A Comparative Evaluation of Multi-Criteria Analysis Methods for Sustainable Transport

Abstract

The article pertains to the utilization of the application potential of MCDM/MCDA (Multi-Criteria Decision Making/Multi-Criteria Decision Analysis) methods in decision-making problems in the field of transport in light of sustainable development. The article consists of a theoretical and an empirical part. As part of the literature studies, a review was carried out on the latest applications of MCDM/MCDA methods for decision-making problems in the field of transport. In the empirical part, a multi-criteria analysis of the placement selection for a strip of expressway located in north-eastern Poland was carried out. For this purpose, a hybrid approach was used, consisting of three selected MCDM/MCDA methods: DEMATEL, REMBRANDT, and VIKOR. The ranking was compared with the results achieved in the EIA report of the investment and the results were obtained by using a different set of MCDM/MCDA methods that were proposed in the first part of the research, i.e., AHP, Fuzzy AHP, TOPSIS, and PROMETHEE. The performed multi-criteria analyses allowed for an eventual multi-dimensional evaluation of the most popular MCDM/MCDA methods currently applied in the field of transport.

1. Introduction

Transport is an important branch of the economy and science, in which the decisions made are often multi-criteria in character. The choice of the investment location, route for road infrastructure, and public transport development scenario are just some of the decision-making problems in the field of transport [1] that require considering multiple environmental, economic, and social factors according to the idea of sustainable transport. The concept of sustainable transport appeared almost parallel with the definition of sustainable development in the Report of the World Commission on Environment and Development: Our Common Future [2]. In the early 1990s, Daly [3] and Pearce [4] defined sustainable transport as transport with non-declining capital, in which the capital includes human capital, monetary capital, and natural capital. Multi-criteria decision-making support in the area of sustainable transport is significant due to the multitude of aspects that should be considered.

Multi-criteria decision-making support in the area of sustainable transport is significant due to the multitude of aspects that should be considered.

There is, after all, a clear relationship between the level of development of transport systems and the area’s economic development. Transport performs many functions in the economy: it is an essential instrument for the exchange of goods and services, it determines the distribution of investments, and it is also an important factor in the location of settlements. Transport also affects the development of other branches of the economy and is an essential factor in GDP growth. Sustainable development requires considering environmental issues in economic decisions. As a component of the economy significantly influencing the environment, transport requires paying particular attention to environmental issues. The social function of transport is also worth emphasizing. Efficient transport satisfies society’s communication needs and shapes the spatial availability of the necessary functions [5].

Among the main features of transport infrastructure, one should include: a long period of its implementation and operation; high capital demands with a long return-on-investment period; immobility; and the possibility for various external influences to exist (especially in the context of the natural environment) [5]. Due to the functions of transport in the modern world and its features, the decision-making process in this field is an important and responsible task that often requires multi-criteria analyses.

The main goal of this work is to perform a comparative analysis of the results obtained while applying various MCDM/MCDA methods to a selected decision-making problem in the field of transport. The specific objectives of the study pertain to:

• updating the review of the current applications of the MCDM/MCDA methods to the selected decision-making problems in the field of transport, along with indication of the latest trends;
• carrying out a multi-criteria analysis of design variants using a hybrid approach that includes the DEMATEL, REMBRANDT, and VIKOR methods;
• carrying out a comparative analysis of the results obtained with previous tests, which used a combination of the AHP, Fuzzy AHP, TOPSIS, and PROMETHEE methods; and
• an assessment of the methods applied in both studies in the context of their advantages, disadvantages, and limitations in solving contemporary problems in the field of transport.

The research questions the authors try to answer are as follows: (1) whether the assessment of road construction investment options differs with the use of several multi-criteria methods and (2) which of the MDCA methods are most helpful in assessing this type of decision problem.

Therefore, this paper is devoted to the study of the applicability potential of selected MCDM/MCDA methods in the field of transport. In the beginning, a review of the literature on the latest applications of the selected methods in various decision-making processes was performed. Then, computational examples were developed through a case study on routing an expressway section in north-eastern Poland. Finally, based on multi-criteria and comparative analyses, the selected methods that have been used most commonly in the field of transport were assessed in detail. They were used to select the variant of the expressway section in north-eastern Poland and compare the result of the analysis with the choice made in the analyzed environmental impact report.

In the available literature on the subject, you can find many items devoted to the analysis of the application potential of MCDM/MCDA methods in various areas of life and science (see, e.g., [6,7,8,9]). However, this work focuses only on decision-making problems in the field of transport: the most popular research topics have been identified and, on the basis of several dozen previous scientific papers, the most popular methods from the MCDM/MCDA family have been identified. In addition, a proprietary hybrid approach was proposed, supporting multi-criteria decision support in environmental impact assessments of transport investments.

