Background
Methods
Alternative anesthesia methods and criteria for evaluation
Criteria | Sub-criteria |
---|---|
Convenience (C1) | Convenience for patient (C11) |
Convenience for doctor (C12) | |
Reliability (C2) | Condition of penis (C21) |
Vital function (C22) | |
Duration (C3) | Duration of anesthesia method (C31) |
Duration of recovery (C32) | |
Psychology (C4) | Psychology of parent (C41) |
Psychology of patient (C42) |
Fuzzy multi-attribute decision making methods
Fuzzy AHP
-
Step 1. Establish an expert team. The quality of the evaluation process depends on experts’ knowledge and experiences.
-
Step 2. Determine the evaluation criteria and construct the hierarchy including alternatives. Literature or questionnaires assists the expert to determine evaluation criteria.
-
Step 3. Construct pairwise comparison matrix and evaluate the relative importance of the criteria. The experts are expected to provide their judgments on the basis of their knowledge.For any expert the pairwise comparison matrix is given by Eq. (1) as:where \( n \) is the number of criteria, \( {\tilde{C}}_k \) is a pairwise comparison matrix belongs to k th expert for k = 1, 2,.., K. Arithmetic mean is used to aggregate experts’ opinion as given in Eq. (2).$$ {\tilde{C}}_k=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill {\tilde{c}}_{12}\dots \hfill & \hfill {\tilde{c}}_{1n}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {\tilde{c}}_{n1}\hfill & \hfill {\tilde{c}}_{n2}\cdots \hfill & \hfill 1\hfill \end{array}\right] $$(1)$$ \tilde{C}=\frac{1}{K}\left({\tilde{C}}^1+{\tilde{C}}^2+\dots +{\tilde{C}}^K\right) $$(2)
-
Step 4. Transform the linguistic terms into triangular fuzzy numbers. The following linguistic terms provided in Table 2 are utilized for the evaluation procedure [60].Table 2Fuzzy evaluation scale for the weightsLinguistic termsTriangular fuzzy scaleEqual (E)(1,1,1)Slightly Important (SI)(1,1,3)Fairly Important (FI)(1,3,5)Highly Important (HI)(3,5,7)Very Important (VI)(5,7,9)Extremely Important (EI)(7,9,9)
-
Step 5. Calculate the fuzzy weight matrix using Buckley’s method as follows using Eqs. (3) and (4).$$ {\tilde{r}}_i={\left({\tilde{c}}_{i1}\otimes {\tilde{c}}_{i2}\otimes \dots \otimes {\tilde{c}}_{in}\right)}^{\frac{1}{n}} $$(3)where \( {\tilde{r}}_i \) is the geometric mean of fuzzy comparison value, \( {\tilde{w}}_i \) is the fuzzy weight of the i th criterion. After the fuzzy relative weight matrix is calculated, fuzzy numbers are defuzzied into crisp values using a common method, centroid method [61], and then apply the normalization procedure as provided in Eq. (5).$$ {\tilde{w}}_i={\tilde{r}}_i\otimes {\left({\tilde{r}}_1+{\tilde{r}}_2+\dots +{\tilde{r}}_n\right)}^{-1} $$(4)$$ {w}_i=\frac{{\tilde{w}}_i}{{\displaystyle {\sum}_{j=1}^n}{\tilde{w}}_j}=\frac{L_i+{M}_i+{U}_i}{{\displaystyle {\sum}_{j=1}^n}{\tilde{w}}_j} $$(5)
-
Step 6. Check the consistency of the pairwise comparison matrices by calculating the consistency ratios.
-
Step 7. Select the best alternative using the weights of criteria and alternatives.
