Heterogeneous multi-attribute case retrieval method based on group decision making using incomplete weight information

2.1 Neutrosophic set

Definition 1. [20] Let X be a space of points (objects), with a generic element in X denoted by x. An NS A in X is characterized by a truth-membership function T _A (x), an indeterminacy-membership function I _A (x), and a falsity-membership function F _A (x). T _A (x), I _A (x), and F _A (x) are real standard or nonstandard subsets of ] 0^-, 1⁺ [, that is, T _A (x) :  X →]0^-, 1⁺ [, I _A (x) :  X →]0^-, 1⁺ [, and F _A (x) :  X →]0^-, 1⁺ [. There is no restriction on the sum of T _A (x), I _A (x), and F _A (x); therefore 0^- ⩽ sup T _A (x) + sup I _A (x) + sup F _A (x) ⩽3⁺.

Definition 2. [31] Let X be a space of points (objects), with a generic element in X denoted by x. An NS A in X is characterized by T _A (x), I _A (x), and F _A (x), which are singleton subintervals/subsets in the real standard [0, 1]; i.e., T _A (x) :  X → [0, 1], I _A (x) :  X → [0, 1], and F _A (x) :  X → [0, 1]. Then, a simplification of A is denoted by A = { < x , T A ( x ) , I A ( x ) , F A ( x ) | x ∈ X > } which is called a single-valued NS and is a subclass of NSs. If, in particular, X has only one element, A is called a single-valued NN, which can be denoted by A =< T _A , I _A , F _A  >.

The operations of NNs are also defined by Ye [31].

Definition 3. [31] Given two single-valued NNs h₁ and h₂,

h₁ ⊕ h₂ = h₂ ⊕ h₁

Definition 4. [31] Let A _j (j = 1, 2, …, n) be a single-valued NS. The simplified neutrosophic weighted geometric average operator is defined by G w ( A 1 , A 2 , … , A n ) = < 1 - ∏ j = 1 n ( 1 - T A j ) w j , ∏ j = 1 n ( I A j ) w j , ∏ j = 1 n ( F A j ) w j > where W =(w₁, w₂, …, w _n ) is the weight vector of A _j , w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1 .

2.2 Incomplete weight information

In the real world, decision-making problems are often complex and uncertain, and the weights of the attributes or/and the weights of relative importance of each expert are incomplete. Several studies have analyzed this incomplete weight information [32–34], which can be expressed in the following five forms, Δ _t (t = 1, 2, 3, 4, 5), which are the subsets of the weight vectors in Δ₀. Let W = {w₁, w₂, …, w _j , …, w _h } be a weight vector of relative importance of each expert or attributes, H = {1, 2, …, h}, j ∈ H, such that ∑ j = 1 h w j = 1 and 0 ⩽ w _j  ⩽ 1.

(1) Weak ranking:

Δ₁ = {W ∈ Δ₀|w _λ  ⩾ w _j ,  for all λ ∈ I₁ and j ∈ J₁}, where I₁ and J₁ are two disjoint subsets of H.

(2) Strict ranking:

Δ₂ = {W ∈ Δ₀|μ _λj  ⩾ w _λ  - w _j  ⩾ θ _λj ,  for all λ ∈ I₂ and j ∈ J₂}, where μ _λj  > 0 and θ _λj  > 0 are constants satisfying μ _λj  > θ _λj , I₂ and J₂ are two disjoint subsets of H.

(3) Difference ranking:

Δ₃ = {W ∈ Δ₀|w _λ  - w _j  ⩾ w _γ  - w _θ ,  for all λ ∈ I₃ , j ∈ J₃, γ ∈ K₃ and θ ∈ L₃}, where I₃, J₃, K₃, and L₃ are four disjoint subsets of H.

(4) Ranking with multiples:

Δ₄ = {W ∈ Δ₀|w _λ  ⩾ ρ _λj w _j ,  for all λ ∈ I₄ andj ∈ J₄}, where ρ _λj  > 0, and I₄ and J₄ are two disjoint subsets of H.

(5) Interval ranking:

Δ₅ = {W ∈ Δ₀|ζ _λ  ⩽ w _λ  ⩽ ξ _λ  + ɛ _λ ,  for all λ ∈ I₅ }, where ζ _λ  > 0 and ɛ _λ  > 0 are two constants, and I₅ is the subset of H.

