Introduction

Multi-criteria decision-making (MCDM) method is referred as a method used for scoring or ranking a finite number of alternatives by considering multiple criteria attached to the alternatives. MCDM concerns with evaluating and selecting alternatives that fit with the goals and necessity. There are many MCDM methods available in the literature whereby PROMETHEE is one of the MCDM methods. The PROMETHEE is the abbreviation of preference ranking organization method for enrichment evaluation. It is a ranking method which is considered as simple in conception and computation compared to many other MCDM methods. The biggest difference between PROMETHEE and other MCDM methods is the inner relationship of PROMETHEE during the decision-making process (Murat et al. 2015). It is well adapted to the decision problems where a finite set of alternatives is to be outranked subjected to multiple conflicting criteria (Bilsel et al. 2006; Albadvi et al. 2007; Tuzkaya et al. 2010). The PROMETHEE method is based on pairwise comparisons of alternatives with respect to each criterion. According to Ulengin et al. (2001), the PROMETHEE has at least three advantages. The first advantage is its user-friendly outranking method. The second advantage is the success of PROMETHEE in applications to real-life planning problems. Another advantage of PROMETHEE lies on completeness of ranking. The PROMETHEE I and PROMETHEE II allow partial and complete ranking of alternatives, respectively. The PROMETHEE I is used to obtain partial ranking while PROMETHEE II is used for complete ranking. These two methods were developed by Brans et al. (1984, 1986).

The PROMETHEE method has been widely used in many applications in regards to its feasibility in outranking of alternatives and also its availability in multiple versions. Murat et al. (2015) used PROMETHEE I and PROMETHEE II to evaluate performance in schools. Tavana et al. (2013) used PROMETHEE method to identify and evaluate the alternative pipeline routes for transporting oil and gas from the Caspian basin to the world market. The PROMETHEE was applied to evaluate five possible pipeline routes. Besides, SWOT analysis was combined with the Delphi method to capture the decision makers' beliefs in which the PROMETHEE method was used to integrate these beliefs with subjective judgements. The PROMETHEE II method was used to compare five types of tires for mine machine mutually. Seven criteria have been used during the evaluation. As a consequence, the alternative—Bridgestone VMTP—was the most suitable for mine machine as the index of alternative is the highest among all alternatives (Tomic et al. 2013). Smet and Liduoh (2013) introduced a special emphasis on PROMETHEE and GAIA methods used to evaluate ten universities. More applications of PROMETHEE can be seen in a research conducted by Hu and Chen (2011). They proposed PROMETHEE method with concordance and discordance for financial decision-making problem regarding bankruptcy prediction. Very recently, Vasić (2018) combined the PROMETHEE and a multi-criteria analysis to participate in renewable energy sources assessment. Another combination of the PROMETHEE with two MCDM methods was applied to a research for achieving lean attributes in automotive industry (Roghanian and Alipour 2014). Nikouei et al. (2017) conducted a selection of membrane prepared from sulfonated poly and either sulfone for proton exchange membrane fuel cell based on weights of criteria using the PROMETHEE. It seems that the PROMETHEE has not yet been fully used to evaluate suppliers’ selection.

Apart from direct applications of PROMETHEE, it is also good to discuss the preference functions of PROMETHEE underpinned by several types. The PROMETHEE method introduces a preference function to describe the decision maker’s preference between pairs of alternatives for each criterion. In PROMETHEE method, different preference functions can be defined for criteria (Dagdeviren 2008). For example, Shi et al. (2016) used an extended S-shaped preference functions to express qualitative criteria such as risk preferences. There are at least six distinct types of generalized preference functions in the literature. The type I of preference function is usual criterion. It is a linear piecewise function where its range takes values of 0 to 1 and limit from the right is zero. Type II is Quasi-criterion, which is almost similar to usual criterion except its limit from the right. The other types of preference functions are type III: criterion with linear preference, type IV: level criterion, type V: criterion with linear preference and indifference area and type VI: Gaussian criteria. It is noticed that all these functions have their own features where the differences among them are inevitable. Type VI: Gaussian criteria, for example, is nonlinear function and definitely differs from type V: criterion with linear preference and indifference area. Most of the research to date has tended to focus on combination of these six types preference functions rather than one single preference function. Therefore, in the present study, a preference of green suppliers using the PROMETHEE under type I usual criterion is proposed. However, the effect of types of preference functions of PROMETHEE on the final preferences, particularly for the case of selecting suppliers, is not immediately known. There is no general agreement about the choices of preference functions and its effect on the complete ranking. In addition, rather than the direct application of PROMETHEE under type 1 usual criterion function, comparable results are also presented to check the effect of the preference functions on the final preference for green supplier selection in a food organic supply chain.

