# Comparative Analysis of TOPSIS and TODIM for the Performance Evaluation of Foreign Players in Indian Premier League

Authors: Vaishnudebi Dutta; Subhomoy Haldar; Prabjot Kaur; Yuvraj Gajpal
Journal: Complexity
 (2022)

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## Abstract

Sports officials, players, and fans are concerned about overseas player rankings for the IPL auction. These rankings are becoming progressively essential to investors when premium leagues are commercialized. The decision-makers of the Indian Premier League choose cricketers based on their own experience in sports and based on performance statistics on several criteria. This paper presents a scientific way to rank the players. Our research examines and contrasts different multicriteria decision-making algorithms for ranking foreign players under various criteria to assess their performance and efficiency. The paper uses three MCDM algorithms, TOPSIS, TODIM, and NR-TOPSIS, for foreign players ranking in the Indian Premier League. Our analysis is limited to the batsmen and bowlers only. We perform the analysis using Python language, a popular high-level programming language. Finally, we perform a sensitivity analysis to determine the stability of each method when the weights of the criterion or the value of a parameter was changed. Our analysis exhibits the superiority of TODIM over TOPSIS and NR-TOPSIS.


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## Body

## 1. Introduction

The Indian Premier League (IPL), established in 2007 by the BCCI (Board of Control for Cricket in India), is a professional twenty-team cricket league in India, with eight teams from eight different cities. Each team in the Indian Premier League has only four foreign players in its starting eleven for any match according to the IPL Player Regulations [1], as well as a maximum of eight overseas players in the entire team [2]. A team can get players in one of three ways: through the annual player auction, exchanging players with other teams during trading periods, or signing substitutes for players who are unavailable. Players sign up for the auction and establish their starting price, following which the franchise with the highest offer buys them. For proper bidding, each player’s price is determined by their individual outcomes; franchisee owners have access to all the statistics. They invest in an appropriate team of players, aiming to earn a profit through the cricket match as prize money, sponsorships, and other forms of income. The decision-maker’s job of selecting the best players in a conflicting situation is often challenging, since many qualitative and quantitative qualities must be included in the player selection process. Several sporting activities have been commercialized for decades, yet IPL is India’s richest sport, and it is only getting richer. We considered the selection of overseas players for a few reasons. First, there might be a regional preference for domestic players making their debut. These preferences complicate the selection process. Second, comparing overseas players mitigates internal biases of the authors. Our approach assumes no preference for any team for any of the players being ranked. At the auction, it is entirely up to the decision-makers and team owners.In this paper, we have studied three different multicriteria decision-making (MCDM) algorithms, namely, TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), an improvement on it called NR-TOPSIS, and TODIM (TOmada de DecisãoInterativa e Multicritério, Portuguese acronym for Interactive multicriteria decision-making). There are various other methods available such as ELimination Et Choix Traduisant la REalité (ELECTRE), Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE), and VIseKriterijumskaOptimizacija I KOmpromisnoResenje (VIKOR), whose comparative analysis can be performed later.The principles of the two MCDM algorithms are different, as TODIM employs Prospect Theory, whereas TOPSIS makes use of normalisedn-dimensional Euclidean distance. We proved the two approaches to be contrasting enough to make an interesting comparison. TOPSIS uses a compensatory approach that allows trade-offs between criteria, where a bad outcome in one criterion may be offset by a favourable outcome in another as studied here [3]. It is preferred over ELECTRE I because of better consistency in the results [4]. Although TOPSIS gives various benefits [5], including simplicity, logic, comprehensibility, computing efficiency, and the ability to measure relative performance for each alternative in a simple mathematical form, in our study, we employed a crisp and accurate dataset. Consequently, we did not make use of fuzzy TOPSIS, which is a superior strategy for imprecise or ambiguous performance assessments [6].Our article shows how TODIM outperforms TOPSIS in multicriteria decision-making. This approach may be used by decision-makers of all levels of experience over TOPSIS whose applications in IPL performance analysis were made by various authors. The sensitivity analysis of TODIM provided in this article does not depict any major change in rank of the foreign players when the weights of the criteria (or preferences of criteria) are shuffled, which makes the algorithm more feasible than TOPSIS. For robustness and stability, we have proved that TODIM is a better and more trustworthy MCDM approach.Compared to the other methodologies, TODIM, which was formed by Gomes and Lima in 1992 [7], has the benefit of considering the subjectivity of decision-makers’ (DM) actions and giving the dominance of one alternative over others with specific operation formulae. TODIM is claimed to be more rational and scientific in the application of MCDM problems and various extensions of TODIM have been studied in fuzzy environment [8, 9]. To the best of our knowledge, this is the first study to employ TODIM for evaluating the performance of international players in the IPL and prove the superiority of the algorithm over TOPSIS.In our article, we compared TOPSIS, which is a more popular MCDM algorithm than TODIM. We claim so with the help of “Google Trends,” which allows us to compare various keywords searched over the Internet [10]. We inferred from the findings that TODIM applications have been examined far less than TOPSIS since 2004. For a crisp dataset, however, our study clearly shows that TODIM outperforms TOPSIS for evaluating alternatives based on many criteria. Other than that, we have also explored several other existing articles where research scholars have studied TOPSIS and TODIM from various aspects such as Choquet based TOPSIS and TODIM [11], TOPSIS and TODIM with Z-numbers [12], and fuzzy TOPSIS-TODIM hybrid method [13]. Some also proved in their article that TOPSIS, TODIM, and PROMETHEE are methods that have higher utility in the selection of mining method in the coal mining industry [14]. Comparative studies have also been made for fuzzy AHP (TOPSIS is an additive variant of AHP) and fuzzy TODIM [15].As both TOPSIS and TODIM are applied at their most basic stages in this paper, without implementation of any extension, decision-makers can readily understand them. Future studies and sensitivity analyses of other MCDM approaches can be conducted to determine the superiority of other methods over TODIM. One fact to note is that both TOPSIS and TODIM are said to have rank reversal problem, which is a shift in the rank ordering of the preferability of alternative feasible decisions. Therefore, we tried to implement NR-TOPSIS [16], which in the article is proved to solve the rank reversal problem in TOPSIS. However, because we have numerous alternatives and criteria and a different approach to calculate criteria weights, NR-TOPSIS does not address the problem in our situation. Instead, we observed that the approach is significantly less stable than both TOPSIS and TODIM. All calculations and results in this paper are done by implementing the algorithm in Python, which is provided with this article so that other authors can use them as well. Finally, while TOPSIS has historically been used to rank players in the IPL, TODIM is not employed as frequently. As a result, we picked these two approaches and used sensitivity analysis to show how TODIM is superior to both TOPSIS and NR-TOPSIS.We employed rank-sum method for generating weights for the criteria. Because of this, the weights vary significantly based on the preferences of the decision-maker. It is also simpler for a regular user to rank the criteria in order of their preference rather than calculating the numeric value of the weights manually. Owing to this significant variance, the results generated by TOPSIS do, in fact, vary based on subjective preferences. The same holds true for TODIM. Personal preferences are not included in TOPSIS, but they are in TODIM, since it employs the idea of prospect theory. However, as we used the rank-sum weight approach that relies on the preferences of decision-makers, as a result, the TOPSIS outcome will be influenced, exposing it to subjective preferences such as TODIM.The following is how the rest of the paper is organised: Section2 contains a review of the literature on the MCDM methodologies used in this work, and Section 3 describes the methodology, which explains the processing done in a step-by-step fashion. The application of the methodology explained in Section 3 is explained through the case study in Section 4. In Section 5, we discuss results we had obtained from the case study. The next section, that is, Section 6, is dedicated to discussion and interpretation of the results, followed by future perspectives, and we conclude the paper in Section 7. At last, we also provide Supplementary Materials, containing Python code and intermediate steps for the calculations performed, followed by a list of references used in this paper.


## 2. Literature Review

The fundamental concept for selecting a foreign player for any club originates from an article by Xing [17], in which a vast amount of data demonstrates that different decision-makers base their selections on prior competition scores and games played. The technical data supplied is frequently used to judge several international players. Multicriteria decision-making (MCDM) is a common method in performance analysis. Thus, several studies on various MCDM methods to analyse IPL team performance analysis have been conducted. Multicriteria group decision method studied in [18] gives a practical way to evaluate a team’s success while producing consistent performance ratings. Multicriteria decision tree approach [19] can classify all-rounders in the Indian Premier League for accurately and efficiently classifying data based on the output of players. It is also shown how multicriteria decision tree method provided a good image of the players in several categories, including performer, batting all-rounder, bowling all-rounder, and underperformer, aiding franchisee owners in picking all-rounders in the auction and compensating them depending on their performance.Application of MGDA (Modified Group Decision Algorithm) is studied further [20] to analyse batsman, fast bowler, and spin bowler statistics from IPL sessions IV, V, and VI separately, and it consistently produced reliable results. The consistency testing property of AHP is used to calculate the weights of the criterion, proving its correctness. The hybridization of two well-known multicriteria decision-making (MCDM) approaches with their classifications and characteristics has been explored [21]. The goal was divided into two parts: First, prioritize the weight of selected parameters for evaluating the players’ output using Analytic Hierarchy Process (AHP). Second, TOPSIS is used to select the best alternative strategy for ranking the teams. Performance-based analysis is studied among IPL batsmen in seasons I (2008), II (2009), and III (2010) using statistical multicriteria decision-making [22]. They concluded that international players performed well in season II, but their performance varied from season to season. In case of IPL, a team has the liberty to retain their previously auctioned players, but it lowers the funds for the owner to enter the auction. A decision tree is made use of to compute the “most valuable player” for a team by player’s batting and bowling points and experiences [23]. Prediction of match is calculated through various match statistics using TOPSIS in multiple studies based on World Cup 2019 [24].Other related recent works in sports include the applications of Bayesian BWM (Best Worst Method) and rough DEMATEL (Decision-MAking Trial and Evaluation Laboratory), which is a type of MCDM algorithm, investigating the impact of Sustainable Sports Tourism Criteria in Taichung City [25]. Applications of fuzzy-ANP and DEMATEL [26], ANP-DEMATEL, and VIKOR (VIseKriterijumskaOptimizacija I KOmpromisnoResenje) [27] are also studied in various sports business applications and helped in creating a management strategy.Another outranking approach known as TODIM (TOmada de DecisãoInterativa e Multicritério, an acronym in Portuguese for iterative multicriteria decision-making) is studied in this article. We may use this MCDM technique to discover foreign player rankings in IPL 2019 as it uses prospect theory studied by two Israeli psychologists in 1979 [28, 29] as well to create a multiattribute value function. The purpose of their study was to evaluate human behaviour during decision-making and in high-risk situations. The two psychologists then discovered that, in scenarios involving advantages, humans have a propensity to be more conservative in terms of risk; that is, individuals prefer to select for a smaller, more secure gain than taking a risk to earn a larger benefit.The systematic flowchart diagram and highlighted formulae in article [30] were used to determine the weight, four normalization techniques, and other details for TODIM, making the algorithm clearer. The comparison between TODIM and modified TODIM on recycled water alternatives based on a range of parameters were mainly studied. The formulae and tabular procedures are likewise based on research work mentioned in article [31]. The weights of the criteria and an assessment matrix that aided us in our implementation were explained. We also referred to another article [32] where applications of TODIM are shown to evaluate broadband Internet services. For TODIM sensitivity analysis, we used the study from article [33], where the authors looked at different values of θ (loss of attenuation factor) in the space curved surfaces of distributor dominance in four distance equations. As a result, how changes in the value of θ have an impact on the alternatives was evaluated. For TOPSIS, we have shuffled the ranks of the criteria, thereby changing the weight for which we referred to related article [34].Although TODIM is an emerging method of MCDM which is not as popularly used as TOPSIS and its extensions, there are recent applications outside the sports world [35–38], where the implementation is based on MCGDM, stock investment selection and assessment of hydro energy storage plant, and green supplier selection problems. Other studies include applications of failure mode and effect analysis (FMEA) and TODIM to demonstrate risk ranking of wind turbine systems [39], extended fuzzy TODIM with dual connection numbers [32], and Pythagorean fuzzy TODIM based on cumulative prospect theory to assess the risk in science and technology project [40]. Progressive studies have also been made on development of TODIM with different fuzzy sets [41]. But there are a handful of research articles where TODIM is implemented to analyse the performance, directly based on the scores achieved by the players in any sport. In our article, we studied the IPL 2019 dataset, but similar applications can also be made in other areas of sports, such as football clubs, golf clubs, and gymnastics.When a choice alternative is added or removed, rank reversal occurs, which implies that the relative ranks of two decision alternatives can be reversed. Belton and Gear were the first to point out such a phenomenon [42]. Although the rank reversal problem may not always have a detrimental influence on many datasets, the decision-makers in our situation may not like to have such an issue when removing, adding, or replacing a player can affect the entire rank list of players. As a result, we examined various recent and previous studies in order to improve the MCDM technique and avoid the problem of rank reversal. Several methods exist such as a combination of Characteristic Objects Method (COMET) and TOPSIS or PROMETHEE II [43], improvement of VIKOR method using R-VIKOR [44], and G-TOPSIS (Gaussian TOPSIS) method for rank reversal problem [45]; analytical studies have also been made [16, 46–49] to find a solution to the problem for various applications. Rank reversal problem for TODIM was first discussed in 1990 [50], where the author compared the problem with AHP and tried to provide a solution for the problem. But few modifications or developments have been made on TODIM rank reversal problem.In this paper, we tried to implement the work of Yang, where R-TOPSIS was modified [51] and the new method was named NR-TOPSIS [52] to remove the rank reversal problem of TOPSIS and compare the results with TODIM. Unfortunately, the approach did not work, and we were still dealing with rank reversal. Instead, we could show and infer that when players are added or deleted from the list, TODIM’s ranking changes relatively little. As a result, the MCDM method becomes more appealing, demonstrating the necessity of focusing on solving the rank reversal problem in TODIM in order to get supremum outcomes.


