Informatica

Multiple-Criteria Decision Making (MCDM) is a systematic approach used to make decisions in complex scenarios where multiple conflicting criteria or objectives need to be considered simultaneously. In many real-world situations, decisions cannot be made solely based on a single criterion; instead, multiple factors with different levels of importance and often incommensurable units need to be taken into account. MCDM provides a structured framework to evaluate and compare different alternatives in a way that reflects the preferences and priorities of decision makers.

During the MCDM process, the data representing different criteria might come in various formats, units, and scales. These criteria could include quantitative measurements like costs, benefits, or performance metrics, as well as qualitative factors like risk levels or expert opinions. Due to these differences, directly comparing and combining these criteria for decision-making purposes can be challenging and lead to misleading results. To deal with the problem, normalization has become a fundamental process in MCDM that enables decision makers to effectively compare and evaluate alternatives across multiple criteria (Pavličić, 2001). It generally supports the goal of making well-informed, objective, and consistent decisions that accurately reflect the preferences and priorities of decision makers.

Distance-based models in MCDM play a crucial role in quantifying the dissimilarity between alternatives, and the issue of rank reversal (RR) has drawn significant attention within this context. RR denotes a phenomenon where the rank order of alternatives undergoes a change due to alterations in the evaluated alternatives or related parameters. This phenomenon contradicts our intuitive expectations and is formally termed the axiom of independence of irrelevant alternatives (IIA) in the multi-attribute utility theory (von Winterfeldt and Edwards, 1986). Belton and Gear (1983) initially identified RR in the Analytic Hierarchy Process (AHP), attributing it to AHP’s relative measurement mode. Salo and Hämäläinen (1997) extended this finding, linking RR to AHP’s mode of measurement. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), applied to MCDM, was later reported to exhibit RR by Triantaphyllou (2000). Subsequent debates in major operations research journals, including works by Saaty (1990a, 1990b, 1991), Saaty and Vargas (1984), Triantaphyllou (2001), and Kulakowski et al. (2019), further underscored the significance of RR.

TOPSIS stands out as a straightforward and intuitive method that accommodates both the positive and negative aspects of each alternative, making it a widely adopted approach in decision-making processes dealing with multiple, conflicting criteria. It has been still well received until recently, e.g. Biswas et al. (2024) and Kousar et al. (2024). Despite its popularity, TOPSIS is not immune to RR, although its occurrence is notably less pronounced compared to AHP (Zanakis et al., 1998). The RR phenomenon in TOPSIS arises from the sensitivity of the distance metric to variations in the attribute values of alternatives. The normalization process in TOPSIS involves dividing the performance values by their respective weights. Furthermore, the choice of distance metrics and the determination of the positive-ideal solution (PIS) and negative-ideal or anti-negative solution (NIS) can introduce subjectivity, influencing the final rankings. Therefore, careful consideration of these factors is crucial to mitigate the RR problem and enhance the robustness of the TOPSIS method in decision-making processes.

To address the issue of RR, previous researchers such as Senouci et al. (2016) and Yang (2020) have suggested approaches like linear normalization and using absolute positive-ideal and negative-ideal solutions. However, many studies have relied on simplistic, fabricated examples or simulations to illustrate the RR phenomenon, leaving its real-world impact and significance unclear (Shih and Olson, 2022). This research is motivated by the need for a more comprehensive understanding of RR in the TOPSIS method, driving an in-depth mathematical analysis of the underlying formulae to better quantify and explain the occurrence of RR.

Distance metrics, such as Manhattan, Euclidean, and Chebyshev, measure the dissimilarity or proximity between alternatives in different ways, each introducing variations in rankings due to their sensitivity to the scale and distribution of data. These variations can trigger RR by affecting the computed distances between alternatives. To mitigate RR, it is crucial to examine how these distance metrics influence the final rankings, providing insight into the underlying mechanics of RR. This study aims to fill a gap in the literature by exploring the role of these distance measures in the TOPSIS framework, offering a more detailed understanding of their influence on ranking reversals.

While the Chebyshev distance in TOPSIS offers robust ranking results, it has limitations in capturing subtle performance differences across multiple criteria. This lack of sensitivity to nuanced variations can lead to less precise distinctions between alternatives. Additionally, although Euclidean distance has been shown to outperform Manhattan distance in reducing RR, it may not fully eliminate the phenomenon. The study supports these conclusions through a case example, providing empirical validation for the effectiveness of different distance metrics in preventing RR, while acknowledging the limitations inherent to each approach.

