Technical note on balanced solutions in goal programming, compromise programming and reference point method
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Technical note on balanced solutions in goal programming, compromise programming and reference point method
Balanced solutions in GP, CP and RPM Romero et al1 present a GP formulation equivalent to the CP and RPM models:
Journal of the Operational Research Society (2002) 53, 927–931. doi:10.1057/palgrave.jors.2601368
Minimize Z ¼ D Subject to:
Introduction In their paper, Romero et al1 describe the linkages between goal programming (GP), compromise programming (CP) and reference point method (RPM). Basically, these approaches are distance function methods. These models take into account several objectives simultaneously where the distance between a certain point and the achievement level of each objective is to be minimized. Usually the decision maker expresses his preferences, chooses the type of the distance function and sets the other parameters regarding the objectives and the system constraints. The GP, CP and RPM models constitutes the most commonly known models of the Multi-Objective Programming. The vast popularity of these models is due, in part, to the fact that they are easy to understand and the fact that they are easy to apply since they constitute an extension of linear programming for which efficient algorithms are available.2 Romero et al1 show that the CP and RPM models are equivalent to the GP formulations. The authors use the results obtained by Balestero and Romero3 to conclude that the solution obtained by the equivalent formulation of the CP and RPM represents a perfectly balanced allocation among the achievement of the different objectives. Moreover, they also state that if the reference levels are set at their anchor values, the weighted deviation of the reference values from achievement level of the objective will be equal. The example that the authors used to illustrate their development shows clearly that if the initial reference points are the anchor values, the solution will be unbalanced. This is in contradiction with their conclusions. We contend that while there are special circumstances when such equilibrium holds, as a general statement this conclusion is not correct. The aim of this viewpoint is to show analytically that the conclusion of Romero et al1 regarding the balanced equations is not correct. We will present a numerical example demonstrating this point.
Wi n 4D Ki i
ðfor i ¼ 1; 2; . . . ; qÞ;
fi ðxÞ þ ni pi ¼ bi
ð1Þ
ðfor i ¼ 1; 2; . . . ; qÞ;
x 2 Cs ; ni and pi 5 0 ðfor i ¼ 1; 2; . . . ; qÞ; where x Cs fi(x) bi Ki Wi q ni pi
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
n dimensional vector of decision variables; feasible set of constraints; mathematical expression of objective i; anchor value associated with objective i; normalization constant of objective i; the importance coefficient of objective i; number of objectives under consideration; negative deviation associated with objective i; positive deviation associated with objective i.
The authors mention that t
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