## 3. Methodology

### 3.1. Weighting through Rank-Sum Weight Method

This subjective method establishes weights solely based on the decision-makers’ considerations or judgments [53]. It may be easier to rank order the relevance of criteria than it is to describe other inaccurate weights, such as bounded weights, for example, in instances involving time constraints, the nature of the criteria, a lack of expertise, imprecise or partial information, and the decision-maker’s limited attention or information processing skills. Because a group of decision-makers may not agree on a set of exact weights, it may be reasonable to assume agreement merely on a ranking of weights in such a case as stated by article [54]. This rank order weight approach entails two steps: first, ranking the criteria by significance and then weighting the criteria using the formula. In this paper, we have used the rank-sum weight method which was proposed by Stillwell [55]. In this method, individual ranks are normalised by dividing the sum of the rankings in the rank-sum (RS) technique. The weights are calculated using the following formula:(1)wjRS=2n+1−rjnn+1.Here,rj is the rank of the jth criteria, j=1,2,…,n.If there are multiple decision-makers, the ranks can be determined through discussion. Alternatively, rankings of the individual decision-makers can be added together and averaged to get the final rank values. It is not strictly necessary for the “ranks” to be integers, but it keeps the process easily understandable.


### 3.2. TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution)

TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a multicriteria decision-making approach created by [56]. It is founded on the principle that the best option should be the one with the least geometric distance from the positive ideal solution and the greatest geometric distance from the negative ideal solution. The steps for this method are listed as follows:Step 1: create an evaluation matrix withm alternatives, with xij representing the score for the ith candidate in the jth criteria in an m×n matrix of the form given below as decision matrix D.(2)D=x11⋯x1n⋮⋱⋮xm1⋯xmn.Step 2: calculate the normalised matrix using the following formula:(3)rij=xij∑i=1mxij2.In the previous formula,i=1,2,…,m and j=1,2…,n.Step 3: construct the weighted normalised matrixvij. This is calculated by multiplying the weight wj we had calculated before by each column of matrix rij. The formula is(4)vij=wjrij.Step 4: identify the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−), where(5)A∗=v1∗,v2∗,…,vn∗,A−=v1−,v2−,…,vn−.We define(6)vj∗=maxvij,ifjϵJ1minvij,ifjϵJ2,vj−=minvij,ifjϵJ1maxvij,ifjϵJ2.In the previous formula,J1 and J2 are attributes related to criteria that have a positive impact and attributes related to criteria that have a negative impact, respectively.Steps 5 and 6: then-dimensional Euclidean distance can be used to calculate the separation or distance between the alternatives. The separations from the PIS A∗ and NIS A− are S∗ and S−, respectively.(7)Si∗=∑j=1nvij−vj∗2,Si−=∑j=1nvij−vj−2.In the previous formula,i=1,2…,mandj=1,2…,n. After this, in Step 3.2.5, we calculate the similarity using(8)Ci∗=Si−Si∗+Si−.The terms have usual meaning mentioned previously.Step 7: we choose the candidate with the maximumCi∗ or rank all the alternatives in a descending order according to their Ci∗ values. This process is done for both the Si∗ and Si− values. The flowchart of these steps is given in Figure 1.Figure 1
Flowchart of TOPSIS.


### 3.3. NR-TOPSIS (Improved R-TOPSIS)

This updated version of TOPSIS method is abbreviated as NR-TOPSIS method because it can (supposedly) handle rank reversal problems based on the historical maximum value of indicator data. In article [52], the author claimed and proved the method to be effective on the dataset, which had 4–7 alternatives and removed the rank reversal problem in original TOPSIS method. The steps of the algorithm are as follows:Step 1: find the minimum valuemj and maximum value Mj of each attribute Cj. So, for any attribute xij, the condition mj≤xij≤Mj is satisfied. Also, the condition mj≤xij≤Mj must be satisfied when the scheme is increased, decreased, or replaced.Step 2: the original decision-making matrixX=xijm×n is standardized and transformed to generate standardized decision-making matrix Y=yijm×n, where yij are normalised attribute values. This will eliminate the influence of dimension on data decision-making.IfCj is benefit attribute, then(9)yij=xij−mjMj−mj.IfCj is cost attribute, then(10)yij=Mj−xijMj−mj.Step 3: calculate the weighted normalised decision-making matrix=vijm×n. The weighted normalised attribute value vij has the same calculation as that of TOPSIS, vij=wjyij,i=1,2,…,m;j=1,2,…,n.Step 4: identify the Positive Ideal Solution PIS (A∗) and Negative Ideal Solution NIS (A−), where(11)A∗=v1∗,v2∗,…,vn∗,A−=v1−,v2−,…,vn−and(12)vj∗=wj,vj−=0.j=1,2,…,n.Step 5: compute the Euclidean distancesSi∗ and Si− for every alternative Ai between the positive ideal solution and the negative ideal solution, respectively.(13)Si∗=∑j=1nvij−vj∗2,Si−=∑j=1nvij−vj−22.In the previous formula, i=1,2…,mandj=1,2…,n. After this, we calculate the similarity using(14)Ci∗=Si−Si∗+Si−.The terms have usual meaning mentioned previously. The logic of the two instances in which the indicator is a benefit type or a cost type is examined as follows:(1)
Two extreme cases are considered whenCj is a benefit attribute: xij=mj and xij=Mj, which means if the value of an attribute is the same as the maximum or minimum value in that column. When xij=mj consider yij=mj−mj/Mj−mj=0 and vj−=vij=0, and when xij=Mj consider yij=Mj−mj/Mj−mj=1, where vj+=vij=wj.(2)
In case of cost attributeCj, again two extreme cases can occur. When xij=mj consider yij=Mj−mj/Mj−mj=1, where vj+=vij=wj, and when xij=Mj consider yij=Mj−Mj/Mj−mj=0 and vj−=vij=0.


### 3.4. TODIM (TOmada de DecisãoInterativa e Multicritério, an Acronym in Portuguese for Interactive Multicriteria Decision-Making)

The TODIM technique we studied from [9] is based on PT (prospect theory), as its value gain/loss function is defined in [29]. Gains and losses are always calculated in relation to a reference point in this situation. As a result, while this technique acknowledges the possibility of decision-makers, it does not incorporate their actual involvement. The following are the stages that a TODIM application would take in algorithmic form:Step 1: we will reuse the initial evaluation matrix considered in the previous section. We directly normalize the ratings and weights using the following formula:(15)Pij=xij∑k=1mxkjifj∈J1,1/xij∑k=1m1/xkjifj∈J2.For the weighting factor or trade-off rate between the reference criteriar and the generic criteria c, here, wr determines the most relevant reference criterion for the decision-maker. Often, it is the maximum weight. In general, any criterion can be used as the reference criterion, and this decision has no effect on the final findings. So, the formula we have is(16)wrc=wcwr,wherewr=maxwc|c=1,2,…,n and i=1,2…,m;j=1,2…,n.Step 2: for calculating the dominance degree, we need to first check the contribution of each criterion using a formula, whereφc is the contribution of criterion c to the function δAi,Aj and θ is the loss of attenuation factor whose value we considered as 1. In our case,(17)φcAi,Aj=Pic−Pjcwrc∑c=1nwrc,ifPic−Pjc≥0−1θ∑c=1nwrcPjc−Picwrc,ifPic−Pjc<0,and combining all contributions, we get the dominance degrees from the measurement of dominanceδAi,Aj as(18)δAi,Aj=∑c=1nφcAi,Aj,wherei,j=1,2…,m;c=1,2…,n.Step 3: finally, compute the values ofξi, which are the normalised global performances of alternatives compared to others, such that the largest value is picked as more significant than the other alternative’s values:(19)ξi=∑j=1nAi,Ajδ−mini∑j=1nδAi,Ajmaxi∑j=1nAi,Aj−mini∑j=1nδAi,Aj.In the previous formula,i=1,2,…,n. The flowchart of these steps is in Figure 2.Figure 2
Flowchart of TODIM.


## 3.1. Weighting through Rank-Sum Weight Method

This subjective method establishes weights solely based on the decision-makers’ considerations or judgments [53]. It may be easier to rank order the relevance of criteria than it is to describe other inaccurate weights, such as bounded weights, for example, in instances involving time constraints, the nature of the criteria, a lack of expertise, imprecise or partial information, and the decision-maker’s limited attention or information processing skills. Because a group of decision-makers may not agree on a set of exact weights, it may be reasonable to assume agreement merely on a ranking of weights in such a case as stated by article [54]. This rank order weight approach entails two steps: first, ranking the criteria by significance and then weighting the criteria using the formula. In this paper, we have used the rank-sum weight method which was proposed by Stillwell [55]. In this method, individual ranks are normalised by dividing the sum of the rankings in the rank-sum (RS) technique. The weights are calculated using the following formula:(1)wjRS=2n+1−rjnn+1.Here,rj is the rank of the jth criteria, j=1,2,…,n.If there are multiple decision-makers, the ranks can be determined through discussion. Alternatively, rankings of the individual decision-makers can be added together and averaged to get the final rank values. It is not strictly necessary for the “ranks” to be integers, but it keeps the process easily understandable.


