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
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
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 variab
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
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
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
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 `s
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 de
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 solut
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 n
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
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
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 rema
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 = . ,
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
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
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 prob
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 pri
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 a
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
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 4.2
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
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 val
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
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
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 pro
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,