A Multiple Attribute Decision Making problem (MADM) is a procedure to find the best alternative that has the highest degree of satisfaction from a finite set of alternatives characterized with multiple criteria (Biswas et al., 2019). MADM problems have been established as an effective methodology for solving a wide variety of decision making issues. As shown in equation (1), a MADM matrix is generally displayed by m alternatives A 1 , … , A m , and n criteria C 1 , … , C n , such that entry x i j indicates the performance measure of alternative A i for criterion C j . The weight attached to criterion C j is represented by w j in a way that 0 ⩽ w j ⩽ 1 and ∑ j = 1 n w j = 1. (1) A criterion is either quantitative or qualitative. Quantitative criteria are stated in numerical performance measures, but qualitative criteria are expressed in ordinal qualitative data. There are many instructions for conversion of qualitative data into numerical values. Anyway, from now on, we assume that all of performance measures x i j are numerical. Moreover, a criterion is either benefit type or cost type. The former receives high performance measures for preferable alternatives, and the latter receives low performance measures for desirable alternatives. Let’s, x j max = max i = 1 , … , m x i j and x j min = min i = 1 , … , m x i j , thus for a benefit criterion the former is the ideal performance measure, and the latter shows the anti-ideal performance measure. Absolutely, these terms are vice versa for a cost criterion.

In the literature, there are several methods to evaluate the alternatives in a MADM problem. These methods can be broadly classified into two major types: compensatory and non-compensatory (Hwang and Yoon, 1981). The former assumes compensation among the criteria, while the latter rejects this assumption. The compensatory methods are classified as scoring (or weighting or aggregating), compromising (or ideal point) and concordance (or outranking) (Hwang and Yoon, 1981). The concentration point of the current paper is the scoring methods. In the scoring methods, an aggregate score for each alternative is calculated (from now on, for short, we call it score), indicating total performance of that alternative. Generally, an alternative that has the largest score is selected. In Section 2, the existent scoring methods are reviewed. Let’s explain two motivations of the current research, as follows:

The aim of the current paper is to extend scoring methods to evaluate the alternatives in an MADM problem, by taking the two above notions into consideration. The paper is organized as follows. Section 2 reviews the existent scoring methods in the related literature. Section 3 defines the YT concept, and justifies why any MADM evaluation method has to take this concept into account. Section 4 explicates the IT concept, and enhances it by giving a new definition. In sections 5 and 6, two proposed extended scoring methods are explained. Section 7 is to present the procedure and results of a simulation study carried out to compare the ranking structures of the alternatives by different scoring methods. Section 8 illustrates a real-world study case performed in the oil industry. Finally, discussion and conclusion are pointed out in the last section.

The seminal and traditional scoring method is Weighted Sum Model (WSM) (Churchman and Ackoff, 1954). In this method, after normalization of the performance measures by the linear normalization (equations (2) and (3) respectively for benefit and cost criteria), the score for alternative A i is calculated by equation (4). (2) r i j = x i j x j max , i = 1 , … , m , j = 1 , … , n , (3) r i j = x j min x i j , i = 1 , … , m , j = 1 , … , n , (4) S i WSM = ∑ j = 1 n w j r i j , i = 1 , … , m . The WSM formula is defined in additive form. The multiplicative pair for the WSM is called Weighted Product Model (WPM) (Miller and Starr, 1969). Equation (5) shows the WPM function to calculate the scores. The normalization technique of the WPM is just like the WSM. (5) S i WPM = ∏ j = 1 n ( r i j ) w j , i = 1 , … , m . In Additive Ratio Assessment (ARAS) (Zavadskas and Turskis, 2010a), firstly, a hypothetical alternative, namely, and ideal alternative including the ideal performance measures is added to the MADM matrix at row zero ( i = 0 ). Then, the method benefits from the linear normalization sum-based, as follows from equations (6) and (7) respectively for benefit and cost criteria. Next, equation (8) is used to work the primary scores of the alternatives out. As it can be seen from this equation, the ARAS score calculation is like the WSM. At the end, final scores are determined by comparison of the primary scores with the primary score of the ideal alternative, as follows from equation (9). (6) r i j = x i j ∑ k = 0 m x k j , i = 0 , … , m , j = 1 , … , n , (7) r i j = 1 / x i j ∑ k = 0 m ( 1 / x k j ) , i = 0 , … , m , j = 1 , … , n , (8) S i = ∑ j = 1 n w j r i j , i = 0 , … , m , (9) S i ARAS = S i / S 0 , i = 1 , … , m . Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) was recently introduced by Stevic et al. (2020). This method is a generalized version of the ARAS. The MARCOS starts with adding two hypothetical alternatives1¹