## 3.2. TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution)

TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a multicriteria decision-making approach created by [56]. It is founded on the principle that the best option should be the one with the least geometric distance from the positive ideal solution and the greatest geometric distance from the negative ideal solution. The steps for this method are listed as follows:Step 1: create an evaluation matrix withm alternatives, with xij representing the score for the ith candidate in the jth criteria in an m×n matrix of the form given below as decision matrix D.(2)D=x11⋯x1n⋮⋱⋮xm1⋯xmn.Step 2: calculate the normalised matrix using the following formula:(3)rij=xij∑i=1mxij2.In the previous formula,i=1,2,…,m and j=1,2…,n.Step 3: construct the weighted normalised matrixvij. This is calculated by multiplying the weight wj we had calculated before by each column of matrix rij. The formula is(4)vij=wjrij.Step 4: identify the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−), where(5)A∗=v1∗,v2∗,…,vn∗,A−=v1−,v2−,…,vn−.We define(6)vj∗=maxvij,ifjϵJ1minvij,ifjϵJ2,vj−=minvij,ifjϵJ1maxvij,ifjϵJ2.In the previous formula,J1 and J2 are attributes related to criteria that have a positive impact and attributes related to criteria that have a negative impact, respectively.Steps 5 and 6: then-dimensional Euclidean distance can be used to calculate the separation or distance between the alternatives. The separations from the PIS A∗ and NIS A− are S∗ and S−, respectively.(7)Si∗=∑j=1nvij−vj∗2,Si−=∑j=1nvij−vj−2.In the previous formula,i=1,2…,mandj=1,2…,n. After this, in Step 3.2.5, we calculate the similarity using(8)Ci∗=Si−Si∗+Si−.The terms have usual meaning mentioned previously.Step 7: we choose the candidate with the maximumCi∗ or rank all the alternatives in a descending order according to their Ci∗ values. This process is done for both the Si∗ and Si− values. The flowchart of these steps is given in Figure 1.Figure 1
Flowchart of TOPSIS.


## 3.3. NR-TOPSIS (Improved R-TOPSIS)

This updated version of TOPSIS method is abbreviated as NR-TOPSIS method because it can (supposedly) handle rank reversal problems based on the historical maximum value of indicator data. In article [52], the author claimed and proved the method to be effective on the dataset, which had 4–7 alternatives and removed the rank reversal problem in original TOPSIS method. The steps of the algorithm are as follows:Step 1: find the minimum valuemj and maximum value Mj of each attribute Cj. So, for any attribute xij, the condition mj≤xij≤Mj is satisfied. Also, the condition mj≤xij≤Mj must be satisfied when the scheme is increased, decreased, or replaced.Step 2: the original decision-making matrixX=xijm×n is standardized and transformed to generate standardized decision-making matrix Y=yijm×n, where yij are normalised attribute values. This will eliminate the influence of dimension on data decision-making.IfCj is benefit attribute, then(9)yij=xij−mjMj−mj.IfCj is cost attribute, then(10)yij=Mj−xijMj−mj.Step 3: calculate the weighted normalised decision-making matrix=vijm×n. The weighted normalised attribute value vij has the same calculation as that of TOPSIS, vij=wjyij,i=1,2,…,m;j=1,2,…,n.Step 4: identify the Positive Ideal Solution PIS (A∗) and Negative Ideal Solution NIS (A−), where(11)A∗=v1∗,v2∗,…,vn∗,A−=v1−,v2−,…,vn−and(12)vj∗=wj,vj−=0.j=1,2,…,n.Step 5: compute the Euclidean distancesSi∗ and Si− for every alternative Ai between the positive ideal solution and the negative ideal solution, respectively.(13)Si∗=∑j=1nvij−vj∗2,Si−=∑j=1nvij−vj−22.In the previous formula, i=1,2…,mandj=1,2…,n. After this, we calculate the similarity using(14)Ci∗=Si−Si∗+Si−.The terms have usual meaning mentioned previously. The logic of the two instances in which the indicator is a benefit type or a cost type is examined as follows:(1)
Two extreme cases are considered whenCj is a benefit attribute: xij=mj and xij=Mj, which means if the value of an attribute is the same as the maximum or minimum value in that column. When xij=mj consider yij=mj−mj/Mj−mj=0 and vj−=vij=0, and when xij=Mj consider yij=Mj−mj/Mj−mj=1, where vj+=vij=wj.(2)
In case of cost attributeCj, again two extreme cases can occur. When xij=mj consider yij=Mj−mj/Mj−mj=1, where vj+=vij=wj, and when xij=Mj consider yij=Mj−Mj/Mj−mj=0 and vj−=vij=0.


## 3.4. TODIM (TOmada de DecisãoInterativa e Multicritério, an Acronym in Portuguese for Interactive Multicriteria Decision-Making)

The TODIM technique we studied from [9] is based on PT (prospect theory), as its value gain/loss function is defined in [29]. Gains and losses are always calculated in relation to a reference point in this situation. As a result, while this technique acknowledges the possibility of decision-makers, it does not incorporate their actual involvement. The following are the stages that a TODIM application would take in algorithmic form:Step 1: we will reuse the initial evaluation matrix considered in the previous section. We directly normalize the ratings and weights using the following formula:(15)Pij=xij∑k=1mxkjifj∈J1,1/xij∑k=1m1/xkjifj∈J2.For the weighting factor or trade-off rate between the reference criteriar and the generic criteria c, here, wr determines the most relevant reference criterion for the decision-maker. Often, it is the maximum weight. In general, any criterion can be used as the reference criterion, and this decision has no effect on the final findings. So, the formula we have is(16)wrc=wcwr,wherewr=maxwc|c=1,2,…,n and i=1,2…,m;j=1,2…,n.Step 2: for calculating the dominance degree, we need to first check the contribution of each criterion using a formula, whereφc is the contribution of criterion c to the function δAi,Aj and θ is the loss of attenuation factor whose value we considered as 1. In our case,(17)φcAi,Aj=Pic−Pjcwrc∑c=1nwrc,ifPic−Pjc≥0−1θ∑c=1nwrcPjc−Picwrc,ifPic−Pjc<0,and combining all contributions, we get the dominance degrees from the measurement of dominanceδAi,Aj as(18)δAi,Aj=∑c=1nφcAi,Aj,wherei,j=1,2…,m;c=1,2…,n.Step 3: finally, compute the values ofξi, which are the normalised global performances of alternatives compared to others, such that the largest value is picked as more significant than the other alternative’s values:(19)ξi=∑j=1nAi,Ajδ−mini∑j=1nδAi,Ajmaxi∑j=1nAi,Aj−mini∑j=1nδAi,Aj.In the previous formula,i=1,2,…,n. The flowchart of these steps is in Figure 2.Figure 2
Flowchart of TODIM.


## 4. Case Study

We used the IPL 2019 dataset and rated 15–16 randomly picked cricketers. In this section, we will first calculate the weights for the 12 and 9 criteria selected to rank batsman and bowlers accordingly. Then we will implement them in TOPSIS, NR-TOPSIS, and TODIM to check the rank generated by these algorithms, respectively. We have highlighted the important calculations only and thus the intermediate steps can be found in the Supplementary Materials provided with this paper.


### 4.1. Criteria Selection

We used the rank-sum weight approach to determine the weights of the criteria specified for batsmen and bowlers, making it easier for decision-makers to grasp and rank the criteria based on their preferences. The criteria for batsmen are described in Table1, and those for bowlers are stated in Table 2. It should be noted that the criteria rankings are adjusted for the T20 format and are subject to change based on the decision-makers’ preferences. The batsmen and bowlers chosen here are randomly selected with no regional preferences from the list given in the two following links: https://www.iplt20.com/stats/2019/most-runs and https://www.iplt20.com/stats/2019/most-wickets. As we can see, our data is collected from a trustworthy and easily available to public source; thus we do not need to implement a fuzzy algorithm here.Table 1
Batting performance criteria.
Batting criteriaSymbolDefinitionRankWeightsIdeallyStrike rateSRSR=runs/BF×10010.1538462HigherBatting averageAvgAvg=runs/outs20.1410256HigherRunsRunsThe total runs scored in the season30.1282051HigherInningsInnThe number of innings played in the season40.1153846HigherNot outsNOThe number of innings undefeated50.1025641Higher6s6sThe number of sixes hit60.0897436Higher4s4sThe number of fours bit70.0769231Higher100s100sThe number of centuries scored80.0641026Higher50s50sThe number of half-centuries scored90.0512821HigherMatchesMatThe number of matches played, might be more than the number of innings played100.0384615HigherHighest scoreHSThe highest individual score110.025641HigherBalls facedBFThe number of balls faced in the tournament120.0128205HigherTable 2
Bowling performance criteria.
Bowling criteriaSymbolDefinitionRankWeightsIdeallyStrike rateSRSR=BB/wkts10.200000LowerEconomyEconEcon=runs/overs=runs/BB×620.177778LowerBowling averageAvgAvg=runs/wkts30.155556LowerWicketsWktsThe number of wickets taken in the season40.133333HigherRunsRunsTotal runs conceded50.111111LowerInningsInnsThe number of innings played60.088889HigherBalls bowledBBTotal number of balls bowled70.066667Higher4 wicket hauls4wFrequency of times 4 wickets were taken80.044444HigherMatchesMatThe number of matched played90.022222HigherThe ranking of criteria is subjective, and the decision-maker is expected to arrange them according to their preferences. Keeping in mind the T20 format, we assume that strike rate is more important than batting average for batsmen courtesy of the limited playtime. Similarly, a good batsman may have scored fewer runs overall but may have helped finish more run chases successfully. In case of bowlers, both strike rate and economy are important. However, bowling out all batsmen ends the game quicker, so strike rate takes priority. This form of reasoning applies to ranking the remaining criteria as well. The rankings we present here are not authoritative, but they are certainly close to what would be used in practice. We illustrate an example that differs from the current context: in One-Day Internationals (ODIs), strike rate may have a lower preference than batting average and total runs scored for batsmen. Similarly, the number of balls faced becomes more important in a test match. Therefore, the decision-makers need to rank the criteria based on the format of the game being played.


