Linear Programing Problem LPP Examples

Q1. A Company produces two types of watches one for gents and another for ladies, both are to be processed on three machines, the processing time required and the total time available per week on each machine are as follows:

MACHINE          MODEL                 AVAILABLE TIME
                    GENTS    LADIES

M1                 3                3                    36
M2                 5                2                    50
M3                 2                6                    60

The contribution of profit for each unit of gents watch is Rs.20 and each unit of ladies watch is Rs.30. How should the company schedule the production to the maximum profit formulate the problem as LPP.

Soln:
Maximize z = 20x1+30x2
Subjected to :
3x1+3x2 <= 36
5x1+2x2 <= 50
2x1+6x2 <= 60
and decision variables x1, x2 >=0.
Explanation:
As per the problem given we have two variables
(i)Gents watch--x1 (ii)Ladies watch --x2
They are processed in three machines therefore we will have 3 sets of constraints.
We have to maximize the profit, hence the problem should maximize the profit therefore the objective function should be maximize z.

Q2. A leading leather good company manufactures two types of cricket ball A and B. Each type of ball requires work by both skilled and semi skilled employees , the available time per month and time required for each ball is given below.

TYPES OF EMPLOYES             MANUF. TIME                AVAILABLE TIME
                                               A                    B
SEMI SKILLED                      2                     3                       320
SKILLED                                4                     6                       600

The cost of the hour of semi skilled labor is Rs.10 & skilled labor is  Rs.15. To meet the monthly demands for at least 60 balls of type A and at least 40 balls of type B must be manufactured. Formulate LPP to minimize the cost of production.

Soln:
Minimize z = (2hrs)(10Rs)x1+(4hrs)(15Rs)x1+(3hrs)(10Rs)x2+(6hrs)(15Rs)x2
                 =80x1+120x2
Subjected to
2x1+3x2 <= 320
4x1+6x2 <= 600
x1>=60
x2>=40

and decision variables x1,x2 >=0

Explanation:
As per the problem given we havve two types of variables
(i) Ball A--x1 (ii)Ball B --x2
They are processed by two types of employees Skilled & semi skilled hence two sets of constraints.
We have to minimize the cost of production hence the objective function Minimize z.
Cost of labor and number of balls to be manufactured should be calculated as the objective functions.
Here the number of balls to be manufactured is provided as at least therefore the decision variables are given as >=