2. Literature Review Materials and Methods

We considered a total of 52 scientific articles [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59] in which MCDM/MCDA methods were applied to decision-making problems in the field of transport in 2020 and 2021 worldwide (Previous publications from 2000–2019 were analyzed in the first part of the paper: Broniewicz, E., Ogrodnik, K., 2020. Multi-criteria analysis of transport infrastructure projects, Transportation Research Part D: Transport and Environment 83, 102351, https://doi.org/10.1016/j.trd.2020.102351, accessed on 15 May 2021). Articles indexed in Scopus and Web of Science were considered, with the leading search criteria being the following keywords: MCDM, MCDA, and transport. Appendix A presents the types of MCDM/MCDA methods used, the research problem, and the temporal and territorial delimitation of the research. As in the previous study [1], six areas of activity related to transport infrastructure were identified: quality and safety of public transport, scenarios for the development of public transport systems, choice of investment location, road, air, rail and sea transport, and electric vehicles, among other areas of activity. It can be observed that between 2020 and 2021, AHP, TOPSIS, PROMETHEE, Fuzzy AHP, and DEA are still the most popular methods of multi-criteria decision making. There are also multiple new, hitherto unobserved multi-criteria decision-making methods such as CODAS, COMET, CRITIC, EDAS, MARCOS, PIPRECIA, and MEW PROMETHEE II, among others (see Appendix A).

Data on multi-criteria decision-making methods were summarized to obtain a database from 2000 to 2021 (Appendix B). Figure 1 shows the methods that have been used in at least two studies over the past 20 years.

Based on the data in Figure 1, it can be observed that—in the last 20 years—the most frequently used MCDM/MCDA methods in the field of transport were the AHP, TOPSIS, and PROMETHEE methods. It should be emphasized that these exact methods were used in previous studies [1]. In the second part, a decision was made to use the selected methods included in the following items of the carried out review. The hybrid approach proposed in this article is the result of the literature review. In the case study, alternative methods were used to those from the first part of the study [1], which were also at the forefront of the MCDM/MCDA method ranking. A hybrid approach was again used including the following methods:

• DEMATEL; the method placed fifth in the ranking. This method makes it possible to perform dependency analyses at the level of decision factors, making it possible to conduct a multi-criteria analysis in a new dimension.
• VIKOR; the method was placed, along with other methods, in the seventh position, selected as an alternative to the TOPSIS method from the first research part.
• REMBRANDT; although this method was placed in a distant nineth place, it is one of the few methods enabling the weighting of decision factors. It was selected as an alternative to the AHP and Fuzzy AHP methods from the first part of the study.

3. Research Methods

Figure 2 shows the hybrid approach proposed in the article and the scope of multi-criteria and comparative analyses.

3.1. DEMATEL Method (Decision-Making Trial and Evaluation Laboratory)

The DEMATEL method was developed in the second half of the 1970s in Switzerland. Generally, it allows for the analysis of cause-and-effect relationships at the level of decision factors. The algorithm of the classic DEMATEL method is relatively simple, as is the case with the AHP method, and is based on comparing factors in pairs using a point scale. The basic assumptions of the method’s algorithm are characterized below [60] (pp. 64–65) [61] (pp. 125–129) [62] (pp. 67–68) [7] (pp. 63–65). The first step in the DEMATEL method is to develop a matrix of direct relations. This matrix is created using a point scale. The classical scale (by Fontel and Gabus, the authors of the method) consists of the following five levels: 0 meaning no direct influence; 1 meaning low impact; 2 meaning medium impact; 3 meaning high impact; and 4 meaning very high impact. There are also other scales. An increased interest in fuzzy scales is observed (see, e.g., [63]). The assessment of the direct interrelationships between the elements is made in pairs (by a single decision-maker or a group of decision-makers). The effect of these comparisons is the so-called matrix of direct relations M. A matrix of direct relations is a square matrix in which the elements on the main diagonal are equal to 0.

In contrast, the other matrix elements show direct relations between the factors within a given pair. The next step in the DEMATEL method is the normalization of elements in the developed matrix of direct relations. The normalization procedure in the DEMATEL method consists of dividing the elements in the matrix of direct relations by the highest value of its row or column sums, thus creating a normalized matrix of direct relations M’. Then, the following formula creates a matrix of integral relations:

N = M′(I − M′)⁻¹

(1)

where N is the matrix of total relations, M′ is the normalized matrix of direct relations, and I is the unit matrix (the dimension that relates to the dimension of the direct relation matrix).

The elements of the n matrix represent both direct and indirect relations between decision factors. The final stage in the DEMATEL method is the calculation of two key indicators, the so-called significance index and relation index. Based on their values, the final cause-and-effect diagram is created, illustrating both the activity of the decision factors and their nature in the created cause–effect chain. The factors with positive values of the relation index are causal, i.e., they have the most significant impact on the remaining elements in the data set under consideration. Conversely, factors with negative values of the dependency index affect the created cause–effect chain.

It is worth adding that the DEMATEL and AHP methods have common features including similar initial matrices, namely the matrix of comparisons and matrix of direct relations. Both matrices are square with the constant form of a diagonal (in the AHP comparison matrix, it is 1; in the DEMATEL matrix, the values on the main diagonal are 0). Both methods also utilize point scales [61]. Examples of approaches to linking the classic Saaty scale with the Fontel and Gabus scale (both classic and its modification) are presented in

Table 1. An example of how to combine the scales used in the family of AHP and DEMATEL methods.

Classic Saaty Scale	Fuzzy Triangular Scale	Classic Fontela and Gabus Scale	Modification Fontela and Gabus Scale
1	1, 1, 1	0	0
2	1, 2, 3	1	1
3	2, 3, 4		2
4	3, 4, 5	2	3
5	4, 5, 6		4
6	5, 6, 7	3	5
7	6, 7, 8		6
8	7, 8, 9	4	7
9	9, 9, 9		8

Source: author’s work based on [61,64,65].