### 4.2. Application of TOPSIS

After computing the weights of the criteria, we will rank the alternatives, which are batsmen and bowlers, using TOPSIS whose algorithm is described in the previous section. The evaluation matrix is the same for all algorithms and we will be dealing with the crisp dataset.Step 1: the evaluation matrix for Foreign Batsmen of IPL 2019 (retrieved from the official website mentioned at the beginning of the section) is represented in Table3.The same for bowlers is presented in Table4 and all the data are taken form the website mentioned at the beginning of the section.Step 2: the normalised matrix is calculated.Step 3: the weighted normalised matrix is calculated.Step 4: the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−) are identified.Step 5:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Step 6:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Then, we proceed to rank the candidates according to the distance values. These values for the batsmen and bowlers are provided in Tables5 and 6.Step 7: we choose the candidate with the maximumCi∗ or rank all the alternatives in a descending order according to their Ci∗ values. This process is done for both the Si∗ and Si− values.Table 3
Batsmen statistics.
NameNationalityMatInnNORunsHSAvgBFSR100s50s4s6sAB de VilliersSouth African131334428244.20287154.00053126Andre RussellWest Indian141345108056.67249204.81043152Ben StokesEnglish9931234620.5099124.240084Chris GayleWest Indian131314909940.83319153.60044534Chris LynnAustralian131304058231.15290139.65044122David WarnerSouth African1212269210069.20481143.86185721Faf du PlessisSouth African121213969636.00321123.36033615Jonny BairstowEnglish1010244511455.63283157.24124818Jos ButtlerEnglish8803118938.88205151.70033814Kane WilliamsonNew Zealander9921567022.29130120.0001125Kieron PollardWest Indian161462798334.88178156.74011422Marcus StoinisAustralian101062114652.75156135.25001410Moeen AliEnglish111022206627.50133165.41021617Quinton de KockSouth African161615298135.27398132.91044525Shane WatsonAustralian171703989623.41312127.56034220Steve SmithAustralian121023197339.88275116.0003304Table 4
Bowlers statistics.
NameNationalityMatInnOvBBRunsWktsAvgEconSR4wAndre RussellWest Indian141230.11812871126.099.5116.450Ben StokesEnglish9616.5101189631.511.2316.830Chris MorrisSouth African9933.01983061323.549.2715.230Dwayne BravoWest Indian121241.124733011308.0222.450Imran TahirSouth African171764.23864312616.586.714.852Jofra ArcherEnglish111143.02582911126.456.7723.450Kagiso RabadaSouth African121247.02823682514.727.8311.282Keemo PaulWest Indian8827.1163237926.338.7218.110Lasith MalingaSri Lankan121244.52694381627.389.7716.812Moeen AliEnglish11925.0150169628.176.76250Mohammad NabiAfghan8829.1175194824.256.6521.881Rashid KhanAfghan151560.03603771722.186.2821.180Sam CurranEnglish9933.01983231032.39.7919.81Sunil NarineWest Indian121244.22663471034.77.8326.60Trent BoultNew Zealander5519.0114163532.68.5822.80Table 5
Distance values for batsmen.
BatsmenSi∗Si−Ci∗AB de Villiers0.0701960.0551120.439813Andre Russell0.0568880.0811650.587927Ben Stokes0.1065260.0272940.203959Chris Gayle0.0761690.0551950.420169Chris Lynn0.0902960.0408400.311430David Warner0.0516790.0907780.637231Faf du Plessis0.0888620.0366060.291756Jonny Bairstow0.0628140.0696480.525796Jos Buttler0.0958100.0326340.254072Kane Williamson0.1057480.0191200.153120Kieron Pollard0.0796070.0628850.441323Marcus Stoinis0.0870820.0619240.415581Moeen Ali0.0932270.0293650.239533Quinton de Kock0.0802780.0530220.397761Shane Watson0.0943070.0419460.307855Steve Smith0.0925510.0339530.268397Table 6
Distance values for bowlers.
BowlersSi∗Si−Ci∗Andre Russell0.056820.040470.41596Ben Stokes0.075090.034800.31667Chris Morris0.053260.043990.45232Dwayne Bravo0.062330.033810.35169Imran Tahir0.026770.081950.75376Jofra Archer0.060680.039070.39169Kagiso Rabada0.024790.079580.76251Keemo Paul0.062160.036590.37049Lasith Malinga0.048950.050300.50679Moeen Ali0.073080.037870.34136Mohammad Nabi0.061350.042460.40905Rashid Khan0.047660.055150.53646Sam Curran0.063260.030360.32433Sunil Narine0.072040.030810.29958Trent Boult0.078510.031140.28400


### 4.3. Application of NR-TOPSIS

We will utilize the same evaluation matrix and weights for the criteria as calculated previously. Here,Mj and mj are the maximum and minimum values selected from each column of Tables 3 and 4, respectively:Step 1: the evaluation matrix for Foreign Batsmen of IPL 2019 (obtained from the official website mentioned at the beginning of the section) is represented in Table3 and that for bowlers is presented in Table 4 and all the data are taken form the website mentioned at the beginning of the section.Step 2: we calculate the normalised matrix.Step 3: the weighted normalised matrix is then calculated using a formula. These results are all available in the Supplementary Materials.Step 4: the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−) are identified.Step 5:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Step 6:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Then we proceed to rank the candidates according to the distance values. These values for the batsmen and bowlers are provided in Tables7 and 8.Table 7
Distance values for batsmen.
BatsmenSi∗Si−Ci∗AB de Villiers0.1743820.1601630.478748Andre Russell0.1167000.2487760.680691Ben Stokes0.3027460.0549200.153552Chris Gayle0.1823000.1644250.474222Chris Lynn0.2263520.1262210.357999David Warner0.1541950.2387260.607567Faf du Plessis0.2378100.1109550.318137Jonny Bairstow0.1755570.1794800.505525Jos Buttler0.2451610.1080040.305817Kane Williamson0.3003920.0406030.119072Kieron Pollard0.2025680.1643770.447960Marcus Stoinis0.2360550.1472920.384226Moeen Ali0.2416910.1057190.304306Quinton de Kock0.2045390.1699530.453822Shane Watson0.2444310.1537020.386057Steve Smith0.2576820.0944430.268208Table 8
Distance values for bowlers.
BowlersSi∗Si−Ci∗Andre Russell0.2076210.1853580.471674Ben Stokes0.2888520.1648040.363279Chris Morris0.1915660.2034920.515094Dwayne Bravo0.2396560.1579350.39723Imran Tahir0.1197270.3201590.727823Jofra Archer0.2249610.1997510.470322Kagiso Rabada0.1097680.3204030.744827Keemo Paul0.2139230.1806450.457831Lasith Malinga0.2218580.1832650.452369Moeen Ali0.2610220.20440.439172Mohammad Nabi0.2137790.2210730.508387Rashid Khan0.1818350.249050.577996Sam Curran0.2601040.1265410.327279Sunil Narine0.2955740.1470270.332188Trent Boult0.2846330.1553750.353119Andre Russell0.2076210.1853580.471674


### 4.4. Application of TODIM

In this part, we will be evaluating the rank generated using TODIM. We have considered the value of attenuation factorθ as 1. However, decision-makers can change the value accordingly. Changing this value might affect the final ranking and is therefore discussed in the Sensitivity Analysis section.Step 1: we can determine the trade-off rate for batsmen and bowlers using the list of weights we calculated before. They are presented in Table9.Step 2: we calculate the matrixφcAi,Aj and then the matrix δAi,Aj for θ=1.Step 3: then, we compute the values ofξi. The values of ξi are calculated in Table 10.Table 9
Table of updated weights.
Batting criteriawrcBowling criteriawrcSR1SR1Avg0.916667Econ0.888889Runs0.833333Avg0.777778Inn0.75Wkts0.666667NO0.666667Runs0.5555566s0.583333Inns0.4444444s0.5BB0.333333100s0.4166674w0.22222250s0.333333Mat0.111111Mat0.25HS0.166667Table 10
ξi for batsmen and bowlers.
BatsmenξiBowlersξiAB de Villiers0.804597Andre Russell0.561531Andre Russell0.818468Ben Stokes0.155654Ben Stokes0Chris Morris0.426346Chris Gayle0.848149Dwayne Bravo0.524013Chris Lynn0.687784Imran Tahir1David Warner1Jofra Archer0.525132Faf du Plessis0.667418Kagiso Rabada0.91773Jonny Bairstow0.777412Keemo Paul0.279925Jos Buttler0.427361Lasith Malinga0.738363Kane Williamson0.130708Moeen Ali0.374627Kieron Pollard0.552321Mohammad Nabi0.428831Marcus Stoinis0.247835Rashid Khan0.719623Moeen Ali0.320078Sam Curran0.398341Quinton de Kock0.823886Sunil Narine0.472208Shane Watson0.703334Trent Boult0Steve Smith0.508533


### 4.5. Results

The rankings generated by TOPSIS for batsmen and bowlers, who are international players in the IPL 2019, are tabulated in a descending order of preference and summarised in Table11. Similarly, the ranks generated by NR-TOPSIS and TODIM are summarised in Tables 12 and 13, respectively.Table 11
Ranks generated by TOPSIS.
RankBatsmenBowlers1David WarnerKagiso Rabada2Andre RussellImran Tahir3Jonny BairstowRashid Khan4Kieron PollardLasith Malinga5AB de VilliersChris Morris6Chris GayleAndre Russell7Marcus StoinisMohammad Nabi8Quinton de KockJofra Archer9Chris LynnKeemo Paul10Shane WatsonDwayne Bravo11Faf du PlessisMoeen Ali12Steve SmithSam Curran13Jos ButtlerBen Stokes14Moeen AliSunil Narine15Ben StokesTrent Boult16Kane WilliamsonTable 12
Ranks generated by NR-TOPSIS.
RankBatsmenBowlers1Andre RussellKagiso Rabada2David WarnerImran Tahir3Jonny BairstowRashid Khan4Ab de VilliersChris Morris5Chris GayleMohammad Nabi6Quinton de KockAndre Russell7Kieron PollardJofra Archer8Shane WatsonKeemo Paul9Marcus StoinisLasith Malinga10Chris LynnMoeen Ali11Faf du PlessisDwayne Bravo12Jos ButtlerBen Stokes13Moeen AliTrent Boult14Steve SmithSunil Narine15Ben StokesSam Curran16Kane WilliamsonTable 13
Ranks generated by TODIM.
RankBatsmenBowlers1David WarnerImran Tahir2Chris GayleKagiso Rabada3Quinton de KockLasith Malinga4Andre RussellRashid Khan5AB de VilliersAndre Russell6Jonny BairstowJofra Archer7Shane WatsonDwayne Bravo8Chris LynnSunil Narine9Faf du PlessisMohammad Nabi10Kieron PollardChris Morris11Steve SmithSam Curran12Jos ButtlerMoeen Ali13Moeen AliKeemo Paul14Marcus StoinisBen Stokes15Kane WilliamsonTrent Boult16Ben StokesFrom the tables, we see that David Warner is ranked as the best batsman (according to our criteria) considering the criteria strike rate, batting average, runs, and so forth in TOPSIS and TODIM. Meanwhile, in case of NR-TOPSIS, Andre Russell is the best batsman. For bowlers, Kagiso Rabada and Imran Tahir are the top two choices for all the three approaches based on criteria like strike rate, bowling average, and wickets. However, their relative ranks are swapped in case of TODIM. For the lowest performers (out of the chosen candidates), we find that Kane Williamson and Ben Stokes appear near the end of the list for batsmen, and Trent Boult is the analogue for bowlers. These rankings are relative to the number of candidates chosen. The rankings of the players in the middle of the lists vary slightly. We elaborate on the application of these findings in the Conclusion.The rankings for both batsmen and bowlers are similar enough in the middle of the list as is evident from our Spearman Rank Correlation calculations. It is generally the top ranked players that shuffle their positions greatly across the methods. We tried to minimize the effect of rank reversal using NR-TOPSIS over TOPSIS but failed to see any improvement at all. TODIM performed the best in this regard out of the three methods by producing the least number of rank reversals during the sensitivity analysis.