Stevic et al. (2020), in their paper, used the terms ‘ideal solution’ and ‘anti-ideal solution’. We know that the term ‘solution’ indicates a feasible point which can be chosen to apply. But the above points are actually artificial and cannot be selected. Hence, we prefer to use the term ‘hypothetical alternative’ instead of ‘solution’.

In the methods WSM, WPM, ARAS, and MARCOS, the relevant procedures to compute the scores do not distinguish between benefit criteria and cost criteria, and therefore, during the normalization step, cost criteria must be converted into benefit criteria. On the contrary, in the procedure of some other methods, benefit criteria are differentiated from cost criteria. The seminal method by such a procedure is Complex Proportional Assessment (COPRAS), proposed by Zavadskas and Kaklauskas (1996). Equation (14) depicts the COPRAS primary formula, where S i + and S i − are determined by equations (12) and (13). Both these equations use only equation (11) to get the weighted normalized performance measures. Lastly, the COPRAS score is calculated by equation (15). (11) r i j = w j x i j ∑ k = 1 m x k j , i = 1 , … , m , j = 1 , … , n , (12) S i + = ∑ j = 1 n r i j | j is a benefit criterion , i = 1 , … , m , (13) S i − = ∑ j = 1 n r i j | j is a cost criterion , i = 1 , … , m , (14) S i = S i + + min l = 1 , … , m S l − ∑ k = 1 m S k − S i − ∑ k = 1 m min l = 1 , … , m S l − S k − , i = 1 , … , m , (15) S i COPRAS = S i max k = 1 , … , m S k , i = 1 , … , m .

Multi-Objective Optimization on the basis of Ratio Analysis (MOORA) was introduced by Brauers and Zavadskas (2006). In the ratio system part of this method, it makes use of equation (16), called vector normalization, to normalize all the MADM matrix entries. Then, equations (17) and (18) are applied to calculate the MOORA score as equation (19). A minor variant of the MOORA is called Multi-Objective Optimization on the basis of Simple Ratio Analysis (MOOSRA) (Das et al., 2012) in which equation (20) is used instead of equation (19). (16) r i j = x i j ∑ k = 1 m ( x k j ) 2 , i = 1 , … , m , j = 1 , … , n , (17) S i + = ∑ j = 1 n w j r i j | j is a benefit criterion , i = 1 , … , m , (18) S i − = ∑ j = 1 n w j r i j | j is a cost criterion , i = 1 , … , m , (19) S i MOORA = S i + − S i − , i = 1 , … , m , (20) S i MOOSRA = S i + / S i − , i = 1 , … , m . Brauers and Zavadskas (2010) suggested the multiplicative form of the MOORA (called MULTIMOORA or MMOORA). The normalization step is similar to the MOORA. Then, S i + and S i + are determined by equations (21) and (22). As it can be seen, these elementary functions act in a similar way to the WPM method. Finally, equation (23) shows the MMOORA function to reach the scores. (21) S i + = ∏ j = 1 n ( r i j ) w j | j is a benefit criterion , i = 1 , … , m , (22) S i − = ∏ j = 1 n ( r i j ) w j | j is a cost criterion , i = 1 , … , m , (23) S i MMOORA = S i + / S i − , i = 1 , … , m . There are some methods which take a strategy to mix additive and multiplicative forms (often the WSM and the WPM). A convex linear combination of the WSM and WPM formulas is Weighted Aggregated Sum Product Assessment (WASPAS) proposed by Zavadskas et al. (2012). In the WASPAS function (equation (24)) μ is an adjusting parameter between 0 and 1. Usually μ = 0.5 is chosen by the Decision Maker (DM). (24) S i WASPAS = μ S i WSM + ( 1 − μ ) S i WPM , i = 1 , … , m . Combined Compromise Solution (COCOSO) (Yazdani et al., 2018) includes an idea to mix additive and multiplicative forms of the scores. Firstly, the COCOSO uses the linear max-min normalization (equations (25) and (26) respectively for benefit and cost criteria) to convert the performance measures into dimensionless values. Secondly, three appraisal formulas as equation (27) to equation (29) are used to get the mixtures. Equation (29) contains a parameter μ that similarly to the WASPAS is an adjusting parameter on interval [ 0 , 1 ], with default value of 0.5. Finally, equation (30) is the proposed COCOSO model to determine the scores. (25) r i j = x i j − x j min x j max − x j min , i = 1 , … , m , j = 1 , … , n , (26) r i j = x j max − x i j x j max − x j min , i = 1 , … , m , j = 1 , … , n , (27) k i a = ∑ j = 1 n w j r i j + ∏ j = 1 n ( r i j ) w j ∑ k = 1 m ( ∑ j = 1 n w j r k j + ∏ j = 1 n ( r k j ) w j ) , i = 1 , … , m , (28) k i b = ∑ j = 1 n w j r i j min k = 1 , … , m ∑ j = 1 n w j r k j + ∏ j = 1 n ( r i j ) w j min k = 1 , … , m ∏ j = 1 n ( r k j ) w j , i = 1 , … , m , (29) k i c = μ ∑ j = 1 n w j r i j + ( 1 − μ ) ∏ j = 1 n ( r i j ) w j μ max k = 1 , … , m ∑ j = 1 n w j r k j + ( 1 − μ ) max k = 1 , … , m ∏ j = 1 n ( r k j ) w j , i = 1 , … , m , (30) S i COCOSO = ( k i a k i b k i c ) 1 / 3 + 1 3 ( k i a + k i b + k i c ) , i = 1 , … , m . Equation (31) was used by Keshavarz Ghorabaee et al. (2021) to determine the alternative scores. This formula was employed in a criteria weighting model called Method based on the Removal Effects of Criteria (MEREC), in which r i j are dimensionless values of the performance measures obtained by linear normalization, i.e. equations (2) and (3). It seems equation (31) originates in the famous Entropy method (Hwang and Yoon, 1981) and a logarithmic normalization formula suggested by Zavadskas and Turskis (2008). Since the MEREC disregards the criteria weights, it may be inapplicable in many situations. (31) S i MEREC = Ln ( 1 + ( 1 n ∑ j = 1 n | Ln ( r i j ) | ) ) , i = 1 , … , m .