.

Despite the possibility of transforming the initial comparison matrix within the AHP method onto the matrix of direct relations within the DEMATEL method, it should be noted that the procedure of comparing elements in the classic DEMATEL variant concerns the mutual relationships between factors, not the estimation of their importance. Therefore, a multi-criteria analysis using the classic DEMATEL method should be performed regardless of the pairwise comparison procedure present in the AHP method. As part of this article, a cause-and-effect chain of decision-making factors has been developed. It should be emphasized that this is the only multi-criteria analysis developed as part of the research, not related to the level of variants. The REMBRANDT method, an alternative to the AHP method, was used to estimate the weight of decision factors.

3.2. REMBRANDT Method

The REMBRANDT method (Ratio Estimation in Magnitudes or deciBells to Rate Alternatives which are Non-DominaTed) was developed in the Netherlands to respond to the selected solutions used in the AHP method that have been subject to criticism. Among the debatable elements of the algorithm of the AHP method, the authors Forman [66] and Moghtadernejad et al. [67] indicated, among others, that the independence of the examined elements, the measurable subjectivity within pairwise comparisons, or the difficulty in accounting for uncertainties related to judgments. Although the REMBRANDT method, similarly to the AHP method, enables the weighting of decision factors, there are significant technical differences between the methods [68,69,70]:

• in the REMBRANDT method, a logarithmic scale is used instead of a Saaty scale;
• a logarithmic method of least squares is used to derive scale vectors; and
• the geometric mean aggregation is used for the generalized evaluation of variants.

A comparison of the scales used in the AHP and REMBRANDT methods and the grades definition is presented in

Table 2. The classic Saaty scale and the scale used in the REMBRANDT method.

Definition	Classic Saaty Scale	REMBRANDT Scale
Equal importance	1	0
Weak or slight	2	1
Moderate importance	3	2
Moderate plus	4	3
Strong importance	5	4
Strong plus	6	5
Very strong	7	6
Very, very strong	8	7
Extremely strong	9	8
“If activity i has one of the above non-zero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i”	1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2	−8, −7, −6, −5, −4, −3, −2, −1

Source: [64] (p. 86), [68] (p. 524), and [69] (p. 339).

.

The first step in the REMBRANDT method is the comparing of pairs of elements at a given decision stage using a numerical scale. The grades are then converted into values on a geometric scale. The conversion is done according to the following formula [69]:

r_ij = exp(γδ_ij)

(2)

where γ is the ln$\sqrt{2}$ and δ_ij is the figure from the scale.

In the second stage, a geometric mean for each row is calculated and then normalized to give the weight vector [68,69,71].

It should be emphasized that in this study, to be able to conduct a full-scale comparative analysis, the ratings from the initial comparison matrix in the AHP method were transformed into ratings that are in accordance with the REMBRANDT scale.

3.3. VIKOR Method

The VIKOR method (Serb. VIsekrzterijumska Optimizacija i Kompromisno Resenje) was developed in the 1990s and then disseminated thanks to the work of Opricovic and Tzeng [72], in which a comparative analysis of the VIKOR and TOPSIS methods (the latter being characterized by similar assumptions) was performed [70]. Similar to TOPSIS, the VIKOR method enables assessing decision alternatives based on their position to defined reference points [62]. The VIKOR method algorithm assumes the following calculation steps [70,72,73,74]. In the first stage, the best and worst grades are determined against the given criteria; these are so-called ideal and anti-ideal points. Then, for individual decision variants, the sum of the weighted, normalized distances from the ideal solution (metric S) and the maximum distance of the weighted, normalized assessment (metric R) are calculated. The lower the value of these metrics, the better. In the next step, the value of the comprehensive indicator is determined (metric Q) according to the following formula [72,73]:

$$Q_i=v \frac{S_i-S^{+}}{S^{-}-S^{+}}+(1-v) \frac{R_i-R^{+}}{R^{-}-R^{+}}$$

(3)

where v is the weight of the strategy of “the majority of criteria”, S_i is the S metric (the sum of the weighted, normalized distances of the i-variant from the ideal solution), and R_i is the R metric, the maximum weighted, normalized rating distance.

Three ranking lists constitute the results of the above calculations: variant evaluations in light of S, R, and Q metrics. The lower the values of these metrics, the higher the position in the rankings is awarded to a given variant. The final stage consists of comparing the rankings obtained in terms of an acceptable advantage and acceptable decision stability. Importantly, rankings in the VIKOR method can be carried out with different values of criteria weights, which enables the analysis of the impact of the weights of these criteria on the proposed compromise solution.

4. Case Study

This article once again takes advantage of the data from the report on the environmental impact of a project comprising the construction of an expressway and the construction of a national road in north-eastern Poland. The data was made available by the General Directorate for National Roads and Motorways, Bialystok Branch [75]. Detailed information on the decision-making factors and variants can be found in this article [1].

The planned project involves the construction of an expressway on the Choroszcz-Ploski section and a national road on the section between the towns of Kudrycze and Grabówka, together with the construction of the necessary technical infrastructure. The planned investment is essential for the economic development of this region of Poland. Additionally, it will contribute to improving the quality and safety of road traffic. This road belongs to the international transport corridor network. Six location variants were adopted for the analysis, including two variants of the expressway and three variants of the national road connected to it. The road route variants differ in the location and number of engineering structures [75]. The characteristics of the analyzed variants are presented in

Table 3. Variants’ characteristics.

