ITARA and ELECTRE III three-way decision model in the spherical fuzzy environment and its application in customer selection

3.1 Spherical fuzzy sets and spherical fuzzy numbers

Definition 1. For any universal set X, an IFS is defined as A = { < X , μ ( x ) , η ( x ) > | x ∈ X }(1)

where μ _A  (x) : X → [0, 1] , η _A  (x) : X → [0, 1] represent the degree of membership and non-membership, respectively, which satisfy the condition 0 ≤ μ _A  (x) + η _A  (x) ≤1. When η _A  (x) =0 it degenerates into fuzzy sets (FS). The hesitation is v _A  (x) =1 - μ _A  (x) - η _A  (x). For simplification, we denote the A =< μ _A , η _A  > as intuitionistic fuzzy number (IFN).

Definition 2. For any universal set X, an PyFS is defined as A = { < X , μ ( x ) , η ( x ) > | x ∈ X }(2)

where μ _A  (x) : X → [0, 1] , η _A  (x) : X → [0, 1] represent the degree of membership and non-membership, respectively, which satisfy the condition 0 ≤ μ 2 A ( x ) + η 2 A ( x ) ≤ 1 . The hesitation is v A ( x ) = 1 - μ 2 A ( x ) - η 2 A ( x ) . For simplification, we denote the A =< μ _A , η _A  > as Pythagorean fuzzy number (PyFN).

Definition 3. For any universal set X, an PFS is defined as A = { < X , μ ( x ) , η ( x ) , ν ( x ) > | x ∈ X }(3)

where μ _A  (x) : X → [0, 1] , η _A  (x) : X → [0, 1] and ν _A  (x) : X → [0, 1] represent the degree of membership, neutral membership, and non-membership, respectively, which satisfy the condition 0 ≤ μ _A  (x) + η _A  (x) + ν _A  (x) ≤1. When η _A  (x) = ν _A  (x) =0 it degenerates into FS. When η _A  (x) =0 it degenerates into IFS. The refusal membership is π _A  (x) =1 - μ _A  (x) - η _A  (x) - ν _A  (x). For simplification, we denote the A =< μ _A , η _A , ν _A  > as picture fuzzy number (PFN).

Definition 4. For any universal set X, an SFS is defined as A = { < X , μ ( x ) , η ( x ) , ν ( x ) > | x ∈ X }(4)

where μ _A  (x) : X → [0, 1] , η _A  (x) : X → [0, 1] and ν _A  (x) : X → [0, 1] represent the degree of membership, neutral membership, and non-membership, respectively, which satisfy the condition 0 ≤ μ 2 A ( x ) + η 2 A ( x ) + ν 2 A ( x ) ≤ 1 . The refusal membership is π A ( x ) = 1 - μ 2 A ( x ) - η 2 A ( x ) - ν 2 A ( x ) . For simplification, we denote the A =< μ _A , η _A , ν _A  > as SFN.

Definition 5. Let A =< μ _A , η _A , ν _A  >, A₁ =< μ _{A 1} , η _{A 1} , ν _{A 1}  > and A₂ =< μ _{A 2} , η _{A 2} , ν _{A 2}  > be any three SFNs, and λ ≻ 0, the algorithm are as follows: A 1 ⊕ A 2 = < ( μ 2 A 1 + μ 2 A 2 - μ 2 A 1 μ 2 A 2 ) 1 / 2 , η A 1 η A 2 , ( ( 1 - μ 2 A 2 ) ν 2 A 1 + ( 1 - μ 2 A 1 ) ν 2 A 2 - ν 2 A 1 ν 2 A 2 ) 1 / 2 >(5) A 1 ⊗ A 2 = < μ A 1 μ A 2 , ( η 2 A 1 + η 2 A 2 - η 2 A 1 η 2 A 2 ) 1 / 2 , ( ( 1 - η 2 A 2 ) ν 2 A 1 + ( 1 - η 2 A 1 ) ν 2 A 2 - ν 2 A 1 ν 2 A 2 ) 1 / 2 >(6) λ · A = < ( 1 - ( 1 - μ 2 A ) λ ) 1 / 2 , η λ A , ( ( 1 - μ 2 A ) λ - ( 1 - μ 2 A - ν 2 A ) λ ) 1 / 2 >(7) A λ = < μ λ A , ( 1 - ( 1 - η 2 A ) λ ) 1 / 2 , ( ( 1 - η 2 A ) λ - ( 1 - η 2 A - ν 2 A ) λ ) 1 / 2 >(8)

Definition 6. Let A =< μ _A , η _A , ν _A  >, the score function and accuracy function are as follows: S ( A ) = ( 2 μ A - ν A 2 ) 2 - ( η A - ν A 2 ) 2(9) Ac ( A ) = μ A 2 + η A 2 + ν A 2(10)

Definition 7. For any two SFNs A₁ =< μ _{A 1} , η _{A 1} , ν _{A 1}  > and A₂ =< μ _{A 2} , η _{A 2} , ν _{A 2}  >.

