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

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

Artificial Variable Techniques Big M-method
Lecture 6 Abstract If in a starting simplex tableau, we don't 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

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 constrain

Problem 5 Maximize
Problem Set 3.4B Pages 101-102
z = 2 x1 + 2 x2 + 4 x3 2 x1 + x2 + x3 2 3 x1 + 4 x2 + 2 x3 8 x1 , x2 , x3 0
Subject to the constraints
We shall solve this problem by two phase method.
Phase I: Minimize
r = R2
=2
Subject to the constraint

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 m m nonsingular submatrix of the contraint matrix of th

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 entrie

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

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

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

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
x1 2 x2 x3 3
4
2
1
x1 +2 x2 + x3 1
2
x1 , x2 , x3 0
Le

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

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

QUADRATIC PROGRAMMING
Quadratic Programming
A quadratic programming problem is a non-linear programming problem of the form Maximize Subject to
z = c X + X DX
T
A X b, X 0
Here
x1 b1 x b 2 2 X = . , b = . , c = [ c1 c2 . . . cn ] . . xn bm
a11 a12 a a2

CLASSICAL OPTIMIZATION THEORY Quadratic forms
x1 x 2 X = . . xn
Let
be a n-vector.
Let A = ( aij) be a nn symmetric matrix. We define the kth order principal minor as the k k determinant
a11 a12 . a1k a21 a22 . a2 k . . ak 1 ak 2 . akk
Then the q

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

Problem 10 Problem Set 10.3A Page 414
Maximize z = y1y2.yn 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

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

Dualit y t heor ems Finding t he dual opt imal solut ion fr om t he pr imal opt imal t ableau
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 so

Dual simplex method for solving the primal
I n this le cture we de scribe the im portant Dual S ple m thod and illustrate the m thod by im x e e doing oneor two proble s. m
Dual Simplex Method
Suppose a basic solution satisfies the optimality conditions b

Some problems illustrating the principles of duality
I n this le cturewelook at som proble s that use e m s the re sults from Duality the (as discusse in ory d C hapte 7). r
Problem 7. Problem Set 4.2D Page 130 Consider the LPP Maximize z = 5 x1 + 2 x2 +

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

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 1 3 2 -2 2
2 -1
Use the graphical procedure to determine the value of the game and the optimal strategy for each player

Proble 6 Proble S t 2.3A Page26(Modified) m m e 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 com

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

CPM and PERT
CPM and PERT 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 activi

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