Intro Class 1

8/18/2019 Intro Class 1

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ME F344/MF F344

ENGINEERING OPTIMIZATIONLect:

Rajesh P Mishra,

Assistant Professor,

Mechanical Engineering,

1228A

[email protected]

Tutorials:

Kailash Choudhary

Narpat Ram Sangwa

Nitesh Sihag


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Mute ur call


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Scope and Objective of the Course • Engineers, scientists, analysts and managers

are often faced with the challenge of makingtrade-offs between different factors in orderto achieve desirable outcomes.

• Optimization is the process of choosing thesetrade-offs in the best way.

• Optimization problems, having reached adegree of maturity over the past several years,are encountered in physical sciences,engineering, economics, industry, planning,and many other areas of human activity.


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Scope and Objective of the Course • Objective of the course is set to familiarize the students

with standard methods of solving optimization

problems.• This course deals with details of various aspects

associated with optimization.

• These include description of optimization techniques,

namely, Linear Programming and NonlinearProgramming, and their applications to variousengineering and science disciplines including economicsand finance.

• Multi-objective optimization which handles optimization

aspects of more than one objective is also discussed.• A brief and informative description of Nontraditional

optimization techniques such as Genetic Algorithms,Differential Evolution, etc. is also provided.


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Text Book & Reference Books • T1 HA Taha, Operations Research: An Introduction,

Pearson Education/PHI, 9/E, 2012.• R1 SS Rao, Engineering Optimization: Theory and Practice,New Age International (P) Limited, Third Edition, 1996

• R2 FS Hillier and GJ Lieberman, Introduction to OperationsResearch, TMH, 8/E, 2006.

• R3 WL Winston, Operations Research: Applications and Algorithms, Thomson Learning, 4th Edition, 2004

• R4 JC Pant, Introduction to Optimization: OperationsResearch, Jain Brothers, New , 6/E, 2004.

• R5 A Ravindran, DT Philips and JJ Solberg, OperationsResearch: Principles and Practice, John Wiley & Sons,Singapore, Second Edition, 1987

• R6 GC Onwubolu and BV Babu, New OptimizationTechniques in Engineering, Springer-Verlag, Heidelberg,Germany, First Edition, 2004.


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Course Plan

Topics to be CoveredIntroduction to optimization

Two variable LP model, Graphical LP solution, Selected LP applications,

Convex Set

LP model in equation form, Transition from graphical to algebraic solution

The Simplex Method, Generalized simplex tableau in matrix form,

Artificial starting solution

Special cases in the simplex method

Definition of Dual Problem, Duality, Primal-Dual Relationships,

Economic Interpretation of Duality, Additional simplex algorithms (Dual

Simplex Method, Generalized Simplex Algorithm),

Post optimal Analysis


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Course Plan Definition of transportation problem, The transportation Algorithm,

The Assignment Model

Goal Programming Formulation,

Goal Programming Algorithms: The Weights Method and The Preemptive

Method

Formulation of IP problem

Branch and Bound method for solving IPPCutting Plane method

Unconstrained problems, Convex and concave functions,

Elimination Methods: Fibonacci Method and Golden Section Method,

Gradient of a Function, Descent Methods: Steepest Descent Method and

Conjugate Gradient Method,

Karush-Kuhn-Tucker (KKT) Conditions,

Quadratic Programming,


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Evaluation Scheme

Component Duration MarksWeightage

(%)

Date &

TimeRemarks

Mid Semester 1.5 hours 30 30 10/3 CB

Evaluative Tutorial - 20 20 Tut class OB

Comprehensive 3 hours 50 50 6/5/2015 CB


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and other important information

• Make-Up Policy: Only genuine cases will be entertained (Prior

permission will be needed for make up, usually make-up will beheld within a week after the regular test)

• Problems: Students are strongly advised to work out all theproblems in the text-book and do similar problems from thereference books. It is also strongly recommended that the studentsshould try out the algorithms on computers to get a better

understanding of the subject.• Chamber Consultation Hours: To be announced in the class by the

respective Instructors.

• Notice: Notices concerning this course will be displayed on FD INotice Board.

• INSTRUCTOR-IN-CHARGE

• (ME F344/MF F344)


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What is management sciences/

Operation Research (OR)?

• In general, the organization goal is to optimize

the use of available resources.

