Introduction to
Design
A good control system should:
operate with least possible error
exhibits suitable damping
less sensitive to small changes in
parameters
be able to mitigate the effect of
undesirable disturbances
Why Compensation?
To achieve the spec
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
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
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
4 x1 2 x2 x3 3
2
1
x1 2 x2 x3 1
2
x1 , x2 , x3 0
Let x4, x5, x6
Some problems
illustrating the principles
of duality
In this lecture we look at some
problems that uses the results
from Duality theory (as discussed
in Chapter 7).
Problem 7. Problem Set 4.2D Page 130
Consider the LPP
Maximize z 5 x1 2 x2 3 x3
subject to
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
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
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
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
Problem 7.5-3 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
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
Problem 7 Problem Set 8.1A Page 351
Two products are manufactured on two sequential
machines. The following table gives the machining
times in minutes per unit for the two products:
Machine
1
2
Machining Time in min
Product 1
Product2
5
3
6
2
The daily pr
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.
Problem 6 Problem Set 2.3A Page
26(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 compon
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
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
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 activities