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LINEAR PROGRAMMINGPOST OPTIMALITY ANALYSIS

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GROUP MEMBERS

Prasad Nadkarni 61

Ashish Nakade 62 Ankur Nandu 63

Kishore Nannaware 64

Deepti Nayak 65

Sweedisha N. 66 Sunny Parekh 67

Satish Parmar 68

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Introduction

Today many of the resources needed as inputs to

operations are in limited supply. Operations managers must understand the

impact of this situation on meeting theirobjectives.

Linear programming (LP) is one way thatoperations managers can determine how best toallocate their scarce resources.

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Definition

A Linear Programming model seeks to maximize

or minimize a linear function, subject to a set oflinear constraints.

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Linear Programming (LP) in OM

There are five common types of decisions in

which LP may play a role Product mix

Production plan

Ingredient mix

Transportation Assignment

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Recognizing LP Problems

A well-defined single objective must be stated.

There must be alternative courses of action.

The total achievement of the objective must beconstrained by scarce resources or other

restraints.

The objective and each of the constraints mustbe expressed as linear mathematical functions.

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LP Model Formulation

Decision variables Mathematical symbols representing levels of activity of an

operation.

Objective function A linear relationship reflecting the objective of an operation. Most frequent objective of business firms is to maximize profit.

Most frequent objective of individual operational units (such as aproduction or packaging department) is to minimize cost.

Constraint A linear relationship representing a restriction on decision

making.

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Cavi Electrici Affini Torino (Electrical Cables

and Allied Products of Turin).

Founded in Italy as CEAT Tyres (1942) byVirginio Bruni Tedeschi.

Manufactured cables for telephones andrailways.

1958 : CEAT Tyres of India Ltd was establishedin collaboration with the TATA Group.

1990 : Renamed the companyCEAT Ltd.

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Current operations

Over 6 million tyres produced every year

Operations in Mumbai and Nasik plants

Exports to USA, Africa, America, Australia and

other parts of Asia Network of 34 regional offices, 7 Zones, over

3,500 dealers.

Dedicated customer service, with customer

service managers in all four divisional offices,assisted by 50 service engineers.

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Problem

CEAT Tyres manufactures produces bike and car tyres.

Each bike tyre requires 1.5 kg of rubber and 700 g of

carbon black, while car tyre requires 1 kg of rubber and 3

kg of carbon black. The total availability of rubber is 120kg and of carbon black is 180 kg per batch. The company

makes a profit of Rs. 250 per bike tyre and Rs. 350 per

car tyre. Formulate the LPP to optimize production mix in

order to maximize profit.

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Solution Let Bike tyre be x

Let Car tyre be y

Objective : Max Z = 250x + 350y

Subject to : 1.5x + 1y 120

0.7x + 3y 180

x, y 0 (non- negativity)

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1.5x + 1y 120

When x = 0 , y = 120

Co-ordinates : (0, 120)

When y = 0 , x = 80

Co-ordinates : (80, 0 )

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0.7x + 3y 180

When x = 0 , y = 180 / 3 = 60

Co-ordinates : (0, 60)

When y = 0 , x = 180 / 0.7 = 257. 14

Co-ordinates : (257.14 , 0)

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Graphical Solution Method1.1. Draw the axis taking the nos. to beDraw the axis taking the nos. to be

manufactured.manufactured.

2.2. Plot model constraint on a set of coordinatesPlot model constraint on a set of coordinatesin a planein a plane

3.3. Identify the feasible solution space on theIdentify the feasible solution space on the

graph where all constraints are satisfiedgraph where all constraints are satisfied

simultaneously.simultaneously.

4.4. Find the point on boundary of this space thatFind the point on boundary of this space that

maximize value of objective function.maximize value of objective function.

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Graphical Solution

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Optimal solution

Z (0,0) = 0

Z (80,0) = Rs. 20,000

Z ( 0, 60) = Rs. 21,000

Z (47.36, 48.95) = Rs. 28,927

Z (47, 49) = Rs. 28,900

Z (48, 48) = Rs. 28,800

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EXCEL SOLVERAn Alternate MethodAn Alternate Method

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Sensitivity Analysis

Over what ranges can these prices change without affecting

the optimality of the present solution?

Will the present solution remain the optimum solution if

the amount of raw materials, production time issuddenly changed because of shortages, machine

failures, or other events?

The amount of each type of resources needed to

produce one unit of each type of product can be eitherincreased or decreased slightly. Will such changes

affect the optimal solution ?

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Summary of output from Excel solver

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Shadow Prices

Assuming there are no other changes to theinput parameters, the change to the objective

function value per unit increase to a righthand side of a constraint is called theShadow Price

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Other Post - Optimality Changes

Addition of a constraint.

Deletion of a constraint.

Addition of a variable.

Deletion of a variable.

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Special thanks to

Prof. Bhide

Operation Research

Mr. Vijay PanzaleAssistant Manager

Industrial Engineering Department

CEAT Ltd, Nahur

Mumbai

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THANK YOU