We face decision-making processes at every moment of our lives. In the decision-making process, people express their knowledge and thoughts via their personal opinions and comments. Decision makers (DMs) often use expressions containing doubt and uncertainty in their judgments. Expressions such as “not very clear”, “likely”, etc., show the uncertainty of human thought and are frequently used in daily or business life. Zadeh (1965) introduced fuzzy set theory in order to model this ambiguity and subjectivity of human judgments and to use linguistic terms in the decision-making process. Thus, fuzzy set theory enables DMs to incorporate their uncertain information in the decision model.

DMs who have knowledge and experience are often not exactly sure of their assessments when they are making a decision. The probability of correct diagnosis of even a doctor is not one hundred percent (Xian et al., 2019). For example, one doctor can say “you likely have anemia”. In the medical world, tests and investigations can be performed to confirm this diagnosis. However, in many fields that need decision-making, subjective judgments cannot be confirmed in that way. Moreover, when quantitative data are used in decision making, they are treated to be exactly accurate since the sources’ reliability level is not questioned. However, it would not be correct to assume the numerical data with 100% certainty due to factors such as the concept of time and measurement accuracy. The possible variations that may occur in numerical data can be modelled with different extensions of fuzzy set theory. However, when qualitative data consisting of uncertain judgments is used in decision making, it would be most logical to explicitly ask people about their confidence level in their judgments. In these cases, the reliability of the experts’ fuzzy judgments must be considered and incorporated to the decision model. As a result, it is clear that restrictive information must be integrated with reliability information especially when linguistic expressions, which represent subjective judgments, are employed in the decision model.

After the introduction of fuzzy set theory, fuzzy versions of classical multi criteria decision making (MCDM) methods have emerged to capture the DMs’ uncertain expressions (Chatterjee et al., 2018a). These methods have been expanded by ordinary fuzzy sets and their several extensions, such as type-2 fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, Pythagorean fuzzy sets, and neutrosophic sets, to find the best representation of human thinking structure. Although the extensions of fuzzy sets are highly beneficial and suited to deal with vague information, their capabilities are limited to represent the reliability of the assigned fuzzy data. In order to overcome this limitation and to reach more accurate and effective results, reliability information must be incorporated into the decision processes.

Z-fuzzy numbers have been proposed by Zadeh (2011) in order to deal with the vagueness and impreciseness of membership functions by incorporating a reliability function to the evaluation system as a complementary element. This can be commented as a similar effort by Zadeh to his type-2 fuzzy sets for preventing the criticisms that membership functions themselves are not fuzzy. Thus, the requirement of reliability information in the decision-making can be satisfied by the use of Z-fuzzy numbers. Z-fuzzy numbers reflect the uncertainty in DMs’ mind through a reliability function, which express how confident they are about their evaluations. In the doctor example, whereas the word “anemia” represents restrictive information, the word “likely” represents reliability information.

Evaluation Based on Distance from Average Solution (EDAS) is one of the recently developed MCDM methods. The EDAS method has been integrated with various fuzzy set extensions to better define the DMs’ uncertain judgments. However, these versions of the EDAS method such as intuitionistic fuzzy EDAS or picture fuzzy EDAS do not fully include the reliability information. To the best knowledge of the authors, the EDAS method has not been extended with Z-fuzzy numbers by any researcher. In the literature, there is only one paper trying to use linguistic Z-numbers in EDAS method, different from our study, for quality function deployment (Mao et al., 2021). In this study, EDAS method is extended to Z-fuzzy EDAS method using ordinary Z-fuzzy numbers to strengthen the reliability degree of the given decisions.

Main objectives of the study are as follows:

This study contributes to the literature in four aspects:

The rest of the paper is organized as follows. Section 2 presents a literature review on EDAS and Z-fuzzy MCDM. Section 3 includes the preliminaries of Z-fuzzy numbers. Section 4 presents the proposed Z-fuzzy AHP method and Section 5 gives the steps of the proposed Z-fuzzy EDAS method. Section 6 presents the application on wind turbine technology selection. Section 7 gives a comparative analysis using Z-fuzzy AHP&TOPSIS methodology. The last section presents the conclusions and future research directions.