### 4.6. Spearman Rank Correlation Coefficient Test

The Spearman Rank Correlation Coefficient [57] is a nonparametric measure of the consistency and control of the relationship between two ranking variables. There are several research articles where authors have evaluated various ranking methods, such as ELECTRE and TOPSIS [58], TOPSIS and modified TOPSIS [59], and MOTV algorithms [60]. In our article, we have implemented the approach to study the correlation between TOPSIS and NR-TOPSIS, TOPSIS and TODIM, and TODIM and NR-TOPSIS. The formula used is(20)ρ=1−6∑i=1mdi2nn2−1,−1≤ρ≤1,where ρ is the spearman rank correlation coefficient, di is the difference between the two ranks of each method we used, and n is the number of alternatives or players. The values of the correlation can vary from −1 to +1 which are further categorized as follows:(i)
Ifρ=1, then correlation is absolute(ii)
If0.8≤ρ0.999, then correlation is very strong(iii)
If0.6≤ρ≤0.79, then correlation is strong(iv)
If0.4≤ρ≤0.59, then correlation is moderate(v)
If0.2≤ρ≤0.39, then correlation is weak(vi)
If0.19≥ρ, then correlation is very weakTherefore, the values ofρ we obtained from results in Tables 11–13 are given in Table 14, and as we can see TOPSIS and TODIM are strongly positively correlated, whereas TOPSIS and NR-TOPSIS or TODIM and NR-TOPSIS are very strongly positively correlated.Table 14
Spearman Rank Correlation Coefficient results of TOPSIS, TODIM, and NR-TOPSIS.
TOPSIS and TODIM, TOPSIS and NR-TOPSIS, and TODIM and NR-TOPSISρ for batsmen: 0.7676, 0.9529, and 0.8647ρ for bowlers: 0.8179, 0.9143, and 0.6464


### 4.7. Rank Reversal

Rank reversal is a phenomenon that occurs when a decision-maker is choosing an alternative from a group of options, and they are presented with additional options that were not considered when the selection process began. We proved rank reversal by deleting one and two alternatives from the list of batsmen in Table3 and then performing the ranking. Among the alternatives, we have seen substantial rank reversals. When we compared the three approaches in this study, we discovered that TODIM had the least amount of variation in the ranking. TOPSIS and NR-TOPSIS exhibit a substantial number of reversals. Table 15 demonstrates the comparison when one alternative is removed and Table 16 demonstrates the comparison when two of the alternatives are removed.Table 15
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when one alternative is removed from the original rank list of batsmen.
Removed alternativesRank reversals for NR-TOPSISRemoved alternativesRank reversals for TOPSISRemoved alternativesRank reversals for TODIM23, 4, 5, 625, 6, 7, 8, 9, 10, 1234, 536, 7, 13, 14312, 1394, 544, 567, 8, 12, 13104, 554, 51511, 12124, 5Table 16
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when two alternatives are removed from the original rank list of batsmen.
Removed alternativesRank reversals for NR-TOPSISRemoved alternativesRank reversals for TOPSISRemoved alternativesRank reversals for TODIM6, 16, 7, 11, 123, 15, 6, 12, 133, 14, 57, 25, 6, 7, 8, 9, 10, 116, 41, 2, 3, 4, 5, 66, 52, 35, 311, 1210, 36, 7, 9, 10, 13, 147, 62, 310, 25, 6, 7, 8, 9, 10, 1214, 26, 7, 8, 9, 119, 14, 5In case of NR-TOPSIS, when we removed the batsman who was ranked 2 (Andre Russell) in Table3, we saw a change in ranking of alternatives present in ranks 3 (Ben Stokes), 4 (Chris Gayle), 5 (Chris Gayle), and 6 (David Warner). They are not moving positions by one spot owing to the elimination of one player; rather they are switching positions with one another. NR-TOPSIS was not meant to operate in this way, as it was not supposed to have a rank reversal problem. In case of TOPSIS, however, when we again removed the batsman present in position 2 (Andre Russell), others in positions 5 (Chris Lynn), 6 (David Warner), 7 (Faf du Plessis), 8 (Jonny Bairstow), 9 (Jos Buttler), 10 (Kane Williamson), and 12 (Marcus Stoinis) changed their respective ranks as well. Similar cases were observed when we removed two alternatives. Finally, we noticed that, in TODIM, only two players’ rankings are switched. When one alternative (3, 9, 10, or 12) or two alternatives ((3, 1), (6, 5), (7, 6), or (9, 1)) are removed, the 4th (Chris Gayle) and 5th (Chris Gayle) players, or 2nd (Andre Russell) and 3rd (Ben Stokes), exchanged positions.We added an alternative mentioned in Table17 to the existing list in Table 3 to observe the problem of rank reversal in all the three methods. Because the new alternative was ranked last in all the three methods, we removed one player from the original list in Table 3 and checked how the rank reversal occurs. With TODIM, we discovered an exceptional phenomenon. Only when the player from the 6th position is removed is there an exchange of position between 2nd and 3rd. But, for all other players, when removed, we see no occurrence of rank reversal issue. For the other two methods, namely, TOPSIS and NR-TOPSIS, we see usual occurrence of rank reversal as we have observed previously.Table 17
New batsman to be added.
NameNationalityMatInnNORunsHSAvgBFSR100s50s4s6sShimron HetmyerGuyanese550907518.0073123.280147Table 18
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when one alternative is removed from the new list of batsmen.
Added alternativeRank reversals for NR-TOPSISAdded alternativeRank reversals for TOPSISAdded alternativeRank reversals for TODIM25, 6, 7, 8, 9, 10, 1261, 2, 3, 43No change312, 1377, 89No change67, 8, 12, 1397, 862, 31511, 12115, 6, 12, 1314No change


## 4.1. Criteria Selection

We used the rank-sum weight approach to determine the weights of the criteria specified for batsmen and bowlers, making it easier for decision-makers to grasp and rank the criteria based on their preferences. The criteria for batsmen are described in Table1, and those for bowlers are stated in Table 2. It should be noted that the criteria rankings are adjusted for the T20 format and are subject to change based on the decision-makers’ preferences. The batsmen and bowlers chosen here are randomly selected with no regional preferences from the list given in the two following links: https://www.iplt20.com/stats/2019/most-runs and https://www.iplt20.com/stats/2019/most-wickets. As we can see, our data is collected from a trustworthy and easily available to public source; thus we do not need to implement a fuzzy algorithm here.Table 1
Batting performance criteria.
Batting criteriaSymbolDefinitionRankWeightsIdeallyStrike rateSRSR=runs/BF×10010.1538462HigherBatting averageAvgAvg=runs/outs20.1410256HigherRunsRunsThe total runs scored in the season30.1282051HigherInningsInnThe number of innings played in the season40.1153846HigherNot outsNOThe number of innings undefeated50.1025641Higher6s6sThe number of sixes hit60.0897436Higher4s4sThe number of fours bit70.0769231Higher100s100sThe number of centuries scored80.0641026Higher50s50sThe number of half-centuries scored90.0512821HigherMatchesMatThe number of matches played, might be more than the number of innings played100.0384615HigherHighest scoreHSThe highest individual score110.025641HigherBalls facedBFThe number of balls faced in the tournament120.0128205HigherTable 2
Bowling performance criteria.
Bowling criteriaSymbolDefinitionRankWeightsIdeallyStrike rateSRSR=BB/wkts10.200000LowerEconomyEconEcon=runs/overs=runs/BB×620.177778LowerBowling averageAvgAvg=runs/wkts30.155556LowerWicketsWktsThe number of wickets taken in the season40.133333HigherRunsRunsTotal runs conceded50.111111LowerInningsInnsThe number of innings played60.088889HigherBalls bowledBBTotal number of balls bowled70.066667Higher4 wicket hauls4wFrequency of times 4 wickets were taken80.044444HigherMatchesMatThe number of matched played90.022222HigherThe ranking of criteria is subjective, and the decision-maker is expected to arrange them according to their preferences. Keeping in mind the T20 format, we assume that strike rate is more important than batting average for batsmen courtesy of the limited playtime. Similarly, a good batsman may have scored fewer runs overall but may have helped finish more run chases successfully. In case of bowlers, both strike rate and economy are important. However, bowling out all batsmen ends the game quicker, so strike rate takes priority. This form of reasoning applies to ranking the remaining criteria as well. The rankings we present here are not authoritative, but they are certainly close to what would be used in practice. We illustrate an example that differs from the current context: in One-Day Internationals (ODIs), strike rate may have a lower preference than batting average and total runs scored for batsmen. Similarly, the number of balls faced becomes more important in a test match. Therefore, the decision-makers need to rank the criteria based on the format of the game being played.


## 4.2. Application of TOPSIS

After computing the weights of the criteria, we will rank the alternatives, which are batsmen and bowlers, using TOPSIS whose algorithm is described in the previous section. The evaluation matrix is the same for all algorithms and we will be dealing with the crisp dataset.Step 1: the evaluation matrix for Foreign Batsmen of IPL 2019 (retrieved from the official website mentioned at the beginning of the section) is represented in Table3.The same for bowlers is presented in Table4 and all the data are taken form the website mentioned at the beginning of the section.Step 2: the normalised matrix is calculated.Step 3: the weighted normalised matrix is calculated.Step 4: the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−) are identified.Step 5:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Step 6:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Then, we proceed to rank the candidates according to the distance values. These values for the batsmen and bowlers are provided in Tables5 and 6.Step 7: we choose the candidate with the maximumCi∗ or rank all the alternatives in a descending order according to their Ci∗ values. This process is done for both the Si∗ and Si− values.Table 3
Batsmen statistics.
NameNationalityMatInnNORunsHSAvgBFSR100s50s4s6sAB de VilliersSouth African131334428244.20287154.00053126Andre RussellWest Indian141345108056.67249204.81043152Ben StokesEnglish9931234620.5099124.240084Chris GayleWest Indian131314909940.83319153.60044534Chris LynnAustralian131304058231.15290139.65044122David WarnerSouth African1212269210069.20481143.86185721Faf du PlessisSouth African121213969636.00321123.36033615Jonny BairstowEnglish1010244511455.63283157.24124818Jos ButtlerEnglish8803118938.88205151.70033814Kane WilliamsonNew Zealander9921567022.29130120.0001125Kieron PollardWest Indian161462798334.88178156.74011422Marcus StoinisAustralian101062114652.75156135.25001410Moeen AliEnglish111022206627.50133165.41021617Quinton de KockSouth African161615298135.27398132.91044525Shane WatsonAustralian171703989623.41312127.56034220Steve SmithAustralian121023197339.88275116.0003304Table 4
Bowlers statistics.
NameNationalityMatInnOvBBRunsWktsAvgEconSR4wAndre RussellWest Indian141230.11812871126.099.5116.450Ben StokesEnglish9616.5101189631.511.2316.830Chris MorrisSouth African9933.01983061323.549.2715.230Dwayne BravoWest Indian121241.124733011308.0222.450Imran TahirSouth African171764.23864312616.586.714.852Jofra ArcherEnglish111143.02582911126.456.7723.450Kagiso RabadaSouth African121247.02823682514.727.8311.282Keemo PaulWest Indian8827.1163237926.338.7218.110Lasith MalingaSri Lankan121244.52694381627.389.7716.812Moeen AliEnglish11925.0150169628.176.76250Mohammad NabiAfghan8829.1175194824.256.6521.881Rashid KhanAfghan151560.03603771722.186.2821.180Sam CurranEnglish9933.01983231032.39.7919.81Sunil NarineWest Indian121244.22663471034.77.8326.60Trent BoultNew Zealander5519.0114163532.68.5822.80Table 5
Distance values for batsmen.
BatsmenSi∗Si−Ci∗AB de Villiers0.0701960.0551120.439813Andre Russell0.0568880.0811650.587927Ben Stokes0.1065260.0272940.203959Chris Gayle0.0761690.0551950.420169Chris Lynn0.0902960.0408400.311430David Warner0.0516790.0907780.637231Faf du Plessis0.0888620.0366060.291756Jonny Bairstow0.0628140.0696480.525796Jos Buttler0.0958100.0326340.254072Kane Williamson0.1057480.0191200.153120Kieron Pollard0.0796070.0628850.441323Marcus Stoinis0.0870820.0619240.415581Moeen Ali0.0932270.0293650.239533Quinton de Kock0.0802780.0530220.397761Shane Watson0.0943070.0419460.307855Steve Smith0.0925510.0339530.268397Table 6
Distance values for bowlers.
BowlersSi∗Si−Ci∗Andre Russell0.056820.040470.41596Ben Stokes0.075090.034800.31667Chris Morris0.053260.043990.45232Dwayne Bravo0.062330.033810.35169Imran Tahir0.026770.081950.75376Jofra Archer0.060680.039070.39169Kagiso Rabada0.024790.079580.76251Keemo Paul0.062160.036590.37049Lasith Malinga0.048950.050300.50679Moeen Ali0.073080.037870.34136Mohammad Nabi0.061350.042460.40905Rashid Khan0.047660.055150.53646Sam Curran0.063260.030360.32433Sunil Narine0.072040.030810.29958Trent Boult0.078510.031140.28400