The contributions of this paper are threefold: (1) The use of type I usual criterion of the PROMETHEE is proposed for providing a complete ranking of green suppliers. (2) Two types of preference functions are dynamically chosen to check the consistency of complete ranking in green supplier selection. (3) Finally, the supplier that optimizes the use of green criteria in supplier chain management is identified. This paper is organized as follows. Section 2 briefly reviews green supplier selection as MCDM method and related research on the MCDM methods used in green supplier selection. Section 3 presents the methodology of the research including the evaluation model. In Sect. 4, computational procedures for the case study of supplier selection and results are presented. Section 5 provides the comparative results. Finally, Sect. 6 concludes.

Literature review

The rising concern for sustainability has forced managers to incorporate environmental criteria along with economical criteria in supply chains management. In other words, green supply chain management is a set of managerial practices that integrate the environmental criteria into supply chain management. It seems like a measure to balance between economic, business and environmental issues. There are at least six main criteria in the green supply chain management such as green purchasing and green raw material procurement, green design, green product development, green manufacturing, green transportation and green process planning. Among all these criteria, the most critical criteria are green purchasing and green raw material procurement (Luthra et al. 2014). Suppliers and customers need to work mutually in order to reduce the environmental impact in production processes (Tseng 2011). The effective way to protect the environment is by controlling the source through green purchasing. According to Lee et al. (2009), green purchasing is the first step in the green supply chain that applies environmental criteria to the selection of services and products. Green supplier selection can be defined as an integration of environmental criteria into conventional supplier selection (Hashemi et al. 2015). In order to earn profit and achieve an effective supply chain, selection of supplier in green purchasing is a critical issue. Standard supplier selection normally includes issues such as cost of products, services provided, delivery of products, quality and quantity of products and equipment used by supplier and so on. On the other hand, green supplier selection considers some additional green criteria which are pollution control, green product and environmental management (Paulina 2014). Green supplier selection is considered as an MCDM because of the existence of multi-criteria, multiple alternatives and finite numbers of decision makers (Bali et al. 2013).

A considerable amount of literature has been published on the MCDM methods used for evaluating green supplier selection. This section provides a short literature review of green supplier selection that had been published in Scopus database from the year 2014 to 2018. The keywords ‘green supplier selection’ and ‘decision making’ are used in this literature survey. It is summarized in Table 1.

Table 1 Review of MCDM methods used for green suppliers research
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These reviews show the importance of green criteria in selecting supplier. It can be seen that there were wide ranges of methods used in green supply chain management. So far, however, there has been little discussion about the applications of PROMETHEE to green supply chain management. To bridge the literature gap between PROMETHEE and other MCDM methods, this paper proposes an application of PROMETHEE to the case of green supplier selection problem. Different types of preference functions of PROMETHEE are used to observe its effect on the selection.

Methodology

The alternatives, criteria, decision makers and evaluation model are presented into three subsections. Preference functions used in this research are briefly explained in Sect. 3.4.

Linguistic data collection

In this research, data were collected via personal communication with a group of senior managers at an organic farm in Malaysia. In order to fit with the evaluation model, the words ‘a group of senior managers’ are now substituted with ‘a group of decision makers’ and will be used throughout this paper. The carefully selected companies offer a wide range of organic products and services to consumers through their outlets located all over Malaysia. They were asked to evaluate in terms of weight of importance to a set of criteria in green supplier selection and also weight of importance of alternatives with respect to criteria. The group of decision makers were requested to evaluate the criteria using a five-point Likert scale ranged from ‘unimportant’ to ‘very important.’ Table 2 shows the scales and their respective weight of importance.

Table 2 Five-point Likert scale and its description
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Criteria and alternatives

In this case study, the criteria for green supplier selection are defined based on the works of Gurel et al. (2015) and Mousakhani et al. (2017). The seven criteria include cost of products (C1), quality of products (C2), service provided (C3), capable of delivering on time (C4), technology level (C5), environmental management systems (C6) and green packaging (C7). Four suppliers denoted by A1, A2, A3 and A4 were investigated in this study. The supplier A1 is MVG Food Marketing Sdn Bhd. They supplied vegan organic frozen food. The wide variety of products is frozen dumpling and frozen meal box like lemongrass chicken rice. Supplier A2 is CF org Noodle Sdn Bhd, which is a noodle manufacturer company. The products they offer are whole wheat noodle, spinach noodle and spirulina stick noodle. The supplier A3 is Hexa Food Sdn Bhd which is a spice, herb and seasoning manufacturer. They sell a variety of ground spices and seasoning to use in preparing dishes. Supplier A4 is SCS Food Manufacturing Sdn Bhd, which is a sugar and salt manufacturer. They supply variety of salts such as organic salt, fine salt and coarse salt.

