Linear Multiobjective Programming

1.1. The origin of the multiobjective problem and a short historical review The continuing search for a discovery of theories, tools and c- cepts applicable to decision-making processes has increased the complexity of problems eligible for analytical treatment. One of the more pertinent criticisms of current decision-making theory and practice is directed against the traditional approximation of multiple goal behavior of men and organizations by single, technically-convenient criterion. Reins- tementof the role of human judgment in more realistic, multiple goal se,ttings has been one of the ma~or recent developments in the literature. Consider the following simplified problem. There is a large number of people to be transported daily between two industrial areas and their adjacent residential areas. Given some budgetary and technological c- straints we would like to determine optimal transportation modes as well as the number of units of each to be scheduled for service. What is the optimal solution? Are we interested in the cheapest transportation? Do we want the fastest, the safest, the cleanest, the most profitable, the most durable? There are many criteria which are to be considered: travel times, consumer's cost, construction cost, operating cost, expected fatalities and injuries, probability of delays, etc.

1. Introduction.- 1.1 The Origin of the Multiobjective Problem and a Short Historical Review.- 1.2. Linear Multiobjective Programming.- 1.3. Comment on Notation.- Linear Multibojective Programming I.- 2. Basic Theory and Decomposition of the Parametric Space.- 2.1. Basic Theory - Linear Case.- 2.2. Reduction of the Dimensionality of the Parametric Space.- 2.3. Decomposition of the Parametric Space as a Method to Find Nondominated Extreme Points of X.- 2.4. Algorithmic Possibilities.- 2.5. Discussion of Difficulties connected with the Decomposition Method.- 2.5.1. Some Numerical Examples of the Difficulties.- Linear Multiobjective Programming II.- 3. Finding Nondominated Extreme Points - A Second Approach (Multicriteria Simplex Method).- 3.1. Basic Theorems.- 3.2. Methods for Generating Adjacent Extreme Points.- 3.3 Computerized Procedure - An Example.- 3.4. Computer Analysis.- Linear Multiobjective Programming III.- 4. A Method for Generating All Nondominated Solutions of X..- 4.1. Some Basic Theorems on Properties of N.- 4.2. An Algorithm for Generating N from Known Nex.- 4.3. Numerical Examples.- 4.3.1. An Example of Matrix Reduction.- 4.3.2. An Example of Nondominance Subroutine.- 5. Additional Topics and Extensions.- 5.1. Alternative Approach to Finding Nex.- 5.1.1. The Concept of Cutting Hyperplane.- 5.1.2. Nondominance in Lower Dimensions.- 5.2. Some Notes on Nonlinearity.- 5.3. A Selection of the Final Solution.- 5.3.1. Direct Assessment of Weights.- 5.3.2. The Ideal Solution.- 5.3.3. Entropy as a Measure of Importance.- 5.3.4. A Method of Displaced Ideal.- Appendix:.- A1. A Note on Elimination of Redundant Constraints.- A.2. Examples of Output Printouts.- A.3. The Program Description and FORTRAN Printout.