• Management science/OR is a discipline that

adapts the scientific approach to problem

solving to decision making.


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Mathematical Modeling and the

Management Science/Operation

Research Process

• Mathematical Modeling

– A process that translates observed phenomenainto mathematical expressions.


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• The Management Science/OR Process

– By and large, the Management Science/OR

process can be described by the following steps

procedure.

Problem definition

Mathematical modeling

Solution of the model


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• Basic Steps of the Management Science/OR

Process

1- Defining the Problem.

2- Building a Mathematical Model.

3- Solving a Mathematical Model.

4- Validate

5- Implementation


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Phases Of An OR Study

Definition of the problem

Construction of the model

Solution of the model

Validation of the model

Implementation of the solution

Linear programming

Dynamic programming

Network programming

Integer programming

Nonlinear programming


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1- Defining the Problem

• Management Science/ OR is Applied When -

– Designing and implementing new operations.

– Evaluating ongoing Operations and Procedures.

– Determining and recommending corrective

actions.


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How to Start and How to Proceed

– Identify the problem.

– Observe the problem from various points of view.

– Keep things simple.

– Identify constraints.

– Work with management, get feedback .


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2- Building a Mathematical Model

• Identify Decision Variables

–Which factors are controllable?

• Quantify the Objective and Constraints

– Formulate the function to be optimized (profit,

cost).

– Formulate the requirements and/or restrictions.

• Construct a Model Shell – Help focus on the exact data required.

• Gather Data --

– Consider time / cost issues.


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Mathematical modeling formulation: The general OR model

can be organized in the following general format:-

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Maximize or minimize Objective Function

subject to

Constraints


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Example: The ABC garment company

manufactures men's shirts and

women’s Salwar suit for WalmarkDiscount stores. Walmark will accept

all the production supplied by ABC.

The production process includes

cutting, sewing and packaging. ABC

employs 25 workers in the cuttingdepartment, 35 in the sewing

department and 5 in the


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packaging department. The factory

works one 8-hour shift, 5 days a week.The following table gives the time

requirements and the profits per unit

for the two garments:


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Garment Cutting Sewing Packaging Unitprofit($)

Shirts 20 70 12 8.00

Salwar 60 60 4 12.00

Minutes per unit

Determine the optimal weeklyproduction schedule for Burroughs.


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Solution: Assume that ABC produces x 1

shirts and x 2

salwar suit per week.

8 x 1 + 12 x 2

Time spent on cutting =

Profit got =

Time spent on sewing = 70 x 1

+ 60 x 2

mts

Time spent on packaging = 12 x 1 + 4 x 2 mts

20 x 1 + 60 x 2 mts


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The objective is to find x 1, x 2 so as to

maximize the profit z = 8 x 1 + 12 x 2

satisfying the constraints:

20 x 1 + 60 x 2 ≤ 25 40 60

70 x 1 + 60 x 2 ≤ 35 40 60

12 x 1 + 4 x 2 ≤ 5 40 60

x 1, x 2 ≥ 0, integers


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This is a typical optimization problem.

Any values of x 1, x 2 that satisfy all

the constraints of the model is called

a feasible solution. We areinterested in finding the optimum

feasible solution that gives the

maximum profit while satisfying allthe constraints.


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More generally, an optimization

problem looks as follows:Determine the decision variables

x 1, x 2, …, x n so as to optimize anobjective function f ( x 1, x 2, …, x n)

satisfying the constraints

gi ( x 1, x 2, …, x n) ≤ bi (i=1, 2, …, m).


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Linear Programming Problems(LPP)

An optimization problem is called aLinear Programming Problem (LPP) when

the objective function and all the

constraints are linear functions of thedecision variables, x1, x2, …, xn. We also

include the “non-negativity restrictions”,

namely x j ≥ 0 for all j=1, 2, …, n.Thus a typical LPP is of the form:


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Optimize (i.e. Maximize or Minimize)

z = c1 x1 + c2 x2+ …+ cn xnsubject to the constraints:

a11 x1 + a12 x2 + … + a1n xn ≤ b1a

21 x

1 +a

22 x2 + … +a

2n x

n ≤ b

2

. . .

am1 x1 + am2 x2 + … + amn xn ≤ bm

x1, x2, …, xn 0


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• Thanks

Intro Class 1

Documents

Transcript of Intro Class 1