(621b) Mixed-Integer Multi-Objective Optimization through Multiparametric Programming

Optimization problems involving more than one objective function are referred to as Multi-Objective Optimization (MOO) problems. Conflicting objectives can arise in cases where the decision makers are concerned with more than one objective (i.e. economic and environmental) or when multiple stakeholders are involved in decision making. Different approaches have been proposed for the development of the pareto front of continuous MOO problems including the ε-constraint approach [1], the weight sum approach [2] and data driven and evolutionary approaches [3, 4]. The explicit derivation of the pareto front for continuous MOO problems through multi-parametric programming has been presented by [5, 6, 7], although very few approaches have been developed for the exact explicit derivation of the pareto front for mixed-integer MOO problems [8].

In this work we present an algorithm for the exact explicit derivation of the pareto front of mixed-integer linear MOO problems based on multi-parametric programming. The ε-constraint approach is used to transform the MOO problem into a single objective multi-parametric mixed-integer linear problem where the tunable suboptimality variables (resulting from the ε-constraints) are considered as parameters. The reformulated problem can be solved using already developed multi-parametric mixed-integer algorithms through the POP toolbox [9], supplying the decision makers with the explicit form of the pareto front in terms of the tunable variables ε. The proposed approach is illustrated through a set of numerical examples and its capabilities are demonstrated in a computational study.

References:

[1] Mavrotas G. Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems 2009, 213, 455-465.

[2] Marler, R. T.; Arora, J. S. The weighted sum method for multi-objective optimization: new insights. Structural and Multidisciplinary Optimization 2010, 41 (6), 853-862.

[3] Beykal, B.; Boukouvala, F.; Floudas, C. A.; Pistikopoulos, E. N. Optimal design of energy systems using constrained grey-box multi-objective optimization. Computers & Chemical Engineering 2018, 116, 488-502.

[4] Zitzler, E.; Thiele, L. Multi-objective optimization using evolutionary algorithms - A comparative case study, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 1498 LNCS, 1998, 292-301.

[5] Oberdieck, R.; Pistikopoulos, E. N. Multi-objective optimization with convex quadratic cost functions: A multi-parametric programming approach. Computers & Chemical Engineering 2016, 85, 36-39.

[6] Hugo, A.; Ciumei, C.; Buxton, A.; Pistikopoulos, E. N. Environmental impact minimization through material substitution: A multi-objective optimization approach. Green Chemistry 2004, 6 (8), 407-417.

[7] Charitopoulos, V.M.; Dua, V. A unified framework for model-based multi-objective linear process and energy optimisation under uncertainty. Applied Energy 2017, 186, 539-548.

[8] Rasmi, S.A.B.; Turkay, M. GoNDEF: an exact method to generate all non-dominated points of multi-objective mixed-integer linear programs. Optimization and Engineering 2019, 20, 89-117.

[9] Oberdieck, R.; Diangelakis, N. A.; Papathanasiou, M. M.; Nascu, I.; Pistikopoulos, E. N. POP - Parametric Optimization Toolbox. Industrial & Engineering Chemistry Research 2016, 55 (33), 8979-8991.