Number	Name	Variant of the Expressway Route	Variant of the National Road Route	Length (km)	Number of Engineering Objects
V1	Variant 1	1	1	53 856.37	80
V2	Variant 2	1	2	53 651.39	82
V3	Variant 3	1	3	53 517.19	77
V4	Variant 4	2	2	52 249.97	78
V5	Variant 5	2	1	53 230.57	79
V6	Variant 6	2	3	52 891.39	76

Source: author’s work based on [75].

.

4.1. Decomposing a Decision-Making Problem

As part of this multi-criteria analysis, six different variants of the road route and 13 decision-making factors reflecting the impact of this investment on the environment and society at large were considered. The criteria for the evaluation of variants are each time individually selected depending on the nature and subject of the analysis. The decision criteria in this case study were adopted from the environmental impact report [75]. The hierarchical structure of the decision problem under consideration is shown in Figure 3.

4.2. Numerical Example

The results of the basic calculations performed using each of the selected methods are presented below: DEMATEL and then REMBRANDT, used to estimate the factor weights, which were later used with the VIKOR method.

The first stage concerned the analysis of dependencies at the level of decision factors. First, the decision criteria were compared in pairs using the classic DEMATEL scale, obtaining the matrix of direct relations M (

Table 4. The matrix of direct relations for decision factors M (DEMATEL method).

M	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13
C1	0.000	4.000	4.000	4.000	4.000	4.000	4.000	4.000	0.000	0.000	0.000	0.000	0.000
C2	3.000	0.000	3.000	3.000	3.000	3.000	3.000	3.000	0.000	0.000	0.000	0.000	0.000
C3	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C4	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C5	0.000	0.000	4.000	4.000	0.000	4.000	4.000	4.000	0.000	0.000	0.000	0.000	0.000
C6	0.000	0.000	0.000	0.000	0.000	0.000	0.000	3.000	0.000	0.000	0.000	0.000	0.000
C7	0.000	0.000	0.000	0.000	0.000	0.000	0.000	3.000	0.000	0.000	0.000	0.000	0.000
C8	0.000	0.000	0.000	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	0.000	0.000
C9	0.000	0.000	4.000	4.000	0.000	4.000	4.000	4.000	0.000	0.000	0.000	0.000	4.000
C10	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C11	0.000	0.000	3.000	3.000	0.000	3.000	3.000	3.000	0.000	0.000	0.000	0.000	0.000
C12	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C13	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Source: author’s work.

). Then, the grades were normalized (

Table 5. Normalized matrix of direct relations M (DEMATEL method).

M’	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13
C1	0.000	0.143	0.143	0.143	0.143	0.143	0.143	0.143	0.000	0.000	0.000	0.000	0.000
C2	0.107	0.000	0.107	0.107	0.107	0.107	0.107	0.107	0.000	0.000	0.000	0.000	0.000
C3	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C4	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C5	0.000	0.000	0.143	0.143	0.000	0.143	0.143	0.143	0.000	0.000	0.000	0.000	0.000
C6	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.107	0.000	0.000	0.000	0.000	0.000
C7	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.107	0.000	0.000	0.000	0.000	0.000
C8	0.000	0.000	0.000	0.000	0.000	0.071	0.071	0.000	0.000	0.000	0.000	0.000	0.000
C9	0.000	0.000	0.143	0.143	0.000	0.143	0.143	0.143	0.000	0.000	0.000	0.000	0.143
C10	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C11	0.000	0.000	0.107	0.107	0.000	0.107	0.107	0.107	0.000	0.000	0.000	0.000	0.000
C12	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C13	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Source: author’s work.

) and the matrix of total N relations was determined (

Table 6. Matrix of total relations (direct and indirect) N with row and column sums (DEMATEL method).

n	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13	Score L
C1	0.016	0.145	0.184	0.184	0.161	0.200	0.200	0.226	0.000	0.000	0.000	0.000	0.000	1.314
C2	0.109	0.016	0.142	0.142	0.124	0.155	0.155	0.175	0.000	0.000	0.000	0.000	0.000	1.017
C3	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C4	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C5	0.000	0.000	0.143	0.143	0.000	0.155	0.155	0.176	0.000	0.000	0.000	0.000	0.000	0.773
C6	0.000	0.000	0.000	0.000	0.000	0.008	0.008	0.109	0.000	0.000	0.000	0.000	0.000	0.124
C7	0.000	0.000	0.000	0.000	0.000	0.008	0.008	0.109	0.000	0.000	0.000	0.000	0.000	0.124
C8	0.000	0.000	0.000	0.000	0.000	0.073	0.073	0.016	0.000	0.000	0.000	0.000	0.000	0.161
C9	0.000	0.000	0.143	0.143	0.000	0.155	0.155	0.176	0.000	0.000	0.000	0.000	0.143	0.916
C10	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C11	0.000	0.000	0.107	0.107	0.000	0.117	0.117	0.132	0.000	0.000	0.000	0.000	0.000	0.580
C12	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
C13	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Score C	0.124	0.161	0.719	0.719	0.285	0.870	0.870	1.119	0.000	0.000	0.000	0.000	0.143

Source: author’s work.

