Linear programming problem (LPP) is a mathematical modeling technique used to determine optimal levels of activities given constraints and an objective to maximize or minimize. It allocates limited resources like labor, materials, machines among competing activities based on an optimality criterion. An LPP consists of decision variables, an objective function to describe the goal, constraints on the decision situation, feasible and optimal solutions. As an example, a dealer wants to maximize profits by purchasing fans and air conditioners within a Rs. 5760 budget and 20 item space limit. The LPP models this as maximizing profits from fan and AC sales subject to cost and space constraints. A second example models a company producing two items to maximize profit within assembly

1. Course Instructor: Dr. Swati Singh
Course: BBA III
Amity Business School

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.

4. General Structure of LPP
 Maximize (or Minimize) Z = c1x1 + c2x2 + ---- + cnxn
 Subject to,
a11x1 + a12x2 + -------- + a1nxn (≤, =, ≥ ) b1
a11x1 + a12x2 + -------- + a1nxn (≤, =, ≥ ) b2
an1x1 + an2x2 + -------- + annxn (≤, =, ≥ ) bn
where, x1 ≥ 0, x2 ≥ 0 ---- xn ≥ 0

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