Linear Programming
and Extensions
Duality in Linear Programming
Dual of an LP Model
0 Primal
Problem (P)
where ARmxn , cnx1, xnx1, bmx1.
0 Every constraint is associated with a dual variable wmx1, and
Linear Programming
and Extensions
Initialization of the Simplex Method
Simplex Initialization
0
P (already converted into the standard form)
If an IBFS is not immediately available we can add:
xa: Art
Linear Programming
and Extensions
Review
Vectors and Matrices
Vectors
0Vector:
A directed line segment in the n-dimensional space.
u = [a b] = [1 2]
2
Starts at
1
0Scalar Product of Vectors (Inner Pro
Linear Programming
and Extensions
Optimality Conditions
Farkas Lemma
0
Farkas
Lemma
0 Let and . Suppose and , then exactly one of the following
systems of inequalities has a solution:
1. or,
Graphical
Linear Programming
and Extensions
Review
The Simplex Algorithm
Caratheodory Theorem(CT)
0 CT tell us that any xS can be written as:
i=1, , k
j=1, , l
where for i=1,.k are the extreme points and for j=
Linear Programming
and Extensions
Simplex Method
Simplex Method
0
TABLEAU
(Table) FORM
P
Rewrite P as
Max Z
St (1)
(2)
Table Form of Simplex
0
From
(2):
(Multiplying (2) by B inverse)
(Multiplying the
Linear Programming
and Extensions
Review
Convexity
Convex Sets
0 Convex Set: is a convex set when for all
0 In linear programming we are maximizing/minimizing in
over a convex set.
0 Feasible region o
Linear Programming
HW. # 1
Question 1:
Part A:
x1 : number of items of laser printer produced over one week
x2 : number of items of jet-ink printer produced over one week
Formulation is as follows:
IE 563: Supply Chain Management
Assignment 1
Assignment 1
IE 563
Supply Chain Management
Due: October 15, 2015 (start of class)
You may want to read Lecture Notes Day 4 or Chapter 11 from the course t
Constrained Optimization of a
Function of Multiple Variables
1
Constrained Optimization with
Equality Constraints
Consider NLPs of the following type:
max (or min) f ( x1 , x2 ,., xn )
s.t.
g1 ( x1 ,
Dynamic Programming
1
Description
Dynamic Programming (DP) is a technique that
can be used to solve many optimization
problems.
It provides a systematic procedure for
determining the optimal combina
Markov Chains
(Classification of States and
Steady State Probabilities)
1
Classification of States of a MC
Definition: Given two states of i and j, state j is
accessible from state i if p(n)ij>0 for
Markov Chains
(Passage Times and
Absorbing Chains)
1
First Passage Times
The number of transitions made by the process
going from state i to state j is the first passage
time from state i to state j.
The Wagner-Whitin Algorithm
Wagner and Whitin have developed a
method that greatly simplifies the
computation of optimal production
schedules when storage and production
capacities are infinite.
1
The
Markov Chains
1
Description
Sometimes we are interested in how a
random variable changes over time.
The study of how a random variable
evolves over time includes stochastic
processes.
We will study
some examples
for the last weeks
course
1
Computer Example
A computer is inspected at the end of every hour. It
is found to be either up or down. If it is found to be
up, the probability of its remai
Continuous Time
Markov Chains
1
Continuous Time Markov Chain
So far, we looked at the evolution of a process
cfw_Xt=cfw_X0,X1,X2, at discrete times
Frequently, the evolution for a process is
observe
Probability Review
1
Basic Rules of Probability
Definition: Any situation where the outcome is
uncertain is called an experiment.
Definition: For any experiment, the sample space S
of the experiment
IE 302
Instructor: O. Orsan
Ozener
Recitation 1, Fall 2016
1
October 3, 2016
Forecasting
Question 1
Shoreline Park has the following data on the number of visitors since its opening (given in Table 1)
IE 302
Instructor: O. Orsan
Ozener
Recitation 4, Fall 2016
1
December 1, 2016
Aggregate Planning
Question 1
St. Clair County Hospital is attempting to assess its needs for nurses over the following fo
IE 302
Instructor: O. Orsan
Ozener
Recitation 7, Fall 2016
1
December 15, 2016
Operations Scheduling
Question 1
In a container terminal, five container carrier vessels are waiting for the unloading th
IE 302
Instructor: O. Orsan
Ozener
Recitation 2, Fall 2016
1
October 11, 2016
EOQ & ABC Analysis
Question 1
Number 2 pencils at the campus bookstore are sold at a fairly steady rate of 60 per week. Th
IE 302
Instructor: O. Orsan
Ozener
Recitation 5, Fall 2016
1
December 11, 2016
Supply Chain
Question 1
Bose is a manufacturer of home theater and sound systems fore households. Bose has four manufactu
IE 302
Instructor: O. Orsan
Ozener
Recitation 3, Fall 2016
1
November 24, 2016
Newsvendor
Question 1
A supermarket must decide how many loaves of bread to purchase every day. Daily demand for the
brea