Grey-based PROMETHEE II with application to evaluation of source water protection strategies

Abstract

An approach to handling uncertainty in Multiple Criteria Decision Analysis (MCDA) using grey numbers is proposed. The grey-based PROMETHEE II methodology is designed to represent and analyze decision problems under uncertainty, which are characterized by limited input data and uncertain preferences of Decision Makers (DMs). To begin with, the basic structure of a grey decision system is developed, including definitions, notation, and detailed calculation procedures. By integrating continuous grey numbers with linguistic expressions, each DM’s uncertain preference according to multiple criteria can be expressed. The new methodology takes account of both quantitative and qualitative data, first aggregating the DMs’ individual judgements on the performance of alternatives according to each criterion, and then integrating the criteria in order to determine the relative preference of any two alternatives. These preferences are then incorporated into the PROMETHEE II method to generate a complete ranking of alternatives. The methodology is illustrated using a case study in which source water protection strategies are ranked for Waterloo Region, Ontario, Canada.

Introduction

Modern decision processes extend beyond the classical objectives in that the aim is to search not for an optimal solution, but a satisfactory one. No decision methodologies can be appropriate to all decision situations, but we can design proper methods to a specific decision problem based on its characteristics [26]. In decision analysis, one of the main difficulties is to incorporate uncertainty into decision processes. In real-world applications, practitioners must deal with deficient understanding of problem structure caused by limited human cognition, poorly understood interactions among criteria, alternatives with limited data, and the interplay of stakeholders holding vague preferences. The natural complexity of multiple criteria assessment calls for the development of effective and reliable techniques for handling decision problems under uncertainty [32], [63].

Hipel et al. [30] suggested four factors that affect the circumstances in which a decision problem must be addressed: (i) whether the context includes uncertainty; (ii) whether the courses of actions can be completely assessed in quantitative terms; (iii) whether multiple objectives must be taken into account; and (iv) whether a single DM or multiple DMs are involved. Based on these factors, decision methodologies are classified into four categories: single participant—single criterion, single participant—multiple criteria, multiple participant—single criterion, and multiple participant—multiple criteria.

In principle, Multiple Criteria Decision Analysis (MCDA) is a single participant - multiple criteria decision making technique, although it can be adopted for use by a group of DMs [4]. One of the most effective and influential sub-disciplines of Operation Research, MCDA is a methodology that includes several valuable techniques to guide DMs in identifying and structuring decision problems, and explicitly aggregating and evaluating multiple alternatives in decision environments [26], [47], [56]. Through more than 40 years, scientists and practitioners not only accelerated the theoretical and technical development of MCDA, but also gained valuable experience through applications to decision problems in many areas including environmental sciences, social sciences, education, and health care [20].

Conventional MCDA treats ratings of performance of alternatives on qualitative criteria as precisely known numerical values. In these methodologies, experts subjectively score all alternatives on criteria using specific numerical values, usually within the interval [0, 1]. However, it is sometimes difficult for DMs to assign exact numerical values to alternatives on certain criteria, or to provide precise preference information over criteria, because of limited information and the uncertainties of human judgement. Thus, linguistic expressions may be more suitable than numerical values to articulate DMs’ opinions; for example, “the performance of alternative p on criterion j is good” is more convincing than “the performance of alternative p on criterion j is 0.6”. Much research has been conducted in linguistic modeling, such as that by Ekel et al. [17] who presented research results related to multicriteria decision making with uncertain information; Agell et al. [1] who developed a qualitative reasoning approach using linguistic labels corresponding to ordinal values to represent DMs’ judgements in the context of multi-attribute and group decision-making; Alonso et al. [2] who introduced a web-based consensus support system based on fuzzy, linguistic and multi-granular linguistic preference relations; and Jiang et al. [35] who proposed a method focusing on multi-granularity linguistic modeling in a fuzzy environment. In this research, DMs’ judgements are expressed using linguistic expressions, which are then mapped to grey numbers.

The main objective of this paper is to incorporate grey numbers into PROMETHEE II (Preference Ranking Organization METHod for Enrichment Evaluation II), a multiple criteria outranking methodology. This methodology assists DMs with different perspectives to achieve a consensus on alternatives ranked on both qualitative and quantitative criteria. This approach uses the concept of grey numbers to represent information that is uncertain or ill-defined, and then combines them with PROMETHEE II. The grey-based PROMETHEE II methodology can produce results that integrate the uncertainties associated with vague input data. To help accomplish this objective, this paper presents the notation, definitions and detailed calculation processes of the grey-based PROMETHEE II methodology. The feasibility and usefulness of the proposed methodology is demonstrated using an illustrative case study. This paper is an extension of work originally presented at the 2012 IEEE International Conference on Systems, Man, and Cybernetics [37].