Using the Manhattan distance for computing the multiobjective Markov chains problem

ABSTRACT

This paper presents a novel method for computing the multi-objective problem in the case of a metric state space using the Manhattan distance. The problem is restricted to a class of ergodic controllable finite Markov chains. This optimization approach is developed for converging to an optimal solution that corresponds to a strong Pareto optimal point in the Pareto front. The method consists of a two-step iterated procedure: (a) the first step consists on an approximation to a strong Pareto optimal point and, (b) the second step is a refinement of the previous approximation. We formulate the problem adding the Tikhonov's regularization method to ensure the convergence of the cost-functions to a unique strong point into the Pareto front. We prove that there exists an optimal solution that is a strong Pareto optimal solution and it is the closest solution to the utopian point of the Pareto front. The proposed solution is validated theoretically and by a numerical example considering the vehicle routing planning problem.

KEYWORDS:

• Multi-objective optimization
• Pareto front
• strong Pareto policies
• Pareto optimality
• Markov chains

2010 AMS SUBJECT CLASSIFICATIONS:

• 90C29
• 90C40
• 90C30
• 47A52
• 47J06

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Julio B. Clempner http://orcid.org/0000-0002-5918-4671