## 4.3. Application of NR-TOPSIS

We will utilize the same evaluation matrix and weights for the criteria as calculated previously. Here,Mj and mj are the maximum and minimum values selected from each column of Tables 3 and 4, respectively:Step 1: the evaluation matrix for Foreign Batsmen of IPL 2019 (obtained from the official website mentioned at the beginning of the section) is represented in Table3 and that for bowlers is presented in Table 4 and all the data are taken form the website mentioned at the beginning of the section.Step 2: we calculate the normalised matrix.Step 3: the weighted normalised matrix is then calculated using a formula. These results are all available in the Supplementary Materials.Step 4: the Positive Ideal Solution (PIS) (A∗) and Negative Ideal Solution (NIS) (A−) are identified.Step 5:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Step 6:Si∗, Si−, and Ci∗ are calculated for all players (batsmen and bowlers).Then we proceed to rank the candidates according to the distance values. These values for the batsmen and bowlers are provided in Tables7 and 8.Table 7
Distance values for batsmen.
BatsmenSi∗Si−Ci∗AB de Villiers0.1743820.1601630.478748Andre Russell0.1167000.2487760.680691Ben Stokes0.3027460.0549200.153552Chris Gayle0.1823000.1644250.474222Chris Lynn0.2263520.1262210.357999David Warner0.1541950.2387260.607567Faf du Plessis0.2378100.1109550.318137Jonny Bairstow0.1755570.1794800.505525Jos Buttler0.2451610.1080040.305817Kane Williamson0.3003920.0406030.119072Kieron Pollard0.2025680.1643770.447960Marcus Stoinis0.2360550.1472920.384226Moeen Ali0.2416910.1057190.304306Quinton de Kock0.2045390.1699530.453822Shane Watson0.2444310.1537020.386057Steve Smith0.2576820.0944430.268208Table 8
Distance values for bowlers.
BowlersSi∗Si−Ci∗Andre Russell0.2076210.1853580.471674Ben Stokes0.2888520.1648040.363279Chris Morris0.1915660.2034920.515094Dwayne Bravo0.2396560.1579350.39723Imran Tahir0.1197270.3201590.727823Jofra Archer0.2249610.1997510.470322Kagiso Rabada0.1097680.3204030.744827Keemo Paul0.2139230.1806450.457831Lasith Malinga0.2218580.1832650.452369Moeen Ali0.2610220.20440.439172Mohammad Nabi0.2137790.2210730.508387Rashid Khan0.1818350.249050.577996Sam Curran0.2601040.1265410.327279Sunil Narine0.2955740.1470270.332188Trent Boult0.2846330.1553750.353119Andre Russell0.2076210.1853580.471674


## 4.4. Application of TODIM

In this part, we will be evaluating the rank generated using TODIM. We have considered the value of attenuation factorθ as 1. However, decision-makers can change the value accordingly. Changing this value might affect the final ranking and is therefore discussed in the Sensitivity Analysis section.Step 1: we can determine the trade-off rate for batsmen and bowlers using the list of weights we calculated before. They are presented in Table9.Step 2: we calculate the matrixφcAi,Aj and then the matrix δAi,Aj for θ=1.Step 3: then, we compute the values ofξi. The values of ξi are calculated in Table 10.Table 9
Table of updated weights.
Batting criteriawrcBowling criteriawrcSR1SR1Avg0.916667Econ0.888889Runs0.833333Avg0.777778Inn0.75Wkts0.666667NO0.666667Runs0.5555566s0.583333Inns0.4444444s0.5BB0.333333100s0.4166674w0.22222250s0.333333Mat0.111111Mat0.25HS0.166667Table 10
ξi for batsmen and bowlers.
BatsmenξiBowlersξiAB de Villiers0.804597Andre Russell0.561531Andre Russell0.818468Ben Stokes0.155654Ben Stokes0Chris Morris0.426346Chris Gayle0.848149Dwayne Bravo0.524013Chris Lynn0.687784Imran Tahir1David Warner1Jofra Archer0.525132Faf du Plessis0.667418Kagiso Rabada0.91773Jonny Bairstow0.777412Keemo Paul0.279925Jos Buttler0.427361Lasith Malinga0.738363Kane Williamson0.130708Moeen Ali0.374627Kieron Pollard0.552321Mohammad Nabi0.428831Marcus Stoinis0.247835Rashid Khan0.719623Moeen Ali0.320078Sam Curran0.398341Quinton de Kock0.823886Sunil Narine0.472208Shane Watson0.703334Trent Boult0Steve Smith0.508533


## 4.5. Results

The rankings generated by TOPSIS for batsmen and bowlers, who are international players in the IPL 2019, are tabulated in a descending order of preference and summarised in Table11. Similarly, the ranks generated by NR-TOPSIS and TODIM are summarised in Tables 12 and 13, respectively.Table 11
Ranks generated by TOPSIS.
RankBatsmenBowlers1David WarnerKagiso Rabada2Andre RussellImran Tahir3Jonny BairstowRashid Khan4Kieron PollardLasith Malinga5AB de VilliersChris Morris6Chris GayleAndre Russell7Marcus StoinisMohammad Nabi8Quinton de KockJofra Archer9Chris LynnKeemo Paul10Shane WatsonDwayne Bravo11Faf du PlessisMoeen Ali12Steve SmithSam Curran13Jos ButtlerBen Stokes14Moeen AliSunil Narine15Ben StokesTrent Boult16Kane WilliamsonTable 12
Ranks generated by NR-TOPSIS.
RankBatsmenBowlers1Andre RussellKagiso Rabada2David WarnerImran Tahir3Jonny BairstowRashid Khan4Ab de VilliersChris Morris5Chris GayleMohammad Nabi6Quinton de KockAndre Russell7Kieron PollardJofra Archer8Shane WatsonKeemo Paul9Marcus StoinisLasith Malinga10Chris LynnMoeen Ali11Faf du PlessisDwayne Bravo12Jos ButtlerBen Stokes13Moeen AliTrent Boult14Steve SmithSunil Narine15Ben StokesSam Curran16Kane WilliamsonTable 13
Ranks generated by TODIM.
RankBatsmenBowlers1David WarnerImran Tahir2Chris GayleKagiso Rabada3Quinton de KockLasith Malinga4Andre RussellRashid Khan5AB de VilliersAndre Russell6Jonny BairstowJofra Archer7Shane WatsonDwayne Bravo8Chris LynnSunil Narine9Faf du PlessisMohammad Nabi10Kieron PollardChris Morris11Steve SmithSam Curran12Jos ButtlerMoeen Ali13Moeen AliKeemo Paul14Marcus StoinisBen Stokes15Kane WilliamsonTrent Boult16Ben StokesFrom the tables, we see that David Warner is ranked as the best batsman (according to our criteria) considering the criteria strike rate, batting average, runs, and so forth in TOPSIS and TODIM. Meanwhile, in case of NR-TOPSIS, Andre Russell is the best batsman. For bowlers, Kagiso Rabada and Imran Tahir are the top two choices for all the three approaches based on criteria like strike rate, bowling average, and wickets. However, their relative ranks are swapped in case of TODIM. For the lowest performers (out of the chosen candidates), we find that Kane Williamson and Ben Stokes appear near the end of the list for batsmen, and Trent Boult is the analogue for bowlers. These rankings are relative to the number of candidates chosen. The rankings of the players in the middle of the lists vary slightly. We elaborate on the application of these findings in the Conclusion.The rankings for both batsmen and bowlers are similar enough in the middle of the list as is evident from our Spearman Rank Correlation calculations. It is generally the top ranked players that shuffle their positions greatly across the methods. We tried to minimize the effect of rank reversal using NR-TOPSIS over TOPSIS but failed to see any improvement at all. TODIM performed the best in this regard out of the three methods by producing the least number of rank reversals during the sensitivity analysis.


## 4.6. Spearman Rank Correlation Coefficient Test

The Spearman Rank Correlation Coefficient [57] is a nonparametric measure of the consistency and control of the relationship between two ranking variables. There are several research articles where authors have evaluated various ranking methods, such as ELECTRE and TOPSIS [58], TOPSIS and modified TOPSIS [59], and MOTV algorithms [60]. In our article, we have implemented the approach to study the correlation between TOPSIS and NR-TOPSIS, TOPSIS and TODIM, and TODIM and NR-TOPSIS. The formula used is(20)ρ=1−6∑i=1mdi2nn2−1,−1≤ρ≤1,where ρ is the spearman rank correlation coefficient, di is the difference between the two ranks of each method we used, and n is the number of alternatives or players. The values of the correlation can vary from −1 to +1 which are further categorized as follows:(i)
Ifρ=1, then correlation is absolute(ii)
If0.8≤ρ0.999, then correlation is very strong(iii)
If0.6≤ρ≤0.79, then correlation is strong(iv)
If0.4≤ρ≤0.59, then correlation is moderate(v)
If0.2≤ρ≤0.39, then correlation is weak(vi)
If0.19≥ρ, then correlation is very weakTherefore, the values ofρ we obtained from results in Tables 11–13 are given in Table 14, and as we can see TOPSIS and TODIM are strongly positively correlated, whereas TOPSIS and NR-TOPSIS or TODIM and NR-TOPSIS are very strongly positively correlated.Table 14
Spearman Rank Correlation Coefficient results of TOPSIS, TODIM, and NR-TOPSIS.
TOPSIS and TODIM, TOPSIS and NR-TOPSIS, and TODIM and NR-TOPSISρ for batsmen: 0.7676, 0.9529, and 0.8647ρ for bowlers: 0.8179, 0.9143, and 0.6464