3.1 Determination of case similarity

In this study, heterogeneous problem attribute X _j (j ∈ {1, 2, …, n}) is described using six formats of attributes, i.e., crisp numbers, interval numbers, linguistic variables, intuitionistic fuzzy numbers, single-valued NNs, and interval-valued NNs. Let Ω₁, Ω₂, Ω₃, Ω₄, Ω₅, and Ω₆ denote the subsets of the attribute vectors corresponding to the crisp numbers, interval numbers, linguistic variables, intuitionistic fuzzy numbners, single-valued NNs, and interval-valued NNs, respectively. Additionally, Ω ^α ∩ Ω ^β  = ø (α, β = 1, 2, …, 6, α ≠ β). Let S = {s₀, s₁, …, s _t , …, s_L-1} be the basic linguistic term set, t ∈ {0, 1, …, L - 1}. Let x _ij denote the jth attribute value of the historical case C _i (i ∈ {1, 2, …, m}), and x_0j denotes the jth attribute value of target case C₀. For the sake of convenience, the six formats of x _ij and x_0j are expressed as follows: x ij = { x ij , ( x ij ∈ Ω 1 ) [ x ij - , x ij + ] , ( x ij ∈ Ω 2 ) s ij , ( x ij ∈ Ω 3 ) < u ij , v ij > , ( x ij ∈ Ω 4 ) < T ij , I ij , F ij > , ( x ij ∈ Ω 5 ) < [ T ij - , T ij + ] , [ I ij - , I ij + ] , [ F ij - , F ij + ] > , ( x ij ∈ Ω 6 )

Let w j P denote the jth attribute weight, such that ∑ j = 1 n w j P = 1 , and 0 ⩽ w j P ⩽ 1 . Let d _j (C₀, C _i ) denote the attribute distance between target case C₀ and historical case C _i , with respect to attribute X _j . Then, attribute distances d _j (C₀, C _i ) for the six types of attributes are defined as follows: d j ( C 0 , C i ) = { | x ij - x 0 j | d j ′ max , X j ∈ Ω 1 ( x ij + - x 0 j + ) + ( x ij - - x 0 j - ) d j ′ ′ max , X j ∈ Ω 2 1 ( N 2 ) ( Δ - 1 ( s ij ) - Δ - 1 ( s 0 j ) ) , X j ∈ Ω 3 ( u ij - u 0 j ) 2 + ( v ij - v 0 j ) 2 + ( π ij - π 0 j ) 2 d j ′ ′ ′ max , X j ∈ Ω 4 1 3 ( ( T ij - T 0 j ) 2 + ( I ij - I 0 j ) 2 + ( F ij - F 0 j ) 2 ) d ˜ j ′ max , X j ∈ Ω 5 1 6 ( ( T ij - - T 0 j - ) 2 + ( T ij + - T 0 j + ) 2 + ( I ij - - I 0 j - ) 2 + ( I ij + - I 0 j + ) 2 + ( F ij + - F 0 j + ) 2 + ( F ij - - F 0 j - ) 2 ) d ˜ j ′ max X j ∈ Ω 6(1) where d j ′ max = max { | x ij - x 0 j | | i ∈ { 1 , 2 , … , m } } , d j ′ ′ max = max { ( x ij + - x 0 j + ) 2 + ( x ij - - x 0 j - ) 2 | i ∈ { 1 , 2 , … , m } } , Δ^-1(s _ij ) = ρ(s _t ) = t(L - 1)/(N - 1), d j ′ ′ ′ max = max { ( u ij - u 0 j ) 2 + ( v ij - v 0 j ) 2 + ( π ij - π 0 j ) 2 | i ∈ { 1 , 2 , … , m } } , π _ij  = 1 - u _ij  - v _ij , π_0j = 1 - u_0j - v_0j, d ˜ j max ′ = { 1 3 ( ( T ij - T 0 j ) 2 + ( I ij - I 0 j ) 2 + ( F ij - F 0 j ) 2 ) | i ∈ { 1 , 2 , … , m } } , d ˜ j ′ ′ max = { 1 6 ( ( T ij - - T 0 j - ) 2 + ( T ij + - T 0 j + ) 2 + ( I ij - - I 0 j - ) 2 + ( I ij + - I 0 j + ) 2 + ( F ij - - F 0 j - ) 2 + ( F ij + - F 0 j + ) 2 ) | i ∈ { 1 , 2 , … , m } } .