), which allowed for the ranking of factors under consideration in the form of a cause-and-effect chain (

Table 7. The value of the position indicator and relation indicator (DEMATEL method).

	Score L	Score C	Prominence	Relation
C1	1.314	0.124	1.439	1.190
C2	1.017	0.161	1.178	0.857
C3	0.000	0.719	0.719	−0.719
C4	0.000	0.719	0.719	−0.719
C5	0.773	0.285	1.058	0.488
C6	0.124	0.870	0.994	−0.746
C7	0.124	0.870	0.994	−0.746
C8	0.161	1.119	1.280	−0.959
C9	0.916	0.000	0.916	0.916
C10	0.000	0.000	0.000	0.000
C11	0.580	0.000	0.580	0.580
C12	0.000	0.000	0.000	0.000
C13	0.000	0.143	0.143	−0.143

Source: author’s work.

and Figure 4).

It should be emphasized that the classic DEMATEL method enables the analysis of dependencies (direct and indirect) between decision factors. This should not be identified directly with the weights of these factors. Therefore, a multi-criteria analysis using DEMATEL helps rank the factors in terms of their mutual relations to identify those that have the most significant impact on the others. In the case of the analyzed data set, the causal factors, i.e., the ones that have the most significant impact on the others, are: factors related to the location of NATURA 2000 areas pertaining to the planned investment (C1, C2, and C5), as well as linear factors, i.e., length sections with a high degree of pollution risk (C9) and the length of the intersection of the complex of soils 2 and 2z (C11). Conversely, the following factors were classified as the resultant of the impact of the above-mentioned criteria: the number of herpetofauna locations in the test buffer (C8); the impact on snail habitats (C6); the impact on the habitats of insects (C7); the number of vascular plant species destroyed (C3); the number of fungi (lichens) species destroyed (C4); and the number of buildings exposed to excessive noise (C13). Two factors remained neutral (C10 and C12), which may indicate their different nature. The classic DEMATEL method facilitates ordering at the level of decision factors and their analysis regarding their mutual interactions. Therefore, it can be an instrument supporting the selection of factors at the stage of the decomposition of the decision-making problem.

Alternatively, the REMBRANDT method was used to estimate the weights of the criteria. It should be noted that the matrix of comparisons in the REMBRANDT method Based on the primary comparison matrix of the AHP method (

Table 8. Matrix of comparisons, criteria weights, and consistency assessment estimated using the classical AHP method.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13	Weight
C1	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C2	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C3	0.333	0.333	1.000	1.000	0.333	0.333	0.333	0.333	3.000	1.000	3.000	0.333	0.333	0.039
C4	0.333	0.333	1.000	1.000	0.333	0.333	0.333	0.333	3.000	1.000	3.000	0.333	0.333	0.039
C5	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C6	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C7	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C8	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C9	0.200	0.200	0.333	0.333	0.200	0.200	0.200	0.200	1.000	0.333	1.000	0.200	0.200	0.019
C10	0.333	0.333	1.000	1.000	0.333	0.333	0.333	0.333	3.000	1.000	3.000	0.333	0.333	0.039
C11	0.200	0.200	0.333	0.333	0.200	0.200	0.200	0.200	1.000	0.333	1.000	0.200	0.200	0.019
C12	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
C13	1.000	1.000	3.000	3.000	1.000	1.000	1.000	1.000	5.000	3.000	5.000	1.000	1.000	0.106
	Consistency check λmax = 13.009 and CR = 0.005

Source: [1].

), which was used in the first part of the study [1], it was developed (

Table 9. Decision factor comparison matrix (REMBRANDT method).

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13
C1	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C2	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C3	−2.000	−2.000	0.000	0.000	−2.000	−2.000	−2.000	−2.000	2.000	0.000	2.000	−2.000	−2.000
C4	−2.000	−2.000	0.000	0.000	−2.000	−2.000	−2.000	−2.000	2.000	0.000	2.000	−2.000	−2.000
C5	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C6	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C7	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C8	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C9	−4.000	−4.000	−2.000	−2.000	−4.000	−4.000	−4.000	−4.000	0.000	−2.000	0.000	−4.000	−4.000
C10	−2.000	−2.000	0.000	0.000	−2.000	−2.000	−2.000	−2.000	2.000	0.000	2.000	−2.000	−2.000
C11	−4.000	−4.000	−2.000	−2.000	−4.000	−4.000	−4.000	−4.000	0.000	−2.000	0.000	−4.000	−4.000
C12	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000
C13	0.000	0.000	2.000	2.000	0.000	0.000	0.000	0.000	4.000	2.000	4.000	0.000	0.000

Source: author’s work.

). Then, according to Formula 2, the matrix elements were transformed into values on the geometric scale. The decision factor weights were estimated using the geometric mean and additive normalization (

Table 10. Transformed comparison matrix of decision factors: the geometric mean of the rows and weights of decision factors (REMBRANDT method).

