CombinAtive distAnCe bAsed Assessment (CodAs) FrAmework using LogArithmiC normALizAtion For muLti-CriteriA deCision mAking

The purpose of this paper is to present an extended Combinative Distance based Assessment (CODAS) framework using logarithmic normalization (LN) scheme. LN is useful in the situations where criteria values differ significantly. This framework is used to carry out a comparative performance based ranking of the popular smartphones in India. The result obtained from this extended version of CODAS method (CODAS-LN) shows consistency with that generated by using some other established multi-criteria decision making (MCDM) approaches. The sensitivity analysis shows considerable stability in the result. Further, it is observed that CODAS-LN is free from rank reversal phenomenon and follows the transitivity property. Findings of the case study suggest that the smartphones with higher computational capability and features rank in top brackets

various kinds of emergent issues. These algorithms or techniques differ from each other in terms of the alternatives (based on the nature of the problems), features or attributes (also known as criteria) and their weights or priorities, and computational logic (Jahan & Edwards, 2015). The attributes or features (also known as criteria) play a central role in every MCDM framework. However, these criteria are of different types, scales, and measurement units and optimization directions such as maximization or minimization. The objective of normalization is to bring various criteria into a non-dimensional form for comparing the alternatives (Jassbi et al., 2014;Zolfani et al., 2020). Thus, the choice of appropriate normalization scheme is of paramount importance for any MCDM framework as it posits notable variations in the outcomes and subsequently impacts the decision-making (Pavličić, 2001;Chatterjee & Chakraborty, 2014;Jahan & Edwards, 2015;Precup et al., 2020).
The literature shows shreds of evidence of several schemes that have been formulated by the researchers for normalization. For example, some of the normalization techniques are vector normalization, linear normalization, non-monotonic normalization, Weitendorf's linear normalization (WLN) method, the Jüttler-Körth normalization (JKN) method and the Peldschus non-linear normalization (NLN) method (Eftekhary et al., 2012;Zavadskas & Turskis 2008;Zavadskas et al., 2006). In addition to these basic normalization schemes, authors have also attempted to bring in new methods. Dehghan-Manshadi et al. (2007) introduced a novel non-linear scheme for normalization, which follows the weighting factor approach and is based on a modified digital logic. In 2008, Zavadskas and his collaborator proposed a new LN (Zavadskas & Turskis, 2008). LN is a useful transformation for normalizing a significantly skewed data which finds its wide applications in data analysis (Changyong et al., 2014). Moving further, Sarraf et al. (2013) tested the efficacy of the technique for order of preference by similarity to ideal solution (TOPSIS) method by using statistical normalization. In their research, Jahan and Edwards (2015) highlighted many normalization methods. In this context, one question has been alluring the researchers: which one is the best normalization scheme? While there is no consensus, several researchers have tried to figure out the impact of change in the normalization approach on the final result obtained by using an MCDM technique. For instance, Çelen (2014) applied the TOPSIS method for comparing the financial performance of selected Turkish banks, wherein the author considered four normalization techniques. The author observed that vector normalization gives better results, while max-min and max methods are close alternatives. For the problem of industrial robot selection using a weighted aggregated sum product assessment method (WASPAS) framework, Mathew et al. (2017) found linear normalization (max-min) as the best normalization scheme. In this regard, Vafaei et al. (2018) used six normalization methods for the TOPSIS framework. However, none of these works used LN as a normalization scheme. Kosareva et al. (2018) mentioned that the linear min-max method performs comparatively better than its counterparts, but LN is particularly useful than the others in some specific cases. In tune with the work of Kosareva et al. (2018) andZolfani et al. (2020) used LN as a normalization scheme for TOPSIS and VIKOR (Vise Kriterijumska Optimizacija Kompromisno Resenje) algorithms as they found the usefulness of LN in the situations where criteria values differ significantly.
In this context, this work uses LN as a normalization scheme for applying a popular distance-based MCDM algorithm, such as CODAS, in solving the smartphone selection problem. This work contributes to the growing literature by providing an alternative approach for CODAS using LN. With the limited search it may be concluded that LN has not been used for the CODAS method. In fact, in the literature, a rare use of LN is observed. Given the relevance of the problem of smartphone selection, it is found that the CODAS-LN framework is useful as the criteria for selecting a smartphone given a price bracket (suitable for the middleincome group) differ significantly from each other's in terms of nature, values, and measurement units.
The rest of this paper is organized as follows. In the next section (section 2), the proposed framework is elaborated. In section 3, the problem considered here is discussed. Section 4 exhibits the results and includes a discussion on the findings. Finally, section 5 concludes the paper and highlights some of the future scopes.