Decision making problems arise when there is a need for comparison or selection from a set of alternatives, taking into account the impact of multiple conflicting criteria. For this purpose, various multiple criteria decision making (MCDM) methods are constructed to determine the best alternative with respect to all relevant criteria (Chatterjee et al., 2018b). Decisions taken in daily life or business life may have different degrees of difficulty due to the factors such as the considered criteria, the relationship between them and the number of alternatives. However, when DMs need to evaluate the alternatives by considering many criteria; many factors such as the number of criteria and alternatives, criteria weights and conflicts between criteria further complicate the problem and need to be evaluated with more comprehensive methods. Therefore, multi-criteria decision making (MCDM) methods are used in order to get more accurate decisions in solving more complex decision problems.

EDAS method has been introduced to the literature by Keshavarz Ghorabaee et al. (2015) as a MCDM method. It is based on the measurement of the positive and negative distances from the average solution rather than calculating the negative ideal solution (NIS) and positive ideal solution (PIS) as in TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) (Chatterjee and Kar, 2016) and VIKOR (Vise Kriterijumska Optimizacija I Kompromisno Resenje) methods. Thus, unlike the TOPSIS and VIKOR methods, EDAS offers a solution based on how far the alternatives are from the average solution instead of PIS and NIS.

After the introduction of EDAS method to the literature, it has been used in many application areas such as supplier selection, project selection, personnel selection, material selection and drug selection. Due to the fact that fuzzy set theory in decision making better defines human thoughts, various fuzzy extensions of EDAS method have been used more frequently than classical EDAS method in the literature. Table 1 presents the classical, stochastic, neutrosophic, and fuzzy EDAS papers published in the literature and their application areas in historical order.

Table 1 shows that the classical EDAS method has been developed by many extensions of ordinary fuzzy sets such as type-2 fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets. However, since it was only put forward in 2015, there is still a gap in the literature about the method and its usage areas.

Since the fuzzy versions of the EDAS method proposed so far do not fully reflect the reliability information, another possible extension of the classical EDAS method is realized in this study through Z-fuzzy numbers, which represent the natural language with better descriptive ability. Thus, apart from the fuzzy extensions in Table 1, the EDAS method has been extended with Z-fuzzy numbers, which are composed of trapezoidal restriction function and triangular fuzzy reliability function.

After Z-fuzzy numbers were introduced to the literature, they have been integrated with several MCDM methods such as AHP (Azadeh et al., 2013; Sergi and Sari, 2021; Tüysüz and Kahraman, 2020a; Kahraman and Otay, 2018), TOPSIS (Krohling et al., 2019), VIKOR (Shen and Wang, 2018), and WASPAS (Sergi and Sari, 2021). Table 2 presents the Z-fuzzy number integrated MCDM methods based on their publication years.

As can be seen in Table 2, Z-fuzzy numbers are integrated with different MCDM methods, and they are used in different application areas. However, there is still a significant literature gap regarding the combined use of Z-fuzzy numbers and MCDM methods. This study contributes to fill this literature gap by integrating the EDAS method with Z-fuzzy numbers.

DMs are often not 100% confident in their assignments for membership degrees. Hence, in addition to assigning a membership degree/function μ A ˜ ( x ), it makes sense to also assign a reliability degree μ B ˜ ( x ) so that DMs can reflect their confidence to the membership. The corresponding pairs ( μ A ˜ ( x ), μ B ˜ ( x )) are known as a Z-fuzzy number which was introduced by Zadeh (2011).

A Z-fuzzy number is an ordered pair of fuzzy numbers Z ( A ˜ , B ˜ ), as given in Fig. 1. The first component A ˜ is a restriction function whereas the second component B ˜ is a measure of reliability for the first component.

The concept of a Z-fuzzy number is intended to provide a basis for computation with ordinary fuzzy numbers which are not reliable.

The AHP method is one of the most widely used MCDM methods to calculate the criteria weights and there are several versions of it (Chatterjee and Kar, 2017). Due to the nature, it is usual for DMs to have hesitation while making pairwise comparisons, and in these situations, it is expected that they will not be absolutely sure about their evaluations. These preferences can be included in the decision methods by modelling the DMs’ thinking structure under the concept of Z-fuzzy numbers. Therefore, in this study, to obtain criteria weights, it is suggested to collect DMs’ judgments using Z-fuzzy numbers integrated AHP method rather than commonly used fuzzy versions of AHP method.

To calculate criteria weights, the steps of the Z-fuzzy AHP method are presented in the following:

Step 1. Determine the criteria set of the decision problem. Fig. 4 can be used to establish the hierarchical structure of goal, main criteria and sub-criteria. Level 1 of the hierarchy represents a goal whereas Level 2 and Level 3 are composed of main-criteria and sub-criteria, respectively.