Evaluation model

The PROMETHEE method begins with an evaluation of alternatives with respect to the criteria. These evaluations essentially need numerical data where their implementation needs information on the relative importance of the criteria and also information on the decision maker’s preference functions. The information is obtained when the decision makers compared the contribution of the alternatives with respect to each criterion. The computational procedures of PROMETHEE need several steps, and this paper has summarized seven steps based on the works of Polat (2015), Geldermann et al. (2000), Behzadian et al. (2010) and Brans et al. (1986).

Step 1 Determine the criteria (j =1,…, k) and the set of possible alternatives in a decision problem.

Step 2 Determine the weight wj of the criteria. It shows the relative importance of each of the criteria and notes that \( \sum\limits_{j = 1}^{k} {w_{j} = 1} \)

Step 3 Normalize the decision matrix to range 0–1 by using

$$ R_{ij} = \frac{\lfloor{X_{ij} - \hbox{min} (X_{ij} )}\rfloor}{[{\hbox{max} (X_{ij} ) - \hbox{min} (X_{ij} )}]}\quad \left( {i = 1,2 \ldots ,n\;{\text{and}}\;j \, = \, 1,2, \, \ldots ,m} \right), $$
(1)

where Xij is evaluation values provided by decision makers i = 1……, n, and numbers of criteria j = 1,….,m.

Step 4 Determination of deviation by pairwise comparison.

$$ d_{j}^{{}} \left( {a, \, b} \right) \, = \, g_{j} \left( a \right) \, {-} \, g_{j} \left( b \right) $$
(2)

dj (a, b) denotes the difference between the evaluations of a and b on each criterion.

Step 5 Define the preference function Pj (a, b)= Fj [dj (a, b)],

where Pj (a, b) represent the function of the difference between the evaluations of alternative a regarding alternative b on each criterion into a degree ranging from 0 to 1. The smaller number of the functions denotes the indifference of the decision maker. On the contrary, the closer to 1 indicates greater the preference.

Step 6 Determine the multi-criteria preference index.

$$ \pi (a,b) = \sum\limits_{j = 1}^{k} {P(a,b)w_{j} } $$
(3)

where wj > 0 are the weights associated with each criterion. The symbol π(a, b) shows that the degree of a is preferred to b over all the criteria.

π(a, b) \( \approx \) 0 implies a weak preference of a over b.

π(a, b) \( \approx \) 1 implies a strong preference of a over b.

Step 7 Obtain the preference order

In this step, ranking can be made either partially or completely. Partial ranking can be obtained using PROMETHEE I, and in case complete ranking is needed, then the computation must proceed to one more step in PROMETHEE II.

  1. (a)

    Ranking the actions by partial ranking (PROMETHEE I).

    $$ \phi^{ + } (a) = \frac{1}{n - 1}\sum\limits_{x \in A} {\pi (a,x)} \;{\text{and}}\;\phi^{ - } (a) = \frac{1}{n - 1}\sum\limits_{x \in A} {\pi (a,x)} . $$
    (4)

\( \phi^{ + } (a) \) represents positive outranking flow or is known as leaving flow (how a dominates all the other alternatives), and \( \phi^{ - } (a) \) represents the negative outranking flow or is known as entering flow (how a is dominated by all the other alternatives).

The alternative with a higher value of \( \phi^{ + } (a) \) and the lower value of \( \phi^{ - } (a) \) is the best alternative. The preference relation and partial ranking are derived as follows:

$$ aP^{ + } b:\left\{ {\begin{array}{*{20}c} {P\;iff\phi^{ + } (a) \succ \phi^{ + } (b),\quad \forall a,b \in A} \\ {I\;iff\phi^{ + } (a) = \phi^{ + } (b),\quad \forall a,b \in A} \\ \end{array} } \right. $$
$$ aP^{ - } b:\left\{ {\begin{array}{*{20}c} {P\;iff\phi^{ - } (a) \prec \phi^{ - } (b),\quad \forall a,b \in A} \\ {I\;iff\phi^{ - } (a) \succ \phi^{ - } (b),\quad \forall a,b \in A} \\ \end{array} } \right. $$

However, not all alternatives are comparable. Thus, we need to calculate the net outranking flow in the following step.