Fuzzy TOPSIS
-
Step 1. Define a decision matrix, D as in Eq. (6). Evaluate the ratings of the alternatives using the linguistic variables and determine the criteria weights.where x ij may be crisp or fuzzy. If it is fuzzy, it is represented by a trapezoidal number as x ij = (a ij , b ij , c ij , d ij ). The fuzzy weights are described by Eq. (7).$$ D=\begin{array}{c}\hfill {A}_1\hfill \\ {}\hfill \dots \hfill \\ {}\hfill {A}_m\hfill \end{array}\left[\begin{array}{ccc}\hfill {x}_{11}\ \hfill & \hfill {x}_{1j}\dots \hfill & \hfill {x}_{1n}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {x}_{m1}\hfill & \hfill {x}_{mj}\cdots \hfill & \hfill {x}_{mn}\hfill \end{array}\right] $$(6)$$ {w}_j=\left({\alpha}_j,{\beta}_j,{\gamma}_j,{\delta}_j\right) $$(7)
-
Step 2. Normalize the decision matrix using the linear scale transformation as in Eq. (8):$$ {r}_{ij}=\left\{\begin{array}{c}\hfill {x}_{ij}/{x}_j^{*},\ \forall j,\ {x}_j is\ a\ benefit\ attribute\hfill \\ {}\hfill {x}_j^{-}/{x}_{ij},\forall j,\ {x}_j is\ a\ cost\ attribute\ \hfill \end{array}\right. $$(8)Then, the normalized decision matrix in Eq. (6) can be written in Eq. (9) by applying Eq. (8) as follows:$$ D=\left[\begin{array}{ccc}\hfill {r}_{11}\hfill & \hfill {r}_{1j}\dots \hfill & \hfill {r}_{1n}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {r}_{m1}\hfill & \hfill {r}_{mj}\dots \hfill & \hfill {r}_{mn}\hfill \end{array}\right] $$(9)When x ij is crisp, its corresponding r ij must be crisp; when x ij is fuzzy, its corresponding r ij must be fuzzy.Let x j * = (a j * , b j * , c j * , d j * ), x j − = (a j − , b j − , c j − , d j − ), and \( {\tilde{x}}_{ij}=\left({a}_{ij},\ {b}_{ij},{c}_{ij},{d}_{ij}\right) \). Replace the Eq. (8) by these fuzzy operations, then obtain Eq. (10):$$ {\tilde{r}}_{ij}=\left\{\begin{array}{c}\hfill {\tilde{x}}_{ij\ }\left(\div \right){\tilde{x}}_j^{*}=\left(\frac{a_{ij}}{d_j^{*}},\frac{b_{ij}}{c_j^{*}},\frac{c_{ij}}{b_j^{*}},\frac{d_{ij}}{a_j^{*}}\right),\ for\ benefit\ attributes\hfill \\ {}\hfill {\tilde{x}}_j^{-}\left(\div \right){\tilde{x}}_{ij} = \left(\frac{a_j^{-}}{d_{ij}},\frac{b_j^{-}}{c_{ij}},\frac{c_j^{-}}{b_{ij}},\frac{d_j^{-}}{a_{ij}}\right),\ for\ cost\ attributes\hfill \end{array}\right. $$(10)
-
Step 3. Construct the fuzzy weighted normalized decision matrix using Eq. (11):where \( {\tilde{w}}_j=\left({\alpha}_j,{\beta}_j,{\gamma}_j,{\delta}_j\right) \).$$ {\tilde{v}}_{ij}={\tilde{r}}_{ij}\otimes {\tilde{w}}_j $$(11)When both r ij and w j are crisp, v ij is crisp. On the other hand, when either r ij or w j (or both) are fuzzy, Eq. (11) can be rewritten as follows in Eq. (12):$$ {\tilde{v}}_{ij}=\left\{\begin{array}{c}\hfill \left(\frac{a_{ij}}{d_j^{*}}{\alpha}_j,\frac{b_{ij}}{c_j^{*}}{\beta}_j,\frac{c_{ij}}{b_j^{*}}{\gamma}_j,\frac{d_{ij}}{a_j^{*}}{\delta}_j\right),\ for\ benefit\ attributes\hfill \\ {}\hfill \left(\frac{a_j^{-}}{d_{ij}}{\alpha}_j,\frac{b_j^{-}}{c_{ij}}{\beta}_j,\frac{c_j^{-}}{b_{ij}}{\gamma}_j,\frac{d_j^{-}}{a_{ij}}{\delta}_j\right),\ for\ cost\ attributes\hfill \end{array}\right. $$(12)The result of Eq. (12) can be summarized as a Ṽ = [ṽ ij ] m × n matrix in Eq. (13):$$ \tilde{V}=\left[\begin{array}{ccc}\hfill {\tilde{v}}_{11}\hfill & \hfill \dots \hfill & \hfill {\tilde{v}}_{1n}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {\tilde{v}}_{m1}\hfill & \hfill \cdots \hfill & \hfill {\tilde{v}}_{mn}\hfill \end{array}\right] $$(13)
-
Step 4. Calculate the fuzzy positive ideal solution (FPIS, ṽ*) and the fuzzy negative ideal solution (FNIS, ṽ −) as defined in the Eqs. (14) and (15) as follows:$$ {A}^{*}=\left[{\tilde{v}}_1^{*},\dots, {\tilde{v}}_n^{*}\right] $$(14)where ṽ j * = max i ṽ ij and ṽ j − = min i ṽ ij .$$ {A}^{-}=\left[{\tilde{v}}_1^{-},\dots, {\tilde{v}}_n^{-}\right] $$(15)
-
Step 6. Compute the closeness coefficient (CC i ), which is defined to determine the ranking order of all alternatives. Once the crisp numbers S i * and S i − , which can be combined, the CC i of each alternative is calculated using the Eq. (20). Then, the alternatives are ranked in descending order of the CC i .$$ C{C}_i=\frac{S_i^{-}}{S_i^{*}+{S}_i^{-}} $$(20)
Participants
Results
Application: anesthesia methods in circumcision surgery
Fuzzy AHP
-
Step 1. The expert team consists of three experts, pediatric surgeons.
