Keywords

1 Introduction

Decision-making rarely involves a single evaluation criterion: indeed, most of our decisions have at least economic, social and environmental consequences, and can involve multiple stakeholders as well. As a result, the optimization of a single and most often economic objective function using classical Operational Research techniques (e.g., Linear Programming, Integer and Mixed Integer Programming) is not providing solid grounds to assist decision-makers. Decision aid emerged as an alternate approach to mathematical optimization by contributing to the analysis of the entire process and of its structure, ensuring that coherent, formal procedures are used to propose solutions and to justify the rationale of the decision [1].

More precisely, decision aid methods can be used in four particular problematics of decision-making:

α: Choosing the best option among a set of potential alternatives.

β: Sorting a small set of alternatives into a number of predefined categories.

γ: Ranking a set of alternatives with the purpose of comparing them.

δ: Describing the consequences of the possible alternatives.

Within the field of decision aid, a set of formal, quantitative approaches deals specifically with the multicriteria nature of decision-making: it is called multi-criteria decision aid (MCDA) and its specific purpose is to assist decision-makers to structure a multi-criteria decision problem [2] and to reach an informed decision [3]. There are many different approaches in MCDA and they can be broadly categorized into the multi-attribute utility and value (MAUT) approach, the outranking methods and the non-classical methods [4].

One of the most known and widely used MCDA outranking methods is PROMETHEE (the acronym stands for Preference Ranking Organization METHod for Enrichment of Evaluations). Two basic variants of the method (PROMETHEE I and II) were developed by J.P. Brans and were presented for the first time in 1982. Over the years, the core method was extended and enriched with the works of J.P. Brans and B. Mareschal.

The purpose of this chapter is to introduce the basic elements of PROMETHEE and to illustrate how it can be applied to various problem settings as a decision support methodology.

The rest of the chapter is organized as follows:

Section 2 introduces the methodological bases of the PROMETHEE and GAIA methods by introducing their mathematical foundations and explaining in a step-by-step fashion how they can be applied. Section 3 is focused on the available software solutions that can assist decision-makers to apply the PROMETHEE methods. Section 4 provides an overview of the applications of the methods in the literature, while finally, conclusions and future avenues for research are presented in the last section of the chapter.

2 Methodology

2.1 PROMETHEE I and II

Most MCDA methods share similar notations and structures. As such, an alternative can be regarded as the object of a decision or a representation of an action that can be put into operation.

A criterion is a tool that represents a perception of the decision-maker (or different perceptions by different decision-makers) on how an alternative can be evaluated or compared to other potential alternatives.

As a result, an MCDA problem can be mathematically defined as follows:

$$ \operatorname{Max}\ \left\{{g}_1(a),{g}_2(a),\dots {g}_j(a),\dots {g}_m(a)|a\in A\right\} $$
(1)

A is a finite set of possible alternatives {a 1, a 2, …a i, …a n} and {g 1(∗), g 2(∗), … g j(∗), …g m(∗)} a set of criteria on which the alternatives are evaluated upon. The criteria might require maximization or minimization without an effect on the process. The MCDA problem thus attempts to identify an alternative that optimizes all the criteria.

Nonetheless, the case where an ideal alternative optimizes all the criteria at once is a rare occurrence and MCDA attempts to find a compromise solution taking into account the preferences of the decision-maker(s). For that purpose, all the information and data is transformed to an evaluation table (Table 1).

Table 1 Evaluation table for MCDA
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The preferences of the decision-maker are translated into a set of natural relations. These are preference, indifference, and incomparability and they are analyzed in Table 2.

Table 2 Relations in MCDA
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The notion of incomparability means that no decision can be made between two alternatives without additional information and/or data either from the decision-maker or the analysis of the problem under study. Consequently, at their core, MCDA methods attempt to reduce or eliminate the incomparabilities in any given problem [5].

The differentiation of the PROMETHEE methods relies on valued preferences, keeping some of the incomparabilities and allowing for a partial compensation among the criteria. These aspects of PROMETHEE are especially important for various classes of decision problems, since incomparabilities can occur naturally, thus making them desirable to be incorporated in the decision-making process because they can reveal more insights regarding the decisions. Furthermore, the partial compensation in the values of the criteria is particularly useful for problems that are unstructured or for those that include several dimensions and no dimension should be easily replaced by another (e.g., assessment of sustainability) ([6, 7].