Step 2. Determine the linguistic terms and their corresponding Z-fuzzy restriction and reliability numbers. Collect the linguistic pairwise comparison evaluations from each DM for the main criteria and sub-criteria by using questionnaires. Then, Z-fuzzy pairwise comparison matrices are constructed based on these evaluations. Each DM can use Z-fuzzy linguistic scales given in Tables 3–4 for his/her assessments, respectively.

Let each decision maker ( DM k ) assign an independent assessment for any pairwise comparison as shown in Eq. (10): (10) Z DMk = ( A ˜ , B ˜ ) = ( ( a 1 DMk , a 2 DMk , a 3 DMk ) , ( b 1 DMk , b 2 DMk , b 3 DMk ) ) .

Step 3. Calculate the consistency ratio (CR) of each Z-fuzzy pairwise comparison matrix obtained by the DMs’ assessments. Defuzzify the restriction functions of Z-fuzzy numbers in the pairwise comparison matrix using Eq. (2) and obtain the crisp pairwise comparison matrix. Apply Saaty’s classical consistency procedure and check if CR is less than 0.1, which is accepted as the consistency limit in the literature (Saaty, 1980).

Step 4. Apply the aggregation procedure for DMs’ Z-fuzzy assessments. Each element of restriction and reliability functions of Z-fuzzy assessments is aggregated by using geometric mean and one Z-fuzzy decision matrix is obtained.

Assume three DMs assign the following terms: Z ˜ DM 1 = ( A ˜ , B ˜ ) = ( ( a 1 DM 1 , a 2 DM 1 , a 3 DM 1 ) , ( b 1 DM 1 , b 2 DM 1 , b 3 DM 1 ) ) , Z ˜ DM 2 = ( A ˜ , B ˜ ) = ( ( a 1 DM 2 , a 2 DM 2 , a 3 DM 2 ) , ( b 1 DM 2 , b 2 DM 2 , b 3 DM 2 ) ) , Z ˜ DM 3 = ( A ˜ , B ˜ ) = ( ( a 1 DM 3 , a 2 DM 3 , a 3 DM 3 ) , ( b 1 DM 3 , b 2 DM 3 , b 3 DM 3 ) ) . Aggregation of these three DMs’ assessments is made by using the geometric mean operator given in Eqs. (11)–(12): (11) Z ˜ A g g = ( A ˜ A g g , B ˜ A g g ) = c ˜ 11 c ˜ 12 … c ˜ 1 m c ˜ 21 c ˜ 22 … c ˜ 2 m ⋮ ⋮ ⋱ ⋮ c ˜ m 1 c ˜ m 2 … c ˜ m m , where (12) c ˜ i j = ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 , a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 , a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 ) , ( b 1 , i j DM 1 ∗ b 1 , i j DM 2 ∗ b 1 , i j DM 3 3 , b 2 , i j DM 1 ∗ b 2 , i j DM 2 ∗ b 2 , i j DM 3 3 , b 3 , i j DM 1 ∗ b 3 , i j DM 2 ∗ b 3 , i j DM 3 3 ) , i = 1 , 2 , … , m ; j = 1 , 2 , … , m .

Step 5. Calculate the alpha (α) from the reliability components of the aggregated pairwise comparison matrix by using Eq. (13). The reciprocal reliability values are the multiplicative inverse of the calculated α values. (13) α i j = ( b 1 , i j DM 1 ∗ b 1 , i j DM 2 ∗ b 1 , i j DM 3 3 + 2 ∗ b 2 , i j DM 1 ∗ b 2 , i j DM 2 ∗ b 2 , i j DM 3 3 + b 3 , i j DM 1 ∗ b 3 , i j DM 2 ∗ b 3 , i j DM 3 3 ) 4 , i = 1 , 2 , … , m ; j = 1 , 2 , … , m .

Step 6. Convert the Z-fuzzy numbers ( Z ˜ A g g ) to ordinary fuzzy numbers ( O ˜) using the matrix obtained in Step 5 by using Eqs. (14) and (15): (14) O ˜ = o ˜ 11 o ˜ 12 … o ˜ 1 m o ˜ 21 o ˜ 22 … o ˜ 2 m ⋮ ⋮ ⋱ ⋮ o ˜ m 1 o ˜ m 2 … o ˜ m m , where (15) o ˜ i j = a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j , a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j , a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j .

Step 7. Apply the ordinary fuzzy AHP method using Buckley’s method (Buckley, 1985).