ProPosed FrAmework
The CODAS method considers the relative importance of separating each possible solution from the positive ideal or optimistic, and negative ideal or pessimistic points. The fundamental philosophy of the CODAS method considers a combination of two distance measures, such as Euclidean (primary measure related to l 2 -norm indifference space) and Taxicab (secondary measure related to l 1 -norm indifference space) for comparing the alternatives (Ghorabaee et al., 2016). A threshold value is used to combine the distance measures mentioned earlier. The decision rule is dependent on the distance from the extreme negative solution (the higher is, the better). Since its introduction the CODAS framework has been used considerably by several researchers in various domains like supplier selection (Ghorabaee et al., 2017), maintenance management (Panchal et al., 2017), organizational performance assessment (Badi et al., 2018a), location selection problem (Badi et al., 2018b;Bolturk & Kahraman, 2018), comparison of energy storage technologies (Ren, 2018), personnel selection (Tuş & Adalı, 2018;Yeni & Özçelik, 2019), renewable energy selection (Boltürk & Karaşan, 2018), investment decision-making (Seker, 2019), material selection (Maghsoodi et al., 2019), evaluation of banking performance (Laha & Biswas, 2019), and strategic decisionmaking for financial management (Despic et al., 2019;Zhou et al., 2020).
There has been a gradual extension of the original framework. For example, Ghorabaee et al. (2017) extended the CODAS framework by incorporating the fuzzy logic theory. Panchal et al. (2017) used a combined fuzzy analytic hierarchy process (AHP) and CODAS framework in a group decision-making setup. Ren (2018) applied an integrated interval AHP and intuitionistic fuzzy CODAS framework to contribute to the state of the art. Bolturk and Kahraman (2018) further extended the work by using the interval-Valued Intuitionistic Fuzzy logic. Moving further, Boltürk and Karaşan (2018) introduced the neutrosophic fuzzy logic-based CODAS framework. The work of Yeni and Özçelik (2019) reported the use of interval-valued Atanassov intuitionistic fuzzy CODAS for group decision-making purposes. Adding to the growing strand of literature, Ijadi Maghsoodi et al. (2019) presented a framework of step-wise weight assessment ratio analysis (SWARA) and CODAS, which considers target based attributes. As a further development, Zhou et al. (2020) formulated a linguistic Pythagorean fuzzy (LPF) based CODAS method. However, despite these gradual and consistent extension works, it is noticed that none have used LN. Instead, the researchers mostly relied on min-max and max type of normalization.
In this paper, the CODAS algorithm is used for the smartphone selection problem in which the fundamental steps are unchanged except the normalization scheme. LN is used as an alternative approach to examine the performance of the CODAS method.

CodAs method with logarithmic normalization (CodAs-Ln)
The computational steps are described below.
Step 1: Construction of the decisionmatrix (DM) X= [x ij ] m×n where, m is the number of alternatives and n is the number of criteria.
Step 2: Normalization Instead of the linear normalization used in the original CODAS algorithm in this paper LN is used as proposed by Zavadskas and Turskis (2008). The authors observed more consistent result while using LN when the criteria values differ significantly. The work of Zolfani et al. (2020) reflected the observations made by Zavadskas and Turskis (2008).
Suppose, R= [r ij ] m×n is the normalized decision matrix. Then, r ij is calculated as follows.
Note that the sum of the normalized values for each criterion is zero.
Step 3: Derive the weighted normalized decision matrix Weighted normalized decision matrix is given by R * = [r * ij ] m×n where the values are given by r * ij = w j r ij ; where w j denotes the weight of the j th criterion.
Step 4: Find out the negative ideal or most pessimistic solution.
Step 5: Measure of separation from the negative ideal solution 324 S. Biswas / SJM 16 (2) As stated earlier, CODAS method uses two distance measures such as Euclidean (E i ) and Taxicab (T i ) for calculating the distances of the alternatives from the negative ideal points. Accordingly, the separations are calculated as Step 6: Formation of relative assessment matrix R a = [h ik ] m×m where Where, k = 1,2,…m; ψ denotes a threshold function representing the equality of the Euclidean distances of two alternatives as d is the difference between Euclidean distances of the two alternatives and τ is a threshold parameter which determines the use of distance measure (τ=0.02 as suggested by Ghorabaee et al., (2016).
Step 7: Calculation of assessment score (H i ) decision rule: The alternative with higher H i value is ranked first than others.
In order to calculate the criteria weights, the entropy method is used which is described in the subsequent sub-section.