(1) If S (A₁) ≻ S (A₂), then A₁ ≻ A₂,

(2) If S (A₁) = S (A₂), then

if Ac (A₁) ≻ Ac (A₂), then A₁ ≻ A₂,

if Ac (A₁) ≺ Ac (A₂), then A₁ ≺ A₂.

Definition 8. For two SFNs A₁ and A₂ defined above in x = {x₁, x₂, . . . , x _n }, the normalized Hamming distance is described as follows: D H = 1 n ∑ i = 1 n ( | μ 2 A 1 ( x i ) - μ 2 A 2 ( x i ) | + | η 2 A 1 ( x i ) - η 2 A 2 ( x i ) | + | ν 2 A 1 ( x i ) - ν 2 A 2 ( x i ) | )(11)

Mohammad proposed a method for attribute weight determination by a combination of data itself and Indifferent threshold (IT _j ). IT _j is generally given by experts. Its meaning is the level of variation between schemes under attribute j. Within the given value, it is considered as no variation between schemes. Proceed as follow:

Step 1. Construct the decision matrix that includes m schemes and n attributes, and determine its IT _j . a _ij represents the jth attribute of the ith scheme, and IT _j represents the IT of the jth attribute.

Step 2. Normalize the data and get the normalized indifference threshold (NIT _j ). α i j = a ij ∑ i = 1 m a ij , ∀ i ∈ m , j ∈ n(12) NI T j = I T j ∑ i = 1 m a ij , ∀ j ∈ n(13)

Step 3. Sort α _{i j} in ascending order, mark as β _{i j} .

Step 4. Calculate the difference between two adjacent values and get difference matrix γ _ij . γ ij = β i + 1 , j - β ij(14)

Step 5. Compare γ _ij and NIT _j to determine the weight adjustment factor τ _j . τ i j = { γ ij - NI T j , γ ij ≻ NI T j 0 , γ ij ≤ NI T j , ∀ i ∈ m , j ∈ n(15)

If the difference between two adjacent decision information (γ _ij ) is greater than the NIT _j , it has an impact on the attribute weight determination and should be attributed to the attribute weight adjustment factor τ _j .

Step 6. Calculate the total weight adjustment factor ν _ij with weight w _j . ν ij = ∑ i = 1 m τ ij , w j = ν ij ∑ j = 1 n ν ij(16)

ELECTRE III is a MADM method in the European School, which inherits the European School’s idea [34] of hierarchical relationships in the decision-making process. It is based on the comparison of the schemes under each attribute to determine the dominant relationship and ultimately the scheme ranking.

The solution process consists of four parts, which are determining the threshold value (strictly better than the threshold p _j , non-differential threshold q _j , and veto threshold v _j ), calculating the consistency index (C (i, k)), calculating the non-consistency index (d _j  (i, k)), and calculating the credibility (S (i, k)) and ranking.

Firstly, assume that there are m schemes and n attributes with attribute weights w _j . a _ij denotes the evaluation value of the ith scheme under the jth attribute.

Step 1. Calculate the threshold values p _j , q _j and v _j .

The threshold is usually given by an experienced expert and satisfies the relation q _j  ≤ p _j  ≤ v _j .

It can also be obtained computationally. Corazza et al. [35] proposed the relation for determining the threshold from the extreme difference (S _j ). S j = max a j - min a j , q j = 1 6 S j , p j = 2 3 S j , v j = 5 6 S j(17)

Step 2. Calculate consistency index (C (i, k)).

We define C (i, k) as the degree of dominance of scheme i over scheme k. For the degree of dominance under a single attribute weight we define c _j  (i, k). Thus C ( i , k ) = ∑ j = 1 n w j c j ( i , k ) ∑ j = 1 n w j(18) where c j ( i , k ) = { 0 , a k j - a ij ≥ p j 1 , a k j - a ij ≤ q j a ij - a kj + p j p j - q j , otherwise(19)

For schemes i and k, the larger the difference, the smaller c _j  (i, k). When c _j  (i, k) =0, it indicates that scheme i is superior to k to the extent of 0 under attribute j and scheme k is strictly superior to scheme i. When c _j  (i, k) =1, it indicates that there is no significant difference between scheme i and scheme k under attribute j.

