Max. Marks: 80
Birla Institute of Technology and Science, Pilani (Raj.)
Second Semester, 2009-10
AAOC C312 (Operations Research)
Comprehensive Examination (Closed Book) PART II
Max. Time: 120 mins
Date:10 May, 2010 (Monday)
Note : (i) Attempt all the ques

MATRIX FORMULATION
OF THE LPps
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
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

Sensitivity Analysis
The optimal solution of a LPP is based on the
conditions that prevailed at the time the LP model
was formulated and solved. In the real world, the
decision environment rarely remains static and it
is essential to determine how the opt

Dual Problem of an LPP
Given a LPP (called the primal problem),
we shall associate another LPP called the dual
problem of the original (primal) problem. We
shall see that the Optimal values of the primal
and dual are the same provided both have
finite fea

Duality theorems
Finding the dual optimal
solution from the primal
optimal tableau
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
Dual problem in Matrix form
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
Dual

Explanation of the
entries in any simplex
tableau in terms of the
entries of the starting
tableau
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
*How the starting Simplex tableau (in matrix
form) gets transformed after some iteratio

Dual simplex method for
solving the primal
In this lecture we describe the
important Dual Simplex method
and illustrate the method by doing
one or two problems.
Dual Simplex Method
Suppose a basic solution satisfies the optimality
conditions but not feasi

BITS Pilani
Pilani Campus
Consider the platform system shown in fig used for
scaffolding. Cable 1 can support 120N, cable 2 can
support 160N, and cables 3 and 4 can support 100N
each. Determine the maximum total load that the
system can support.
BITS Pila

Addition of a new constraint
The addition of a new constraint to an existing
model can lead to one of two cases:
1. The new constraint is redundant, meaning that it
is satisfied by the current optimal solution and
hence can be dropped altogether from the

The Transportation Model
Formulations
The Transportation Model
The transportation model is a special class of LPPs
that deals with transporting(=shipping/ passengers/
solid waste/ water/ material, etc.) a commodity
from sources (e.g. factories/ terminals/

Determination of
Starting Basic Feasible
Solution
Determination of the starting Solution
In any transportation model we determine a starting
BFS and then iteratively move towards the optimal
solution which has the least shipping cost.
There are three meth

Iterative
computations of the
Transportation
algorithm
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 sol

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

The Simplex algorithm
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 solu

In this lecture we shall look at
some miscellaneous LPPs. Each
problem will illustrate a certain
idea which will be explained when
the problem is discussed.
Problem 6 Problem set 3.4A Page 97
Maximize
z 2 x1 4 x2 4 x3 3 x4
Subject to the constraints
x1 x2

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

Miscellaneous Problems
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
Maximize
z 2 x1 4 x2 4 x3 3 x4
Subject to the constraints
x1 x2 x3
x1 4 x2
4
x4 8
x1 , x2 , x3 , x4 0
BITS Pilani, Deemed to be University under Section 3 of UGC

CE F411
Operation Research for
engineers
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Rajiv Gupta, Senior Professor, Civil Engg Dept
rajiv@pilani.bits-pilani.ac.in, 1110 B
8741959971
51, Paschim Marg
Application of scientific methods,
techniques an

Implicitenumeration
Implicit enumeration
Objectivesofthetopic:
Presentandapplytheimplicitenumeration
methodforthesolutionof01integer
h df h
l
f
programmingmodels.
S th h d t
Seethehandout.
DrMuhammadAlSalamah,IndustrialEngineering,KFUPM
The method of i

If the payoff matrix of a game is given as below:
B B B
1
2
3
A -2 0
7
1
A 3
1
3
2
A 6 -1 -3
3
Then the value of the game is equal to _.
The following game gives As pay off. The values of p and q that will make the entry (2,2) of
game a saddle point are r

BITS Pilani
Pilani Campus
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, and type II mo

CLASSICAL OPTIMIZATION THEORY
Quadratic forms
Let
x1
x
2
X .
.
xn
be a n-vector.
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

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

The Assignment Model
" The best person for job" is an apt description of
the assignment model.
The general assignment model with n workers
and n jobs is presented below:
Jobs
1 2 .
n
1 c11 c12
c1n
Workers 2 c21 c22
c2n
n
cn1 cn2
cnn
The element cij is

Genetic Algorithm
BITS Pilani
Pilani Campus
Evolutionary algorithms are a promising alternative for problems where
adaptivity to problems, complex computations, innovation and large
scale parallelism is required.
Heuristic search algorithm belonging to th

EXTENDED USE OF LINEAR GRAPH
THEORY (ELGT) FOR ANALYSIS OF
PIPE NETWORKS
BITS Pilani
Pilani Campus
INTRODUCTION
Pipe network analysis methods consist mainly of
a constitutive relation (nonlinear equation);
the formulation of system equations; and
a soluti

The Simplex algorithm
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
LPP is put into a simplex tableau.
Starting from a BFS,
Optimal solution.
Simplex method by examples.
BITS Pilani, Deemed to be University under Section 3 of UGC A

GAME THEORY
Life is full of conflict and competition.
Numerical examples involving adversaries in
conflict include project bid, parlor games,
military battles, political campaigns,
advertising and marketing campaigns by
competing business firms and so for