Mathematics 88-369
Operations Research
Course Summary
Bar Ilan University
Semester A - 2008
Mathematics 88-369
Operations Research
Syllabus
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Formulate the
Linear Programming Problem
OR math models have three elements:
1. Decision Variable (DVs) — things you can control
2. Constraints — limit of control
3. Objective Function — measure of effectiveness
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Definitions and Theory
Conduct Parametric Programming



A  S
A  S
A  S
is CLOSED if it Contains Each of its Limit Points
is BOUNDED if
$ e > 0 ' A  Ne ( X)
is COMPACT if it is Both Closed and Bounded
Weierstrass Theorem
A FUNCTION f ( X)  C DEFINED ON A COMPACT SET, S , HAS AN OPTIMUM, x *  S
Linear Program
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Definitions and Theory
Conduct Parametric Programming
–
LINEAR PROGRAMMING
•
•
•
•
Formulate the
Linear Programming Problem
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Solve using the
Simplex Algorithm
Algorithm Structure
)
f(x
Δ
Solve using the
Simplex Algorithm
d1
f(x)
d2
Optimality Condition
≥0
Feasibility Condition
bi
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Solve using the
Simplex Algorithm
Algorithm
Conduct Parametric Programming
T=
≤0
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Solve using the
Simplex Algorithm
Algorithm – Two Phase Method
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
•
•
•
•
•
Solve using the
Simplex Algorithm
Algorithm – Special Cases
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Solve using the
Simplex Algorithm
Algorithm – Revised Simplex / Implementation
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Solve using the
Simplex Algorithm
Algorithm – Computational Efficiency
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Definitions – Duality
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Interpretation
Conduct Post – Optimality Analysis
Theory – Duality
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Theory – Duality
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Theory – Duality
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Theory – Duality
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Determine “Shadow Prices”
“SHADOW PRICES”
•
•
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Determine “Ranges of Optimality”
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Determine “Ranges of Feasibility”
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Post – Optimality Analysis
Add New DVs and / or Constraints
Formulate the
Linear Programming Problem
Solve using the
Simplex Algorithm
Conduct Post-Optimality Analysis
Determine “Shadow Prices”
Determine “Ranges of Optimality”
Determine “Ranges of Feasibility”
Add new “Decision Variables”
and / or “Constraints”
Conduct Parametric Programming
Conduct Parametric Programming
LINEAR PROGRAMMING
SUMMARY

DEFINITIONS AND FUNDAMENTAL THEOREMS
1.
2.
3.
4.
5.
LP MODEL
BASIC FEASIBLE SOLUTION (BFS)
CONVEXITY
EXTREME POINTS
FUNDAMENTAL THEOREMS OF OPTIMALITY
-

SIMPLEX ALGORITHM
1.
2.
3.
4.
5.

PIVOT OPERATIONS
FEASIBILITY CONDITION
OPTIMALITY CONDITION
SIMPLEX ALGORITHM
MATRIX FORM OF SIMPLEX TABLEAU
MODIFICATIONS TO SIMPLEX
1.
INITIAL BFS
-
2.
3.
4.
5.
REVISED SIMPLEX
COMPUTER IMPLEMENTATIONS OF THE SIMPLEX ALGORITHM
COMPUTATIONAL EFFICIENCY OF THE SIMPLEX ALGORITHM
DUALITY THEORY
-
2.
3.
DEFINITIONS
THEOREMS
APPLICATIONS
COMPLIMENTARY SLACKNESS
DUAL SIMPLEX ALGORITHM
-
OPTIMALITY CONDITION
FEASIBILITY CONDITION
PIVOT OPERATION
POST OPTIMALITY ANALYSIS
1.
2.
TABLEAU / LP PARAMETER RELATIONSHIPS
SENSITIVITY ANALYSIS
-

ALTERNATE OPTIMA
UNBOUNDEDNESS
INFEASIBILITY
DEGENERACY AND CYCLING
DUALITY AND DUAL SIMPLEX
1.

SURPLUS AND ARTIFICIAL VARIABLES
TWO PHASE METHOD
SPECIAL CASES
-

CONVEXITY OF FEASIBLE REGION
IDENTITY OF BFS AND EXTREME POINTS
OPTIMUM OF LP OCCURS AT AN EXTREME POINT
RANGE OF OPTIMALITY
RANGE OF FEASIBILITY
SIMULTANEOUS CHANGES
PARAMETRIC PROGRAMMING
1.
2.
OBJECTIVE FUNCTION COEFFICIENTS
CONSTRAINT EQUATION RHS VALUES