Sensitivity Analysis
Solving an LP model is not a one-shot affair
Change in original data
What if Analysis
For what range of a particular data value is the current
solution valid?
Types of Sensitivity Analysis
Data Changes
1. Objective Function Coefficien
Network Flow Problems
Consider a Network with some commodity flowing through it.
The flow originates at SOURCE NODE : s
terminates at SINK NODE : t
each arc (i,j) has following parameters associated with it
cij capacity (in terms of flow per unit time)
ai
Goal Programming
Limitation of LP
Permits only one objective function.
Constraints are hard. No flexibility.
Real World Situation
Several objectives/goals
Achieve high profits
Limit Costs
Gain/maintain Market Share
Ensure stable work force
Need to expa
Transportation Problem
Transport a commodity from several sources to several
destinations.
No. of sources : m, No. of destinations : n
Fixed supply at each source : si
Fixed demand at each destination : dj
Unit cost of transportation from source i to dest
Review
Simultaneous linear equations (three cases)
m = n (unique solution)
m > n (either unique solution, or no solution)
m < n (infinite number of solutions)
(This case is the only one of interest in LP)
How to solve
Set excess variables (n-m) to value
Minimum Wt. Spanning Tree
Need to select n-1 edges
Total weight of edges is minimized.
There are no cycles.
Greedy Approach
Sort edges by weight.
Select n-1 edges with least weight.
Whats wrong with this approach?
There may be cycles.
Modified Greedy Appr
Finding initial basic feasible solution
An initial bfs is automatically available when
the all constraints are type, and
rhs is non-negative (why?)
What if
a)rhs for some constraint is negative?
multiply the constraint by -1 : changes the inequality type.
Duality in Linear Programming
Every LP has a closely related LP, called its DUAL.
Very different looking representations of the same problem.
Like two sides of a coin.
Management
Study of this relationship yields fresh insights into the problem
structure
Shadow Price
Constraints represent resources.
RHS represents quantity of resource available (e.g. raw material)
More Resource : More Profit (better Z value)
Question: How much improvement in objective function is possible if one
extra unit of resource is
Standard Form
An LP is said to be in Standard Form if
All constraints are equality constraints.
All variables are non-negative.
Objective function is Minimization.
Any linear program can be converted into
standard form. How?
Add Slack/surplus variables
Solving a System of Equations
m equations, n unknowns:
Case 1 : m = n
There exists a unique solution.
Case 2 : m > n (more equations
than unknowns)
either m-n equations are redundant,
or no solution.
Case 3 : m < n
No unique solution exists
Infinite No.
Problem 2: New Venture Finance
Need to raise 50 lakhs
Upto 20 lakhs in bonds @ 12%
Fin. Corp A : 30 lakhs @ 18%
Condition: Bond debt + loan from B < loan from A
Fin. Corp. B : (@ 16% )
Condition: loan amt. not to exceed the loan from A
x1: bond debt,
x2:
Binomial Distribution
Binomial Distribution
Discrete Distribution
Only 2 outcomes
Throw of coin
Pass / Fail a course
Defective / Non defective
Characteristics
Only 2 outcomes
The Probability of getting success is same in
each trial
The outcomes of the
Sampling
Why we resort to Sampling?
Complete enumeration not possible
Destroyed
Time
Expensive
Glossary
Population people , all items that have been
chosen for study
Sample Subset of the population
Population Parameters
Sample Statistic
Types of S
Random Variables & Probability
Distributions
Frequency Distribution
A frequency distribution is a listing of the
observed frequencies of all outcomes of an
experiment.
Frequency Distribution
First Toss
Second Toss
No. of Tails in 2 tosses of
a coin
H
T
1
Confidence Intervals for Population
Mean & Population Proportion
= significance level = combined area under
both tails
/2 = area under one tail
(1-) = confidence level
Confidence Intervals for Population
Mean
Drawing up Interval estimates
CI = sample
PROBABILITY
Chance
Probability = no. of favorable outcomes/total
no. of outcomes
Collectively Exhaustive
A list that includes every possible outcome
Similar to Set Theory
Listing of Sample Space
Mutually Exclusive
Events are said to be mutually excl
Normal Distribution
Characteristics of the Normal
Distribution
Bell Shaped Curve
Curve has single peak unimodal
Curve is symmetrical
Peak corresponds to mean, median,mode
Tails extend indefinitely and never touch the
horizontal axis
Special Case of Normal
Probability
Addition Rule for Mutually Exclusive
Events
P(A or B) = P(A) + P(B)
P (A U B)
Venn Diagram
Addition Rule for Non -Mutually
Exclusive Events
P(A or B) = P(A) + P(B) P (A B)
Venn Diagram
Marginal Probability
What is the probability of draw
Hypothesis Testing
Hypothesis Testing
Hypothesis means making a statement based on
gut feel
For value about Population Parameter like the
population mean or proportion
Hypothesis is a claim about the Population
parameter
Collect Data
To validate or i
Correlation & Regression
The prices of Gold and Silver have been rising
steadily. Is there a correlation between them?
Is there a correlation between the number of
study hours put by the students and the
scores obtained by them in a particular
subject?