Step 3. Calculate the non-conformity index (d _j  (i, k)).

In contrast to consistency, non-consistency measures the degree to which scheme i is inferior to scheme k. d j ( i , k ) = { 0 , a k j - a ij ≤ p j 1 , a k j - a ij ≥ v j a kj - a ij - p j v j - p j , otherwise(20)

For schemes i and k, the larger the difference, the larger d _j  (i, k). When d _j  (i, k) =0, indicates that the rejection of scheme i is better than the rejection of scheme k to the extent of 0, scheme i is not better than scheme k. When d _j  (i, k) =1, indicating that rejection of scheme i is better than a rejection of scheme k to the extent of 1, scheme i is inferior to scheme k.

Step 4. Calculate credibility index (S (i, k)). S ( i , k ) = { C ( i , k ) ∏ 1 - d j ( i , k ) 1 - C ( i , k ) , j ∈ n , d j ( i , k ) ≻ C ( i , k ) C ( i , k ) , max { d j ( i , k ) } ≤ C ( i , k )(21)

From S (i, k) we can see that the role of d _j  (i, k) is mainly to make the poor data uncompensable. When scheme i is compared with scheme k, if d _j  (i, k) =0, then S (i, k) =0 and the confidence that scheme i is better than scheme k is 0. We can get that even though most of the indicators are better than other schemes, a defect directly negates the scheme’s advantage. The d _j  (i, k) makes decision more cautious.

Finally, we derive the net credibility and the ranking of the programs.

Calculate consistent confidence ( B i + ) B + i = ∑ k = 1 , k ≠ i m S ( i , k )(22)

Calculate non-consistent confidence ( B i - ) B - i = ∑ k = 1 , k ≠ i m S ( k , i )(23)

Calculate net confidence (B _i ) B i = B i + - B i -(24)

Suppose that X has a set Ω = {A, ¬ A} loaded with two states and a set A _C  = {a _P , a _B , a _N } with three actions acceptance, delay and rejection. Respectively, x ∈ POS (x), x ∈ BND (x) and x ∈ NEG (x) indicate that x belongs to the positive region, the boundary region, and the negative region, indicating the attitude to the program of the acceptance of decision, delayed decision and rejection of decision.

Then we can derive the loss function matrix as in Table 1, where λ _pp , λ _BP and λ _NP denotes the loss of taking action a _P , a _B and a _N when x belongs to A, λ _PN , λ _BN and λ _NN denotes the loss of taking action a _P , a _B and a _N when x does not belong to A.

Pr(C| [x]) is the possibility of each action, and Pr(C| [x]) + Pr(¬ C| [x]) =1. For x, the expected loss is: R ( a P ∣ [ x ] ) = λ PP Pr ( C ∣ [ x ] ) + λ PN Pr ( ¬ C ∣ [ x ] ) R ( a B ∣ [ x ] ) = λ BP Pr ( C ∣ [ x ] ) + λ BN Pr ( ¬ C ∣ [ x ] ) R ( a N ∣ [ x ] ) = λ NP Pr ( C ∣ [ x ] ) + λ NN Pr ( ¬ C ∣ [ x ] )(25)

According to Bayesian minimum risk theory, people always choose the action with the least expected loss to make the decision result. At the same time, the loss of the delayed strategy is generally located between the acceptance strategy and rejection strategy. There is the following relationship: λ PP ≤ λ BP ≤ λ NP λ PN ≥ λ BN ≥ λ NN(26)

Therefore, we can draw the following conclusions: ( P ) R ( a P | [ x ] ) ≤ R ( a B | [ x ] ) ∧ R ( a P | [ x ] ) ≤ R ( a N | [ x ] ) , x ∈ POS ( x ) ( B ) R ( a B | [ x ] ) ≤ R ( a P | [ x ] ) ∧ R ( a B | [ x ] ) ≤ R ( a N | [ x ] ) , x ∈ BND ( x ) ( N ) R ( a N | [ x ] ) ≤ R ( a P | [ x ] ) ∧ R ( a N | [ x ] ) ≤ R ( a B | [ x ] ) , x ∈ NEG ( x )(27)