The rest of the paper runs as follows. Section 2 reviews the existing resolutions for RR in TOPSIS. Section 3 delves into an examination of the ranking index of TOPSIS, presenting a mathematical perspective on TOPSIS using different distance metric methods for handling RR. Section 4 examines a case study. Section 5 offers concluding remarks on RR solutions for TOPSIS.

RR can occur in an MCDM process when a new alternative is introduced or an existing one is removed from the candidate list, all without any adjustments to the criteria weights. This phenomenon is rooted in the intricate interactions between alternatives and their relative positions in the ranking. With the addition of a new alternative, the dynamic relationships among all options might shift, causing some alternatives to be reevaluated in the context of the changed landscape. Similarly, the removal of an alternative can lead to redistributions in the ranking as the absence of that option could amplify the importance of others. Rank reversal serves as a reminder of the delicate balance within decision frameworks, highlights the non-intuitive nature of multi-criteria scenarios, and underscores the need for a thoughtful and adaptable approach to decision-making.

Building on the observations of the RR phenomenon in TOPSIS, Ren et al. (2007) presented pioneering research in addressing RR within TOPSIS. They introduced a novel modified synthetic evaluation method wherein two separation measures are transformed into a two-dimensional plane, and the ranking index is determined by the distance to PIS minus the minimum of all alternatives to PIS. However, it appears that their proposal lacks a systematic approach to avoiding the RR phenomenon, and the examples presented, such as those in the public health system and student evaluation, serve primarily for illustrative purposes.

Wang and Luo (2009) contributed to the literature by showcasing an example highlighting the occurrence of RR in the TOPSIS algorithm. However, they did not specifically identify the impact arising from non-dominated alternatives.

Zavadskas et al. (2006) studied the impact of vector and max-min normalization on TOPSIS ranking accuracy. Chakraborty and Yeh (2009) simulated comparisons of one vector and three linear normalization methods for TOPSIS, advocating for vector normalization based on ranking consistency and weight sensitivity. Çelen (2014) compared four normalization methods from Chakraborty and Yeh (2009) in evaluating the Turkish deposit banking market, affirming vector normalization’s reliability and noting that max-min and max normalization also produced consistent results. Kong (2011) aimed to mitigate RR by introducing linear normalization and incorporating extreme fictitious alternatives to stabilize PIS and NIS. However, Kong’s example was limited to five alternatives with two criteria.

García-Cascale and Lamata (2012) similarly considered the approaches introduced by Kong (2011). They further elaborated on numerous situations, presenting an example with four alternatives and two criteria.

İç (2014) explored the integration of design of experiments (DoE) into TOPSIS for the financial performance evaluation of companies. The study concluded with a five-alternative case involving five criteria to illustrate the DoE-TOPSIS method, aiming to preserve ranking. However, despite the belief that this combination could avoid RR, the method did not demonstrate improvement in the steps of TOPSIS.

Cables et al. (2016) proposed a reference ideal method, redefining the distance to the reference ideal as an interval and linearly normalizing performance data into a value. They also modified the separation measures, making the reference ideal as the vector of weights. The method was illustrated using a personnel selection case with five candidates and six criteria, claiming independence from the type of data. However, verification of this illustration is challenging due to the assumed interval values and reliance on a single example.

Senouci et al. (2016) suggested a modified TOPSIS for selecting a mobile network interface, incorporating linear normalization with four types and dynamically setting max/min values against RR. Their case involved seven alternatives and was evaluated by five criteria through simulation. Despite achieving a zero RR ratio with two types of normalization, we will later demonstrate that their case is unable to avert the RR phenomenon.

In addition to the developments above, Kuo (2017) investigated weighting on the elements of the TOPSIS ranking index, showing that the proposed modified ranking index outperforms the traditional TOPSIS in ranking consistency. Mufazzal and Muzakkir (2018) proposed a proximity index to overcome RR, utilizing Manhattan distance to count separation measures and using the sum of distances as the ranking index. Although they provided seven historical cases as evidence for the proposed approach, they were unable to clarify the ranking differences observed in their outcomes or elucidate the properties of the index.

Authors de Farias Aires and Ferreira (2019) introduced two forms of linear normalization to modify the traditional TOPSIS. They demonstrated the robustness of their approach through a student selection case involving twenty alternatives with three criteria and their subsets.