Reviewing the scoring methods displays that different methods employ various techniques to remove the difference in scale of each criterion. For review of different kinds of normalization techniques, see Brauers and Zavadskas (2006), Zavadskas and Turskis (2008), and Jahan and Edwards (2015).

The Aspiration Level (AL) idea was introduced by Lotfi et al. (1992). For a given criterion, the AL is a value that represents the minimum satisfactory outcomes for the DM. Lotfi et al. (1992) have stated that the AL for a criterion should not exceed the ideal performance measure of that criterion, and should be less than the anti-ideal performance measure of that criterion. For example, in a house selection MADM problem with three alternatives, if areas are 60, 40, and 80 square meters, the DM’s AL for house area should be a number between 40 to 80. A drawback to such assumption is that the DM’s willingness actually may be beyond this range, e.g. 200 square meters. In fact, confining the DM’s desire to performance measure range is not reasonable. To overcome the AL limitation, this paper proposes the Yearning Threshold (YT) concept as follows.

Now, having the above YT definition, let us discuss this fact that incorporation of the YT concept into the MADM evaluation models is not an option, but rather a requirement to ensure the accuracy of the decisions. Again, consider the example of house selection problem by the following MADM matrix. The criteria are ( C 1 :) area and ( C 2 :) the ratio of the area of windows to the house area. The YT is 200 for C 1 , and 0.4 for C 2 .

We know that in mathematics, | Y | shows the absolute value of Y, such that | Y | = max { Y , − Y }. In fact, a number and its opposite have the same absolute value. This operator is defined in additive form of numbers. Now, let’s generalize it into multiplicative form, by the following definition.

In what follows, two definitions are introduced. The former depicts the preliminary and traditional statement of the IT concept, and the latter is a novel statement for the IT concept that is firstly introduced in the current study.

Notably, the DIT values are not homogeneous (because dimensional unit of DIT j is the same as dimensional unit of criterion C j ); on the contrary, all the RIT values range from 0 to 1, thus the RIT has an advantage over the DIT from this point of view.

