اولویت گزینه فازی برای مدل گراف برای تحلیل تعارض

A fuzzy option prioritization technique is developed to efficiently model uncertain preferences of DMs in strategic conflicts as fuzzy preferences by using the decision makers' (DMs') fuzzy truth values of preference statements at feasible states within the framework of the Graph Model for Conflict Resolution. The preference statements of a DM express desirable combinations of options or courses of action, and are listed in order of importance. A fuzzy truth value is a truth degree, expressed as a number between 0 and 1, capturing uncertainty in the truth of a preference statement at a feasible state. A fuzzy preference formula is introduced based on the fuzzy truth values of preference statements, and it is established that the output of this formula is a fuzzy preference relation. It is shown that fuzzy option prioritization can also be used when the truth values of preference statements at feasible states are completely based on Boolean logic, thereby generating a crisp preference over feasible states that is the same as would be found by employing the existing crisp option prioritization, making the crisp option prioritization technique a special case of the fuzzy option prioritization methodology. To demonstrate how this methodology can be employed to represent fuzzy preferences in real-world decision problems, fuzzy option prioritization is applied to an actual dispute over groundwater contamination that took place in Elmira, Ontario, Canada.

Decision making is a common activity in everyday life. To make decisions easier, a number of methodologies have been developed including linear and non-linear optimization [1] and [35], multiple-criteria decision analysis [13], game theory [38], fuzzy decision making [10] and [30], and the Graph Model for Conflict Resolution [11] and [28]. In these decision making techniques, decision makers' (DMs') preference information, expressed implicitly or explicitly, is an essential component. Preference information may be given in various forms, for instance as utilities (as in classical game theory [38]), as fuzzy utilities (as in fuzzy decision making [10] and [30]), via option prioritization (as in the Graph Model for Conflict Resolution [12], [22], [32] and [33]), or simply as pairwise relative preferences over the feasible states or scenarios (as in a crisp or fuzzy Graph Model [2], [3], [4], [11], [20] and [28]). In whatever form a DM's preference information is provided, the objective is always to represent a crisp or fuzzy preference relation over the states or scenarios. A crisp or ordinary preference relation is composed of the binary relations “is (strictly) preferred to” and “is indifferent to”. A crisp preference describes the certainty of the preference for one state over another which, in general form, may or may not be transitive. A fuzzy preference relation is expressed using numerical values between 0 and 1, interpreted as pairwise preference degrees. A preference degree for one state over another indicates the degree of certainty of the preference for the first state over the second. A degree of certainty of 1 is equivalent to the crisp binary relation “strictly preferred to” while a degree of certainty of 0 is equivalent to equal certainty but in the reverse direction. In multiple participant–multiple objective decision problems, DMs interact through their decisions, and often have opposing (or inconsistent) preferences. Consequently, strategic conflicts are obvious in these decision problems. A number of formal methodologies have been developed to facilitate the analysis of these problems and to advise on possible resolutions. These methodologies include game theory [38], metagame analysis [24], conflict analysis [16], drama theory [25], and the Graph Model for Conflict Resolution [11] and [28]. These various methodologies all share fundamental characteristics: they represent and analyze decision situations with at least two DMs, each of whom has one or more options and distinctive preferences over the outcomes [27]. This paper is specifically addressed to the Graph Model methodology. To apply this methodology, there are two steps: modeling and analysis. In the modeling step, feasible states and moves among them are usually constructed using the option form [11] and [28]. A feasible state is a feasible combination of options, selected or not selected. To make clear how a real-world decision problem is formulated within the Graph Model framework, an environmental conflict in Elmira (a small town in Ontario, Canada) is described here. This conflict began in late 1989 when the Ontario Ministry of the Environment (MoE) found that an underground aquifer in the town was contaminated by a carcinogen. The main suspect was a chemical company in Elmira, Uniroyal Chemical Ltd. (UR), which produced the same carcinogen as a by-product. MoE issued a control order demanding that UR take necessary measures to rectify the contamination. However, UR appealed the control order. The Local Government (LG) was another DM of the conflict as it attempted to represent local interests. These DMs had differing objectives; for example, MoE wanted to require UR to rectify the contamination, while UR wanted the control order lifted or at least modified. The dispute is modeled as a Graph Model, in which each DM has one or more options that it either selects or not. For instance, to attempt to reach a preferable outcome, UR could delay the appeal process, accept the original control order, or abandon its Elmira operation. An important input to the analysis step of the Graph Model, and to GMCR II [12] and [32], a decision support software that implements the Graph Model methodology, is each DM's preference information. Modeling preferences by pairwise comparison is a challenging task for DMs as well as for analysts. Rather, it is easier for a DM to provide a priority sequence of what he or she likes to see about the available courses of action. This idea was formalized as option prioritization [12], [32] and [33]. A DM's preference is modeled using a priority list of preference statements. These statements are composed of options using logical connectives, such as “and”, “or”, and “if–then”, and listed from most preferred to least. The option prioritization methodology relies on the truth values (“true” or “false”) of each preference statement at each state, where the truthfulness of a more important preference statement dominates its falsity in calculating a DM's preferences. For example, in the Elmira model, LG's most important preference statement is “UR does not close its operations in Elmira”; so any state or scenario in which UR closes its operations in Elmira is less preferred than any state in which UR continues its operations. A limitation of preference modeling by option prioritization is that it assesses a preference statement based only on whether it is “true” or “false” at a state. For example, to model its preference in the Elmira dispute, LG may consider the proposition “insist on the application of the original control order” as one of its preference statements. It may be reasonable to restrict the truth value of this preference statement to either “true” or “false” at states in which UR is delaying the appeal process. However, the truth value of this statement at a state in which UR accepts a control order (original or modified) may not be precisely “true” or “false”; rather, it may be more reasonable to assume a fuzzy truth value, a degree of truth, taken from a fuzzy truth space, usually the closed unit interval [0,1][0,1]. A truth degree of 1 for a preference statement at a state indicates that the preference statement is true, while a truth degree of 0 implies that the preference statement is false. Note that fuzzy truth values are the main concept behind fuzzy logic and fuzzy sets, introduced by Lotfi A. Zadeh [41], which have a wide variety of applications in engineering, decision sciences and other areas [6], [7] and [34]. Bashar et al. [5] found a way to use fuzzy truth values of preference statements at feasible states to model a DM's preferences. They developed a formula to calculate a fuzzy score for each feasible state, and then ranked the states based on these scores, resulting in crisp preferences; a state with a higher score is preferred to a state with a lower score. However, since fuzzy scores of states were calculated by using the fuzzy truth values of DMs' preference statements, it is more reasonable to represent the DMs' preferences as fuzzy preferences over the feasible states. The objective of the present research is to extend [5], developing a fuzzy version of the crisp Graph Model option prioritization to model DMs' fuzzy preferences for use in the analysis step of the Fuzzy Preference Framework for the Graph Model for Conflict Resolution. The Fuzzy Preference Framework for the Graph Model for Conflict Resolution is a generalized Graph Model methodology for modeling and analyzing decision problems with DMs' certain or uncertain preference information. A Graph Model fuzzy preference framework takes a DM's fuzzy preferences into account to calculate fuzzy stabilities. But, just as for crisp preferences, modeling fuzzy preferences by pairwise comparison of states is difficult, and even impractical for larger problems. Until now, there has been no suitable technique to model fuzzy preferences by aggregating DMs' uncertain preference information within a Graph Model framework. In this paper, the fuzzy truth values of preference statements for a feasible state are used to generate a fuzzy score interval for the state. One feasible state is then compared to another using their fuzzy score intervals to calculate a fuzzy preference degree for the first state over the second. To perform this comparison, an idea is adapted from [40] in which two intervals of real numbers are examined to measure the degree of possibility that one interval is not less than another. The remainder of the paper is as follows. The next section presents the structure of a Graph Model and its associated preferences. A fuzzy option prioritization methodology is developed in Section 3. In Section 4, the fuzzy option prioritization technique is applied to the Elmira groundwater contamination conflict. In the final section of the paper, some conclusions are drawn.

