Post on 20-Jun-2015
transcript
Course Instructor: Dr. Swati Singh
Course: BBA III
Amity Business School
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.
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.
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
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?
Solution 1. Let us suppose, dealer purchases x1 Fans & x2 ACs.
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
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.
Solution 2.
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
Products Time Required for Producing One Unit
Total hrs. available
x1 x2
Assembly Dept. 4 2 60 Finishing Dept. 2 4 48
Profit Rs. 8 Rs. 6