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 u
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 int
CPMandPERT
CPMandPERT
CPM (Critical Path Method) and PERT
(Program Evaluation and Review Technique)
are network based methods designed to assist
in the planning, scheduling, and control of
projects. A
PERT Networks
In PERT the duration of any activity is indeterministic. It bases the duration of an activity on three estimates: Optimistic Time, a Most Likely Time, m Pessimistic Time, b
The range [a,
QUADRATIC
PROGRAMMING
Quadratic Programming
A quadratic programming problem is a non-linear
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 .
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 campai
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
Wo
Iterativecomputationsofthe
Transportationalgorithm
Iterative computations of the Transportation algorithm
After determining the starting BFS by any one of the
three methods discussed earlier, we use t
Hillier and Lieberman Problem
14.4-2 Page 746
Consider the game having the following
pay-off (to A) table:
Player B
Strategy
Player A
1
2
3
-2
2 -1
2
1
Use the graphical procedure to determine
the val
Deterministic Dynamic Programming
Dynamic Programming (DP) determines the
optimum solution to an n-variable problem by
decomposing it into n stages with each stage
constituting a single-variable sub p
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 prob
PERT Networks
In PERT the duration of any activity is
indeterministic. It bases the duration of an
activity on three estimates:
Optimistic Time, a
Most Likely Time, m
Pessimistic Time, b
The range
BIRLA
INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
K K BIRLA GOA CAMPUS
DEPARTMENT OF MATHEMATICS
_
FIRST SEMESTER
(2012-2013)
OPTIMISATION
(AAOC C222)
TUTORIAL BOOKLET
Instructor-In-Charge
WA cA Wcfw
Solving linear Programming
Problems
Dr. Manoj Kumar Pandey
Problem
Wild West produces two types of cowboy hats. Type I
hat requires twice as much labor time as a Type II hat. If
all the available labo
TRANSPORTATION
PROBLEMS
Source
Destinations
c11 : x11
Units of supply
a1
1
1
b1
a2
2
2
b2
m
n
am
cmn : xmn
m - no. of sources
n - no. of destinations
cij transportation cost per unit
xij amount shippe
Non-linear Programming
Problem
Classical Optimization
Classical optimization theory uses differential
calculus to determine points of maxima and
minima for unconstrained and constrained
functions.
T
Goal programming
In this chapter we discuss the Goal programming
technique for solving multi-objective models (linear).
Goal programming problem is a problem of
finding solution which attains a prede
Integer Linear Programming
Integer Linear Programming
There are many LP problems in which the decision
variables can take only integer values.
If all the decision variables take only integer values it
1
Birla Institute of Technology and Science, Pilani-K. K. Birla Goa Campus
FIRST SEMESTER 2012-2013
Optimization
(AAOC C222 & MATH F212)
Tutorial Sheet-1
1. Taking the xj as variables, and all other s
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 sol
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
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
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
Problem 5
Maximize
Problem Set 3.4B Pages 101-102
z 2 x1 2 x2 4 x3
Subject to the constraints
2 x1 x2 x3 2
3 x1 4 x2 2 x3 8
x1 , x2 , x3 0
We shall solve this problem by two phase
method.
Phase I:
Min
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
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
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