## 4.7. Rank Reversal

Rank reversal is a phenomenon that occurs when a decision-maker is choosing an alternative from a group of options, and they are presented with additional options that were not considered when the selection process began. We proved rank reversal by deleting one and two alternatives from the list of batsmen in Table3 and then performing the ranking. Among the alternatives, we have seen substantial rank reversals. When we compared the three approaches in this study, we discovered that TODIM had the least amount of variation in the ranking. TOPSIS and NR-TOPSIS exhibit a substantial number of reversals. Table 15 demonstrates the comparison when one alternative is removed and Table 16 demonstrates the comparison when two of the alternatives are removed.Table 15
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when one alternative is removed from the original rank list of batsmen.
Removed alternativesRank reversals for NR-TOPSISRemoved alternativesRank reversals for TOPSISRemoved alternativesRank reversals for TODIM23, 4, 5, 625, 6, 7, 8, 9, 10, 1234, 536, 7, 13, 14312, 1394, 544, 567, 8, 12, 13104, 554, 51511, 12124, 5Table 16
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when two alternatives are removed from the original rank list of batsmen.
Removed alternativesRank reversals for NR-TOPSISRemoved alternativesRank reversals for TOPSISRemoved alternativesRank reversals for TODIM6, 16, 7, 11, 123, 15, 6, 12, 133, 14, 57, 25, 6, 7, 8, 9, 10, 116, 41, 2, 3, 4, 5, 66, 52, 35, 311, 1210, 36, 7, 9, 10, 13, 147, 62, 310, 25, 6, 7, 8, 9, 10, 1214, 26, 7, 8, 9, 119, 14, 5In case of NR-TOPSIS, when we removed the batsman who was ranked 2 (Andre Russell) in Table3, we saw a change in ranking of alternatives present in ranks 3 (Ben Stokes), 4 (Chris Gayle), 5 (Chris Gayle), and 6 (David Warner). They are not moving positions by one spot owing to the elimination of one player; rather they are switching positions with one another. NR-TOPSIS was not meant to operate in this way, as it was not supposed to have a rank reversal problem. In case of TOPSIS, however, when we again removed the batsman present in position 2 (Andre Russell), others in positions 5 (Chris Lynn), 6 (David Warner), 7 (Faf du Plessis), 8 (Jonny Bairstow), 9 (Jos Buttler), 10 (Kane Williamson), and 12 (Marcus Stoinis) changed their respective ranks as well. Similar cases were observed when we removed two alternatives. Finally, we noticed that, in TODIM, only two players’ rankings are switched. When one alternative (3, 9, 10, or 12) or two alternatives ((3, 1), (6, 5), (7, 6), or (9, 1)) are removed, the 4th (Chris Gayle) and 5th (Chris Gayle) players, or 2nd (Andre Russell) and 3rd (Ben Stokes), exchanged positions.We added an alternative mentioned in Table17 to the existing list in Table 3 to observe the problem of rank reversal in all the three methods. Because the new alternative was ranked last in all the three methods, we removed one player from the original list in Table 3 and checked how the rank reversal occurs. With TODIM, we discovered an exceptional phenomenon. Only when the player from the 6th position is removed is there an exchange of position between 2nd and 3rd. But, for all other players, when removed, we see no occurrence of rank reversal issue. For the other two methods, namely, TOPSIS and NR-TOPSIS, we see usual occurrence of rank reversal as we have observed previously.Table 17
New batsman to be added.
NameNationalityMatInnNORunsHSAvgBFSR100s50s4s6sShimron HetmyerGuyanese550907518.0073123.280147Table 18
Comparison of rank reversal results of NR-TOPSIS, TOPSIS, and TODIM when one alternative is removed from the new list of batsmen.
Added alternativeRank reversals for NR-TOPSISAdded alternativeRank reversals for TOPSISAdded alternativeRank reversals for TODIM25, 6, 7, 8, 9, 10, 1261, 2, 3, 43No change312, 1377, 89No change67, 8, 12, 1397, 862, 31511, 12115, 6, 12, 1314No change


## 5. Sensitivity Analysis

The impact and stability of the weights (for both batting and bowling criteria) produced by rank-sum weight method are revealed in the sensitivity analysis for TOPSIS and TODIM. We also tweaked the TODIM attenuation factor to see how sensitive the system is for the purpose of selecting international players. For our analysis, we altered the weights for each criterion by shuffling the rank of the criteria. Because decision-makers can select how the criteria are ranked, we looked at how weight changes can affect TOPSIS and TODIM rankings. The sensitivity of the approaches is displayed using the heat map representation method. The ranks that are seen more frequently in this scheme have a darker colour than the ranks that are seen less often, which have a lighter colour. In this paper, the darkest hue is black, and the lightest colour is white.


### 5.1. Criteria Shuffling or Weight Shuffling for TOPSIS

As there are twelve criteria for batsmen and nine criteria for bowlers, we end up with12! permutations for batting performance criteria and 9! for bowling criteria. In order to see how changing the ranking affects the ranking of alternatives, we shuffle the rankings of the criteria (and hence their weights). We uniformly selected 1000 random shuffles for the weights using a Monte Carlo approach. The ranking of international players using TOPSIS is represented using heat map style point graphs in Figures 3 and 4.Figure 3
Heat map for batsmen ranks generated by TOPSIS.Figure 4
Heat map for bowler ranks generated by TOPSIS.The darker (black) dots indicate those ranks which are more frequent in the1000 samples chosen. Grey dots are documented, though they appear less frequently. The ranks that all the players have had for various combinations of rank of the criterion or various values of the weight may be seen on this point graph. For David Warner (batsman), for example, in Figure 3, we may observe two points. The colour of one point differs from that of the other one. The darker dot represents the rank that appeared more frequently than the other. As a result, regardless of changes in weights or criteria, we can show that he has consistently ranked first in most cases. We may also say the same thing about Imran Tahir (bowler) in Figure 4. Kieron Pollard (batsman) has had a rank of 2–13, with 4 being the most common because it is the darkest of all the dots and 2 being the least often because it is the lightest. Even for Mohammad Nabi (bowler) his rank varies between 4 and 12, with the darkest point at rank 5. Players like Lasith Malinga (bowler) and Rashid Khan (bowler) or Andre Russell (batsman) and Ben Stokes (batsman) have a ranking plot with two equal dark spots, indicating that both rankings occur frequently.


### 5.2. Criteria Shuffling or Weight Shuffling for NR-TOPSIS

In the instance of NR-TOPSIS, we used the same definition as in TOPSIS, but the outcome is different. Figure5 represents the sensitivity analysis of NR-TOPSIS for batsmen when the weights are shuffled. With Ben Stokes, Kane Williamson, and David Warner, we observe somewhat consistent results, with their rankings not deviating significantly with the change in weight. In case of TOPSIS, we saw that these players had more grey or lighter coloured dots surrounding the black ones, indicating that they had those ranks for a specific weight value. Other rankings of batsmen like Kieron Pollard, Jonny Bairstow, Shane Watson, and Marcus Stoinis are highly unstable, similar to TOPSIS. There is not a lot of consistency to be seen.Figure 5
Heat map for batsmen ranks generated by NR-TOPSIS.In Figure6, we see the sensitivity analysis of bowlers. The stability is quite feeble overall. Only Kagiso Rabada, Imran Tahir, and Rashid Khan have a reasonable level of consistency. When the weights are shuffled or altered, all the other bowlers exhibit a considerable deal of inconstancy in their rankings. Even though the dark-coloured dots most of the time reflect the incidence of that rank when the weights are shuffled in the player’s direction, the existence of comparable black or lighter coloured dots shows that such positions were likewise achieved for certain sets of criterion weights.Figure 6
Heat map for bowlers ranks generated by NR-TOPSIS.


### 5.3. Criteria Shuffling or Weight Shuffling for TODIM

We repeat the same procedure for generating a heat map of the rankings generated by TODIM when the weights are shuffled. The results are shown in Figures7 and 8.Figure 7
Heat map for batsmen ranks by TODIM through shuffled weights.Figure 8
Heat map for bowler ranks by TODIM through shuffled weights.Unlike TOPSIS, where players like Chris Morris (bowler) and Kieron Pollard (batsman) have a wide range of ranks that shift dramatically with weight changes, the TODIM rankings are stable ranks differing by four positions at most. Even for Sam Curran (bowler), who had the highest rating range of 6–12, the darkest point is at rank 8. We can also notice that several alternatives exhibit stable behaviour, such as the first, second, or last and second last rankers of bowlers and batsmen, who keep their rank regardless of the weight or criteria ranking.


### 5.4. Change in Attenuation Factor (θ) for TODIM

The sensitivity analysis studied from [34] is formulated by taking six randomly selected different values of attenuation factor. In our case, we observe the effects of change in the attenuation factor when it is varied from 1 to 101 with increments of 0.1, providing us around 1000 observations. If the value of θ is too large, the penalty for being inferior in a category becomes insignificant. The results are graphed in Figures 9 and 10.Figure 9
Heat map for batsmen ranks by TODIM through varyingθ.Figure 10
Heat map for bowlers ranks by TODIM through varyingθ.We can see the ranks that all the players have had for various values ofθ, the attenuation factor. David Warner (batsman) and Imran Tahir (bowler), for example, had rank 1 for all values of θ. As a result, there are no grey dots areas for them. Pollard (batsman), on the other hand, had a rank system that ranges from 7 to 9, with 7 being the most common and 9 being the least common. Similarly, for all values of θ, Ben Stokes (batsman) and Kane Williamson (batsman) have been last or second last.


## 5.1. Criteria Shuffling or Weight Shuffling for TOPSIS

As there are twelve criteria for batsmen and nine criteria for bowlers, we end up with12! permutations for batting performance criteria and 9! for bowling criteria. In order to see how changing the ranking affects the ranking of alternatives, we shuffle the rankings of the criteria (and hence their weights). We uniformly selected 1000 random shuffles for the weights using a Monte Carlo approach. The ranking of international players using TOPSIS is represented using heat map style point graphs in Figures 3 and 4.Figure 3
Heat map for batsmen ranks generated by TOPSIS.Figure 4
Heat map for bowler ranks generated by TOPSIS.The darker (black) dots indicate those ranks which are more frequent in the1000 samples chosen. Grey dots are documented, though they appear less frequently. The ranks that all the players have had for various combinations of rank of the criterion or various values of the weight may be seen on this point graph. For David Warner (batsman), for example, in Figure 3, we may observe two points. The colour of one point differs from that of the other one. The darker dot represents the rank that appeared more frequently than the other. As a result, regardless of changes in weights or criteria, we can show that he has consistently ranked first in most cases. We may also say the same thing about Imran Tahir (bowler) in Figure 4. Kieron Pollard (batsman) has had a rank of 2–13, with 4 being the most common because it is the darkest of all the dots and 2 being the least often because it is the lightest. Even for Mohammad Nabi (bowler) his rank varies between 4 and 12, with the darkest point at rank 5. Players like Lasith Malinga (bowler) and Rashid Khan (bowler) or Andre Russell (batsman) and Ben Stokes (batsman) have a ranking plot with two equal dark spots, indicating that both rankings occur frequently.


