intro class 1
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ME F344/MF F344
ENGINEERING OPTIMIZATIONLect:
Rajesh P Mishra,
Assistant Professor,
Mechanical Engineering,
1228A
rpm@pilani.bits-pilani.ac.in
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
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