Simulation
Simulation
Simulation is a technique of manipulating a model of a
system through a process of imitation. In simulation, the performance of the system is simulated by artificially generating a large number of sampling experiments on the model o
Joint DistributionsDiscrete and Continuous
In many statistical investigations, one is frequently interested in studying the relationship between two or more random variables, such as the relationship between annual income and yearly savings per family or
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
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 t he 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 +
Duality theorems Finding the dual optimal solution from the primal optimal 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
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
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
Let
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
Let
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
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
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
I n this le ctureweshall look at som e m llane LPPs. Each proble will illustrate isce ous m a ce rtain ide which will bee a xplaine whe the d n proble is discusse . m d
Problem 6 Problem set 3.4A Page 97 Maximize
z = 2 x1 + 4 x2 + 4 x3 3 x4
Subject to 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
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
QUADRATIC PROGRAMMING
Quadratic Programming Quadratic
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
a1
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 quadratic
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
Problem 10 Problem Set 10.3A Page 414
Maximize z = y1y2yn 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 repres
PERT Networks PERT
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 o
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
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
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 the
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
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
Artificial Variable Techniques Big M-method
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
Probability
Sample Spaces and Events Probability The Axioms of Probability Some Elementary Theorems Conditional probability Bayes Theorem
Sample Space and Events
A set of all possible outcomes of an experiment is called a sample space. It is denoted by