Abstract
Systems can be unstructured, uncertain and complex, and their optimisation often requires operational research techniques. In this study, we introduce AUGMECON-R, a robust variant of the augmented ε-constraint algorithm, for solving multi-objective linear programming problems, by drawing from the weaknesses of AUGMECON 2, one of the most widely used improvements of the ε-constraint method. These weaknesses can be summarised in the ineffective handling of the true nadir points of the objective functions and, most notably, in the significant amount of time required to apply it as more objective functions are added to a problem. We subsequently apply AUGMECON-R in comparison with its predecessor, in both a set of reference problems from the literature and a series of significantly more complex problems of four to six objective functions. Our findings suggest that the proposed method greatly outperforms its predecessor, by solving significantly less models in emphatically less time and allowing easy and timely solution of hard or practically impossible, in terms of time and processing requirements, problems of numerous objective functions. AUGMECON-R, furthermore, solves the limitation of unknown nadir points, by using very low or zero-value lower bounds without surging the time and resources required.
Highlights
Despite rapid technological advancements in software and hardware performance, many problems featuring numerous evaluation criteria or objective functions (Wiedemann 1978), multiple constraints of different nature and hundreds to thousands of decision variables remain challenging to solve (Carrizosa et al 2019)
According to Hwang et al (1980), Multiple-Objective Mathematical Programming (MOMP) solving algorithms can be organised in three groups: a priori methods, in which the decision makers have the capacity to express their preferences or objective function weights prior to solving the problem; interactive methods, which feature an ongoing dialogue between analysts and decision makers, eventually leading to preferences converging with solutions; and a posteriori methods, in which the problem is solved and the effective Pareto solutions are
Another significant novelty of AUGMECON is that it exploits cases where the problem is infeasible, leading to an early exit from the nested loop of the step increase function: the algorithm initially sets lower bounds to the constrained objective functions, which gradually become stricter; if the problem becomes infeasible, i.e. the model cannot be solved for the given constraint of an objective function, after a specific grid point increase, there is no point in strengthening the constraints and the algorithm exits from the innermost loop and continues to the grid point of said objective function
Summary
Despite rapid technological advancements in software and hardware performance, many problems featuring numerous evaluation criteria or objective functions (Wiedemann 1978), multiple constraints of different nature and hundreds to thousands of decision variables remain challenging to solve (Carrizosa et al 2019). Several methods have been developed to solve multi-objective linear programming problems, each of which features strengths and weaknesses (Sylva and Crema 2007). By looking at its main novelties, its core weaknesses are identified and discussed in detail, serving as a motivation for developing a new model that effectively overcomes them. These weaknesses, dependent on various characteristics and processes of the method, can be summarised in the ineffective handling of the true nadir points of the objective functions of a problem and, most notably, in the significant amount of time required to apply it as more objective functions are added to a problem, which can even make a problem practically insolvable.
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