Kolman & Beck <Elementary linear programming with applications> 2nd ed.

Elementary linear programming with applications

By Bernard Kolman, Robert Edward Beck

Contributor Robert Edward Beck

Edition: 2, illustrated

Published by Academic Press, 1995

ISBN 012417910X, 9780124179103

449 pages

Prologue
: Introduction to Operations Research

Definitions

Operations research (OR) is a scientific method for providing a quantitative basis for decision making (that can be used in almost any field of endeavor).

http://en.wikipedia.org/wiki/Operations_research

The techniques of OR give a logical and systematic way of formulating a problem so that the tools of mathematics can be applied to find a solution (when the established way of doing things can be improved.

A central problem in OR is the optimal allocation of scarce resources (including raw materials, labor, capital, energy, and processing time).

eg. T. C. Koopmans and L. V. Kantorovich, the 1975 Nobel Prize awardees

Development

WWII, Great Britain
the United States Air Force
1947, George B. Dantzig, the simplex algorithm (for solving linear programming problems)
programmable digital computer (for solving large-scale linear programming problems)
1952, the National Bureau of Standards SEAC machine



Phases

> Steps
1. Problem definition and formulation

to clarify objectives in undertaking the study
to identify the decision alternatives
to consider the limitations, restrictions, and requirement of the various alternatives

2. Model construction

to develop the appropriate mathematical description of the problem
: limitations, restrictions, and requirements -> constraints
: the goal of the study -> quantified as to be maximized or minimized
: decision alternatives -> variables

3. Solution of the model

: the solution to the model need not be the solution to the original problem

4. Sensitivity analysis

to determine the sensitivity of the solution to changes in the inpuit data
: how the solution will vary with the variation in the input data

5. Model evalution

to determine whether the answers produced by the model are realistic, acceptable and capable of implementation

6. Implementation of the study

The Structure of Mathematical Models

There are two  types of mathematical models: deterministic and probabilistic.

A deterministic model will always yield the same set of output values for a given set of input values, whereas a probabilistic model will typically yield many different sets of output values according to some probability distribution.

Decision variables or unknowns

to describe an optimal allocation of the scarce resources represented by the model

Parameters

: inputs, not adjustable but exactly or approximately known

Constraints

: conditions that limit the values that the decision variables can assume

Objective function

to measures the effectiveness of the system as a function of the decision variables
: The decisin variables are to be determined so that the objective function will be optimized.

Mathematical Techniques

mathematical programming

Mathematical models call for finding values of the decision variables that maximize or minimize the objective function subject to a system of inequality and equality constraints.

Linear programming

Integer programming

Stochastic programming

Nonlinear programming

network flow analysis

standard techniques vs. heuristic techniques (improvised for the particular problem and without firm mathematical basis)


Journals

Computer and Information Systems Abstracts Journal
Computer Journal
Decision Sciences
European Journal of Operational Research
IEEE Transactions on Automatic Control
Interfaces
International Abstracts in Operations Research
Journal of Computer and System Sciences
Journal of Research of the National Bureau of Standards
Journal of the ACM
Journal of the Canadian Operational Research Society
Management Science (published by The Institute for Management Science - TIMS)
Mathematical Programming
Mathematics  in Operations Research
Naval Research Logisics (published by the Office of Naval Research - ONR)
Operational Research Quarterly
Operations Research (published by the Operations Research Society of America - ORSA)
Operations Research Letters
OR/MS Today
ORSA Journal on Computing
SIAM Journal on Computing
Transportation Science