5.1 Numerical example in SF-ITARA-ELECTRE III-3WD

The COVID-19 infestation has had a major impact on both production and life, and the supply of materials in the infected areas has become a concern. For manufacturers, it is crucial to understand the status of customers in time and ensure an adequate supply of materials for customers under the condition of a small inventory. Effective analysis of customer demand and reasonable arrangement of supply becomes an effective means to enhance competitiveness. During COVID-19, emergency logistics became the lifeline for the survival of a city’s inhabitants. For the researches and improvement of emergency logistics, Chang et al. proposed to use emergency center, transportation, demand point, supply point, distance matrix and routing schedule as the index system for calculating post-earthquake emergency logistics. According to the actual situation, it is known that the urban population density is proportional to the smoothness of logistics, and the greater the population density, the greater the probability of COVID-19 transmission, and the stricter the logistics control. Therefore, we have used a system of indicators for assessing local customers in terms of the smoothness of logistics within 3 days, material scarcity, profit and the smoothness since the outbreak of COVID-19. At the moment, there are 8 customers in 8 regions who place orders with a manufacturer at the same time. However, due to the different demands and the current epidemic control policies in their regions, the manufacturer needs to analyze their situation and choose the order of production to ensure the order is completed.

The production company invites several experts to evaluate 4 attributes (a₁ : Smoothness of logistics within 3 days,  a₂ : Material scarcity,  a₃ : Profit level,  a₄ : Smoothness logistics since the outbreak of COVID - 19) of 8 customers (c₁, c₂, c₃, c₄, c₅, c₆, c₇, c₈), forming a spherical fuzzy decision information matrix of 8*4. a _ij  =< μ _ij , η _ij , ν _ij  > denotes the information of experts’ evaluation of the ith customer under the jth attribute. The decision matrix is shown in Table 3.

We use the proposed SF-ITARA-ELECTRE III-3WD model to evaluate 8 customers and determine the production allocation sequence.

Step 1. Find the positive ideal scheme and calculate the distance between each scheme to them by Equation (11) shown in Table 4. The ideal solution is calculated as follows:

a j + = ( < 0.6 , 0.1 , 0.4 > , < 0.8 , 0.1 , 0.3 > , < 1.0 , 0.0 , 0.0 > , < 0.8 , 0.1 , 0.2 > )

Step 2. Sort a _{i j} in ascending order and denote them as β _{i j} . The β _{i j} matrix is in Table 5.

Step 3. Measure the distance of adjacent decision information γ _{i j} by Equation (11) in Table 6 and compare with NIT _j . (Experts believe that NIT₁ = 0.16,  NIT₂ = 0.19,  NIT₃ = 0.30,  NIT₄ = 0.20).

Step 4. Calculate the τ _ij , attribute adjustment v _j , and attribute weight w _j by Equation (15) and (16) in Table 7. We get the w _j  = (0.40, 0.23, 0.21, 0.16).

Step 5. Calculate the threshold values p _j , q _j and v _j .

According to the distance between decision information and the positive ideal scheme in Table 4, we learn that the best and worst schemes under the 1st attribute are a₂ and a₇, therefore, the extreme difference within its scheme (S₁) we denote by D (a₂₁, a₇₁) =0.91. Similarly, we obtain the extreme differences (S _j ) under each attribute. As well as using Equation (31), we get the preference threshold (p _j ), undifferentiated threshold (q _j ), and veto threshold (v _j ) under each attribute, show in Table 8.

Step 6. Calculate consistency index (C (i, k)).

According to Equation (32) and (33), we obtain the consistency index of each attribute (c _j  (i, k)) in Tables 9–12, in turn, the consistency index of each scheme (C (i, k)) in Table 13.

Step 7. Calculate non-consistency index (d _j  (i, k)) by Equation (34) show in Tables 14–17.

Step 8. Calculate credibility index (S (i, k)) by Equation (35) shown in Table 18.

Step 9. Calculate consistent confidence ( B i + ), non-consistent confidence ( B i - ), and net confidence (B _i ) (possibility) by Equations (22)–(24), and (38) show in Table 19. Possibility ( B i ) = B i - B i - B i + - B i -(38)

Step 10. Get the loss matrix for each scheme by Equation (36) in Table 20. (Suppose that σ = 0.3)

Step 11. Calculate the expected loss by Equation (25) and get the expected loss, division, and ranking in Table 21 and Fig. 2.

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Fig. 2

Expected loss and region.

In this section, we compare the division and ranking results of the proposed method with the existing MADM method to test the scientific validity of our method.

We select SF-EVAMIX-3WD, SF-TOPSIS-3WD, SF-VIKOR-3WD, and SF-PROM ETHEE-3WD methods to compare with the proposed method, and the results of the schemes division and ranking are shown in Tables 22, 23 and Figs. 3, 4. (where the risk aversion coefficients σ are 0.30).

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Fig. 3

Division in different methods.

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Fig. 4

Ranking in different methods.