Yang (2020) made modifications to the approach proposed by de Farias Aires and Ferreira (2019), incorporating linear normalization of historical extreme values and assigning weights only to PIS. The study used Wang and Luo’s questionable example (Wang and Luo, 2009) to showcase the rank stability achieved by their approach.

Based on the studies mentioned above, the proposed solutions to mitigate RR are classified into the following five main categories. It is observed that the five categories mentioned above are arranged according to the steps of the TOPSIS procedure. Table 1 summarizes other research efforts, and the proposed strategies aimed at alleviating RR in TOPSIS. The majority of proposed solutions are concentrated in Categories I and II, while fewer solutions are found in the other categories. Categories I, II, and III primarily revolve around the selection of suitable normalization methods and modifications to the original separation measure formulae. The choice of a distance measurements plays a crucial role in assessing the relative proximity of alternatives to PIS and NIS, aside from modifying the determination of PIS and/or NIS. However, there is a lack of literature that explicitly address the relationship between RR and the specific method used for distance measures. As a result, our focus is directed towards exploring the effects of different normalization methods and distance metrics, particularly the Minkowski metric.

This section provides a comprehensive mathematical explanation of how normalization and distance measurements contribute to the phenomenon of RR. This investigation aims to understand the factors that give rise to RR in MCDM, with a particular focus on TOPSIS. The goal is to enhance decision-making by addressing the occurrences of rank reversal, which are comparatively less frequent in TOPSIS. The common algorithm of TOPSIS for ranking and selection includes the following seven steps (Hwang and Yoon, 1981).

In the TOPSIS methodology, when comparing two alternatives, ${A_{p}}$ and ${A_{q}}$, their respective relative closeness values are calculated as follows: ${C_{p}^{\ast }}={S_{p}^{-}}/({S_{p}^{+}}+{S_{p}^{-}})$ for alternative ${A_{p}}$ and ${C_{q}^{\ast }}={S_{q}^{-}}/({S_{q}^{+}}+{S_{q}^{-}})$ for alternative ${A_{q}}$. If alternative ${A_{p}}$ is preferred over alternative ${A_{q}}$ in TOPSIS, then ${C_{p}^{\ast }}$ must be greater than ${C_{q}^{\ast }}$. This can be expressed as: After transposing the terms, the inequation can be rewritten as: The mentioned expression is the fundamental formula employed to determine the preference relationship between alternatives ${A_{p}}$ and ${A_{q}}$ in TOPSIS. By assessing the result of ${S_{p}^{-}}{S_{q}^{+}}-{S_{p}^{+}}{S_{q}^{-}}$, it is possible to establish whether ${A_{p}}$ is preferable to ${A_{q}}$. A positive value indicates that ${A_{p}}$ is preferred over ${A_{q}}$.

The mathematical models are based on a scenario in which a new alternative with the highest performance in a specific criterion, referred to as criterion k, is added to the candidate list. All criteria are considered as benefit criteria. Benefit criteria represent the positive attributes or qualities that decision-makers are trying to optimize or maximize in their decision-making process.

Similar to many MCDM methods, TOPSIS utilizes criteria weights in its aggregation process. These weights, assigned to criteria, play a crucial role in measuring the overall preferences of alternatives. Different sets of weights can significantly impact the ranks of alternatives (Kaliszewski et al., 2018). However, this study solely focuses on the distance measurements and normalization methods, thereby disregarding the weighting effect. Hence, all criteria weights are assumed to be identical and unchanged. Consequently, we assume ${w_{j}}=1/n$ for all criteria j. Figure 1 outlines the scope of this mathematical evaluation. The forthcoming exploration will delve into the mathematical aspects to identify the fundamental causes of RR in TOPSIS, particularly focusing on the impact of the normalization process and the use of different distance metrics. Initially, the max-min normalization method will be employed as a basis for analysis. The study will examine how different distance metrics—namely Euclidean, Manhattan, and Chebyshev—affect the ranking of alternatives, driving a detailed mathematical investigation into the underlying formulae to better quantify and explain the occurrence of RR. In the subsequent section, the analysis will extend to explore the effects of the max normalization method.