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13	Avg.	Weight
C1	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C2	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C3	0.500	0.500	1.000	1.000	0.500	0.500	0.500	0.500	2.000	1.000	2.000	0.500	0.500	0.726	0.050
C4	0.500	0.500	1.000	1.000	0.500	0.500	0.500	0.500	2.000	1.000	2.000	0.500	0.500	0.726	0.050
C5	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C6	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C7	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C8	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C9	0.250	0.250	0.500	0.500	0.250	0.250	0.250	0.250	1.000	0.500	1.000	0.250	0.250	0.363	0.025
C10	0.500	0.500	1.000	1.000	0.500	0.500	0.500	0.500	2.000	1.000	2.000	0.500	0.500	0.726	0.050
C11	0.250	0.250	0.500	0.500	0.250	0.250	0.250	0.250	1.000	0.500	1.000	0.250	0.250	0.363	0.025
C12	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100
C13	1.000	1.000	2.000	2.000	1.000	1.000	1.000	1.000	4.000	2.000	4.000	1.000	1.000	1.452	0.100

Source: author’s work.

).

The final stage focuses on developing a ranking of the considered decision variants in light of the defined criteria. For this purpose, the VIKOR method was used. In addition to the weights calculated by the REMBRANDT method, sets of weights estimated by the AHP and Fuzzy AHP methods were also used, finally obtaining three rankings (

Table 11. Initial data (VIKOR method).

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13
W1	86.1	2425	9	0	39.1	4	11	64	3.1	23	12850	7	7
W2	42.6	953	11	0	44.8	5	10	62	2.9	22	12268	9	7
W3	42.6	937	11	0	41.8	4	11	60	2.9	23	11806	8	7
W4	48.6	1573	9	1	62.9	6	6	65	2.9	33	14353	14	26
W5	92.1	3045	7	1	57.2	5	7	68	3.1	34	15185	17	26
W6	48.6	1557	9	1	59.9	5	7	64	2.9	34	14141	14	26
Character	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.	dest.
Weights (AHP)	0.106	0.106	0.039	0.039	0.106	0.106	0.106	0.106	0.019	0.039	0.019	0.106	0.106
Weights (FAHP)	0.105	0.105	0.041	0.041	0.105	0.105	0.105	0.105	0.019	0.041	0.019	0.105	0.105
Weights (REMBRANDT)	0.100	0.100	0.050	0.050	0.100	0.100	0.100	0.100	0.025	0.050	0.025	0.100	0.100

Source: [1,75].

and

Table 12. The values of S, R, and Q metrics together with a ranking and verification of conditions.

Methods	Variants	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13	S	R	Q	Ranking
VIKOR method, AHP weights	W1	0.093	0.075	0.020	0.000	0.000	0.000	0.106	0.053	0.019	0.003	0.006	0.000	0.000	0.375	0.106	0.662	3
	W2	0.000	0.001	0.039	0.000	0.025	0.053	0.085	0.027	0.000	0.000	0.003	0.021	0.000	0.253	0.085	0.065	1
	W2	0.000	0.000	0.039	0.000	0.012	0.000	0.106	0.000	0.000	0.003	0.000	0.011	0.000	0.171	0.106	0.500	2
	W4	0.013	0.032	0.020	0.039	0.106	0.106	0.000	0.066	0.000	0.036	0.014	0.074	0.106	0.612	0.106	0.850	5
	W5	0.106	0.106	0.000	0.039	0.081	0.053	0.021	0.106	0.019	0.039	0.019	0.106	0.106	0.801	0.106	1.000	6
	W6	0.013	0.031	0.020	0.039	0.093	0.053	0.021	0.053	0.000	0.039	0.013	0.074	0.106	0.555	0.106	0.805	4
	Condition 1. Acceptable advantage, yes; Condition 2. Acceptable stability, yes
VIKOR method, Fuzzy AHP weights	W1	0.092	0.074	0.021	0.000	0.000	0.000	0.105	0.053	0.019	0.003	0.006	0.000	0.000	0.373	0.105	0.660	3
	W2	0.000	0.001	0.041	0.000	0.025	0.053	0.084	0.026	0.000	0.000	0.003	0.021	0.000	0.253	0.084	0.065	1
	W2	0.000	0.000	0.041	0.000	0.012	0.000	0.105	0.000	0.000	0.003	0.000	0.011	0.000	0.172	0.105	0.500	2
	W4	0.013	0.032	0.021	0.041	0.105	0.105	0.000	0.066	0.000	0.038	0.014	0.074	0.105	0.612	0.105	0.851	5
	W5	0.105	0.105	0.000	0.041	0.080	0.053	0.021	0.105	0.019	0.041	0.019	0.105	0.105	0.798	0.105	1.000	6
	W6	0.013	0.031	0.021	0.041	0.092	0.053	0.021	0.053	0.000	0.041	0.013	0.074	0.105	0.556	0.105	0.806	4
	Condition 1. Acceptable advantage, yes; Condition 2. Acceptable stability, yes
VIKOR method, REMBRANDT weights	W1	0.088	0.071	0.025	0.000	0.000	0.000	0.100	0.050	0.025	0.004	0.008	0.000	0.000	0.370	0.100	0.657	3
	W2	0.000	0.001	0.050	0.000	0.024	0.050	0.080	0.025	0.000	0.000	0.003	0.020	0.000	0.253	0.080	0.063	1
	W2	0.000	0.000	0.050	0.000	0.011	0.000	0.100	0.000	0.000	0.004	0.000	0.010	0.000	0.176	0.100	0.500	2
	W4	0.012	0.030	0.025	0.050	0.100	0.100	0.000	0.063	0.000	0.046	0.019	0.070	0.100	0.614	0.100	0.854	5
	W5	0.100	0.100	0.000	0.050	0.076	0.050	0.020	0.100	0.025	0.050	0.025	0.100	0.100	0.796	0.100	1.000	6
	W6	0.012	0.029	0.025	0.050	0.087	0.050	0.020	0.050	0.000	0.050	0.017	0.070	0.100	0.561	0.100	0.811	4
	Condition 1. Acceptable advantage, yes; Condition 2. Acceptable stability, yes

Source: author’s work.