## 5.2. Criteria Shuffling or Weight Shuffling for NR-TOPSIS

In the instance of NR-TOPSIS, we used the same definition as in TOPSIS, but the outcome is different. Figure5 represents the sensitivity analysis of NR-TOPSIS for batsmen when the weights are shuffled. With Ben Stokes, Kane Williamson, and David Warner, we observe somewhat consistent results, with their rankings not deviating significantly with the change in weight. In case of TOPSIS, we saw that these players had more grey or lighter coloured dots surrounding the black ones, indicating that they had those ranks for a specific weight value. Other rankings of batsmen like Kieron Pollard, Jonny Bairstow, Shane Watson, and Marcus Stoinis are highly unstable, similar to TOPSIS. There is not a lot of consistency to be seen.Figure 5
Heat map for batsmen ranks generated by NR-TOPSIS.In Figure6, we see the sensitivity analysis of bowlers. The stability is quite feeble overall. Only Kagiso Rabada, Imran Tahir, and Rashid Khan have a reasonable level of consistency. When the weights are shuffled or altered, all the other bowlers exhibit a considerable deal of inconstancy in their rankings. Even though the dark-coloured dots most of the time reflect the incidence of that rank when the weights are shuffled in the player’s direction, the existence of comparable black or lighter coloured dots shows that such positions were likewise achieved for certain sets of criterion weights.Figure 6
Heat map for bowlers ranks generated by NR-TOPSIS.


## 5.3. Criteria Shuffling or Weight Shuffling for TODIM

We repeat the same procedure for generating a heat map of the rankings generated by TODIM when the weights are shuffled. The results are shown in Figures7 and 8.Figure 7
Heat map for batsmen ranks by TODIM through shuffled weights.Figure 8
Heat map for bowler ranks by TODIM through shuffled weights.Unlike TOPSIS, where players like Chris Morris (bowler) and Kieron Pollard (batsman) have a wide range of ranks that shift dramatically with weight changes, the TODIM rankings are stable ranks differing by four positions at most. Even for Sam Curran (bowler), who had the highest rating range of 6–12, the darkest point is at rank 8. We can also notice that several alternatives exhibit stable behaviour, such as the first, second, or last and second last rankers of bowlers and batsmen, who keep their rank regardless of the weight or criteria ranking.


## 5.4. Change in Attenuation Factor (θ) for TODIM

The sensitivity analysis studied from [34] is formulated by taking six randomly selected different values of attenuation factor. In our case, we observe the effects of change in the attenuation factor when it is varied from 1 to 101 with increments of 0.1, providing us around 1000 observations. If the value of θ is too large, the penalty for being inferior in a category becomes insignificant. The results are graphed in Figures 9 and 10.Figure 9
Heat map for batsmen ranks by TODIM through varyingθ.Figure 10
Heat map for bowlers ranks by TODIM through varyingθ.We can see the ranks that all the players have had for various values ofθ, the attenuation factor. David Warner (batsman) and Imran Tahir (bowler), for example, had rank 1 for all values of θ. As a result, there are no grey dots areas for them. Pollard (batsman), on the other hand, had a rank system that ranges from 7 to 9, with 7 being the most common and 9 being the least common. Similarly, for all values of θ, Ben Stokes (batsman) and Kane Williamson (batsman) have been last or second last.


## 6. Discussion and Future Scope

We do not need to apply our models on a fuzzy environment here because our data is obtained from a reliable and publicly accessible source. When it came to choosing the players, no geographical preferences were made. This is one of the reasons why we chose international players rather than Indian ones. TOPSIS and TODIM were used in this research to analyse the performance of international players during the selection of persons through auction in the IPL, and we could obtain conclusive results. But rank reversal is an issue with both methods. As a result, we implemented NR-TOPSIS [52] to address the problem, which is a modified version of TOPSIS. We did, however, run across a rank reversal issue with NR-TOPSIS. One of the probable reasons might be the selection of weighting method. Although NR-TOPSIS and TODIM are strongly correlated, there is no advantage of implementing NR-TOPSIS as it is more unstable than TODIM. Hence, for ranking, we have proven that TODIM is better than both TOPSIS and NR-TOPSIS. We chose rank-sum weight method for our models instead of implementing our own randomly selected values. We present the differences between the two basic methods TOPSIS and TODIM in Table 19.Table 19
Difference between TOPSIS and TODIM.
TOPSISTODIM1.The essential premise is that the chosen alternative should be closest to the Positive Ideal Solution and farthest from the Negative Ideal Solution. The metric most used is Euclidean distance.The chosen alternative should have better scores for most criteria against all alternatives. Having a lesser score is penalised by an amount determined byθ.2.It has a lower time complexity for comparison,Omn, where m is the number of criteria and n is the number of candidates.It has a higher time complexity for comparison,Omn2, where m is the number of criteria and n is the number of candidates.In this article, our findings proved that, whether players were added or withdrawn, TODIM saw a significantly lower change in rank. After sensitivity analysis of the rankings, we observed that the rankings generated by TOPSIS and NR-TOPSIS are both more susceptible to change of weights than the rankings generated by TODIM. In article [34], it is stated that the final rankings from TOPSIS will vary if any single weight varies in proportion by ±20% to ±50%. Since we are only allowing decision-makers to set the rankings of the judgement criteria and using these rankings to calculate criteria weights, the changes in proportion are larger than 20%, thereby affecting the final rankings and making the outcome unstable. In case of TODIM, the rankings of the criteria are not as important because the final ranks for the candidates change little. The decision-makers can choose any parameter values they like, and the TODIM model will give stable rankings anyway. Sensitivity analysis done for a range of values of attenuator factor θ and weight shuffling for criteria proved the stability of TODIM. Therefore, this method is recommended over TOPSIS and NR-TOPSIS for any level of decision-makers, from experts to less experienced. The ranks for the best and worst performers remain relatively the same for both approaches, with the candidates in the middle hovering near their general regions.Amos Tversky and Daniel Kahneman proposed cumulative prospect theory (CPT) in 1992 [61] as a model for describing judgments under risk and uncertainty. The main difference between prospect theory (PT) and cumulative prospect theory is that, rather than the probabilities themselves, the cumulative probabilities are transformed in CPT. Thereby, CPT removes the limitation of PT which violates first-order stochastic dominance. We did not use CPT-TODIM in our paper since we wanted to demonstrate the superiority of TODIM over TOPSIS for rank alternatives or IPL foreign player performance analysis. We have therefore demonstrated that TODIM is better than TOPSIS in terms of sensitivity and accuracy without having to use any modified version of TODIM. However, the following questions may arise: What if the source we used is deemed untrustworthy by people? What if a circumstance arises in which player performance must be analysed based on more than simply their scores? In such scenario, scholars may want to employ CPT-TODIM in a fuzzy environment. Researchers may also want to study the risk of the economic side of the IPL auction, and in such cases as well CPT- TODIM in a fuzzy environment can be used. Some articles [40, 62, 63] presented implementation of fuzzy CPT-TODIM for diverse decision-making applications in recent studies. Future research can be conducted to compare the findings of TODIM with those of CPT-TODIM or in a fuzzy environment for the IPL dataset. DEA is another quantitative tool that can be used for foreign player ranking in the Indian Premier League. In 1978, Charnes, Cooper, and Rhodes proposed Data Envelopment Analysis (DEA) as a performance evaluation method. It is sometimes known as frontier analysis. It is a strategy for assessing the relative efficiency of decision-making units (DMUs) in organizations. A DMU is a discrete unit inside an organization that has flexibility in some of its decisions but not total freedom in others. In the case of IPL, runs per innings, boundaries per innings, and so on can be commonly used to determine efficiency. The relative efficiencies are then determined using the highest ratio as a reference. Future research might use DEA [64–66] to assess the player’s efficiency in Indian Premier League.We did not pick any additional MCDM techniques for our comparative analysis because of practicality and ease of comprehending the approaches. We proved through this research work that TODIM is superior to TOPSIS and its variations. The simplicity and high usage of the algorithm are the reasons why TOPSIS was chosen over other MCDM approaches in this study. TODIM was chosen because it is the only approach based on prospect theory, the result of which the decision-makers can use to auction overseas players. We have therefore proved in our article that TODIM is better than TOPSIS and NR-TOPSIS through the sensitivity analysis. The ease with which it can be computed in challenging scenarios and its low sensitivity are some other reasons for selecting this method.We had also attempted to alter TODIM by applying the same modification used in NR-TOPSIS, that is, a change in the normalization procedure formulation. However, no meaningful results were obtained, and the rank reversal problem persisted. In fact, the modified TODIM was less stable than the original TODIM and was more sensitive to changes in weight or attenuation factor. Rank reversal may occur in TODIM, as shown in our work; as a future scope, it should be explored further, so that the problem may be eradicated from the approach and the application can be applied on a wide scale. Further, CPT-TODIM and modified variants of TODIM can be applied to the IPL dataset in the future to compare international player rankings for the IPL auction. Lastly, all the player statistics were obtained from the official website (http://www.iplt20.com). We did not take into consideration the statistics from IPL 2020 or IPL 2021 because the matches were played during the COVID-19 pandemic, which could have affected the players’ general confidence and mental health, preventing them from performing as well as expected. 2019 was the latest year unaffected by the effects of the pandemic and, consequently, our year of choice.


## 7. Conclusion

IPL cricket players’ salaries are determined through an auction process. The same rule applies to foreign players as well. As a result, the team owners must make a judgement based on the performance of the players in previous matches of IPL to determine which player to bid for and at what price. So, the models proposed in this paper might assist a franchisee in selecting the right players available. The IPL management releases a list of international players who may draw a little attention for a maximum number of slots available during the auction. For example, according to one article [67], the IPL management had released a list of 292 international players for sixty-one places available for the 2021 auction. We ranked players in this paper based only on statistics from the 2019 season of IPL as an example. During the real-life player enlistment process, decision-makers can change these criteria at their will.David Warner and Andre Russell have a shared first rank among batsmen, while Imran Tahir and Kagiso share the top spot among bowlers, according to our TOPSIS, NR-TOPSIS, and TODIM approaches. Hence, these players can establish higher starting prices during the IPL auction. Since the team owners may need to work on several lists of players simultaneously, our recommended approach TODIM is basic and straightforward to execute. To compute the value of the weights, we use the rank-sum weight approach, which comprises rating the criteria. As a result, it is up to the decision-makers to prioritize the criteria. In this paper, we performed sensitivity analysis on our algorithms to examine how the rank may vary owing to weight changes. TOPSIS and NR-TOPSIS are definitely affected by this weighting scheme, and the subjective preferences of the decision-maker are reflected properly. However, TODIM allows retention of subjectivity along with improved stability. If the decision-makers cannot rank the criteria in order of importance, they can give the same rank to conflicting criteria. Even when the attenuation factor or weight values were changed, the model remained quite stable. The fact that TODIM’s higher time complexity might be viewed as a detractor is overshadowed by the accuracy of the method. As TODIM was never used before to rank players in IPL or any sports, we have demonstrated how this approach can be useful. We also looked at the differences between two well-known MCDM techniques (and a modification) and how their behaviour affects the ranking of alternatives. It is therefore recommended that decision-makers should select TODIM over TOPSIS and NR-TOPSIS based on their requirements.However, while utilising NR-TOPSIS to solve the heavy rank reversal problem of TOPSIS, we observed that the problem still exists. This showed TODIM’s superiority, making it the preferred technique for ranking players. We have included the Python code for TOPSIS, NR-TOPSIS, and TODIM in the Supplementary Materials section of this paper. Therefore, researchers may use the comparative analysis for additional scenarios in the future, not simply rating players, to aid in practicality.



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*Source: 9986137-2022-04-10.xml*