Hwang and Yoon’s fighter selection problem is modified to consider only the first three benefit criteria for interpretation (Hwang and Yoon, 1981). Table 4 shows the decision matrix. Criterion ${X_{1}}$ corresponds to maximum speed, ${X_{2}}$ pertains to ferry range, and ${X_{3}}$ relates to maximum payload. Table 5 shows the normalized decision matrix using the min-max normalization method. Utilizing the TOPSIS method, Tables 6, 7, and 8 showcases the ranking outcomes obtained by employing Manhattan distance, Euclidean distance, and Chebyshev distance to calculate separation measures, respectively, without taking criteria weights into account.

According to Tables 6 and 7, using Manhattan and Euclidean distances yields identical ranking results. The initial ranking result shows ${A_{2}}\succ {A_{4}}\succ {A_{3}}\succ {A_{1}}$. However, when employing Chebyshev distance, the separation measures calculate the distance between the alternative and PIS/NIS, considering only the maximum absolute difference among the three criteria. Under such circumstances, the outcomes are determined solely by the performance of the alternative in a single criterion. Therefore, when an alternative excels in a particular criterion, it secures the optimal ranking. For example, ${A_{3}}$ outperforms in criterion ${X_{3}}$, resulting in a priority 1 ranking, or the same as ${A_{2}}$. Because using Chebyshev distance in calculating separation measures does not simultaneously consider the influence of all criteria, the resulting ranking may be biased. Evaluating the performance of alternatives neglects other criteria and focuses solely on one criterion with extreme values. Therefore, adopting Chebyshev distance to compute separation measures in most MCDM problems might not be entirely appropriate. It could be challenging to differentiate rankings in the performance assessment of many candidate solutions.

To simulate a scenario where RR might arise, alternative A5 is introduced for evaluation based on the criteria $[{X_{1}}=2,{X_{2}}=2,700,{X_{3}}=21,000]$. Despite the addition of A5, PIS remains unchanged. The new rankings, using Manhattan and Euclidean distances, are ${A_{5}}\succ {A_{2}}\succ {A_{4}}\succ {A_{3}}\succ {A_{1}}$., However, when employing Chebyshev distance, the rankings are ${A_{5}}\succ {A_{2}}={A_{3}}\succ {A_{4}}\succ {A_{1}}$. Importantly, the order of ${A_{1}}$, ${A_{2}}$, ${A_{3}}$, and ${A_{4}}$ remains consistent by using the same distance measurement, indicating the absence of RR. The achievement rating of alternative ${A_{5}}$ on criterion ${X_{2}}$ is gradually increased, leading to a change in PIS. In the current scenario, when the achievement rating of alternative ${A_{5}}$ on criterion ${X_{2}}$ continues to increase, the ratio of change ${\Delta _{{X_{2}}}}$ is greater than 1 and increases synchronously. Tables 9, 10, and 11 illustrate the effect of increasing ${\Delta _{{X_{2}}}}$ on alternatives ${A_{2}}$ and ${A_{4}}$ by using various distance measurements.

The results indicate that, initially, ${A_{2}}$ is preferred over ${A_{4}}$ when ${\Delta _{{X_{2}}}}=1$, regardless of the distance measurements employed. The preference order remains consistent with the situation before the introduction of ${A_{5}}$ to the candidate list. As ${\Delta _{{X_{2}}}}$ increases to 2.0, signifying a twofold improvement in ${A_{5}}$’s performance in criterion ${X_{2}}$, RR occurs when Manhattan distance is employed to calculate separation measures (see Table 9). Interestingly, ${A_{4}}$ outperforms ${A_{2}}$ in this scenario, although ${A_{2}}$ is the superior solution before the introduction of ${A_{5}}$. If Euclidean distance is utilized, then RR emerges when ${\Delta _{{X_{2}}}}$ increases to 2.6 (see Table 10).

As discussed in Section 3, it is highlighted that Euclidean distance has the potential to decrease the likelihood of RR compared to using Manhattan distance. In this case, the results are consistent with the previously stated expectations. In Table 10, it is evident that ${A_{2}}$ consistently outperforms ${A_{4}}$, irrespective of the increasing values of ${\Delta _{{X_{2}}}}$. This consistent superiority indicates the absence of RR. ${A_{2}}$ excels in ${X_{1}}$ and has the poorest performance in ${X_{3}}$, but not in ${X_{2}}$, while ${A_{4}}$ does not exhibit the best or worst performance in any criteria. Notably, ${A_{4}}$ has the longest distance to PIS in criterion ${X_{2}}$ both before and after the introduction of ${A_{5}}$. Additionally, Table 4 confirms that ${A_{2}}$ performs better than ${A_{4}}$ in criterion ${X_{2}}$, aligning with condition 6 in Table 11. Hence, using Chebyshev distance to calculate separation measures in this case ensures the absence of RR.