).

4.3. Comparative Analysis

Figure 5 presents individual decision factors’ weights estimated using the following methods: AHP, Fuzzy AHP [1], and REMBRANDT. It should be emphasized that the starting point for the calculations was the AHP comparison matrix evaluations, which were then transformed according to the algorithm of the other two methods.

Comparing the sets of weights estimated with the AHP, Fuzzy AHP, and REMBRANDT methods, it can be noticed that there are specific differences; however, the order of the factors in terms of their importance has not changed. It should be added, though, that the differences between the weights obtained with the AHP method and the weights estimated with the Fuzzy AHP method are much lower than the differences between the sets of AHP and REMBRANDT scales. Therefore, it can be concluded that the modifications proposed in the REMBRANDT method (mainly the form of the scale and the logarithmic method of least squares that is used to estimate the weight vector) have a greater overall impact on the final vector of the weights than the case with a transformation of a classic Saaty scale into a fuzzy triangular scale has (see also [68]).

The assessment of the impact that a change in the value of factor weights has on the final ranking of decision variants serves as yet another important element of the comparative analysis.

Table 13. Ranking of decision variants: summary list.

Method Used	Variant Ranking (from Best to Worst)
Report method	W3 > W2 > W1 > W6 > W4 > W5
TOPSIS (AHP weights)	W2 > W3 > W1 > W6 > W4 > W5
TOPSIS (FAHP weights)	W2 > W3 > W1 > W6 > W4 > W5
TOPSIS (REMBRANDT weights)	W2 > W3 > W1 > W6 > W4 > W5
PROMETHEE (AHP weights)	W3 > W2 > W1 > W6 > W4 > W5
PROMETHEE (FAHP weights)	W3 > W2 > W1 > W6 > W4 > W5
PROMETHEE (REMBRANDT weights)	W3 > W2 > W1 > W6 > W4 > W5
VIKOR (AHP weights)	W2 > W3 > W1 > W6 > W4 > W5
VIKOR (FAHP weights)	W2 > W3 > W1 > W6 > W4 > W5
VIKOR (REMBRANDT weights)	W2 > W3 > W1 > W6 > W4 > W5

Source: author’s work.

presents the final rankings of alternatives developed using the three MCDM/MCDA methods: TOPSIS, PROMETHEE, and VIKOR. Additionally, the ranking of variants developed as part of the environmental impact report for the investment in question was given as a reference point. A detailed description of the obtained rankings in terms of content is presented in [1].

A change in the weighting of decision factors did not affect the final ranking of the variants. For both the TOPSIS and PROMETHEE methods, even after changing the set of weights, the final order did not change. Similar observations apply to the multi-criteria analysis performed using the VIKOR method. It should be emphasized that the final rankings of the decision variants presented in Table 13 are almost identical. However, in the case of receiving different rankings, it is recommended to use, for example, the rank similarity coefficient in decision-making problems (see [76]).

Significantly, the obtained rankings of the decision alternatives differ in regards to the first position. In the case of the method used in the report and the PROMETHEE method, which represents the family of methods based on the height difference, W3 arrived in the first place. However, among the methods based on reference points, identical rankings were obtained, with W2 in the first place. In the case of the rankings obtained using the TOPSIS method, the differences between the first and second place were negligible (e.g., with the use of weights calculated by using the classic AHP and the final index of W2 = 0.777 and W3 = 0.772). In contrast, in the rankings prepared using the VIKOR method, the difference between the first two places was higher and W2 met both the condition of “acceptable advantage” and “acceptable stability”.

It should be emphasized that the comparative analysis of the distance methods VIKOR and TOPSIS (especially in the context of the algorithms used) was the subject of previous research; see, e.g., [72,77].

5. Discussion and Conclusions

The performed multi-criteria analyses allowed to verify the choice of the course of the selected investment made as part of the report [75] and which made it possible to evaluate the usefulness of the selected MCDM/MCDA methods in the field of transport. Based on the performed calculations as well as the comparative analyses of the obtained results and their interpretations,

Table 14. Assessment of the most popular MCDM/MCDA methods used so far in the field of transport with their SWOT analysis and examples of applications.