-
Step 3. After several surveys have been conducted, experts determined the criteria weights by the pairwise comparison matrix. Constructed consensus matrices are provided in Table 3.Table 3Consensus matricesCriteriaC1C2C3C4C1C11C12C2C21C22C3C31C32C4C41C42C1E1/VIFI1/HIC11EHIC21E1/VIC31EVIC41E1/FIC2VIEVIHIC121/HIEC22VIEC321/VIEC42FIEC31/FI1/VIE1/HIC4HI1/HIHIEC11A1A2A3C12A1A2A3C21A1A2A3C22A1A2A3A1E1/HIFIA1E1/HI1/FIA1E1/SIHIA1ESI1/FIA2HIEHIA2HIEVIA2SIEVIA21/SIEHIA31/FI1/HIEA3FI1/VIEA31/HI1/VIEA3FI1/HIEC31A1A2A3C32A1A2A3C41A1A2A3C42A1A2A3A1E1/VI1/VIA1E1/HIFIA1E1/HISIA1E1/FISIA2VIESIA2HIEVIA2HIEFIA2FIEFIA3VI1/SIEA31/FI1/VIEA31//SI1/FIEA31/SI1/FIE
-
Step 4. The linguistic terms are transformed into triangular fuzzy numbers given in Table 2.
-
Step 5. The final weights of the alternatives are calculated using Eqs. (3), (4) and (5). Illustrative examples for weights of sub-criteria C31 and C32 are given as follows:$$ {\tilde{r}}_{C31}={\left({\tilde{c}}_{C31C31}\otimes {\tilde{c}}_{C31C32}\right)}^{\frac{1}{2}} $$$$ {\tilde{r}}_{C31}={\left(\left(1,1,1\right)\otimes \left(5,7,9\right)\right)}^{\frac{1}{2}} $$$$ {\tilde{r}}_{C31}=\left(2.23,2.64,3\right) $$$$ {\tilde{r}}_{C32}={\left({\tilde{c}}_{C32C31}\otimes {\tilde{c}}_{C32C32}\right)}^{\frac{1}{2}} $$$$ {\tilde{r}}_{C32}={\left(1/\left(5,7,9\right)\otimes \left(1,1,1\right)\right)}^{\frac{1}{2}} $$$$ {\tilde{r}}_{C32}=\left(0.33,0.37,0.44\right) $$$$ {\tilde{w}}_{C31}={\tilde{r}}_{C31}\otimes {\left({\tilde{r}}_{C31}+{\tilde{r}}_{C32}\right)}^{-1} $$$$ {\tilde{w}}_{C31}=\left(2.23,2.64,3\right)\otimes {\left[\left(2.23,2.64,3\right)+\left(0.33,0.37,0.44\right)\right]}^{-1} $$$$ {\tilde{w}}_{C31}=\left(0.64,0.87,1.17\right) $$$$ {\tilde{w}}_{C32}={\tilde{r}}_{C32}\otimes {\left({\tilde{r}}_{C31}+{\tilde{r}}_{C32}\right)}^{-1} $$$$ {\tilde{w}}_{C32}=\left(0.33,0.37,0.44\right)\otimes {\left[\left(2.23,2.64,3\right)+\left(0.33,0.37,0.44\right)\right]}^{-1} $$$$ {\tilde{w}}_{C32}=\left(0.09,0.12,0.17\right) $$$$ {w}_{C31}=\frac{{\tilde{w}}_{C31}}{{\displaystyle {\sum}_{j=1}^2}{\tilde{w}}_{C3j}}=\frac{L_{C31}+{M}_{C31}+{U}_{C31}}{{\tilde{w}}_{C31}+{\tilde{w}}_{C32}} $$$$ {w}_{C31}=\frac{\left(0.64+0.87+1.17\right)}{\left(0.64+0.87+1.17+0.09+0.12+0.17\right)}=0.88 $$$$ {w}_{C32}=\frac{{\tilde{w}}_{C32}}{{\displaystyle {\sum}_{j=1}^2}{\tilde{w}}_{C3j}}=\frac{L_{C32}+{M}_{C32}+{U}_{C32}}{{\tilde{w}}_{C31}+{\tilde{w}}_{C32}} $$$$ {w}_{C32}=\frac{\left(0.09+0.12+0.17\right)}{\left(0.64+0.87+1.17+0.09+0.12+0.17\right)}=0.12 $$The similar calculation approach is applied for all pairwise comparisons. The final weights of the alternatives are provided in Table 4. An illustrative example for W A3 is provided as follows:Table 4Final weightsC1C2C3C40.080.620.050.25C11C12C21C22C31C32C41C420.830.170.140.860.880.120.300.70WA10.270.130.620.390.080.230.240.330.36A20.870.800.880.630.570.880.780.730.69A30.150.180.140.350.450.100.240.250.29$$ {W}_{A3}=0.08\times 0.83\times 0.15+0.08\times 0.17\times 0.18 + \dots +0.25\times 0.30\times 0.24+0.25\times 0.70\times 0.25=0.29 $$
-
Step 6. The consistency ratios of the comparison matrices are checked.
