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Linear Programming for Math Students

This document provides an introduction to linear programming problems (LPP). It defines LPP as determining optimal allocation of limited resources to meet objectives, with resources like men, materials, demand, money, and machines. The objective is typically maximizing profit or minimizing cost. The key requirements for LPP are: an objective function, alternative actions to choose from, and constraints that are linear equations or inequalities involving the decision variables. The mathematical formulation of an LPP involves defining an objective function to maximize or minimize as a linear combination of decision variables, subject to linear constraints relating the decision variables. Two example problems are provided to demonstrate formulating an LPP to minimize cost of fruit servings given vitamin and cost constraints

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169 views10 pages

Linear Programming for Math Students

This document provides an introduction to linear programming problems (LPP). It defines LPP as determining optimal allocation of limited resources to meet objectives, with resources like men, materials, demand, money, and machines. The objective is typically maximizing profit or minimizing cost. The key requirements for LPP are: an objective function, alternative actions to choose from, and constraints that are linear equations or inequalities involving the decision variables. The mathematical formulation of an LPP involves defining an objective function to maximize or minimize as a linear combination of decision variables, subject to linear constraints relating the decision variables. Two example problems are provided to demonstrate formulating an LPP to minimize cost of fruit servings given vitamin and cost constraints

Uploaded by

infantgabriel
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Formulation of

Linear Programming Problem

G Infant Gabriel
Assistant Professor of Mathematics,
PG & Research Department of Mathematics,
Sri Ramakrishna College of Arts & Science, Coimbatore
Introduction

Linear Programming Problem [LPP] deals with determining optimal allocation of


limited resources to meet given objectives . The resources may be in the form of men ,
raw material , marked demand , money and machines etc .
•Programming means planning . All the relationship between the variables considered in
this problems are linear . Hence the name Linear Programming Problem [LPP] .
•The objective is maximizing the profit and minimizing the total cost .
Introduction
Types :
•Maximization problems such as maximize the profit
•Minimization problems such as minimize the cost
Requirements for employing LPP technique :
•There must be a well defined objective function
•There must be alternative sources of action to choose
•At least some of the resources must be in the supply , which give rise to constraints
must be linear equations or inequalities
Introduction
Mathematical Formulation of [LPP]
Objective Function :Minimize or Maximize Z
Z = c1x1 + c 2x2 + … + c nxn
Subject to the Constraints
a11x1 + a12x2 + …+ a1nxn ≤ (or) = (or) ≥ bm
a21x1 + a22x2 + …+ a2nxn ≤ (or) = (or) ≥ bm
.
.
am1 x1 + am2 x2 + …+ amnxn ≤ (or) = (or) ≥ bm
and the non – negativity restrictions x1 , x2 , x3 , … xn ≥ 0
Introduction
Formulation of LPP :
Step 1: Identify the unknown decision variables to be determined and assign symbols to them.
Step 2: Identify all the restrictions or constraints (or influence factors) in the problem and express
them as linear equations or inequalities of decision variables .
Step 3: Identify the objective or aim and represent it also as a linear function of decision variables .
Step 4 : Express the complete formulation of LPP as general mathematical model .
Note : We consider only those situations where this will help the reader to put proper inequalities in
the formulation .
Problem 1:
A health-conscious family wants to have a very well controlled vitamin C-rich mixed fruit-
breakfast which is a good source of dietary fibre as well; in the form of 5 fruit servings per
day. They choose apples and bananas as their target fruits, which can be purchased from an
online vendor in bulk at a reasonable price. Bananas cost 30 rupees per dozen (6 servings)
and apples cost 80 rupees per kg (8 servings). Given: 1 banana contains 8.8 mg of Vitamin
C and 100-125 g of apples i.e. 1 serving contains 5.2 mg of Vitamin C. Every person of the
family would like to have at least 20 mg of Vitamin C daily but would like to keep the intake
under 60 mg. How much fruit servings would the family have to consume on a daily basis
per person to minimize their cost?
Solution:
We begin step-wise with the formulation of the problem first.
The constraint variables – ‘x’ = number of banana servings taken and ‘y’ = number of
servings of apples taken.Let us find out the objective function now. ·
Cost of a banana serving = 30/6 rupees = 5 rupees. Thus, the cost of ‘x’ banana servings = 5x
rupees·
Cost of an apple serving = 80/8 rupees = 10 rupees. Thus the cost of ‘y’ apple servings = 10y
rupees
Minimize Cost Z = 5x + 10y
Constraints: 8.8x + 5.2y ≥ 20 (1)
8.8x + 5.2y ≤ 60 (2) and x ≥ 0; y ≥ 0 .
Problem 2:
A company makes 3 product x , y , z which pass through three departments drill , lath ,
assembling the hours available in each department , hours required by each product in
each department and profit contribution of each product is given below . Time
required in hours
Product Drill Lathe Assemble Profit for
units (Rs)

A 3 3 8 9

B 6 5 10 15

C 7 4 12 20

Hours available 210 , 240 , 260 . Formulate the above as an LPP .


Problem 2:
Let x1 be the product of A , x2 be the product of B and x3 be the product of C
Objective function :

Maximize Z = 9x1 + 15x2 + 20x3


Subject to constraints:

3x1 + 6x2 + 7x3 ≤ 210

3x1 + 5x2 + 4x3 ≤ 240

8x1 + 10x2 + 12x3 ≤ 260


Thank You

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