entropy method
The concept of the entropy method was proposed in information theory (Shannon, 1948). Over the years, this method has found its application in many research problems pertaining to various disciplines (Li et al., 2011;Ghosh & Biswas, 2016;Karmakar et al., 2018;Biswas et al., 2019;Gupta et al., 2019). This method suggests that the higher value of entropy for a particular criterion signifies a greater amount information given by it. The procedural steps (Zou et al., 2006) are given below.
Step 1: Formation of normalization matrix The normalization matrix is represented as (R) m×n where, the elements r ij are given by: (When the criterion is having positive effect direction) (When the criterion is having positive effect direction) Step 2: Calculation of entropy values The entropy value for i th alternative for j th criterion is given by: Where, In this context, Zou et al., (2006) mentioned Step 3: Calculation of criteria weight The weight for each criterion is given by

iLLustrAtive CAse study: smArtPhone seLeCtion
In this paper, the proposed framework of CODAS-LN is used for the smartphone selection problem. With the extensive developments in information and communication technology (ICT), post-2010, the world has witnessed a massive increase in the number of smartphone users. Over the years, the average price for purchasing smartphones has also come down significantly. Besides, the cost of accessing the internet has also become within reach of common people. Moreover, there have been an increased number of applications wherein smartphones are used. As a result of that, alongside the old brands like Apple, Nokia, Samsung, and Motorola, some late entrants like Xiaomi, Realme, Vivo, and Oppo have also attracted a notable number of customers able to hold a considerable amount of the total market share. As we move through an age known as Industry 4.0, the competition is getting intensified day by day, and brands are competing mainly on two aspects, such as price and features or applications. Hence, many users are curious about which brand/model to select for purchasing a smartphone for quite an apparent reason. Since the buying intension, level of use, technical awareness, and purchasing capability vary from buyers to buyers, the selection of smartphones depends on multiple criteria or attributes. In other words, smartphone selection is a typical problem for MCDM.
Many researchers have tried to solve the smartphone selection problem using various MCDM algorithms. For example, Hu et al.
technical features. In addition to technical features, Rani et al. (2019) also considered internet connectivity as a criterion. However, Kim et al. (2020) applied a preferential relation model to discriminate smartphones based on attributes and brand loyalty. The following table (see table 1) exhibits a comparative analysis of some recent smartphone selection work. Table 2 describes the criteria considered in this paper. In this paper, the technical features and customer satisfaction measured by using a proxy variable called average rating are considered. As it is seen from table 1 that the criteria used in this study are in tune with past work.
A set of 25 popular smartphone models of different brands like Samsung, Redmi (Xiaomi), Oppo, Honor, Lava, Vivo, Huawei, and Poco are selected. The price range of maximum INR 25000 is considered in the sense that customers belonging to midincome groups can afford to buy these models. Those models with wide popularity (i.e., average customer rating of 4-star and above) on acclaimed e-commerce platforms like Amazon are considered. The relevant information is collected mainly from publicly available data sources like company websites and e-commerce sites. The aim is to compare these 25 models using the proposed Entropy-CODAS-LN framework to suggest possible best option to buy.

resuLts And disCussion
In this section, the results obtained from step by step data analysis by using the proposed framework is presented. Table 4 presents the criteria weights as calculated by using the entropy method (Eq. 11-16).
Next, these criteria weights are used to proceed for comparative analysis of the models selected. Table 5 shows the relative ranking of the models as obtained by using the proposed CODAS-LN framework.
It is seen from table 6 that M6, M19, and M25 secure the first three positions. If we further probe into their specifications, it reveals that these models offer higher processor speed, standard battery backup, larger capacities for RAM, and better display quality at considerable prices compared to the models belonging to the bottom performer group, i.e., M10, M11 and M25. Further, all these models are next-generation 329 S. Biswas / SJM 16 (2) (2021) 321 -340   smartphones, which are launched very recently. This finding indicates the buyers' inclination towards computational power and storage capacities to support high end and exhaustive applications.