-
Step 7. According to Table 4, the best anesthesia method in circumcision surgery is Alternative 2, general anesthesia with penile block.
Fuzzy TOPSIS
-
Step 1. Experts assess the ratings of the alternatives, provided in Table 5, using a linguistic scale.Table 5Linguistic evaluation of alternatives by each expertExpert 1C1C2C3C4C11C12C21C22C31C32C41C42A1VGVGFGVGGVGGA2FPGFVGPFFA3GFFGFFVGFExpert 2C1C2C3C4C11C12C21C22C31C32C41C42A1GFGFFVFFA2VGVGVGGGVGVGGA3FGPFGFFGExpert 3C1C2C3C4C11C12C21C22C31C32C41C42A1FFGFFGPFA2GGVGGGVGFGA3FVGFFFFPFThe details of the linguistic scale for the evaluation of the alternatives are provided in Table 6.Table 6Fuzzy evaluation scale for the alternativesLinguistic termsTriangular fuzzy scalePoor (P)(0,2.5,5)Fair (F)(2.5,5,7.5)Good (G)(5,7.5,10)Very Good (VG)(7,10,10)
-
Step 2. The linguistic terms are converted into triangular fuzzy numbers. Then, normalized values are computed by dividing each value to 10 which is the largest upper value in the evaluation matrix. As an illustration, the normalized matrix of Expert 3 is provided in Table 7.Table 7Normalized fuzzy decision matrix of Expert 3Normalized MatrixExpert 3C1C2C3C4C11C12C21C22C31C32C41C42A1(0.25,0.5,0.75)(0.25,0.5,0.75)(0.5,0.75,1)(0.25,0.5,0.75)(0.25,0.5,0.75)(0.5,0.75,1)(0,0.25,0.5)(0.25,0.5,0.75)A2(0.5,0.75,1)(0.5,0.75,1)(0.7,1,1)(0.5,0.75,1)(0.5,0.75,1)(0.7,1,1)(0.25,0.5,0.75)(0.5,0.75,1)A3(0.25,0.5,0.75)(0.7,1,1)(0.25,0.5,0.75)(0.25,0.5,0.75)(0.25,0.5,0.75)(0.25,0.5,0.75)(0,0.25,0.5)(0.25,0.5,0.75)
-
Step 3. Weighted normalized fuzzy decision matrix of Expert 3 is provided in Table 8.Table 8Weighted normalized fuzzy decision matrix of Expert 3Expert 3C1 (0.08)C2 (0.66)C3 (0.05)C4 (0.21)C11 (0.90)C12 (0.10)C21 (0.17)C22 (0.83)C31 (0.88)C32 (0.12)C41 (0.40)C42 (0.60)A1(0.018,0.036,0.054)(0.002,0.004,0.006)(0.056,0.084,0.112)(0.136,0.273,0.410)(0.011,0.022,0.033)(0.003,0.004,0.006)(0,0.021,0.042)(0.031,0.063,0.094)A2(0.036,0.054,0.072)(0.004,0.006,0.008)(0.078,0.112,0.112)(0.273,0.410,0.547)(0.022,0.033,0.044)(0.004,0.006,0.006)(0.021,0.042,0.063)(0.063,0.094,0.126)A3(0.018,0.036,0.054)(0.005,0.008,0.008)(0.028,0.056,0.084)(0.273,0.410,0.547)(0.011,0.022,0.033)(0.001,0.003,0.004)(0,0.021,0.042)(0.031,0.063,0.094)Note that, here, to determine the criteria weights of main criteria (C1, C2, etc.) and sub-criteria (C11, C12, etc.), which are presented in parentheses, we utilize the weights obtained from the fuzzy AHP method. For example, the criteria weights associated with the Expert 3 are calculated using the pairwise matrices of the expert provided in Table 9.Table 9Pairwise matrices of Expert 