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
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
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
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
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
The Transportation Model
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
det
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
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
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
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
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
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
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
In this presentation
we illustrate the ideas
developed in the
previous presentation
with two more
problems
Consider the following LPP:
Maximize z 6 x1 x2 2 x3
Subject to
1
2 x1 2 x2 x3 2
2
3
4 x1 2 x2 x3 3
2
1
x1 2 x2 x3 1
2
x1 , x2 , x3 0
Let x4, x5, x6
In this presentation we
illustrate the ideas
developed in the
previous presentation
with two more problems
Consider the following LPP:
Maximize z 6 x1 x2 2 x3
Subject to
1
2 x1 2 x2 x3 2
2
3
4 x1 2 x2 x3 3
2
1
x1 2 x2 x3 1
2
x1 , x2 , x3 0
Let x4, x5, x6
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