The underlying idea of the proposed EWPM stands on the relative relations between the alternatives. This idea leads us to a pairwise comparison matrix whose rows and columns are the alternatives. Let’s portray the notion using a simple numerical example as follows, including 3 alternatives A 1 A 2 , and A 3 , and two criteria C 1 (benefit type) and C 2 (cost type) weighted as 0.7 and 0.3, respectively.

If the analyst wishes to compare two alternatives A 1 and A 2 from the point of view of criterion C 1 , then the fraction 6 / 2 can be taken. This fraction is called individual importance of A 1 relative to A 2 . Similarly, the individual relative importance of A 1 to A 2 in view of C 2 is ( 1 8 ) / ( 1 3 ) or 3 / 8 (Note that C 2 is a cost criterion). Now, we need to combine 6/2 and 3/8. These two numbers are ratios; therefore, to combine them the WPM form can be employed. Thus, the combined relative importance of A 1 to A 2 would be ( 6 / 2 ) 0.7 × ( 3 / 8 ) 0.3 . This calculation is sometimes called dimensionless analysis because its mathematical structure removes any units of measure (Triantaphyllou, 2000). By performing similar calculations for the other pairs of the alternatives, the result is the following matrix called Objective Multiplicative Pairwise Comparison (OMPC):

The OMPC matrix is actually in a multiplicative preference relation format, but in spite of preference judgment by the DM, the objective relative importance is replaced. Thus, each entry a i k ( i , k = 1 , … , m ) of the OMPC matrix represents the objective importance of alternative A i relative to alternative A k in terms of a multiplicative reasoning. Equation (36) shows a general formula to determine the entries of the OMPC matrix. In this function, t j is determined by equation (37): (36) a i k = ∏ j = 1 n ( x i j t j x k j t j ) w j , i , k = 1 , … , m , (37) t j = + 1 , if C j is a benefit criterion , − 1 , if C j is a cost criterion , j = 1 , … , n .

According to the following Proposition 1, the OMPC matrix is multiplicative inverse (i.e. reciprocity rule a i k × a k i = 1) and full consistent (i.e. transitivity rule a i k × a k l = a i l ) for all entries of the matrix.

Now we deal with the famous problem of deriving the scores from a pairwise comparison matrix. To extract the alternative scores from the OMPC matrix, with regard to the multiplicative form the entries, the Simple Geometric Mean (SGM) (Crawford and Williams, 1985) is utilized here. In the current situation, this method has been recommended by many researchers (Zhang et al., 2004; Hovanov et al., 2008). Therefore, equation (38) represents the formula for calculating the score of alternative A i . (38) S i = ∏ k = 1 m ∏ j = 1 n ( x i j t j x k j t j ) w j m , i , = 1 , … , m . The scores in the above example are 1.2286, 0.7642, and 1.0651, respectively.

In the previous section, we saw that the EWPM is founded on a multiplicative structure. There is a pair for the EWPM, founded on the additive structure. Suppose the same numerical example of previous section: In the additive form, we can say that in view of C 1 , the importance of A 1 compared with A 2 is 6 − 2 = 4. Indeed, regarding C 1 , the distance between the importance of A 1 and A 2 is 4. In the same way, the relative importance of A 1 to A 2 from the point of view of C 2 is ( 1 / 8 ) − ( 1 / 3 ) = − 5 / 24 (Note that C 2 is a cost criterion). The two numbers 4 and − 5 / 24 are distances, thus the WSM rule directs the current method to additively combine the relative importance. Because the distances may differ in dimensional unit, distance 6 − 2 is transformed to ( 6 − 2 ) / 6 in which denominator 6 is the ideal performance measure of C 1 . In the same way, distance ( 1 / 8 ) − ( 1 / 3 ) is transformed to ( ( 1 / 8 ) − ( 1 / 3 ) ) / ( 1 / 3 ) where denominator 1 / 3 is the reciprocal of the ideal performance measure of C 2 . The combined relative importance of A 1 and A 2 is written as 0.7 × ( 6 − 2 6 ) + 0.3 × ( ( 1 / 8 ) − ( 1 / 3 ) ( 1 / 3 ) ). By running the same calculations for the other pairs of the alternatives, the following matrix is obtained. We name it Objective Additive Pairwise Comparison matrix (OAPC): The OAPC matrix is in additive preference relation format. In this matrix, each entry b i k ( i , k = 1 , … , m ) is interpreted as the objective importance degree of alternative A i over alternative A k in terms of an additive reasoning. Equation (43) displays a general formula to obtain the entries of the OAPC matrix, in which t j is determined like the EWPM (i.e. t j is +1 for benefit criteria, and is −1 for cost criteria). In equation (43), x j ∗ refers x j max for benefit criteria, and x j min for cost criteria. (43) b i k = ∑ j = 1 n w j ( x i j t j − x k j t j x j ∗ t j ) , i , k = 1 , … , m . It can be simply demonstrated that the OAPC matrix is additive inverse (opposition rule), i.e. b i k + b k i = 0, and full consistent (transitivity rule), i.e. usually b i k + b k l = b i l .

