Research on the fire safety assessment of high building with intuitionistic fuzzy TOPSIS method

2.1 Intuitionistic fuzzy sets

Definition 1 [2]. An intuitionistic fuzzy set (IFS) on the universe X is an object of the form

I = { ⟨ x , μ I ( x ) , ν I ( x ) ⟩ | x ∈ X . }(1)

where μ I ⁢ ( x ) ∈ [ 0 , 1 ] is called the “degree of membership of I ” and ν I ⁢ ( x ) ∈ [ 0 , 1 ] is called the “degree of non-membership of I ”, and μ I ⁢ ( x ) , ν I ⁢ ( x ) satisfy the following condition: 0 ⩽ μ I ⁢ ( x ) + ν I ⁢ ( x ) ⩽ 1 , ∀ x ∈ X .

Definition 2 [38]. Let I 1 = ( μ 1 , ν 1 ) and I 2 = ( μ 2 , ν 2 ) be two intuitionistic fuzzy numbers (IFNs), the operation formula of them can be defined:

I 1 ⊕ I 2 = ( μ 1 + μ 2 - μ 1 ⁢ μ 2 , ν 1 ⁢ ν 2 )(2) I 1 ⊗ I 2 = ( μ 1 ⁢ μ 2 , ν 1 + ν 2 - ν 1 ⁢ ν 2 )(3) $\lambda I_1=(1-{(1-{\mu }_1)}^{\lambda },{\nu }_1^{\lambda }),\lambda >0$(4) $I_1^{\lambda }=({\mu }_1^{\lambda },1-{(1-{\nu }_1)}^{\lambda }),\lambda >0$(5)

Definition 3 [39]. Let I 1 = ( μ 1 , ν 1 ) and I 2 = ( μ 2 , ν 2 ) be IFNs, the score and accuracy functions of I 1 and I 2 can be expressed:

S ⁢ ( I 1 ) = μ 1 + μ 1 ⁢ ( 1 - μ 1 - ν 1 ) , (6) S ⁢ ( I 2 ) = μ 2 + μ 2 ⁢ ( 1 - μ 2 - ν 2 ) H ⁢ ( I 1 ) = μ 1 + ν 1 , H ⁢ ( I 2 ) = μ 2 + ν 2(7)

For two IFNs I 1 and I 2 , regarding on the Definition 3, then

(1)

if , then ;

Definition 4 [40]. Let I 1 = ( μ 1 , ν 1 ) and I 2 = ( μ 2 , ν 2 ) be IFNs, the novel Hamming distances between two IFNs can be defined as follows:

𝐼𝐹𝐻𝐷 ⁢ ( I 1 , I 2 ) = ℓ 1 + ℓ 2 + ℓ 3 6(8)

where

ℓ 1 = | μ 1 - μ 2 | + | ν 1 - ν 2 | + | ( μ 1 + 1 - ν 1 ) - ( μ 2 + 1 - ν 2 ) | 2 ℓ 2 = π 1 + π 2 2 ℓ 3 = max ⁡ ( | μ 1 - μ 2 | , | ν 1 - ν 2 | , | π 1 - π 2 | 2 )

Under the context of the intuitionistic fuzzy, some aggregation operators will be introduced in this chapter, including intuitionistic fuzzy weighted averaging (IFWA) operator and intuitionistic fuzzy weighted geometric (IFWG) operator.

Definition 5 [38]. Let I j = ( μ I j , ν I j ) ( j = 1 , 2 , ⋯ , n ) be a collection of IFNs, the IFWA operator can be defined as:

𝐼𝐹𝑊𝐴 ω ⁢ ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j ⁢ I j )(9)

where ω = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of I j ( j = 1 , 2 , … , n ) and ${\omega }_j>0$ , ∑ j = 1 n ω j = 1 .

Derived from the Definition 5, the following result can be obtained:

Theorem 1. The aggregated value by using IFWA operator is also a IFN, where

𝐼𝐹𝑊𝐴 ω ⁢ ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j ⁢ I j ) (10) = ( 1 - ∏ j = 1 n ( 1 - μ I j ) ω j , ∏ j = 1 n ( ν I j ) ω j )

where ω = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of I j ( j = 1 , 2 , … , n ) and ${\omega }_j>0$ , ∑ j = 1 n ω j = 1 .