Let Sim _j (C₀, C _i ) denote the attribute similarity between the target case C₀ and historical case C _i , with respect to attribute X _j . According to the method proposed by Fan et al. [8], the inverse exponential function is used to construct the attribute similarity measurement. The problem attribute similarity Sim _j (C₀, C _i ) can be calculated as follows: Sim j ( C 0 , C i ) = exp ( - d j ( C 0 , C i ) )(2)

To aggregate the attribute similarities, the attribute weights need to be determined first. To calculate the attribute weights, an optimized model based on the method of maximizing deviation was constructed. Wang et al. [35] were the first to propose this method of maximizing deviation. The main concept of this method is that if the deviation value between the different historical cases is significantly large, the attribute should be assigned a large weight because the attribute plays an important role in the decision-making process; if the deviation value between the different historical cases is very small, the attribute should be assigned a small weight because the attribute plays a relatively small role in the decision-making process. Based on this, an optimized model, which maximizes the overall deviation for all the case attributes, was constructed to determine the weights. The detailed processes are described as follows:

Step 1: Let D _ij denote the variance between historical case C _i and a different historical case C _f (f ∈ {1, 2, …, m}), with respect to attribute X _j . Then, D _ij can be defined as follows: D ij = ∑ f = 1 m ( w j P d j ( C i , C f ) )(3)

Step 2: Let D _j denote the deviation values of all the historical cases from the other historical cases, with respect to attribute X _j . Then, D _j can be expressed as follows: D j = ∑ i = 1 m D ij = ∑ i = 1 m ∑ f = 1 m ( w j P d j ( C i , C f ) )(4)

Step 3: Based on this analysis, it is appropriate to maximize the deviation values of all the attributes to determine the optimal attribute weights. Thus, a maximization deviation model is constructed as follows: max D P = ∑ j = 1 n ∑ i = 1 m ∑ l = 1 m ( w j P d j ( C i , C f ) )(5) s . t . ( w 1 P , w 2 P , … , w n P ) ∈ Δ 0 where Δ₀ are the constraints of the incomplete weights. This model can be executed using the Lingo 11.0 optimization modeling software. By solving this model, it is possible to obtain the optimal weights of attributes W P = { w 1 P , w 2 P , … , w n P } .

Let Sim(C₀, C _i ) denote the problem case similarity between historical case C _i and target case C₀. Then, Sim(C₀, C _i ) can be obtained using the simple additive weighting method, as follows: Sim ( C 0 , C i ) = ∑ j = 1 n w j P Sim j ( C 0 , C i )(6)

From this it is clear that Sim(C₀, C _i ) ∈ [0, 1]. The greater Sim(C₀, C _i ) is, the more similar the historical case C _i is to target case C₀.

For EDM based on CBR, when the case similarity reaches a certain threshold, it can effectively serve as a reference for generating the target case’s alternative [34]. Therefore, a similar historical case set should be constructed according to the case similarities. The threshold of case similarity γ is set according to the experience and knowledge of DMs, such that γ ∈ [min {Sim(C₀, C _i ) |i ∈ {1, 2, …, m}}, max {Sim(C₀, C _i ) |i ∈ {1, 2, …, m})}]. The higher the threshold of case similarity is, the higher is the DMs’ requirement for case similarity.

When a historical case C _i satisfies the condition Sim ( C 0 , C i ) ⩾ γ , it is selected. Then, a similar historical case set Z _v (v ∈ {1, 2, …, b}) is constructed, such that Z _v  = {C _i |i ∈ N _v }, where N _v  = {i|Sim(C₀, C _i ) ⩾ γ, i ∈ M}, which represents the subscript of the historical case when it reaches the case similarity threshold.

3.3 Determination of DMs’ evaluation utility for the similar historical case set

Given the evaluation attribute set of the emergency alternative from the similar historical case set Z _v applied to the current emergency R = {R₁, R₂, …, R _l , … R _h }, l ∈ {1, 2, ⋯ , h}, the corresponding evaluation attribute weights set is W R = { w 1 R , w 2 R , … , w h R } , such that ∑ l = 1 h w l R = 1 and 0 ⩽ w l R ⩽ 1 . Let evaluation value of the attribute R _l , with respect to the similar historical case set Z _v , be given by the kth DM DM _k (k = {1, 2, …, g}), expressed as r vl k ; r vl k is described as an single-valued NN, i.e., r vl k = ( T vl k , I vl k , F vl k ) . Let W D = { w 1 D , w 2 D , … , w g D } denote the weights of relative importance of the DMs, such that ∑ k = 1 g w k D = 1 and 0 ⩽ w k D ⩽ 1 .