Besides the evaluation table, the PROMETHEE methods require additional information from the decision-maker(s). The first piece concerns the relative importance of the criteria, which is expressed by their respective weights that must follow:

$$ \sum_{j=1}^m{w}_j=1 $$
(2)

Furthermore, the PROMETHEE methods take into account the level of deviations between the evaluation of the alternatives in each criterion. For small levels of deviation, there might be the option of indifference by the decision-maker, while on the other hand, the bigger the deviation, the larger the preference of one alternative compared to another. This is expressed with a set of functions:

$$ {P}_j\left(a,b\right)={F}_j\left[{d}_j\left(a,b\right)\right],\forall a,b\in A $$
(3)

Where

$$ {d}_j\left(a,b\right)={g}_j(a)-{g}_j(b) $$
(4)

And for which

$$ 0\le {P}_j\left(a,b\right)\le 1 $$
(5)

As it was mentioned above, some of the criteria might require minimization instead of maximization. In that case, the preference function is changed to:

$$ {P}_j\left(a,b\right)={F}_j\left[-{d}_j\left(a,b\right)\right],\forall a,b\in A $$
(6)

Hence, the information about and on the criteria is accompanied by the preference function enriching the process. This expanded criterion is called generalized criterion and six types of preference functions have been proposed. They are illustrated in Fig. 1.

Fig. 1
figure 1

Types of preference functions for the construction of the generalized criteria [5]

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The preference functions introduce a new set of extra parameters that need to be defined. These are:

  • q is the threshold of indifference: it is the largest deviation which is considered as negligible by the DM.

  • p is the threshold of strict preference: it is the smallest deviation which is considered as sufficient to generate a full preference.

  • s is an intermediate value between q and p. It only appears in the case of the Gaussian criterion (type 6).

The PROMETHEE process continues with the aggregation of the preference indices.

Let a, b ∈A and

$$ \pi \left(a,b\right)=\sum_{j=1}^k{P}_j\left(a,b\right){w}_j $$
(7)
$$ \pi \left(b,a\right)=\sum_{j=1}^k{P}_j\left(b,a\right){w}_j $$
(8)

π(a, b) is expressing the degree to which alternative a is preferred to alternative b over all the criteria and π(b, a) the reverse (by how much b is preferred to a). In most cases, there is no alternative dominating any other on all criteria; for some criteria alternative a will be preferred to b and for other the reverse will apply. The following properties hold for all (a, b) ∈A

$$ \pi \left(a,a\right)=0 $$
(9)
$$ 0\le \pi \left(a,b\right)\le 1 $$
(10)
$$ 0\le \pi \left(b,a\right)\le 1 $$
(11)
$$ 0\le \pi \left(a,b\right)+\pi \left(b,a\right)\le 1 $$
(12)

It is clear that π(a, b)~0 implies a weak global preference of alternative a over alternative b and π(a, b)~1 implies a strong global preference of alternative a over alternative b. To gain clear insights on the strengths and weakness of each alternative, the outranking flows are defined.

  • The positive outranking flow:

    $$ {\varphi}^{+}(a)=\frac{1}{n-1}\sum_{x\in A}\pi \left(a,x\right) $$
    (13)

    • It expresses how an alternative a is outranking all the others. The higher the positive outflow the better the alternative

  • The negative outranking flow:

    $$ {\varphi}^{-}(a)=\frac{1}{n-1}\sum_{x\in A}\pi \left(x,a\right) $$
    (14)

    • It expresses how an alternative a is outranked by all the others. The lower the negative outranking flow the better the alternative.

Having all the information and calculation of the outranking flows, we are ready to construct the ranking of the alternatives.

2.1.1 PROMETHEE I Partial Ranking

The PROMETHEE I partial ranking is inferred by the positive and negative outranking flows

$$ {aP}^Ib\kern1em \mathrm{iff}\kern1em {\displaystyle \begin{array}{c}{\varphi}^{+}(a)>{\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)<{\varphi}^{-}(b)\\ {}{\varphi}^{+}(a)={\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)<{\varphi}^{-}(b)\\ {}{\varphi}^{+}(a)>{\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)={\varphi}^{-}(b)\end{array}} $$
(15)
$$ {aI}^Ib\kern1em \mathrm{iff}\kern1em {\varphi}^{+}(a)={\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)={\varphi}^{-}(b) $$
(16)
$$ {aR}^Ib\kern1em \mathrm{iff}\kern1em {\displaystyle \begin{array}{c}{\varphi}^{+}(a)>{\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)<{\varphi}^{-}(b)\\ {}{\varphi}^{+}(a)<{\varphi}^{+}(b)\;\mathrm{and}\;{\varphi}^{-}(a)>{\varphi}^{-}(b)\end{array}} $$
(17)

Where P I, I I, R I stand for preference, indifference, and incomparability, respectively.