We omit the proof here, since the proof is straightforward and similar to that in Proposition 1 (In this point, the reader is advised to see Appendix A).

Because the entries of the OAPC matrix are in the additive form, the Simple Column Sum (SCS) method (Saaty, 1980) is a proper way to derive the scores. Equation (44) shows the relevant formula. (44) S i = ∑ k = 1 m ∑ j = 1 n w j ( ( x i j t j − x k j t j ) / x j ∗ t j ) m , i = 1 , … , m . The scores for alternatives A 1 , A 2 , and A 3 in the above example are 0.1483, −0.1308, and −0.0175, respectively, such that their sum is equal to 0.

This section is to answer two questions. Whether a given scoring method selects the same best alternative as another given scoring method? And, how much is similarity between the overall rank structures of the alternatives obtained by two given scoring methods? To these, we examine a kind of simulation process that is a broadly accepted framework to compare the performance of the methods (Ahn, 2017; Sarabando and Dias, 2010; Ahn and Park, 2008; Barron and Barrett, 1996). The idea is to repeatedly (N times, namely simulation run number) generate a random MADM matrix and random criteria weights, and investigate how well the result of a given method matches another one, in terms of an efficacy measure. To answer the above first question, a measure called Hit Rate (HR) is used as equation (49). (49) HR = B N , where N is a simulation run number, and B is the number of runs in which a method selects the same best alternative as another method does. The HR value ranges from 0 to 1, such that 1 indicates two methods select identical best alternative through all simulation runs. To answer the second question, Rank order Correlation (RC) is employed here. The RC shows the similarity of the overall rank structures of the alternatives constructed by two given methods. The RC is calculated using Kendall’s formula (Winkler and Hays, 1985) as equation (50). (50) RC = 1 − 2 × V A , where A is the sum of alternative numbers through all simulation runs, and V indicates the number of pairwise preference violations in all simulation runs. The RC value ranges from − 1 to + 1, where + 1 means perfect correspondence between the two rank orders in all simulation runs.

The simulation is designed with two levels of the alternatives: Low ( 3 ⩽ m ⩽ 6 ) and High ( 9 ⩽ m ⩽ 12 ), and two levels of the criteria: Low ( 3 ⩽ n ⩽ 6 ) and High ( 9 ⩽ n ⩽ 12 ). For each combination of the levels (i.e. combination 1: low level of the alternatives and low level of the criteria, combination 2: low level of the alternatives and high level of the criteria, combination 3: high level of the alternatives and low level of the criteria, and combination 4: high level of the alternatives and high level of the criteria), the following 10-step procedure is performed.

Step 1: Set N = 10000 (N shows simulation run number), k = 0, and c = 1 (c indicates combination type, i.e. c could be 1, 2, 3, or 4).

Step 2: Set k = k + 1, A = 0, B p q = 0 and V p q = 0 ( p , q = 1 , … , 12 ). Where “12” is the number of the methods to be compared (the WSM, WPM, ARAS, MARCOS, COPRAS, MOORA, MOOSRA, MMOORA, WASPAS, COCOSO, null EWPM, and null EWSM). Thus, p = 1 indicates the WSM, p = 2 indicates the WPM method, and so on.

Step 3: Based on the combination type, generate a random integer number for the number of alternatives ( m ), and an independent random integer number for the number of criteria ( n ). Set A = A + m.

Step 4: Generate a random MADM matrix from the independent uniform distribution on ( 0 , 1 ].