直觉模糊集在多准则决策中的PROMETHEE方法

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"直觉模糊PROMETHEE的多准则决策" 在多准则决策分析（MCDA）中，直觉模糊PROMETHEE是一种将直觉模糊集理论与PROMETHEE方法相结合的决策工具。PROMETHEE（Preference Ranking Organization Method for Enrichment Evaluation）是由比利时运筹学家J. M. Vincke和J. P. Brans提出的，它是一种基于 outranking 的多准则决策方法，主要用于评估和比较不同选项，尤其在信息不完全或偏好不确定的情况下。直觉模糊集（Intuitionistic Fuzzy Set, IFS）是模糊集理论的一个扩展，由Atanassov于1986年提出。IFS考虑了模糊度的两个方面：成员度和非成员度，这使得在处理具有不确定性和不精确信息的问题时更为灵活。在直觉模糊PROMETHEE（IF-PROMETHEE）中，不仅考虑了决策者对备选方案的直觉模糊偏好，还考虑了权重的直觉模糊性，从而提高了决策的精度和适应性。本研究论文深入探讨了PROMETHEE方法在直觉模糊环境下的应用，并提出了IF-PROMETHEE方法。该方法首先定义了直觉模糊偏好关系，然后通过引入流函数和优先流来评估备选方案之间的相对优劣。流函数衡量了方案在各个准则上的性能，而优先流则反映了方案的整体优势。论文通过两个实际案例来展示IF-PROMETHEE方法在解决多准则决策问题中的适用性和效率。第一个案例涉及可再生能源开发项目的评估，展示了如何利用IF-PROMETHEE综合考虑各种复杂因素，如成本、环境影响和技术成熟度，从而选择最优的能源项目。第二个案例可能涉及其他领域，用于进一步验证IF-PROMETHEE的普适性。直觉模糊PROMETHEE提供了一个强大的决策框架，能够处理包含直觉模糊信息的多准则问题，这对于那些在不确定性和复杂性环境下进行决策的领域，如项目管理、环境评价、金融投资等，具有重要的实践价值。此外，通过实例分析，论文证明了IF-PROMETHEE的有效性和实用性，对于未来的研究和应用提供了有益的参考。

H. Liao and Z. Xu / Multi-criteria decision making with intuitionistic fuzzy PROMETHEE 1705

there is no criteria such that b is strongly better than a,

then a will be preferred over b [1, 15, 17, 31]. Based

on the positive outranking ﬂow, the negative outranking

ﬂow and the net outranking ﬂow, the PROMETHEE can

derive partial or complete ranking of alternatives [13].

Some scholars have compared these two methods, and

also with some utility theory based methods (for more

details, readers can see [16, 26, 38]). The ELECTRE

requires the inputs of preference parameters which can

be determined by the decision maker directly or through

inference from answers to a series of questions. All the

information required in the PROMETHEE is clear and

easy to deﬁne; meanwhile, the method is easy to imple-

ment and also useful in handling incomparability. This

paper focuses on the later method, i.e., PROMETHEE,

and tries to investigate its new extension for MCDM

under uncertainty.

Since it was ﬁrstly appeared in 1982 [10], the

PROMETHEE method has been extended into different

forms, especially for the fuzzy input data (see Sec-

tion 2.1 for details). After analyzing all these extended

forms deeply, we can ﬁnd that the central idea to these

extensions is that they are based on different types of

fuzzy inputs respectively, such as interval numbers,

fuzzy numbers, fuzzy intervalnumbers, and generalized

fuzzy numbers. Some limitation may arise due to these

fuzzy sets themselves since all the above mentioned

forms of fuzzy sets are characterized by single-valued

functions of membership, which can’t be used to

express the support and objection evidences simulta-

neously. While, another extended form of fuzzy set,

named intuitionistic fuzzy set (IFS) [5, 6], which is char-

acterized by a membership function, a non-membership

function and a hesitancy function, can solve this

problem perfectly. Once not possessing a precise or

sufﬁcient level of knowledge over the problem domain

due to the complexity, people usually have some uncer-

tainty in assigning the preference evaluation values to

the objects considered, which makes the judgments

of cognitive performance exhibit the characteristics of

afﬁrmation, negation and hesitation. Due to the deﬁ-

nitely powerfulness of modeling humans’ perception

and cognition, the IFS has attracted more and more

scholars’ attention and has been applied to many differ-

ent ﬁelds [5, 6, 23, 24, 28, 32, 33, 35, 36]. As to the two

above-mentioned outranking methods, the integration

of intuitionistic fuzzy logic into ELECTRE seems to be

more advanced [31]. Generally, PROMETHEE could

also assimilate the intuitionistic fuzzy set in express-

ing the preferences of the decision maker. Therefore,

to make the PROMETHEE method more applicable in

handling uncertain information, in this paper we try to

extend the PROMETHEE method within the context of

IFSs, which is the focus of this paper.

The remainder of this paper is set out as follows.

Section 2 brieﬂy reviews the recent advances

in PROMETHEE and describes the classical

PROMETHEE II method. Some basic knowledge

about intuitionistic fuzzy set is also given in this sec-

tion. Section 3 proposes the IF-PROMETHEE method.

Two illustrative examples are given in Section 4 to

show the applicability and efﬁciency of the developed

method. The paper ends with some concluding remarks

in Section 5.

2. Preliminaries

2.1. Extensions of the PROMETHEE methods

The main idea of PROMETHEE is to derive a par-

tial or complete ranking of alternatives based on the

positive outranking ﬂow, the negative outranking ﬂow

and the net outranking ﬂow. Many different types of

PROMETHEE methods used in different circumstances

have been proposed in recent years. The PROMETHEE

I (partial ranking) and the PROMETHEE II (complete

ranking) were developed by Brans [10]. After that, sev-

eral different kinds of PROMETHEE versions were

proposed, such as the PROMETHEE III (ranking based

on intervals) and the PROMETHEE IV (continuous

case). The PROMETHEE V was proposed to han-

dle problems with segmentation constraints [11]. The

PROMETHEE VI was introduced for the representation

of the human brain [12]. The PROMETHEE GDSS was

developed for group decision-making and the visual

interactive module GAIA was established for graphi-

cal representation [13]. Due to the simplicity in both

concept and application, the family of PROMETHEE

methods has been applied successfully in many ﬁelds,

such as environment management [38], business and

ﬁnancial management [3], information technology [2],

project selection [19] and so on. Behzadian et al. [7]

made a literature review on the applications of the

PROMETHEE methods.

Although those PROMETHEE methods are useful in

handling speciﬁc MCDM problem, there are still some

drawbacks arisen from the fact that the above mentioned

PROMETHEE methods are all modeled with the crisp

input data, which is not the usual case since fuzziness

and impreciseness are common in our daily life. The

fuzzy input data involved in a decision making problem

arises from:

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