  1. b.

    Ranking the actions by a complete ranking (PROMETHEE II).

The complete ranking of alternatives can avoid incomparability.

$$ \phi (a) = \phi^{ + } (a) - \phi^{ - } (a), $$
(5)

where \( \phi (a) \) denotes the net outranking flow for each alternative. The preference relations are as follows:

$$ a\;{\text{outranks}}\;{\text{of}}\;b\;(aP^{(II)} b)\quad iff\;\phi (a) \succ \phi (b),\quad \forall a,b \in A $$
$$ a\;{\text{indifferent}}\;{\text{of}}\;b\;aP^{(II)} b\quad iff\;\phi (a) = \phi (b),\quad \forall a,b \in A $$

Thus, all the alternatives are able to be compared based on the values of \( \phi (a) \). The highest values of \( \phi (a) \) denote the most preferred alternative.

In these series of computational procedures, most of the steps are fixed except Step 5. In this step, it is an arbitrary where the choice of preference functions depends very much on the characteristics of criteria and also the preference of decision makers. Attention is paid to the choice of types of preference functions as it may affect the final net outranking values.

Preference functions

The implementation of the PROMETHEE method requires preference functions. Preference function of PROMETHEE is a function used to define deviations between alternatives for each criterion. In this paper, the definitions of preference functions are presented to fulfill the requirement of PROMETHEE algorithm that will be implemented to a case study (Tuzkaya et al. 2010; Ozgen et al. 2011; Brans and Vincle 1985). With reference to Step 5, in Subsect. 3.3, three preference functions are employed. They are defined as follows.

Definition 1

Type I: Usual criteria function is defined as

$$ p(x) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{for}}\;x \le 0} \\ {1,} & {{\text{for}}\;x > 0} \\ \end{array} } \right. $$
(6)

where x represents the deviation between two alternatives.

In type 1, indifference only occurs when f(a)= f(b). It is used when the decision makers cannot allocate importance for the differences between criteria values and only seem to know the formula ‘the more the better.’

Definition 2

Type III: Criterion with linear preference function is defined as

$$ p(x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x < 0} \hfill \\ {\frac{x}{m},} \hfill & {0 \le x \ge m} \hfill \\ {1,} \hfill & {x > m} \hfill \\ \end{array} } \right. $$
(7)

The intensity of preference increases linearly and becomes strict on point m. Parameter m is arbitrary and needs to be defined.

Definition 3

Type IV: Level criterion function is defined as

$$ p(x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {{\text{for}}\;x \le q} \hfill \\ {\frac{1}{2},} \hfill & {{\text{for}}\;q < x \le q + p} \hfill \\ {1,} \hfill & {{\text{for}}\;x > q + p} \hfill \\ \end{array} } \right. $$
(8)

Indifference on the interval [− q, q]. For type IV, it is always used for quantitative criteria.

Proposed selection framework and implementation

The framework applied to green supplier selection is presented in Fig. 1.

Fig. 1
figure 1

Framework for supplier selection

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The framework includes the goal of selection problem, the economical and environmental criteria, the suppliers and also the expected results of partial ranking and net outranking.

The implementation of PROMETHEE to the case is made by identifying the decision makers, green criteria and the suppliers. In personal communication, a group of five managers (D1, D2, D3, D4, D5) was requested to rank and evaluate four suppliers (A1, A2, A3, A4) based on the seven criteria (C1, C2, C3, C4, C5, C6, C7).

Data that were collected are shown in Table 3.

Table 3 Performance rating of the suppliers
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Information in Table 3 and the computational procedures using the usual criterion preference functions (see Sect. 3.3) are computed according to the following steps.

Step 1 Identify the criteria and suppliers

Criteria: cost (C1), quality (C2), services (C3), delivering on time (C4), technology level (C5), environmental management systems (C6), green packaging (C7).

Suppliers: supplier A1, supplier A2, supplier A3, supplier A4.

To understand the relationship between suppliers and criteria, the analysis of GAIA (Graphical Analysis for Interactive Assistance) is made. With a representation value of 97%, the relationship between suppliers and criteria is depicted in Fig. 2.