When the case similarity set is built, the most appropriate historical case should be selected. Currently, the most common method is the GDM method. Therefore, the GDM method was utilized in this study to determine the most suitable historical case. The selection steps are described as follows.

First, normalize the DMs’ evaluation information. For the benefit evaluation attribute, the value of the attribute is kept unchanged, i.e., r vl k = r ¯ vl k = ( T ¯ vl k , I ¯ vl k , F ¯ vl k ) = ( T vl k , I vl k , F vl k ) . For the cost evaluation attribute, the complement of the DMs’ evaluation value is taken, i.e., r vl k = r ¯ vl k = ( T ¯ vl k , I ¯ vl k , F ¯ vl k ) = ( 1 - T vl k , 1 - I vl k , 1 - F vl k ) .

Then, the evaluation attribute distance d ( r vl k , r xl k ) according to evaluation attribute R _l can be expressed as d ( r vl k , r xl k ) = 1 3 ( ( T vl k - T xl k ) 2 + ( I vl k - T xl k ) 2 + ( F vl k - F xl k ) 2 ) d max(7) where d^max = max { 1 3 ( ( T vl k - T xl k ) + ( I vl k - I xl k ) + ( F vl k - F xl k ) ) | v ∈ { 1 , 2 , … , b } } .

According to the analysis in Model (5), a maximization model was constructed to determine the weight of the evaluation attributes w l R , max D = ∑ k = 1 g ∑ v = 1 b ∑ l = 1 h ( w l R d ( r vl k , r xl k ) )(8) s . t . ( w 1 R , w 2 R , … , w h R ) ∈ Δ 0

This model can be executed using the Lingo 11.0 software. By solving this model, it is possible to obtain the optimal weights of evaluation attribute, W R = { w 1 R , w 2 R , … , w h R } .

For the kth DM DM _k and evaluation attribute R _l , the positive and negative values, i.e., ( r ¯ l k ) + , ( r ¯ l k ) - , can be defined as follows.

Positive values: ( r ¯ l k ) + = { max { ( T vl k , I vl k , F vl k ) | v ∈ } , benefit max { ( ( 1 - T vl k ) , ( 1 - I vl k ) , ( 1 - F vl k ) ) | v ∈ } , cost(9) Negative values: ( r ¯ l k ) - = { min { ( T vl k , I vl k , F vl k ) | v ∈ } , benefit min { ( ( 1 - T vl k ) , ( 1 - I vl k ) , ( 1 - F vl k ) ) | v ∈ } , cost(10)

For DM _k , let ( D v k ) + and ( D v k ) - denote the distances between the evaluation of the similar historical case, and the positive and negative values, respectively. Then the calculation can be performed as follows: ( D v k ) + = ∑ l = 1 h w l R d + ( r ¯ vl k , ( r ¯ l k ) + )(11) ( D v k ) - = ∑ l = 1 h w l R d - ( r ¯ vl k , ( r ¯ l k ) - )(12) where d + ( r ¯ vl k , ( r ¯ l k ) + ) and d - ( r ¯ vl k , ( r ¯ l k ) + ) can be obtained using the distance measurement of Equation (7).

Then, for DM _k , the relative closeness CI v k of the evaluation of the similar historical case to the positive values can be defined as CI v k = ( D v k ) + ( D v k ) - + ( D v k ) +(13)

From this it is clear that the larger the value of CI v k , the larger the dispersion measurement. Accordingly, evaluation attribute CI v k becomes more important [36].

According to the above analysis, a maximization model was created to determine the DMs’ weights. max CI = ∑ k = 1 g ∑ v = 1 b w k D CI v k(14) s . t . ( w 1 D , w 2 D , … , w m D ) ∈ Δ 0

This model can be executed using the Lingo 11.0 software. By solving this model, it is possible to obtain the weights of the relative importance of DMs W D = { w 1 D , w 2 D , … , w g D } .