According to the definitions of the outranking flows:

$$ \varphi (a)={\varphi}^{+}(a)-{\varphi}^{-}(a)=\frac{1}{n-1}\sum_{j=1}^k\sum_x\left[{P}_j\left(a,x\right)-{P}_j\left(x,a\right)\right]{w}_j $$
(18)

Consequently,

$$ \varphi \left(\alpha \right)=\sum_{j=1}^m{\varphi}_j(a){w}_j $$
(19)

If

$$ {\varphi}_j(a)=\frac{1}{n-1}\sum_x\left[{P}_j\left(a,x\right)-{P}_j\left(x,a\right)\right]{w}_j $$
(20)

φ j(a) is the single criterion net flow obtained when only criterion g j(∗) is considered (100% of the total weight is allocated to the criterion). It expresses how an alternative a is outranking (φ j(a) > 0) or outranked (φ j(a) < 0) by all the other alternatives on criterion g j(∗). The profile of an alternative consists of the set of all the single criterion net flows.

2.1.2 PROMETHEE II Complete Ranking

Often, a decision-maker might ask for a complete ranking. In that case we define the net outranking flow:

$$ \varphi (a)={\varphi}^{+}(a)-{\varphi}^{-}(a) $$
(21)

The higher the net flow, the better the alternative:

$$ a{P}^{II}b\ \mathrm{iff}\ \varphi \left(\alpha \right)>\varphi (b) $$
(22)
$$ a{I}^{II}b\ \mathrm{iff}\ \varphi \left(\alpha \right)=\varphi (b) $$
(23)

The strength of PROMETHEE II is that the aggregation of the outranking flows means that all alternatives can be compared among one another. This can be easier for a decision-maker to understand and communicate; however, valuable information is lost, especially concerning incomparabilities among alternatives.

A final remark concerning both the PROMETHEE approaches concerns the weighting of the criteria. This process can be considered subjective since each decision-maker may assign unique weight values to the criteria, which could result in differentiations in the final ranking of the alternatives. As a result, both PROMETHEE I and II could benefit greatly and increase the robustness of the results by performing sensitivity analysis [8].

2.2 PROMETHEE Visualizations

One of the important aspects of the PROMETHEE methods is that it allows a comprehensive visualization of the results that communicates the ranking in an intuitive way, provides information about the positive and negative outranking flows, and increases the transparency of the analysis.

In that aspect, several efforts have been performed in the previous years in that direction. For example, Mareschal and De Smet [9] developed the “PROMETHEE diamond,” which can be used to represent the outranking flows along with the net flow without losing information. However, the established visualization approach is the GAIA plane. It is obtained from the unicriterion net flows and its purpose is to analyze the impact of each individual criterion in the final results [5].

In particular, the GAIA plane can be considered complementary to the PROMETHEE II complete ranking. This is an effort to visualize a decision-making problem in a two-dimensional representation and to include all its aspects: alternatives, criteria, weights, and preference parameters. Figure 2 below provides an example of what the GAIA plane looks like.

Fig. 2
figure 2

Example of the GAIA plane

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The alternatives are represented by bullets and the criteria with arrows. Already by representing these in a two-dimensional figure new insights can be gained. The position of the alternative provides the decision-maker with a first picture of existing similarities: the closer the actions, the more similar they are. Similarity and non-similarity are determined by the limit of indifference and preference. This means that the GAIA plane depends on the preference information provided by the decision-maker in the form of the parameters q, p, and/or s.

In a similar manner, the relative position of the criteria indicates the correlation and/or conflict among them. The closer the arrows are, the higher the contribution of the criteria to the decision problem. Information is provided also by the angle of the arrows: the greater the angle between the criteria, the greater the conflict.

As a result, the advantage of using the GAIA plane is that it allows the illustration of contradictory views. In addition, the length of a criterion measures its “discriminating” or “differentiating” power as a function of the data. The more different the actions in a criterion, the greater the arrow and, therefore, the more distinctive criterion. The discretionary power of a criterion depends on the selected limits and its corresponding weight.

Finally, the arrow represented by the letter D, known as the decision stick, illustrates the compromise chosen by the decision-maker as it corresponds to the weight adjustment. The visibility of the alternatives on this line is a clear way to show to the decision-maker their own priorities. The greater the visibility of an alternative on the stick, the better its ranking position. However, since it is only a two-dimensional representation, it can lead to a loss of information. The amount of information held, the so-called delta or D, depends on the data and the number of criteria. As a consequence of the loss of information, the classification resulting from the promotion of the decision stick does not necessarily have exactly the same results with PROMETHEE II. [10].