Fig. 2
figure 2

GAIA visual analysis

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It can be seen that there are two groups of criteria, which can be identified as {C6, C4, C1, C5, C7} and {C2, C3}. These two sets seem to be independent from each other, and therefore, there is no strong conflict between the criteria

Step 2 Compute criteria weight.

Evaluations provided by the decision makers need to be normalized in the range 0–1 and compute the fraction based on each criterion. Using the normalization in Eq (1) and fraction, the normalized data given by each decision maker are summarized in Table 4.

Table 4 Normalized data for each criterion
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Then, aggregate the weight of each criterion using arithmetic mean. Table 5 presents the criteria and their respective weights.

Table 5 Average weights of criteria
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Step 3 Normalize and aggregate the rating of suppliers.

The ratings of suppliers are averaged to aggregate and normalize them to 0–1 scale. The evaluations of four suppliers corresponding to all the criteria are displayed in Table 6.

Table 6 Evaluation of supplier with respect to criteria
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Step 4 Determination of deviation by pairwise comparison.

This step involves the calculation of the differences in criteria values of A1 with respect to other suppliers. The deviations are obtained using Eq (2). The summary of deviations is shown in Table 7.

Table 7 Deviations of any two potential suppliers with respect to criteria Cj
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Step 5 Selection of preference function.

The implementation of the PROMETHEE method requires a preference function. Preference function of PROMETHEE is a function used to define deviations between alternatives for each criteria. The usual criterion preference function (see Eq 6) is employed in this step. Table 8 shows the deviations using usual function.

Table 8 Preference function—usual function
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Step 6 Calculate the preference index.

Calculation of the preference index of each supplier takes into account the criteria weight. It is also a value to show the degree of preference of a supplier over another supplier. The index is calculated using Eq (3). The preference index is presented in Table 9.

Table 9 Preference index value
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Step 7 Determine the positive and negative outranking flows of each supplier (PROMETHEE I partial ranking).

  1. (a)

    Leaving flow and entering flow of suppliers

    Positive outranking flow (leaving flow) shows the degree of the supplier dominated other suppliers. In contrast, negative outranking flow (entering flow) shows the degree of the supplier dominated by other suppliers. Equation (4) is used to calculate these two flows. Leaving flow and entering flow of suppliers are shown in Table 10.

    Table 10 PROMETHEE I flow
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  2. (b)

    Determine the net outranking flow (PROMETHEE II) for each supplier.

    Net flow values are calculated to avoid incomparability. Equation (5) is used to complete the calculation of net outranking flow. It is presented in Table 11.

    Table 11 Net flow value of suppliers
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The ranking of suppliers is arranged in descending order of net flow value. The best supplier is the one having the highest net flow value, \( \phi (a) \). By using PROMETHEE II (complete ranking) method, supplier A1—MVG Food Marketing Sdn Bhd—is selected as the best alternative.

The results of PROMETHEE I and PROMETHEE II are presented in Table 10 and Table 11, respectively. These results could be corroborated by drawing a PROMETHEE Diamond chart as this chart displays the results of PROMETHEE I and PROMETHEE II simultaneously. Figure 3 displays the positions of alternatives (PROMETHEE II—complete ranking) and also the positive and negative outranking flows values (PROMETHEE I partial ranking) of alternatives.

Fig. 3
figure 3

PROMETHEE diamond

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It can be seen that alternative A1 is ranked as the first preference followed by A4. The ranking order of preference of suppliers is obtained as \( A_{1} \succ A_{4} \succ A_{3} \succ A_{2} \), where ‘\( \succ \)’ shows ‘more preferred than.’ It can be concluded that MVG Food Marketing Sdn Bhd is the most preferred supplier.

It is often difficult to get a robust result due to the variability in relative importance of a given criterion. In response to this issue, an interactive tool called walking weights is used to check the sensitivity of the result. For example, the relative importance of the criterion C2 is increased by 33%; thus, new result is shown in Fig. 4.

Fig. 4
figure 4

Walking weights

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It is noticed that the result does not have an impact on the first-ranked alternative. However, a slight inconsistency in ranking can be seen for other three alternatives. The results obtained are further discussed in Sect. 5.