In terms of division, regardless of the method c₁ and c₅ are always in the POS region, and the rest of the schemes vary due to the differences in the possibility obtained by each method. In terms of ranking, c₁ and c₅ also always occupy the top two places, c₃ and c₇ have records of ranking last, and c₆ ranking changes more unstable.

We analyze the ranking of the schemes obtained by several methods by applying the Spearman rank consistency coefficient, as shown in Fig. 5.

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Fig. 5

Speraman rank consistency coefficient in different methods.

By checking the table we know that when correlation coefficient >0.643 at n=8 indicates a strong correlation at the confidence interval 95% level.

The correlation coefficient analysis shows that, except for SF-ELECTRE III-3WD and SF-TOPSIS-3WD, the correlation coefficients between the methods are greater than 0.643, which has a strong correlation, indicating that the results of the proposed methods are more reliable.

In addition, when analyzing the Spearman rank correlation coefficient, it is found that SF-TOPSIS-3WD has the lowest average correlation with the remaining four methods, and a comparison of the program rankings revealed that c₈ is only in the last three in SF-TOPSIS-3WD, with a large gap with the remaining results.

Additionally, SF-ELECTRE III-3WD can partition the alternatives evenly in one go compared with the existing method proposed in this paper. Observing the Table 22, it can be seen that the proposed method can quickly eliminate two obvious inferior solutions c₆ and c₇ compared with SF-EVAMIX-3WD and SF-TOPSIS-3WD, while SF-VIKOR-3WD and SF-PROMETHEE-3WD have 3 and 4 solutions respectively in the POS partition, which still have more obvious selection difficulties.

Also, ELECTRE III itself is a non-compensatory MADM method, and in this case, scheme c₆ performs the worst under attribute a₄, and even though scheme c₆ is ranked highly under the remaining attributes, the final result has the inability to be in the top. And through Table 22 we find that both SF-EVAMIX-3WD and SF-TOPSIS-3WD place it in the third place, and in SF-VIKOR-3WD and SF-PROMETHEE-3WD, scheme c₆ is also ranked ahead of SF-ELECTRE III-3WD.

In summary, the consistency of the results obtained by several methods is high, and the proposed method has strong scientific validity. The combination of ELECTRE III and 3WD can be used to quickly eliminate significantly inferior alternatives, resulting in an accurate and efficient decision process, and extending it to a spherical ambiguous environment, which greatly expands its application area, for example, in the most common supplier decision scenario when a supplier has a particular poor attribute, it should be excluded at the first time.

Due to the different attitudes of firms towards risk, different values of risk aversion coefficients are derived, yet σ is always between [0.0,0.5]. Different risk aversion coefficients can have an impact on the division results of 3WD, which in turn affects the ranking of customers. Therefore, we analyze the division and ranking results under different σ to find the differences and connections that exist.

We take a minimum σ of 0.10, a maximum σ of 0.50, and the step size of 0.05, then obtained 9 sets of sensitivity analysis results, show in Tables 24, 25 and Figs. 6, 7.

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Fig. 6

Division in different σ.

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Fig. 7

Ranking in different σ.

From the division results, it can be seen that as the risk aversion coefficient σ increases from 0.10 to 0.45, the number of customers in the BND region decreases from 5 to 0, indicating that the larger the σ, the closer the result is to the two-way decision. And in the real decision-making situation, due to the uncertainty and fuzziness of information, we generally take σ = 0.3 as the risk avoidance coefficient. The result in this case is experimentally verified to have a more uniform distribution of programs, which can achieve both the decision-making effect and the actual situation.

By observing the decision results under each σ, we find that when σ is at [0.10,0.25], its division and ranking results are the same, indicating that when σ ≤ 0.25, there are highly similar in its results. When σ is at [0.30,0.35], its division is more uniform and the ranking only changes for c₃, and this interval is also the most widely used in real life. When σ is at [0.40,0.50], its ranking is completely consistent, indicating that when the risk aversion level reaches 80%, it can already produce quite reliable results, and when σ reaches 0.45, its division and ranking results are completely consistent with 0.50.

Regardless of the value of σ, c₁, c₅ always be in the POS region, c₆ always be in the NEG region, and c₅ always be the most desirable customer, so the company should consider prioritizing stocking for c₅ and c₁ to issue.

We analyze the ranking of the schemes obtained by different σ by applying the Spearman rank consistency coefficient show in Fig. 8.

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Fig. 8

Ranking in different σ.

By observational analysis we found that the minimum Spearman’s rank correlation coefficient is 0.69, which has reached a confidence level of more than 95%, indicating that the proposed method is insensitive to changes in σ.