Transshipment Problem
Transshipment Problem
Each supply point will have a supply equal to its
original supply, and each demand point will have a
demand to its original demand.
Let s= total available supply.
Then each transshipment point will have a supply
Goal Programming
Slides, courtesy of Dr. Shokri Selim
Motivation (by example)
It is required to promote a product on TV and Radio.
The number of radio minutes 6
Total number of viewers 45 million
Total budget $ 100,000
8 workers are available
Find the adv
Chapter. 3
Simplex Method and Sensitivity
Analysis
Heungjo An (BD22-141)
1
Agenda
I.
II.
III.
IV.
V.
VI.
VII.
Introduction
Standard Form
From Graphical To Algebraic Solution
Simplex Method
Artificial Starting Solution
Special Cases
Sensitivity Analysis
Wh
Chapter. 4
Duality and Post-Optimal Analysis
1
Agenda
I.
II.
III.
IV.
V.
Dual Problem
Primal-Dual Relationships
Economic Interpretation of Duality
Dual Simplex Algorithm
Post-Optimal Analysis
I. Dual Problem
Every LP model has its dual problem.
Original
Modeling with Linear Programming
Chapter. 2
Part 1. Modeling
Heungjo An (BD22-141)
1
Agenda
Linear Programming (LP)
LP Example
Applications
1.
2.
3.
4.
5.
Investment
Production Planning and Inventory Control
Manpower Planning
Urban Development Planning
Chapter. 8
Goal Programming
1
Goal Programming
Only need to satisfy constraints
Seeks only an efficient, rather than optimum,
solution
Definition of a goal
aX1 + bX2 + b
The goal is to make left hand side is less than or equal to
the right hand side
Chapter. 5
Transportation Model and Its
Variants
1
Agenda
I.
II.
The Transportation Model
The Assignment Model
2
The Transportation Model
A product is transported from a number of sources to a number of
destinations at the minimum possible cost.
Each sou
Chapter. 2
Modeling with Linear Programming
Part 2. Graphical Solution
Heungjo An (BD22-141)
1
Ready Mikks Company
R. M. produces interior and exterior paints
from 2 raw materials A and B.
Chap 2 page
Reddy Mikks Formulation
The LP formulation is:
Maxi
What Is Operations Research (OR)
Chapter. 1
Heungjo An (BD22-141)
Agenda
Introduction
OR Models
Solving OR Models
Art of Modeling
Introduction
OR is a discipline that deals with the application of advanced
analytical methods to help make better decisions
Simplex Method CH. 3
Prof. M. Al-Haboubi
Contents
Standard Form of LP
The Primal Simplex Method
Big-M Method
Two-Phase Method
Introduction to TORA
Special cases of LP
Analysis of the Optimum Tableau
Standard Form of LP
Max Z c j x j
j
s.t.
a x
ij
j
bi
j
x
The Transportation Model CH.5
Prof. M. Al-Haboubi
The Transportation Model
The basic model deals with the determination of a
minimum cost plan for transporting a single
commodity ( cars, boxes, etc.) from a number of
sources to a number of destinations
KING FAHD UNIVESITY FOR PE TROLEUM & MINERALS
COLLEGE OF COMPUTER SCIENCE & ENGINEERING
DEPARTMENT OF SYSTEMS ENGINIEERING
(162)
Course
Instructor
Textbook
:
:
:
ISE 303 Operations Research I
Prof. Muhammad Al-Haboubi
operations Research-An Introduction ,
Duality and Post-optimal Analysis
- CH. 4
Prof. M. Al-Haboubi
Contents
Dual problem
Formulation of the Dual problem
Optimal Dual solution
Primal-Dual Theorems
Dual simplex Method
Simplex Tableau Computations
Economic Interpretation of Duality
Post-Optimal
Network Models CH. 6
Prof. M. Al-Haboubi
Network Models
Minimal Spanning Tree Problem
Shortest-Route Problem
Maximal Flow Problem
Minimal Spanning Tree Problem
It involves choosing the branches for a network
that has the shortest total length while pr
Goal Programming CH. 8
Prof. Muhammad Al-Haboubi
Goal Programming ( GP )
LP is based on a single objective function
There are situations with multiple objectives
GP seeks a compromise solution, not necessarily an optimum
solution, based on the relative
Modeling with Linear
Programming (LP) CH.2
Prof. M. Al-Haboubi
Contents
Formulation of the LP models
Graphical Solution of the LP models
Examples of LP Formulation
Tabular Representation of data
General form of the LP model
Assumptions of LP models
Phase
Introduction to Operations
Research (OR ) CH.1
Prof. M. Al-Haboubi
Contents
Definition of OR
History of OR
Factors Played in the Growth of OR
Characteristics of OR
Mathematical Models
Phases of OR Study
Definition of OR
OR seeks finding the optimum solut
Chapter. 6
Network Model
1
Agenda
I.
II.
III.
IV.
Introduction
Shortest Path Problem
Maximal Flow Problem
Critical Path Method & Program Evaluation and
Review Technique
2
I. Introduction
Definition of a network
Consists of a set of nodes linked by arcs
ISE 303, term 142
Instructor: Heungjo An
How to use AMPL
(A Modeling Language for Mathematical Programming)
1. Download the AMPL Student Edition
=> Go to: http:/www.ampl.com/DOWNLOADS/index.html
=> Download the AMPL Student Edition zip archive file, amplc
Algebraic Sensitivity
Analysis
4.6
Motivation
It may be desirable to study how the current optimal solution
changes when the parameters of an LP problem are changed.
The discrete changes in the parameters include
1. changes in the values of a few bi `s or
Introduction to Duality
in Linear Programming
Duality Formulation
Following are the steps adopted to convert primal
problem into its dual.
Step 1. Convert the primal problem into standard
equation form.
Step 2. Assign a dual variable for each primal
co
Special Cases in Simplex Method
Degenerate Optimal Solution
How to Identify in Tableau:
Tie for Minimum Ratio, Next Iteration will have one basic
variable with zero value
Alternative Optima
How to Identify in Tableau:
Non-basic variable with zero coeffici
Artificial Starting
Solutions
Big M-Method
Two Phases Method
Artificial Starting Solutions
If all constraints from the L.P formulation require a
slack variable then we know that the corresponding
basic solution can be used to start the simplex.
? What (to
Simplex Method
Solving linear programming:
the Simplex Method
The simplex method is an algebraic procedure.
Constraint boundary is a line that forms the boundary of what is permitted
by the corresponding boundary.
The points of intersection are the cor
Maximum Flow Problem
Maximum flow problem
A commodity is shipped from a source (supply) to a destination
(demand) through a number of channels and rerouted at certain
intermediate destinations (transshipment nodes). The objective is
to maximize the total
The Shortest Path Problem
The Shortest Path Problem
2
4
4
2
2
1
1
2
3
4
6
2
3
3
5
What is the shortest path from a source node (s) to a sink node, (S)?
What is the shortest path from node 1 to node 6?
Assumptions:
1. There is a path from the source to all
Introduction to Network Modeling
Objectives
Explain the basic concepts of network modeling
Explain the terminology of network modeling
Explain the benefits of network modeling
Network Analysis
Network Analysis.Analysis OR technique(s) for the solution of
Assignment Problem
5.4
1
Assignment Problems
Example: A workstation supervisor has four jobs to be
completed. Each machine must be assigned to complete
one job. The time required to setup each machine for
completing each job is shown in the table below. T