Method	Strengths/Opportunities	Weaknesses Limitations	Examples of Problems Present in the Field of Transport
AHP	◾ relatively easy calculation algorithm ◾ possibility of performing calculations in a regular spreadsheet ◾ possibility of assessing the consistency of comparisons in pairs ◾ possibility of pairwise comparisons by a group of experts (especially important when choosing the location variants for large-scale investments) ◾ possibility of estimating both the weighting of decision factors and the possibility of constructing rankings of the variants under consideration ◾ possibility of including the pair comparison procedure in the form of a questionnaire ◾ possibility of integration with other methods ◾ a considerable amount of software (often available and free) supporting the calculations with the chosen method ◾ a significant number of examples of the method’s application in the field of transport ◾ recommendation of the method by the authorities responsible for maintaining and developing road infrastructure (in Poland, by the General Directorate for National Roads and Motorways)	◾ possible problems with maintaining the consistency of pairwise comparisons when more elements are taken into consideration	◾ assessment of the preferences displayed by users of transport systems (see, e.g., [82]) ◾ assessment and selection of the transport system/systems development scenario (see, e.g., [83,84] ◾ selecting an investment location (see e.g., [85])
Fuzzy AHP	◾ possibility of taking uncertain information into account ◾ the possibility of comparing the elements in pairs ◾ possibility of integration with other methods ◾ development of software supporting complex multi-criteria analyses and also on fuzzy sets	◾ more time-consuming (as compared to the classic AHP method) ◾ more complex algorithm (as compared to the classic AHP method)	◾ analysis of safety factors (see, e.g., [86,87])
REMBRANDT	◾ possibility of pairwise comparisons by a group of experts (especially important when choosing the location variants of large-scale investments) ◾ possibility of decision factors and the possibility of constructing rankings of the variants under consideration ◾ possibility of performing calculations in a regular spreadsheet	◾ more complex algorithm (as compared to the classic AHP method) ◾ no computer software ◾ possibility of taking into account three decision levels	◾ evaluation of transport infrastructure projects (see, e.g., [88,89])
DEMATEL	◾ possibility of analysis at the level of decision factors in terms of mutual dependencies ◾ relatively simple algorithm (compared to, e.g., the ANP method) ◾ development of the method towards weighing factors ◾ possibility of group assessments, estimating the weighting ◾ ability to visualize the results in the form of a cause-and-effect diagram	◾ high subjectivism at the stage of dependency assessment ◾ a classic version of the method does not assume the weighting of factors	◾ development factor analysis (see, e.g., [90])
PROMETHEE	◾ possibility of taking into account both quantitative and qualitative criteria ◾ possibility of taking into account the size of the variant differences in light of subsequent decision criteria ◾ possibility of integration with other methods ◾ ability to perform an analysis in the Visual PROMETHEE program, which enables calculations and offers various visualization functions of the obtained results (available in the academic or business version) ◾ systematically created and publicly available PROMETHEE Bibliographical Database	◾ need to weigh decision factors using other methods ◾ need to select the preference function, the selection of the preference, and equivalence thresholds	◾ selection of an investment location (see, e.g., [91])
VIKOR	◾ possibility of performing calculations in a regular spreadsheet ◾ is based on quantitative data ◾ identification of patterns and anti-patterns ◾ possibility of integration with other methods ◾ possibility of defining a compromise solution, taking into account many conflicting criteria	◾ need to weigh decision factors using other methods	◾ assessment of the quality of public transport systems (see, e.g., [92])
TOPSIS	◾ possibility of performing calculations in a regular spreadsheet ◾ is based on quantitative data ◾ identification of patterns and anti-patterns ◾ possibility of integration with other methods	◾ need to weigh decision factors using other methods	◾ analysis of selected transport systems (see, e.g., [93]) ◾ choice of investment location (see, e.g., [94])

Source: author’s work.

presents the evaluation of the MCDM/MCDA methods applied, their advantages/opportunities, and disadvantages/limitations, including the fields of application in the transport domain. It should be noted that, due to the growing popularity of MCDM/MCDA methods, they have already been studied in terms of their advantages and disadvantages (see, e.g., [8,78,79,80,81]). However, in the framework of this paper, the observations formulated by the authors concern the application of these methods in decision-making problems in the area of transport. Of course, despite such a narrower approach, selected comments on the usefulness of the selected methods can be considered universal.

Based on the literature studies and the conducted analyses, the following conclusions can be drawn:

• Multi-criteria decision support in the field of transport is invariably based mainly on the methods from the AHP, PROMETHEE, and TOPSIS family. This is primarily due to their universal nature, transparent and proven algorithms, and the available software to facilitate the performance of multi-dimensional analyses. However, each year, new methods not yet observed in the field of transport emerge, such as CODAS, COMET, CRITIC, EDAS, MARCOS, PIPRECIA, or MEW.
• In any multi-criteria analysis, the weighting of decision-making factors serves as a crucial stage, especially in investments in the field of transport, which are usually characterized by large dimensions and thus a large scale of potential impacts. Therefore, it seems appropriate to use methods that enable pairwise comparisons of factors as part of the expert research stage (AHP, Fuzzy AHP, or REMBRANDT).
• The DEMATEL method can serve as a support tool at the stage of selecting and analyzing decision factors. To increase objectivity, the analysis of dependencies can be successfully performed as part of expert interviews.
• When ranking decision alternatives, the number of multi-criteria methods available is significant. Concerning decision-making problems in the field of transport, especially in a situation where the considered alternatives are similar, it is recommended to use at least two methods.

Due to the dynamic development of MCDM/MCDA methods, the development of multi-criteria analyses in the field of transport with the use of other methods, especially the “youngest” methods which have so far been characterized by single implementations of the study, should be indicated among the future directions of development. Furthermore, it would be interesting to compare their usefulness with classic methods well known in theory and practice that dominate the developed application overview.