Definition 6 [38]. Let I j ( j = 1 , 2 , ⋯ , n ) be a collection of IFNs, the IFWG operator can be defined as:

𝐼𝐹𝑊𝐺 ω ⁢ ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j(11)

where ω = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of I j ( j = 1 , 2 , … , n ) and ${\omega }_j>0$ , ∑ j = 1 n ω j = 1 .

Derived from the Definition 6, the following result can be obtained:

Theorem 2. The aggregated value by using IFWG operator is also a IFN, where

𝐼𝐹𝑊𝐺 ω ⁢ ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j (12) = ( ∏ j = 1 n ( μ I j ) ω j , 1 - ∏ j = 1 n ( 1 - ν I j ) ω j )

where ω = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of I j ( j = 1 , 2 , … , n ) and ${\omega }_j>0$ , ∑ j = 1 n ω j = 1 .

4.1 An empirical example

Many major malignant fire accident occurred in recent years, more than 80% fire accidents and fire safety evacuation has a direct relationship, due to the high building floors high, large size, many personnel, complex function, when a fire happens, many risk factors affecting the safety evacuation, safety evacuation and escape the preferred method of high-rise building fire. High rise building fire in a burst and come, how to choose the best, the scientific evacuation route, choose the escape routes have scientific significance, whether through calculation and practice, learning, exercise is able to adapt to the evacuation process under different scenarios. Therefore, safety evacuation of high-rise building fire, to further prevent and contain many malignant fire accidents, to ensure the safety of people’s lives and property, has a very important role. In this chapter, an empirical application about fire safety assessment based on the high building will be provided by making use of IF-TOPSIS method. There are five potential high buildings F i ( i = 1 , 2 , 3 , 4 , 5 ) preparing to evaluate their fire safety. In order to assess these high buildings fairly, three experts H = { H 1 , H 2 , H 3 } (expert’s weight h = ( 0.30 , 0.50 , 0.20 ) are invited. All experts give their assessment information depending on the four subsequently attributes: ⟀ R 1 is fire resistance of buildings; ⟁ R 2 is the cost of extinguishing a building; ⟂ R 3 is safe evacuation capability of the building; ⟃ R 4 is safety management level of the building. Evidently, R 2 is the cost attribute, while R 1 ⁢ R 3 and R 4 are the benefit attributes and the decision making matrix are given in Tables 1–3.

Step 1. Based on the decision making information Q ( k ) = ( q i ⁢ j k ) m × n ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ) given in Tables 1–3 and the expert’s weights h = ( 0.35 , 0.32 , 0.33 ) , we can derive the overall intuitionistic fuzzy evaluation information Q = ( q i ⁢ j ) m × n ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ) according to Eq. (16), the computing results are list in Table 4.

Step 2. Normalize the intuitionistic fuzzy evaluation matrix Q = [ q i ⁢ j ] m × n to Q N = [ q i ⁢ j N ] m × n .

Step 3. Decide the attribute weights r j ( j = 1 , 2 , … , n ) by making use of the CRITIC method as recorded in Table 6.

Step 4. Calculate the intuitionistic fuzzy positive ideal solution (IF-PIS) A j + and the intuitionistic fuzzy negative ideal solution (IF-NIS) A j - according to Eqs (20) and (21);

A j + = { ( 1.0000 , 0.0000 ) , ( 0.8666 , 0.0808 ) , ( 0.7703 , 0.1000 ) , ( 0.7287 , 0.1000 ) } A j - = { ( 0.2131 , 0.6355 ) , ( 0.2828 , 0.6127 ) , ( 0.2176 , 0.5785 ) , ( 0.3213 , 0.5186 ) }

Step 5. Compute the distances d i + and d i - , the results are list as follows.

d 1 + = 0.1380 , d 2 + = 0.0542 , d 2 + = 0.3987 , d 2 + = 0.3431 , d 2 + = 0.1907 ; d 1 - = 0.2952 , d 2 - = 0.3633 , d 2 - = 0.0302 , d 2 - = 0.0858 , d 2 - = 0.2363 .

Step 6. Compute each alternative’s correlation coefficient C i by Eq. (24), the results are list as:

C 1 = 0.6814 , C 2 = 0.8702 , C 3 = 0.0705 , C 4 = 0.2001 , C 5 = 0.5535

Step 7. Relying on the calculating results of C i , all the alternatives can be ranked, the higher value of C i is, the optimal alternative selected will be. Evidently, the rank of all alternatives is $F_2>F_1>F_5>F_4>F_3$ and F 2 is the optimal enterprise.