This document provides an overview of linear programming (LP). It begins with definitions, including that LP is a technique to optimize an objective function subject to constraints. It provides an example LP problem about production allocation. It then discusses key LP concepts like decision variables, objective functions, constraints, feasible and optimal solutions. Matrix notation for representing LP problems is introduced. Finally, it formulates three sample LP problems: production allocation, material blending, and diet formulation.

1. LINEAR
PROGARMMING
By:
Divya K

2. Title and Content
 Meaning
 Example
 General Linear Programming Problem
 Matrix notation
 Definitions
 Formulation of L.P.P

3. Meaning
 Linear programming (LP, also called linear optimization) is a method to achieve the
best outcome (such as maximum profit or lowest cost) in a mathematical
model whose requirements are represented by linear relationships.
 Technique for determining the best allocation of a firm’s limited resources to achieve
optimum goal.
 Optimization (maximization/ minimisation) of profit, sales, costs, labour, etc is
subjected to “Set of constraints”
 Constraints – demand for commodities, availability of raw material, storage space,
variation of cost, etc
 1947 – George Danzig along with his associates LP Technique with regards to
military activities

5. 2

6. General Linear Programming Problem
 And the non-negativity restrictions
x1 ≥ 0 , x2 ≥ 0, ………………… , xn ≥ 0

7. Matrix notation
LPP can be represented in matrix form also
OPTIMIZE
Z = CX
Subjected to constraint
AX (≤= ≥)b
And the non negativity restriction
X ≥ O Where O= zero

8. Definitions
Consider LPP :-
OPTIMIZE Z = CX
Subject to AX (≤= ≥)b
And X ≥ O
1. Objective Function:
The function Z= CX, which is to be (maximized or minimized)
2. Decision Variables:
Whoes value we need to determine. (x1, x2, …. , xn)
3. Cost(Profit) Coeffecients:
c,1, c2, ….., cn is
4. Requirements:
Constants b1, b2, ….bm

9. 5. Solution:
a set of real values X=(x1, x2, ….., xn) which satisfies the constraints AX (≤= ≥)b
6. Feasible Soultion:
a set of real values X=(x1, x2, ….., xn) which
i. satisfies the constraints AX (≤= ≥)b
ii. Satisfies non negativity constraint X ≥ O
7. Optimal Solution:
a set of real values X=(x1, x2, ….., xn) which
i. satisfies the constraints AX (≤= ≥)b
ii. Satisfies non negativity constraint X ≥ O
iii. Optimizes the objective function Z = CX
8. Results: Based on no of Optimal solution LPP is classified as
a) 2 or more - multiple Solution
b) One – unique solution
c) No feasible solution – No solution
d) If Z value is infinity – unbounded Solution

10. Formulation of LPP – expressing the optimization subjected to
certain constraints using mathematical expression
Excersie 1: (Production allocation Problem)
Manufacturer produces 2 models M1 & M2 of a product. Each unit of model M1
requires 4hrs grinding and 2hrs of polishing. Each unit of model M2 requires 2hrs
grinding and 5hrs polishing. The manufacturer has 2 grinders each of which works for
40hrs a week. There are 3 polishers each of which works for 60hrs per week. Profit of
model M1 is Rs. 300 per unit and profit of model M2 is Rs. 400 per unit. The
manufacturer has to allocate his production capacity so as to maximize his profit.
Formulate the LPP
Model 1 Model 2 Requirement
No of Units x y
Profits (Rs)
Grinding Time(Hrs)
Polishing Time(Hrs)
Solution:
Let x = no of units of M1 to be produced
y= no of units of M2 to be
produced

11. Model 1 Model 2 Requirement
No of Units x y
Profits (Rs) 300x 400y maximize
Grinding Time(Hrs) 4x 2y ≤80 (2piece*40hrs)
Polishing Time(Hrs) 2x 5y ≤180 (3piece*60hrs)
The LPP is: -
Maximize
Z=300x+400y
Subject to (s.t)
4x+2y≤80
2x+5y≤180
And x≥0 , y≥0

12. Exercise 2: (Blending problem)
 A firm produces an alloy having the following specifications.
i. Specific gravity ≤ 0.98
ii. Chromium content ≥ 8%
iii. Melting Point ≥ 4500c
Raw material A and B are used to produce the alloy. Raw material has specific
gravity 0.92, Chromium content 7% and melting point 4400c . Raw material B has
specific gravity 0.99, Chromium content 13% and melting point 4900c. If raw
material A cost Rs 90 /Kg and raw material B cost Rs. 280/Kg, find
ratio(proportion) should be blended keeping cost in mind
Solution:
Let x = proportion of raw material A
y= proportion of raw material
B
RM1 RM2 Requirement
Proportions x y
Cost(Rs)
Specific Gravity
Chromium content
Melting point

13. The LPP is: -
Minimization
Z=90x+280y
Subject to (s.t)
0.92x+0.99y≤0.98
7x+13y ≥ 8
440x+490y≥450
And x≥0 , y≥0
Raw Material1 Raw Material 2 Requirement
Proportions x y
Cost(Rs) 90x 280y Minimization
Specific Gravity 0.92x 0.99y ≤ 0.98
Chromium content 7x 13y ≥ 8
Melting point 440x 490y ≥ 450

14. Exercise 3: (Diet problem)
 A person wants to decide the constituents of a diet which will fullfill his daily
requirements of protients, fats and carbhohydrates at minimium cost. The
combination is made among two food products whose contents are indicate
below.
Formulate LPP
Food Content Cost/ Unit
(Rs)
Proteins Fats Carbohydrate
s
A 3 2 6 45
B 4 2 4 40
Minimum
requirements
800 200 700

15.  Solution:
Let x = Number of units food A needed
y= Number of units food B needed
The LPP is: -
Minimization
Z=45x+40y
Subject to (s.t)
3x+4y ≥ 800
2x+2y ≥ 200
6x+4y ≥700
And x≥0 , y≥0