The research in visualizing PROMETHEE has not stalled in the later years. Schröder et al. [11] presented a new visualization tool which allows the joint representation of PROMETHEE I and II results. Furthermore, the new technique can perform in a group decision-making context by illustrating in a transparent way the differences of the decision-makers as they are represented by the different weighting schemes of the criteria. Finally, the technique offers a clustering of the alternatives under all these representations, thus increasing the transparency of the decision-making process.

In conclusion, PROMETHEE I and II do not only provide a structured way of ranking alternatives, but they also provide the decision-maker with enriched information that can assist her to better clarify their own preferences. Finally, powerful visualization techniques such as the GAIA plane, prove to be powerful communication tools that can further assist the decision-maker to fully understand how and why the particular ranking of the alternatives occurred.

3 PROMETHEE Software

The popularity and usability of the PROMETHEE methods, naturally led to the development of several software implementations over time. The most important are:

  • PROMCALC was introduced at the end of 1980s. It was an interactive and graphical software running under MS-DOS.

  • Ten years later, Decision Lab 2000 became the first MS-Windows implementation of PROMETHEE.

  • Visual PROMETHEE is the currently available software.

Besides the PROMETHEE rankings and GAIA plane computation, Visual PROMETHEE includes many extensions and sensitivity analysis tools, including:

  • Different graphical representations of the PROMETHEE rankings.

  • PROMETHEE V for portfolio selection under constraints.

  • GDSS-PROMETHEE extension for multiple decision-makers’ analysis.

  • Hierarchical organization of the criteria.

  • Interactive weight sensitivity analysis tools.

The availability of interactive software and sensitivity (“what if”) analysis tools is essential in the context of multicriteria decision aid: decision-makers should be able to easily check the impact of changes of their preference parameters on the results of the analysis. Visualization is also important, either through simple graphics or more elaborated analyses such as GAIA.

Visual PROMETHEE can be downloaded from the http://www.promethee-gaia.net web site. The site also includes additional information related to the PROMETHEE methods.

4 PROMETHEE Applications

The friendliness-of-use, the rich information provided, and the availability of the software have contributed to the increasing use of the PROMETHEE methods in a wide variety of areas. Searches in relevant scientific databases (ScienceDirect, Scopus, EBSCO, DOAJ, SpringerLink, etc.) are used to update the PROMETHEE Bibliographical Database (http://biblio.promethee-gaia.net). Figure 3 illustrates the areas of application as of February of 2020,Footnote 1 including over 2200 papers published by more than 4000 authors from 88 countries. The figure shows that the majority of the papers are related to societal issues. This is not unexpected since these types of problems are not easily quantified and involve multiple stakeholders. PROMETHEE is thus a natural candidate to structure such ill-defined problems.

Fig. 3
figure 3

Areas of application for PROMETHEE

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Furthermore, Fig. 4 illustrates the number of papers that were published per year.

Fig. 4
figure 4

Number of papers published per year (1982–2020)

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The figure reveals an upward trend for PROMETHEE, meaning that more and more authors are steadily using the methods to solve decision-making problems. Of these papers, real-life applications are also increasing, meaning that the method is considered an established one, thus suitable to be used. Finally, PROMETHEE covers problems that are not limited to industrial applications but concern wider societal issues [12].

5 Conclusions and Future Research

The focus of this chapter was to provide an overview of PROMETHEE as a multi-criteria decision aid method, how it can be used in supporting decisions, and where it can be applied. The chapter illustrated the structured way that the methodology proceeds in helping decision-makers, and how it can be used not only to rank alternatives but also to communicate and justify in an intuitive way any decision taken.

One of the constants of the method is that it evolves with new variations emerging to address problems. As a result, future directions for research could include the explicit incorporation of uncertainty in the evaluation of the decisions. Such an effort has been attempted from the very first steps of PROMETHEE, when Mareschal [13] showed that insensitivity intervals can visualize the impact of varying weights on the rankings. Moreover, the existence of big data can stir the developments towards attempting to incorporate them in PROMETHEE analyses, maybe even changing the nature of what is considered a relevant criterion and what its values can be. In that regard, PROMETHEE can not only be combined with, but can also be adapted to act as an Artificial Intelligence (AI) algorithm.

As it was shown in the previous sections, visualization is an intrinsic part of PROMETHEE, with ongoing research efforts. Consequently, we believe that new attempts at more dynamic and interactive visualizations will continue to emerge in the field. Furthermore, decisions are rarely static: they need to be dynamic and constantly evaluated. Hence, the incorporation of the time dimension in the PROMETHEE framework is a promising and imperative research avenue.

Finally, the proliferation of mobile phones could signal a new surge of PROMETHEE applications, provided that software is developed that will be specifically designed for small screens and on-the-fly decision-making.