Behnam Malakooti, Ph.D., P.E.

A computer package has been developed based on Malakooti's paper,"An Interactive Paired Comparison Method for MOLP Problems with an Underlying Quasi-Concave Utility Function." The package has several interesting features which can be used as subroutines. Each of the following features can be used independently or in conjunction with other subroutines.

1. Ranking and assessment through strength of preference; for paired comparison of two alternatives the decision maker can indicate strong or weak preference (has similarities to the Malakooti and Ravindran approach).

2. Assuming an implicit linear utility function (has similarities to the Zionts and Wallenius approach).

4. Search of the weight space (center and upper boundaries) and Generation of their associated alternatives (has similarities to the Steuer interval approach).

7. Trial and error of alternatives; generation of feasible efficient non-extreme points associated with the trial point (has similarities to setting a goal, reference point, or an ideal point; also has similarities to the goal programming and the compromise programming of Zeleny)

8. One-dimensional search, searching the efficient facet (i.e., non-extreme points), (has similarities to the Geoffrion, Dyer, andFeinberg approach).

9. Linear approximation of nonlinear utility functions for finding the best direction based on paired comparison of alternatives.

MOLP.EXE is a program that helps a decision-maker (DM) to determine the optimum solution to a multiple-objective linear program (MOLP). The procedure begins with a current solution, and then picks a new solution, which in also 'efficient'. This new solution could be anywhere in the efficient space (candidate) or an adjacent point to the current (adjacent). The DM in required to make a choice between the current solution and this new (adjacent or candidate) solution, by indicating if it is preferred. Additionally, the DM is asked if the preference is strong or week. After a series of such questions, the program will reach the optimum solution.

e) Every variable 'X' must have a coefficient. For ex.. 4X3 will appear as 4X3, however X7 will appear as 1X7.

f) A variable that does not exist in a constraint need not be typed in. For ex.. 2.3X1-1X3+4X5 = 10 in a valid constraint.

g) A variable with a larger label must come after a variable with a lower label. For ex., 2.3X1+4X5-1X3 = 10 is invalid. However 2.3X1-1X3+4X5 = 10 is valid.

h) Every constraint must end with an equality sign, and a right hand side entry. The right hand side could be positive or negative.

i) There must be no spaces between variables and coefficients and the "+" & "-" signs. However, spaces are optional around the equality sign.

j) All variables are assumed to take on values that are greater than or equal to zero. It is not required to enter this information explicitly.

An example input file for an MOLP with 3 objective functions, 10 constraints, and 16 variables is

C) The excueable and thedatafiles will exist in the current directory or in the directory where you excute MOLP_ZIP.EXE

1. Ranking through strength of preferences for nonlinear utility functions (Section 3) is not included.

2. Ranking through quasi-concave utility functions (Section 2.5) is considered only for two-point cones.

3. In the one-dimensional Search (Step 7), the program does not incorporate the numeration of adjacent points or the generation of efficient points by using problem 5.

4. In the inconsistency resolution (Step 4 of the method), only linear utility modifications (A+) are incorporated. However, inconsistency with respect to nonlinear (A*) is tested.