Comparative results

In PROMETHEE, it is possible to choose a different preference function for each criterion. Differently from the preference function used in Sect. 4, linear preference function with linear preference and indifference area (type V) and level preference function (type IV) are chosen for the green supplier selection problem (Tuzkaya et al. 2010; Ozgen et al. 2011; Brans and Vincke 1985). Both functions are chosen based on the nature of criteria. The PROMETHEE with linear and level function method is assumed to be tailored to the nature of the criteria. For instance, linear preference function was chosen as one of the functions because it is best suited for quantitative criteria such as criterion C1 (cost). However, level preference function is best suited for qualitative criteria such as criterion C2 (quality of products). In addition, the level function works well in small numbers of different levels, such as five-point measure scale.

To begin with the computations, Step 1 to Step 4 in Sect. 4.3 are iterated. For the purpose of comparative analysis, Step 4 is continued as follows.

Step 5 Selection of preference function.

In this step, linear and level functions are chosen corresponding to the criteria used in this study. Linear function and level function are defined in Eqs (7) and (8), respectively.

Both of the linear and level functions consist of indifference and preference thresholds. The indifference threshold represents the largest value of q which is there is no preference for suppliers over another whereas the preference threshold represents the smallest value of p which is sufficient to generate a full preference. In this case study, indifference threshold and preference threshold are set as 0.05 and 0.1, respectively. Table 12 shows the criteria and their respective preference function.

Table 12 Type of preference function corresponding to criteria
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Deviations between alternatives are translated into the domains of preference functions. It is shown in Table 13.

Table 13 Preference function—linear and level function
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Step 6 Calculate the preference index value.

Preference index values are obtained using Eq (3), and the results are presented in Table 14.

Table 14 Preference index value
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Step 7 Determine the positive and negative outranking flows of each supplier (PROMETHEE I).

  1. (a)

    Positive outranking flow and negative outranking flow are obtained using Eq (4). The flows of each supplier are summarized in Table 15.

    Table 15 PROMETHEE I Flow
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  2. (b)

    Determine the net outranking flow (PROMETHEE II) for each supplier.

Net flows are calculated using Eq (5). The results are presented in Table 16.

Table 16 Net flow value of suppliers
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It is shown that supplier A1—MVG Food Marketing Sdn Bhd—is the best supplier by using the combination of linear and level function in the PROMETHEE method. All in all, the preference order can be written as \( A_{1} \succ A_{3} \succ A_{4} \succ A_{2} \)

Discussion and conclusion

This paper has proposed the preference in selecting the suppliers that deal with green organic products. Different types of preference functions have been used in the implementation. The first net outflows and preference order are obtained using the usual criterion preference function, which is considered as the simplest function. The second net outflows and preference order are obtained using the combination of linear preference function and level preference functions. Both of these functions are chosen based on the nature of criteria. The final preference order of suppliers is summarized in Table 17.

Table 17 Summary of preference order of suppliers
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It can be seen that supplier A1 is consistently ranked as the first choice of supplier. Contrarily, the supplier A2 has the weakest performance for both preference functions.

Furthermore, the net flow values of all the suppliers are analyzed and compared. Figure 5 shows the performance (net flows) of each supplier under two different preference functions.

Fig. 5
figure 5

Net flow values of suppliers under usual function and linear and level function

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The net flow values are used to see the suppliers’ performance. Comparing the results in Fig. 2, it is noticed that supplier A1 received the highest net flow value and supplier A2 received the lowest net flow value for both preference functions. It is also good to note that supplier A3 received the third highest net flow value in PROMETHEE (usual function) method and the second highest value in PROMETHEE (linear and level function) method. Finally, supplier A4 received the second highest value in PROMETHEE (Usual Criterion function) method and the third highest value in PROMETHEE (linear and level function) method.

The PROMETHEE is one of the MCDM methods that is based on outranking. This method comprises many steps and choices of preference function as one of the significant steps. Many literatures acknowledged that there are at least six types of preference functions in PROMETHEE. However, the effect of type of preference functions to the final preference order is not fully discussed. This paper has investigated this issue where the usual function, linear function and level function were used as the preference functions of PROMETHEE. This investigation was implemented to the case study of green supplier selection in which four alternatives, five decision makers and seven criteria were the main structures of this MCDM method. It is found that the effect of these functions to the final preference order is not significant.

However, this study has some limitations, particularly in the choice of preference functions, its arithmetic operations and also the comparative analysis used. The choice of preference functions is limited to the three functions. Other preference functions or a modified preference function of PROMETHEE could be investigated in future research. The final preference order of this present study was obtained using subtraction operation which is sometimes looks very straightforward. Other arithmetic operations such as division combined with the concept of distance perhaps shed light on future research. This comparative study could also be implemented to other types of selection problems, such as manufacturing and service sectors.