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
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
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 project is a collection of
interrelated activities wit
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, b] is assumed to enclose all possible estimates of the
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 . , b . , c c1 c2 . . . cn
.
.
xn
bm
a11 a12
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
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
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
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 value of the game and the optimal
strategy for each player
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 problem.
Recursive Nature of Computations in DP
Computat
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
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, b] is assumed to enclose all
possible estimates of
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_ttt
Table of Contents
Tutorial-1 .1
Tutorial-2 .4
Tuto
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 labor time is dedicated to Type II alone,
the company can p
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 shipped
ai - amount of supply available at source i
bj amount
Non-linear Programming
Problem
Classical Optimization
Classical optimization theory uses differential
calculus to determine points of maxima and
minima for unconstrained and constrained
functions.
This chapter develops necessary and sufficient
condition
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 predefined target
for one or more objective function.
If th
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 is
called a pure integer LPP; otherwise the problem is
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 symbols as given constants, determine whether
each of th
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
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
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
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
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:
Minimize
r R2
Subject to the constraints
2 x1 x2 x3 s1
3 x
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
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
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