Fuzzy option prioritization, developed in this paper, is the first formal methodology to model fuzzy preference within the Graph Model for Conflict Resolution structure in order to deal with uncertain preference information. This technique offers flexibility to DMs or analysts who can assume the intensity of truth of a preference statement at a feasible state to be any number between 0 and 1 (inclusive). Note that the crisp option prioritization methodology takes only Boolean logic-based truth values, “true” (equivalent to a truth degree of 1) and “false” (equivalent to a truth degree of 0), into account, and therefore fuzzy option prioritization constitutes a generalization of crisp option prioritization. Moreover, the fuzzy option prioritization methodology includes two parameters, p and α, the values of which can be chosen by DMs or analysts based on their judgements with respect to the uncertainty associated with available preference information. When applied to the Elmira groundwater contamination conflict, the fuzzy option prioritization technique generates fuzzy preferences for UR and LG that are close to the fuzzy preferences modeled by pairwise comparison of states in [4]. Because the truth values of preference statements of MoE at feasible states are all based on Boolean logic, the methodology provides a crisp preference ordering of states for MoE that justifies the claim in the previous paragraph. It follows that the methodology developed in this paper is efficient and generates crisp or fuzzy preference outputs depending on input information. Crisp or fuzzy preferences may be modeled by pairwise comparison of states for a reasonably small problem, but not for a large one. However, the fuzzy option prioritization technique developed in this paper can be applied to a dispute of any size. Since the Fuzzy Preference Framework for the Graph Model for Conflict Resolution was developed in [4] to study multiple participant–multiple objective decision problems by calculating fuzzy stabilities based on DMs' fuzzy preferences, the introduction of fuzzy option prioritization, an efficient tool to model fuzzy preferences, will make the Graph Model fuzzy preference framework more useful.