Step 7.1. Calculate the geometric mean vector ( GM ˜) whose elements are given in Eqs. (16)–(17). Thus, m × 1 matrix is obtained from m × m matrix. (16) GM ˜ = g ˜ 11 g ˜ 21 ⋮ g ˜ m 1 , where (17) g ˜ i 1 = ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m , ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m , ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m , i = 1 , 2 , … , m .

Step 7.2. Sum the values in GM ˜ vector using Eq. (18): (18) S ˜ = ∑ i = 1 m ( ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m ) , ∑ i = 1 m ( ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m ) , ∑ i = 1 m ( ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m ) .

Step 7.3. Apply fuzzy division operation to obtain relative fuzzy weights vector ( R ˜) of criteria as given in Eqs. (19)–(20): (19) R ˜ = r ˜ 11 r ˜ 21 ⋮ r ˜ m 1 = g ˜ 11 / S ˜ g ˜ 21 / S ˜ ⋮ g ˜ m 1 / S ˜ , where (20) r ˜ i 1 = ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m , ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m , ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m , i = 1 , 2 , … , m .

Step 7.4. Defuzzify the relative fuzzy weights vector ( R ˜) using Eq. (21): (21) d j = ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m + 2 ∗ ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 α i j ) m + ∏ j = 1 m ( a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 α i j ) m ∑ i = 1 m ∏ j = 1 m ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 α i j ) m ∗ 4 − 1 , j = 1 , 2 , … , m .

Step 7.5. Normalize the defuzzified weights to satisfy ∑ w j = 1 using Eq. (22). Thus, the weights of the criteria are obtained as crisp values. (22) w j = d j ∑ j = 1 m d j , j = 1 , 2 , … , m .

Step 8. Apply Steps 3–6 for the other Z-fuzzy pairwise comparison matrices of DMs for the sub-criteria under each main criterion and obtain the weight of each sub-criterion j ´, j ´ = 1 , 2 , … , p. w j j ´ ˙ where j = 1 , 2 , … , m and j ´ = 1 , 2 , … , p for each j . Step 9. Combine the local sub-criteria weights ( w j j ´ ˙ ) and main criteria weights ( w j ) in order to obtain global criteria weights ( w j j ´ G ) as in Eq. (23). (23) w j j ´ G = w j ∗ w j j ´ ˙ , j = 1 , 2 , … , m and j ´ = 1 , 2 , … , p for each j .

The first fuzzy EDAS method is introduced by Keshavarz Ghorabaee et al. (2016) for the solution of MCDM problems under uncertainty. It is integrated with various fuzzy set extensions to model the vagueness and impreciseness. In this study, due to the fact that these extensions cannot completely combine the reliability information with the EDAS method, it is extended to Z-fuzzy EDAS method by using ordinary Z-fuzzy numbers. This method allows to define the DMs’ preferences over the alternatives with their degree of confidence, which creates a more comprehensive and flexible decision-making environment. Z-Fuzzy EDAS method is presented as follows:

Step 1. Determine the evaluation criteria C = ( C 1 , C 2 , … , C m ) and alternatives A = ( A 1 , A 2 , … , A n ) for the decision problem.

Step 2. Construct the fuzzy decision matrix ( D ˜) using Z-fuzzy numbers, shown as in Eq. (24): (24) D ˜ = [ x ˜ i j ] n × m = A 1 A 2 ⋮ A n x ˜ 11 x ˜ 12 … x ˜ 1 m x ˜ 21 x ˜ 22 … x ˜ 2 m ⋮ ⋮ ⋱ ⋮ x ˜ n 1 x ˜ n 2 … x ˜ n m , where x ˜ i j ⩾ 0 and it denotes the Z-fuzzy performance value of ith alternative on jth criterion ( i ∈ { 1 , 2 , … , n } and j ∈ { 1 , 2 , … , m } ) . Z-fuzzy linguistic restriction scale presented in Table 5 and the reliability scale in Table 4 are used for DMs’ assessments in the decision matrix.