validation
Next, validation of the results obtained by using the CODAS-LN framework is examined. The results of CODAS-LN based analysis are compared with that of using other commonly used distance-based MCDM algorithms to check whether there are significant deviations in the comparative rankings of the smartphones under comparison (Biswas & Pamucar, 2020). For this purpose, the frameworks like TOPSIS (Hwang & Yoon, 1981), EDAS (Ghorabaee et al., 2015) and original form of CODAS (Ghorabaee et al., 2016) are applied for ranking the same group of smartphones. In addition, the calculation of TOPSIS based on LN (TOPSIS-LN) is also carried out. Table 6 highlights the comparison of ranking results. It is evident from the above table that ranking is quite consistent in nature. The results are compared further statistically by using Spearman's rank correlation and Kendall's correlation test (see table 7). It is observed that the ranking result obtained by using CODAS-LN method is highly consistent with that of using the established frameworks.
Moving further the possibility of the rank reversal phenomenon is checked in case of CODAS-LN framework. One of the major drawbacks that the MCDM methods are suffered from is rank reversal phenomenon (RRP). In many cases it is found that the ranking orders as obtained by using a particular MCDM algorithm gets changed as a consequence of addition or deletion of a particular alternative (de Farias Aires & Ferreira, 2019). In the past several researchers have worked on this issue for examining the effectiveness for several MCDM methods such as TOPSIS (Wang & Luo, 2009;Chatterjee & Stevic, 2019), ELECTRE (Wang & Triantaphyllou, 2008), PROMETHEE (Macharis et al., 2004), ANP (Kong et al., 2016), GRA (Huszak & Imre, 2010  researchers have pointed out that normalization is one of the potential reasons (García-Cascales & Lamata 2012; Pamucar & Ecer, 2020). In this regard, Senouci et al.
(2016) worked on the possibilities to avoid or reduce the effect of RRP for TOPSIS method wherein they applied four normalization schemes. Therefore, for a justified reason for proceeding to examine whether the CODAS-LN framework suffers from any RRP. In this regard, two experiments are carried out in tune with the work of Mousavi-Nasab and Sotoudeh- Anvari (2018). First, the best alternative is removed and the revised ordering is checked. Next, the worst alternative is eliminated and change in the relative ranking is observed.  Sharma et al., 2018). Hence, it is examined whether the framework of CODAS-LN satisfies the transitivity property test as suggested in Wang and Triantaphyllou (2008) and Triantaphyllou and Shu (2001). A non-optimal alternative such as M10 is replaced with another worse one such as M11 and the relative rankings are checked. Table 9 shows the finding which indicates that CODAS-LN follows the transitivity property.

sensitivity analysis
Now the sensitivity analysis is carried out. Any MCDM framework is based on the goal to reduce the bias and ensure reliability of the solution (Pamučar et al., 2017;Mukhametzyanov & Pamučar, 2018). Criteria weights contribute significantly in finding out the final ranking. Hence, changes in the criteria weights may affect the final solution of the MCDM framework. Therefore, a sensitivity analysis is required to be performed for checking the stability of the solution subject to variations in the criteria weights in a given situation (Pamučar & Ćirović, 2015;Gharib, 2020). In this paper, the approach as suggested by Önüt et al. (2009) is followed which is based on exchange of criteria weights. 331 S. Biswas / SJM 16 (2) (2021)   Accordingly, the highest weight is exchanged with that of next three higher weights and three lowest weights subsequently. Table 10 depicts the experimental cases. Table 11 shows the variations in relative rankings of the smartphones under different cases. It is observed from this sensitivity analysis that except few positional variations, the rankings remain consistent. This finding is supported by the correlation among the ranking results obtained under different situations as given in table 12. Figure 1 pictorially represents the results of the sensitivity analysis which reveals the same fact as found in table 11 and 12.

ConCLusion
In this paper an extended version of the fundamental CODAS framework by using LN is proposed. LN is found quite rare in use in the literature but as suggested by Zavadskas and Turskis (2008), this scheme is useful when there is a significant variation among the criteria. This framework is applied for solving smartphone selection problem in Indian context. It is observed that buyers tend to incline on computational 332 S. Biswas / SJM 16 (2) (2021)    Therefore, one future study may consider variations in τ and check the results under different normalization schemes. Third, the impact of LN can be further tested by applying the CODAS method in uncertain environment using fuzzy or grey numbers. Fourth, in this paper a general sensitivity analysis is performed. One may attempt to carry out a statistical sensitivity analysis. Fifth, in CODAS method a combination of two different distance measures is used. One future study may examine the impact of changes in distance measures on the relative rankings while using LN. Sixth, the results obtained by using objective information may be further contrasted with subjective reviews of the users as obtained through natural language processing (NLP). Seventh, the moderation effect of socio-economic factors may also be considered on the purchase intention of the smartphones while using the composite ranking scores as inputs. Nevertheless, it is assumed that this limitation may not undermine the usefulness 333 S. Biswas / SJM 16 (2) (2021) 20  19  19  20  22  18  20  M3  10  15  12  12  15  12  13  M4  4  5  5  5  6  7  5  M5  9  11  11  10  11  11  11  M6  1  3  2  2  1  3  2  M7  16  16  17  16  17  14  16  M8  21  21  21  21  20  21  21  M9  8  10  8  9  10  10  10  M10  23  22  23  23  23  22  23  M11  24  24  24  24  24  23  24  M12  11  14  10  13  12  15  15  M13  18  18  18  18  19  20  18  M14  15  13  16  15  16  16  14  M15  22  23  22  22  21  24  22  M16  5 Original_ Rank

Funding
The authors received no external funding for carrying out this research.