iso – profit line (or lower possible iso – cost) line.

(iv)    Read the co – ordinates of optimum point either directly from the graph or by simultaneous solution two lines intersecting at that point.

(v)     Compute the optimum profit (or cost) values.

(b)     Corner Point Method :

(i)      Identifying each of the corner or extreme points of the feasible regions either directly from the graph or by method of simultaneous equation.

(ii)     Evaluate the objective function at each of the corner points of the feasible region.

(iii)    Identify the optimum solution which corresponds to that corner points which gives the optimum values of the objective function.

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