SUMMARY OF GRAPHICAL METHOD - Quantitative Techniques for management

Step 1: Convert the inequality constraint as equations and find co-ordinates of the line.

Step 2: Plot the lines on the graph.
(Note: If the constraint is ≥ type, then the solution zone lies away from the centre. If the constraint is ≤ type, then solution zone is towards the centre.)

Step 3:
Obtain the feasible zone.

Step 4:
Find the co-ordinates of the objectives function (profit line) and plot it on the graph representing it with a dotted line.

Step 5:
Locate the solution point.
(Note: If the given problem is maximization, Zmax then locate the solution point at the far most point of the feasible zone from the origin and if minimization, Zmin then locate the solution at the shortest point of the solution zone from the origin).

Step 6: Solution type
i. If the solution point is a single point on the line, take the corresponding values of x1 and x2.
ii. If the solution point lies at the intersection of two equations, then solve for x1 and x2 using the two equations.
iii. If the solution appears as a small line, then a multiple solution exists.
iv. If the solution has no confined boundary, the solution is said to be an unbound solution.

Example:
Solve the Geetha perfume company (Example 1.7) graphically using computer.

The formulated LP model is,

Zmax = 7x1 + 5x2

Subject to constraints,
Subject to constraints,
8x1+ 4x2≤ 20 .........................(i)
2x1+ 3x2≤ 8 .........................(ii)
– x1+ x2≤ 2 .........................(iii)
x2≤ 2 .........................(iv)

where x1, x2≥ 0

Solution:
The input values of the problem are given to obtain the output screen as shown in Figure.

Graphical Presentation (Output Screen, TORA)

Graphical Presentation (Output Screen, TORA)

Results:
Perfumes to be produced, x1 = 1.75 litres or 17.5 say 18 bottles of 100 ml each Body sprays to be produced, x2 = 1.50 litres or 15 bottles of 100 ml each Maximum profit, Zmax = Rs. 19.75


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