Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Formulation
The Decision Variables
Variables whose values are under our control and
influence system performance are called
decision variables.
The Objective Function
In most models, there will be a function we wish
to maximize or
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
Feasible Solution: Any values of x1,
x2 that satisfy all the constraints of the
model is called a feasible solution.
Optimum feasible solution that gives
the maximum profit / Minimum cost
while satisfying all the
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
MATRIXFORMULATION
OFTHELPps
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 LPP
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
Convex sets
A point X is called a convex linear combination (CLC)
of points X1 and X2 , if there exists a , 0 1, such
that X = (1 )X1 + X2. or
X = 1X1 + 2X2 , 1 + 2 =1, 1, 2 0.
Graphically: X = (1 )X1 + X2 is t
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
Step 1: convert the LPP into equation form
Step 2: In order to obtain starting BFS, add
non negative artificial variable in all
constraints of and = sign,
Step 3: In order to get rid of the artificial
variable
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
Two Phase Method
Phase I:
Step1: Convert the constraints in Standard form
(add the necessary slack, surplus and artificial
variables to secure starting basic solution).
Step 2: Ignore the Objective function of Or
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
Special Cases in Simplex
Method
Alternative optima
Unbounded Solution
Nonexisting or infeasible solution
Degeneracy
Alternative optima
When the objective function is parallel to a
non redundant binding constrai
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
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
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2012
Optimization Techniques
Linear Programming
If the primal LPP is a minimization problem
Minimize
z = cX
subject to A X = b,
X 0, b 0
the dual LPP in matrix form becomes
Maximize
w = Yb
subject to Y A c,
Y unrestricted in sign
7.4 B:Q3. Consider the followi
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
Problem6ProblemSet2.3APage26(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 component, a
Birla Institute of Technology & Science, Pilani  Hyderabad
Optimisation
MATHEMATIC OPTi 101

Spring 2013
ArtificialVariableTechniques
BigMmethod
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 vari