Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Winter 2016
REVISED SIMPLEX METHOD
The Revised simplex method offers an Efficient
Computational procedure for solving Linear
Programming Problem.
THE ITERATIVE STEPS OF THE REVISED
SIMPLEX METHOD ARE EXACTLY SAME AS IN
THE SIMPLEX METHOD TABLEAU.
THE MAIN DIFFERENCE
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI (Raj.)
FIRST SEMESTER 20072008
COMPREHENSIVE EXAMINATION
COURSE NO. : AAOC C312
DATE: 14.12.2007
MAX. MARKS: 120
COURSE TITLE: OPERATIONS RESEARCH
TIME: 3 HOURS
_
NOTE: 1.The question paper is divided in
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Problem6ProblemSet2.3APage26(Modified)
Electra produces two types of electric motors,
each on a separate assembly line. The respective
daily capacities of the two lines are 150 and 200
motors. Type I motor uses 2 units of a certain
electronic component, a
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Iterativecomputationsofthe
Transportationalgorithm
Iterative computations of the Transportation algorithm
After determining the starting BFS by any one of the
three methods discussed earlier, we use the following
algorithm to determine the optimum solutio
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Problem 7 Problem Set 8.1A Page 351
Two products are manufactured on two sequential
machines. The following table gives the machining
times in minutes per unit for the two products:
Machine
1
2
Machining Time in min
Product 1
Product2
5
3
6
2
The daily pr
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Goal programming
The LPP models discussed so far are based on the
optimization of a single objective function. There
are situations where multiple objectives are to be
met. We now present the goal programming
technique for solving multi objective models.
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
QUADRATIC
PROGRAMMING
Quadratic Programming
A quadratic programming problem is a nonlinear
programming problem of the form
Maximize
Subject to
T
z c X X DX
A X b , X 0
Here
x1
b1
x
b
2 2
X . , b . , c c1 c2 . . . cn
.
.
xn
bm
a11 a12
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
CLASSICAL OPTIMIZATION THEORY
Quadratic forms
Let
x1
x
2
X .
.
xn
be a nvector.
Let A = ( aij) be a nn symmetric matrix.
We define the kth order principal minor as
the kk determinant
a11 a12 . a1k
a21 a22 . a2 k
.
.
ak 1 ak 2 . akk
Then the q
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Problem 7.53 Hillier and Lieberman Page 345
The Research and Development Division of the
Emax Corporation has developed three new
products. A decision now needs to be made on
which mix of these products should be
produced. Management wants primary
consid
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
II Semester 20042005
AAOC C312: Operations Research
Comprehensive Exam.(Open Book)
Date: May 13, 2005
Day: Friday
Max.Marks: 120
Time: 3 hours
Note: The question paper contains two parts: PART A and PART
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Birla Institute of Technology & Science, Pilani
Second Semester, 20072008
Comprehensive Examination
Course No: AAOC C312
Date: 07.05.2008
Time: 3 hours
Course Title: Operations Research
Max.Marks: 120
Note:
1. The question paper is divided into two parts
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Birla Institute of Technology and Science, Pilani.
(II Semester 200506)
Comprehensive Examination QUIZ (Open Book)
Part A
Course No.: AAOC C312
Course
Name: Operations Research Time: 45 min
Dt. 12.05.2006(Friday)
Max. Marks: 30
A
ID Number
Name
Note: (i)
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Winter 2016
Integer Programming
Introduction
In LP problems, decision variables are non
2
negative values, i.e. they are restricted to
be zero or more than zero.
It demonstrates one of the properties of LP
namely, continuity, which means that
fractional values of the
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Winter 2016
Dual
Linear Programming Problem
Dual Linear Programming Problem
With every LPP, there is associated LPP which is called the
dual of the original problem (or the primal problem)
The two problems are so closely related that the optimal
solution of one yield
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Winter 2016
Explanation of the
entries in any simplex
tableau in terms of the
entries of the starting
tableau
In this lecture we explain how the
starting Simplex tableau (in matrix
form) gets transformed after some
iterations. We also give the meaning of
the entries
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Winter 2016
MATRIX FORMULATION
OF THE LPps
In this lecture we shall look at the matrix
formulation of the LPPs. We see that the
Basic feasible solutions are got by solving
the matrix equation BX b
where B is a mm nonsingular submatrix
of the contraint matrix of the L
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Birla Institute of Technology & Science, Pilani
II Semester: 2006 2007
AAOC C 312: Operations Research
Duration: 3Hours
Comprehensive Exam
Max Marks: 120
03.05.2007
Instructions: 1. Answer Part A and Part B in separate answer scripts.
2. Write Null and Al
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
I SEMESTER 20062007 COMPREHENSIVE EXAMINATION
OPTIMIZATION: AAOC C222
Max. Marks: 120
Instructions:
04.12.2006 (Monday)
Time: 3 Hours
Please write Part A and Part B in separate answer scripts, mentioning
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI (Raj.)
FIRST SEMESTER 20052006
COMPREHENSIVE EXAMINATION
COURSE NO. : AAOC C312
DATE: 08.12.2005
MAX. MARKS: 40
COURSE TITLE: OPERATIONS RESEARCH
TIME: 3 HOURS
_
NOTE: 1.The question paper is divided in t
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
INTEGER LINEAR PROGRAMMING
There are many LP problems in which the decision
variables will take only integer values. If all the
decision variables will only take integer values it is
called a pure integer LPP; otherwise the problem is
called a mixed integ
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Algebraic Solution of LPPs  Simplex
Method
To solve an LPP algebraically, we first put it
in the standard form. This means all
decision variables are nonnegative and all
constraints (other than the nonnegativity
restrictions) are equations with nonnegati
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Deterministic Dynamic Programming
Dynamic Programming (DP) determines the
optimum solution to an nvariable problem by
decomposing it into n stages with each stage
constituting a singlevariable sub problem.
Recursive Nature of Computations in DP
Computat
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Problemz 10 y y
Problem Set
Maximize = y1 2 n
10.3A Page 414
subject to y1+y2+yn = c,
yi 0
Thus there are n stages to this problem. At
stage i, we have to choose the variable yi.
The state of the problem at stage i is defined
by the variable xi, which rep
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
GAME THEORY
Life is full of conflict and competition.
Numerical examples involving adversaries in
conflict include parlor games, military battles,
political campaigns, advertising and
marketing campaigns by competing business
firms and so forth. A basic f
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
TheSimplexalgorithm
Abstract: In this lecture we discuss the
computational aspects of the Simplex
algorithm. We shall see how a LPP is put
into a simplex tableau. Starting from a BFS,
we explain how to proceed step by step till
we reach the optimal soluti
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Someproblemsillustratingthe
principlesofduality
Inthislecturewelookatsomeproblemsthatuses
the results from Duality theory (as discussed in
Chapter7).
Problem 7. Problem Set 4.2D Page 130
Consider the LPP
Maximize z 5 x1 2 x2 3 x3
subject to
x1 5 x2 3 x3 b
Birla Institute of Technology & Science, Pilani  Hyderabad
Engineering optimization
ME 344

Fall 2014
Dualitytheorems
Findingthedualoptimal
solutionfromtheprimaloptimal
tableau
Dual problem in Matrix form
In this lecture we shall present the primal
and dual problems in matrix form and
prove certain results on the feasible and
optimal solutions of the prim