After determining the weights for the evaluation attributes and weights of relative importance of each DM, the normalized DMs’ evaluation information was aggregated according to Definition 4. Let r v k denote the utility of the vth similar historical case with respect to the kth DM and r _v denote the evaluation utility for the vth similar historical case. The aggregation process is defined as follows: r v k = ( T v k , I v k , F v k ) = G w l R ( r v 1 k , r v 2 k , … , r vh k )(15) r v = ( T v , I v , F v ) = G w k D ( r v 1 , r v 2 , … , r v g )(16)

The value of r _v is clearly a single-valued NN. Owing to the need to compare the advantages of the historical cases, it is necessary to convert the single-valued NN into a crisp number. Therefore, we used the cosine theorem to obtain the crisp number [31]. Let S(r _v ) denote the evaluation utility, which is a crisp number and can be expressed as follows: S ( r v ) = T v ( T v ) 2 + ( F v ) 2 + ( I v ) 2(17)

From this it is clear that S(r _v ) ∈ [0, 1]. The greater the value of S(r _v ), the more effective historical case Z _v will be.

To select the most suitable historical case, it is important to consider the case similarities between the historical case, target case, and the DMs’ evaluation information. The product theory was adopted to determine the comprehensive utility. Let U _v denote the comprehensive utility of the similar historical case Z _v . The formula to calculate U _v is as follows: U v = Sim ( C 0 , Z v ) × S ( r v )(18)

Here, U _v  ∈ [0, 1] and the greater the value of U _v , the more suitable historical case Z _v will be to target case C₀.

According to Z _v , it is possible to determine the ranking order of similar historical cases and choose the most appropriate case.

In summary, the procedure for ranking based on the GDM approach is as follows:

Step 1: Calculate attribute distance d _j (C₀, C _i ) according to Equation (1); construct the deviation model to determine attribute weights w j P using Equations (3)–(5); then calculate the problem case similarity Sim(C₀, C _i ) using Equation(6).

Step 2: According to the DMs’ experience and knowledge, a similar historical case set Z v is determined.

Step 3: The DMs then evaluate the emergency plan from the similar historical case set. Then, evaluation attribute weights w l R are determined using Equations (7)–(8); the weights of the relative importance of the DMs are determined using Equations (9)–(14).

Step 4: The aggregated evaluation information is obtained using Equations (15)–(16). Evaluation utility S(r _v ) is obtained using Equation (17).

Step 5: Comprehensive utility U _v is obtained using Equation (18).

4.1 Case study

Example 1. Gas explosions are the most serious disasters in coal mining. They occur suddenly and can be significantly destructive. When a gas explosion happens, DMs need to quickly generate an emergency alternative in response. Because of the suddenness and similarity of gas explosion, DMs can refer to the historical emergency response to generate an alternative. The attributes of a gas explosion, as concerned in this case study, are as follows:

X₁: Number of underground personnel

X₂: Area of impact of the gas explosion

X₃: Original ventilation system status

X₄: Damage to the ventilation system

X₅: Degree of landslide

X₆: Scope of the fire

X₇: O₂ concentration

X₈: CO concentration

X₉: CH₄ concentration

Among these, X₁, X₇, X₈, and X₉ are crisp numbers, X₂ is an interval number, X₃ is a linguistic variable, X₄ is an intuitionistic number, X₅ is a single-valued NN, and X₆ is an interval-valued NN. The information from 10 historical cases {C₁, C₂, …, C₁₀} and target case C₀ is presented in Table 1. The attribute weights have the following constraints:

w 1 P ⩾ 1.2 w 2 P , w 3 P - w 2 P ⩾ 0.05 , w 3 P - w 2 P ⩽ 0.2 ; 0.1 ⩽ w 4 P ⩽ 0.3 , w 5 P - w 4 P ⩾ w 1 P - w 2 P , 0.1 ⩽ w 6 P ⩽ 0.2 , 0.1 ⩽ w 7 P ⩽ 0.2 , and w 8 P - w 9 P ⩾ w 6 P - w 7 P .

Using Equation (1), attribute distance d _j (C₀, C _i ) can be calculated, and the results are presented in Table 2. The attribute similarities Sim _j (C₀, C _i ) are then determined using Equation (2).