3CriteriaC1C2C3C4C1C11C12C2C21C22C3C31C32C4C41C42C1E1/VIFI1/HIC11EEIC21E1/HIC31EVIC41E1/SIC2VIEEIVIC121/EIEC22HIEC321/VIEC42SIEC31/FI1/EIE1/HIC4HI1/VIHIEC11A1A2A3C12A1A2A3C21A1A2A3C22A1A2A3A1E1/VIFIA1E1/HI1/FIA1E1/SIHIA1ESI1/FIA2VIEHIA2HIEVIA2SIEVIA21/SIEVIA31/FI1/HIEA3FI1/VIEA31/HI1/VIEA3FI1/VIEC31A1A2A3C32A1A2A3C41A1A2A3C42A1A2A3A1E1/VI1/VIA1E1/HIFIA1E1/HISIA1E1/SISIA2VIESIA2HIEVIA2HIESIA2SIEFIA3VI1/SIEA31/FI1/VIEA31/SI1/SIEA31/SI1/FIERegarding to the Table 7, an illustrative example is given for the A3 under C31 as follows:$$ {\tilde{A}}_3^w={w}_{C3}\times {w}_{C31}\times {\tilde{A}}_3 $$$$ {\tilde{A}}_3^w=0.05\times 0.88\times \left(0.25,0.5,0.75\right) $$$$ {\tilde{A}}_3^w=\left(0.011,0.022,0.033\right) $$
-
Step 4. The distances of each alternative from the fuzzy positive ideal solution and the fuzzy negative ideal solution for Expert 3 are given in Tables 10 and 11. An illustrative example is provided for the value given in the third row of the sixth column of Table 10 as follows:Table 10The distances of each alternative from the fuzzy positive ideal solution for Expert 3FPISExpert 3C1C2C3C4CC*C11C12C21C22C31C32C41C42A10.5570.5730.5450.3460.5650.5760.5650.5414.268A20.5460.5740.5190.3460.5580.5740.5530.5234.194A30.5570.5750.5290.4240.5650.5750.5650.5414.331Table 11The distances of each alternative from the fuzzy negative ideal solution for Expert 3FNISExpert 3C1C2C3C4CC−CCiC11C12C21C22C31C32C41C42A10.0220.0040.0350.2460.0140.0020.0160.0390.3780.081A20.0320.0040.0590.2460.0200.0030.0260.0570.4460.096A30.0220.0020.0500.1710.0140.0030.0160.0390.3170.068$$ FPIS,{A}_3^{*}=1/3{\left[{\left(0.011-1\right)}^2+{\left(0.022-1\right)}^2+{\left(0.033-1\right)}^2\right]}^{1/2} $$$$ FPIS,{A}_3^{*}=0.565 $$The value given in the third row of the sixth column of Table 11 is calculated as follows:$$ FNIS,{A}_3^{-}=1/3{\left[{0.011}^2+{0.022}^2+{0.033}^2\right]}^{1/2} $$$$ FNIS,{A}_3^{-}=0.014 $$
-
Steps 5–6. Once the separation measures are determined, closeness coefficients are calculated, which are presented in Tables 10 and 11.The value of CC * (4.331) for Alternative 3, which is obtained by taking the sum of the third row, is provided in the last column of Table 10. Similarly, the value of CC −(0.317) for Alternative 3, is provided by taking the sum of the related row, is in the last column of Table 11.The value of CC i for Alternative 3,CC 3 , is calculated as follows:$$ C{C}_3=\frac{0.317}{4.331+0.317} $$$$ C{C}_3=0.068 $$
Sensitivity analysis
-
Step 1. The calculation procedure provided in the Methods section is applied for each expert’s preferences.
-
Step 2. Various weighting scenarios, which are provided in Table 12, are applied in order to find a joint decision matrix.Table 12Weighting scenariosScenario 1Scenario 2Scenario 3Scenario 4Expert 140%30%30%33.33%Expert 230%40%30%33.33%Expert 330%30%40%33.33%
-
Step 3. Ranking scores with respect to each expert is computed.