Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Transportation Problems
The Transportation Model
The transportation model is a special class of LPPs that
deals with transporting (shipping) a commodity from
sources (e.g. factories) to destinations (e.g. warehouses).
The objective is to determine the shi
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Constrained optimization
Lagrangean Multiplier Method:
Consider the problem
max/min z = f(X) = f(x1, x2, xn)
subject to g(X) = 0 [g1(X) = 0
g2(X) = 0
.
gm(X) = 0]
(the nonnegativity restrictions, if any, are
Function f(x) and gi(x) are twice
continuously
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
AAOC C222
Goal Programming
Deviational variables are used in goal
programming.
+
si and si are called deviational variables .
They represent the deviation below and
above the RHS of the ith constraint.
They are dependent to each other.
At most one of
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
AAOC C222
Goal Programming
Deviational variables are used in goal
programming.
+
si and si are called deviational variables .
They represent the deviation below and
above the RHS of the ith constraint.
They are dependent to each other.
At most one of
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
Dualsimplexmethodfor
solvingtheprimal
In this lecture we describe the important Dual
Simplex method and illustrate the method by
doingoneortwoproblems.
Dual Simplex Method
Suppose a basic solution satisfies the optimality
conditions but not feasible, we a
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
QUADRATIC
PROGRAMMING
Quadratic Programming
A quadratic programming problem is a nonlinear
programming problem of the form
T
Maximize
z =cX +X DX
Subject to
A X b, X 0
Here
x
b
1 1
x
b
2 2
X = b = c =[ c1 c2 . . . cn ]
. ,
. ,
.
.
n m
x
b
a11
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Classical Optimization Theory
General Classification of Solution
Methods of NLP
Direct Methods
Indirect Methods
qThe idea is to identify
Interval of Uncertainty that
is known to include the
optimum solution point.
qThe original problem is
replaced by an a
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
Goal Programming
Goal Programming is a fancy name for a very simple
idea: the line between objectives and constraints is
not completely solid. In particular, when there are a
number of objectives, it is normally a good idea to
treat some or all of them as
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Evolutionary algorithms:
Genetic Algorithms
GA is computerized search and
optimization algorithm based on the
mechanics of natural genetics and natural
selection.
Prof. John Holland of the university of
Michigan proposed the concept of this
algorithm arou
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
TheTransportationModel
Formulations
Introduction
Transportation problem is a special
kind of LP problem in which goods
are transported from a set of
sources to a set of destinations
subject to the supply and demand
of the source and the destination
respec
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Goal programming
In this chapter we discuss the Goal programming
technique for solving multiobjective models.
Goal programming problem is a problem of
finding solution which attain a predefined target
for one or more objective function
If there exists no
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Classical Optimization Theory
Local maximum/Local minimum
Let f (X)=f (x1, x2,xn) be a realvalued function of the n
variables x1, x2, , xn (we assume that f (X) is at least twice
differentiable ).
A point X0 is said to be a local maximum of f (X) if ther
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
DeterminationofStarting
BasicFeasibleSolution
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 methods
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
TheTransportationModel
Formulations
The Transportation Model
The transportation model is a special class of LPPs
that deals with transporting(=shipping) a
commodity from sources (e.g. factories) to
destinations (e.g. warehouses). The objective is to
deter
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
ArtificialVariableTechniques
BigMmethod
Lecture 6
Abstract If in a starting simplex tableau,
we dont have an identity submatrix (i.e. an
obvious starting BFS), then we introduce
artificial variables to have a starting BFS.
This is known as artificial vari
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
Introduction
In LP problems ,decision variables are non
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 decision variables
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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
MATHEMATIC 222

Fall 2015
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