Figure 4 illustrates the variations in the functions of ${{S_{2}^{-}}^{\prime }}{{S_{4}^{+}}^{\prime }}-{{S_{2}^{+}}^{\prime }}{{S_{4}^{-}}^{\prime }}$ in response to changes in ${\Delta _{{X_{2}}}}$ using different distance measurements. When the values of these functions decrease below 0, it indicates a shift in the preference order between ${A_{2}}$ and ${A_{4}}$, transitioning from ${A_{2}}\succ {A_{4}}$ to ${A_{4}}\succ {A_{2}}$. When utilizing Manhattan distance, it is evident that the decreasing slope of the function is steeper compared to using Euclidean distance. This suggests that using Manhattan distance can make it more susceptible to the occurrence of RR. If the primary concern is to avoid RR, then opting for Chebyshev distance can be the best solution.

For comparison, max normalization is now employed in this example, utilizing a fixed NIS in the process, denoted as ${V^{-}}=[{v_{1}^{-}},\dots ,{v_{n}^{-}}]=[0,\dots ,0]$. Figure 5 depicts the fluctuations in the functions of ${{S_{2}^{-}}^{\prime }}{{S_{4}^{+}}^{\prime }}-{{S_{2}^{+}}^{\prime }}{{S_{4}^{-}}^{\prime }}$ in response to variations in ${\Delta _{{X_{2}}}}$, employing different distance measurements. It is evident that employing max normalization appears to yield a more resilient ranking result compared to using max-min normalization. No instances of RR occurred even after increasing ${\Delta _{{X_{2}}}}$ to 10. Notably, when utilizing Manhattan distance, the decreasing slope of the function is steeper compared to using Euclidean distance. However, the utilization of Chebyshev distance still yields highly robust results.

Max-min normalization scales the performance scores based on the range of values for each criterion. It is sensitive to outliers and can be influenced by extreme values. For instance, consider alternative ${A_{5}}$, which exhibits an extreme performance score in ${X_{2}}$ as ${\Delta _{{X_{2}}}}$ continues to increase. This situation emphasizes the sensitivity of max-min normalization to outliers, potentially leading to a more pronounced impact on the ranking outcomes, especially when certain alternatives demonstrate extreme performance in a particular criterion.

This paper presents a detailed mathematical analysis of the RR phenomenon in the context of TOPSIS, with a focus on its distance metrics and normalization. The objective is to enhance the TOPSIS process by understanding and addressing the occurrence of RR.

The exploration focused on how the introduction of a new alternative excelling in a specific criterion could disrupt the normalization balance and potentially lead to RR. The study analysed the impact of min-max normalization scores on ranking outcomes using Euclidean, Manhattan, and Chebyshev distance measures. Mathematical expressions were examined for separation measures before and after the introduction of a new alternative. Euclidean distance showed scenarios where RR might occur, particularly when the introduced alternative performed well in a specific criterion. Manhattan distance revealed similar trends, while Chebyshev distance highlighted conditions for RR under extreme differences. Overall, Euclidean distance appeared to outperform Manhattan in preventing RR.

Despite offering valuable insights into RR in MCDM processes within the TOPSIS framework, the study is limited to min-max and max normalization and three distance measures, with other normalization methods and dynamic criteria weights left unexplored. Managers should carefully select distance measures and normalization techniques based on the decision-making context, particularly when introducing strong alternatives could alter rankings. Future research could explore additional normalization techniques, distance metrics, dynamic criteria weighting, and hybrid approaches, providing further practical insights and enhancing the robustness of MCDM tools across industries.

In conclusion, the mathematical analysis provides valuable insights into the conditions and factors contributing to the phenomenon of RR in MCDM processes, specifically within the TOPSIS framework. By exploring different normalization methods and distance measures, we contribute to a deeper understanding of decision-making challenges and improve the robustness of MCDM methodologies in practical applications. The findings underscore the importance of careful consideration and customization of methods based on specific decision contexts to mitigate the risk of RR and to enhance the reliability of decision outcomes.