Step 3. Aggregate the Z-fuzzy evaluation matrices of all DMs. Aggregation of three DMs’ assessments is made by using the geometric mean given in Eqs. (25)–(26): (25) Z ˜ D ˜ A g g = x ˜ 11 x ˜ 12 … x ˜ 1 m x ˜ 21 x ˜ 22 … x ˜ 2 m ⋮ ⋮ ⋱ ⋮ x ˜ n 1 x ˜ n 2 … x ˜ n m , where (26) x ˜ i j = ( a 1 , i j DM 1 ∗ a 1 , i j DM 2 ∗ a 1 , i j DM 3 3 , a 2 , i j DM 1 ∗ a 2 , i j DM 2 ∗ a 2 , i j DM 3 3 , a 3 , i j DM 1 ∗ a 3 , i j DM 2 ∗ a 3 , i j DM 3 3 ) , ( b 1 , i j DM 1 ∗ b 1 , i j DM 2 ∗ b 1 , i j DM 3 3 , b 2 , i j DM 1 ∗ b 2 , i j DM 2 ∗ b 2 , i j DM 3 3 , b 3 , i j DM 1 ∗ b 3 , i j DM 2 ∗ b 3 , i j DM 3 3 ) , i = 1 , 2 , … , n ; j = 1 , 2 , … , m .

Step 4. Calculate the Z-fuzzy average values ( AV ˜) by using Eqs. (27)–(28): (27) AV ˜ = [ AV ˜ j ] 1 × m = AV ˜ 1 AV ˜ 2 … AV ˜ j , (28) AV ˜ j = ∑ i = 1 n X ˜ i j n , ∀ j , j = 1 , 2 , … , m .

Step 5. Calculate the Z-fuzzy positive distance from average ( PDA ˜) and Z-fuzzy negative distance from average ( NDA ˜) for each alternative by employing Eqs. (29)–(32): (29) PDA ˜ = [ PDA ˜ i j ] n × m , (30) NDA ˜ = [ NDA ˜ i j ] n × m , (31) PDA ˜ i j = max ( 0 , ( x ˜ i j − AV ˜ j ) ) AV ˜ j M , NDA ˜ i j = max ( 0 , ( AV ˜ j − x ˜ i j ) ) AV ˜ j M , for benefit criteria , (32) PDA ˜ i j = max ( 0 , ( AV ˜ j − x ˜ i j ) ) AV ˜ j M , NDA ˜ i j = max ( 0 , ( x ˜ i j − AV ˜ j ) ) AV ˜ j M , for cost criteria , where PDA ˜ i j and NDA ˜ i j represent the Z-fuzzy positive and negative distances from average value of ith alternative according to jth criterion, respectively.

To determine max ( 0 , ( x ˜ i j − AV ˜ j ) ), Z-fuzzy numbers are defuzzified as in Eqs. (33)–(34) and compared with each other. (33) a j = ( a 1 , i j + 2 ∗ a 2 , i j + 2 ∗ a 3 , i j + a 4 , i j ) 6 , ∀ j , for restriction function , (34) b j = ( b 1 , i j + 2 ∗ b 2 , i j + b 3 , i j ) 4 , ∀ j , for reliability function . After determining the max ( 0 , ( x ˜ i j − AV ˜ j ) ), we still continue with Z-fuzzy numbers. Then, max ( 0 , ( x ˜ i j − AV ˜ j ) ) is divided by AV ˜ j using Z-fuzzy numbers.

Step 6. Use the criteria weights obtained by Z-fuzzy AHP method in Section 4 and calculate the weighted summation of PDA ˜ and NDA ˜ shown as in Eqs. (35)–(36): (35) SP ˜ i = ∑ j = 1 m w j ∗ PDA ˜ i j , (36) SN ˜ i = ∑ j = 1 m w j ∗ NDA ˜ i j , where w j = ( w 1 , w 2 , … , w m ) and it is the weight of jth criterion.

w j ( 0 < w j < 1 ) denotes the weight of jth criterion and ∑ j = 1 m w j = 1.

Step 7. Transform the obtained Z-fuzzy SP ˜ i and SN ˜ i values to positive values if there is any negative value among them for all alternatives shown as in Eqs. (37)–(40). Thus, we obtain the shifted SP ˜ i and SN ˜ i values, SSP ˜ i and SSN ˜ i , respectively.

For restriction function: (37) SSP ˜ i R e s = SP ˜ i R e s + max i | ( SP ˜ i a 1 R e s ) | , if any a 1 < 0 , (38) SSN ˜ i R e s = SN ˜ i R e s + max i | ( SN ˜ i a 1 R e s ) | , if any a 1 < 0 . For reliability function: (39) SSP ˜ i R e l = SP ˜ i R e l + max i | ( SP ˜ i b 1 R e l ) | , if any b 1 < 0 , (40) SSN ˜ i R e l = SN ˜ i R e l + max i | ( SN ˜ i b 1 R e l ) | , if any b 1 < 0 .