2. Linear Programming Problem
LPP is a mathematical modeling technique, used to
determine a level of operational activity in order to
achieve an objective, subject to restrictions.
It is a mathematical modeling technique, useful for
economic allocation of ‘scarce’ or ‘limited’ resources
like labor, material, machine, time, space, energy etc. to
several competing activities like product, services, jobs
etc. on the basis of a given criterion of optimality.
3. LPP Consists of:
Decision Variables: Decision to produce no. of units of
different items.
Objective Function: Linear mathematical relationship
used to describe objective of an operation in terms of
decision variables.
Constraints: Restrictions placed on decision situation
by operating environment.
Feasible Solution: Any solution of general LPP which
also satisfies non negative restrictions.
Optimum Solution : The feasible solution which
optimizes the objective function.
6. Question 1.
A dealer wishes to purchase a no. of fans and Air
Conditioners. He has only Rs. 5760 to invest & space for
at most 20 items.
A fan costs him Rs. 360 & AC Rs. 240. His
expectation is that he can sell a fan at a profit of Rs. 22 &
AC at profit of Rs. 18.
Assuming he can sell all items he can buy, how
should he invest money in order to maximize his profits?
7. Solution 1. purchases x1 Fans & x2 ACs.
Let us suppose, dealer
Since no. of fans & ACs can’t be negative
So, x1 ≥ 0, x2 ≥ 0
Since cost of fan = Rs. 360 & AC = Rs. 240
& Total money to be invested = Rs. 5760
Thus, 360 x1 + 240 x2 ≤ 5760
Also, space is for at most 20 items
So, x1 + x2 ≤ 20
Again, if dealer can sell all his items
Profit is Z = 22 x1 + 18 x2, which is to be maximized
Thus, the required LPP is:
Maximize Z = 22 x1 + 18 x2
Subject to Constraints,
360 x1 + 240 x2 ≤ 5760
x1 + x2 ≤ 20
& x1 ≥ 0, x2 ≥ 0
8. Question 2.
A company produces two articles R & S. Processing
is done through assembly & finishing departments. The
potential capacity of the assembly department is 60 hrs. a
week & that of finishing department is 48 hrs. a week.
Production of one unit of R requires 4 hrs. in
assembly & 2hrs. in finishing.
Each of the unit S requires 2 hrs. in assembly & 4hrs.
in finishing.
If profit is Rs. 8 for each unit of R & Rs. 6 for each
unit of S. Find out the no. of units of R & S to be produced
each week to give maximum profit.
9. Solution 2.
Products Time Required for Producing One Total hrs.
Unit available
x1 x2
Assembly Dept.
4 2 60
Finishing Dept.
2 4 48
Profit
Rs. 8 Rs. 6
Objective Function: Max. Z = 8x1 + 6x2
Subject to Constraints,
4 x1 + 2 x2 ≤ 60 (Time available in assembly dept.)
2 x1 + 4 x2 ≤ 48 (Time available in finishing dept.)
where, x1 ≥ 0, x2 ≥ 0