Using Equations (3)–(5), a deviation model was constructed to determine the attribute weights, whereby it was found that W ^P  = {0.0833, 0.0600, 0.1400, 0.1000, 0.1500, 0.1500, 0.1167, 0.1167, 0.0833}

Based on the attribute similarities Sim _j (C₀, C _i ) and the attribute weights W ^P , the case similarities Sim(C₀, C _i ) can be determined using Equation(6), whereby the following results were found:

Sim ( C 1 , C 0 ) = 0.6202 , Sim ( C 2 , C 0 ) = 0.5864 , Sim ( C 3 , C 0 ) = 0.6665 , Sim ( C 4 , C 0 ) = 0.6431 , Sim ( C 5 , C 0 ) = 0.6517 , Sim ( C 6 , C 0 ) = 0.6028 , Sim ( C 7 , C 0 ) = 0.5566 , Sim ( C 8 , C 0 ) = 0.5874 , Sim ( C 9 , C 0 ) = 0.6400 , Sim ( C 10 , C 0 ) = 0.6868 .

According to these results and their own knowledge and experience, the DMs set the case similarity threshold at γ = 0.65. Additionally, the set of similar historical cases is Z = {Z₁, Z₂, Z₃} = {C₃, C₅, C₁₀}.

Based on the set of similar cases, three DMs evaluated the emergency plans from similar historical cases. Three main evaluation attributes were considered; i.e., the reduction of casualties (R₁), the reduction of property loss (R₂), and the overall rescue effect (R₃). The evaluation information is presented in Table 3. The evaluation attribute weights meet these conditions: w 1 R ⩾ 1.1 w 2 R and 0.05 ⩽ w 3 R - w 2 R ⩾ 0.1 . The weights of the relative importance of DMs satisfy the conditions w 1 D ⩾ 1.2 w 2 D and 0.1 ⩽ w 3 D - w 2 D ⩽ 0.2 .

Using Equations (7)–(8), the evaluation attribute weights w l R were determined as w 1 R = 0.3565 , w 2 R = 0.3065 , and w 3 R = 0.3371 . Using Equations (9)–(14), the weights of DMs were determined as w 1 D = 0.4500 , w 2 D = 0.2500 , and w 3 D = 0.3000 .

Once the evaluation attribute weights and the weights of the DMs were determined, the evaluation information and the aggregate evaluation information were normalized using Equations (15)–(16). The results of the aggregation are as follows: r 1 = < 0.4754 , 0 , 0 > , r 2 = < 0.4969 , 0.4802 , 0 > , and r 3 = < 0.5481 , 0.4548 , 0 > .

Based on this, the single-valued NN was converted into crisp numbers using Equation (17) and the results are as follows: S ( r 1 ) = 1.0000 , S ( r 2 ) = 0.7191 , and S ( r 3 ) = 0.7695 .

Finally, the comprehensive utility, U _v , is obtained using Equation (18), where the results are as follows: U 1 = 0.6868 , U 2 = 0.4686 , and U 3 = 0.5129 .

The greater the comprehensive utility U _v is, the more suitable the historical case will be. Consequently, according to the calculated comprehensive utilities, U₁ is the most suitable historical case; i.e. C₃, can be retrieved.

Example 2. The case information was taken from [37]. The attribute weights are also given according to [37]; i.e, ∑ i = 1 12 w f = 1 , w₁ = w₄ = w₅, w₃ = w₈ = w₉, w₂ = w₆ = w₇, w₁₀ = w₁₂, w₁₀ < w₂ < w₁ < w₃, 0.07 < w₁ < 0.13, 0.04 < w₂ < 0.1, w₃ > 0.1, w₁₀ < 0.07. The evaluation information of Example 1 is also used in Example 2.

According to Equations (1)–(2), the attribute similarities are calculated. Then, the attribute weights are determined by Equations (3)–(5). Furthermore, the case similarities are determined by Equation (6). The DMs set the threshold of case similarity at 0.65, then, the similar historical case set is constructed; i.e., Z = {Z₁, Z₂, Z₃} = {C₁₁, C₁₂, C₁₈}, and the case similarities are calculated as follows: Sim ( C 0 , C 11 ) = 0.7283 , Sim ( C 0 , C 12 ) = 0.7294 , Sim ( C 0 , C 18 ) = 0.6581 .

Based on case similarities and evaluation utilities, comprehensive utility U _v is obtained as follows: U 1 = 0.7283 , U 2 = 0.5254 , U 3 = 0.5064 .

Therefore, the historical case, C₁₁, is retrieved as the most suitable historical case.

Based on the comparative analysis, the superiority of the proposed method can be summarized as follows:

The proposed method considers six formats of case attributes, which used single-valued NNs, and interval-valued NNs to represent the uncertainty and fuzziness of information. The attribute weights are represented incompletely. These are useful for EDM problems characterized by partial or incomplete information, and limited expertise.