Fine–Kinney-Based Occupational Risk Assessment Using Interval-Valued Pythagorean Fuzzy VIKOR

Abstract

This chapter aims at the adaptation of the Fine–Kinney occupational risk assessment concept into the VIKOR multi-attribute decision-making method with an interval-valued Pythagorean fuzzy set. The classical fuzzy set theory has been improved by proposing a number of extended versions. One of them is the Pythagorean fuzzy set. It has been firstly developed by Yager (IEEE Transactions on Fuzzy Systems 22:958–965, [1]). In this chapter, we use this type of fuzzy set with VIKOR since it reflects uncertainty in occupational risk assessment and decision-making better than other fuzzy extensions. To demonstrate the proposed approach applicability, a case study regarding the activities of the surface treatment area in a chrome plating unit of a gun factory is performed. Some additional analysis to test the solidity and validity of the approach is executed. Finally, the Python codes in the implementation of the proposed approach are given for scholars and practitioners for usage in further studies.

4.1 Pythagorean Fuzzy Sets and VIKOR

In this part, before explaining the proposed approach, some preliminaries are presented regarding Pythagorean fuzzy sets and more specifically interval-valued Pythagorean fuzzy sets. Then, the algorithm interval-valued Pythagorean fuzzy VIKOR (IVPFVIKOR) is provided in detail.

4.1.1 General View on Pythagorean Fuzzy Sets

Pythagorean fuzzy sets have been first of all suggested by Yager [1] and have been applied by many scholars to various areas to handle uncertainty such as intuitionistic fuzzy sets. These two of the sets include a membership, nonmembership, and hesitancy degree. On the other hand, the criterion of membership and nonmembership degrees that are larger than 1 cannot be satisfied by intuitionistic fuzzy sets. To overcome this drawback, Pythagorean fuzzy sets are developed [1]. This type of fuzzy sets is considered as flexible and more powerful to resolve issues regarding uncertainty [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].

The sum of membership and nonmembership degrees can exceed 1 while the sum of squares cannot surpass 1 in Pythagorean fuzzy sets [2,3,4,5,6, 15,16,17]. This situation is given in Definition (1) and Fig. 4.1.

Fig. 4.1

Adapted from Ref. [17], with kind permission from Springer Science + Business Media

Pythagorean and intuitionistic fuzzy number comparison according to spaces

Definition 1

Let X be a set in a discourse universe. A Pythagorean fuzzy set P has the form in the following [16]:

$$P = \{ < x, \, P(\mu_{P} (x),v_{P} (x)) > \left| {x \in X} \right.\}$$

(4.1)

where $\mu_{P} (x):X \mapsto [0,1]$ describes the degree of membership and $v_{P} (x):X \mapsto [0,1]$ describes the degree of nonmembership of the element $x \in X$ to P, respectively, and, for every $x \in X$, it takes

$$0 \le \mu_{P} (x)^{2} + v_{P} (x)^{2} \le 1$$

(4.2)

For any Pythagorean fuzzy set P and $x \in X$, $\pi_{P} (x) = \sqrt {1 - \mu^{2}_{P} (x) - v^{2}_{P} (x)}$ is named as the degree of indeterminacy of x to P.

Definition 2

Let $P_{1} = P(\mu_{{P_{1} }} ,v_{{P_{1} }} )$ and $P_{2} = P(\mu_{{P_{2} }} ,v_{{P_{2} }} )$ be two Pythagorean fuzzy numbers, and λ > 0, then the operations on these two Pythagorean fuzzy numbers are determined as shown below [15, 16, 18, 19]:

$$P_{1} \oplus P_{2} = P\left( {\sqrt {\mu _{{P_{1} }}^{2} + \mu _{{P_{2} }}^{2} - \mu _{{P_{1} }}^{2} \mu _{{P_{2} }}^{2} } ,v_{{P_{1} }} v_{{P_{2} }} } \right)$$

(4.3)

$$P_{1} \otimes P_{2} = P\left( {\mu _{{P_{1} }} \mu _{{P_{2} }} ,\sqrt {v_{{P_{1} }}^{2} + v_{{P_{2} }}^{2} - v_{{P_{1} }}^{2} v_{{P_{2} }}^{2} } } \right)$$

(4.4)

$$\lambda P_{1} = P\left( {\sqrt {1 - (1 - \mu _{{P_{1} }}^{2} )^{\lambda } } ,(v_{{P_{1} }} )^{\lambda } } \right),{\text{ }}\lambda > 0$$

(4.5)

$$P_{1} ^{\lambda } = P\left( {(\mu _{{P_{1} }} )^{\lambda } ,\sqrt {1 - (1 - v_{{P_{1} }}^{2} )^{\lambda } } } \right)\lambda > 0$$

(4.6)

Definition 3

Let $P_{1} = P(\mu_{{P_{1} }} ,v_{{P_{1} }} )$ and $P_{2} = P(\mu_{{P_{2} }} ,v_{{P_{2} }} )$ be two Pythagorean fuzzy numbers, a nature quasi-ordering on the Pythagorean fuzzy numbers is determined as shown below [16]:

$$P_{1} \ge P_{2} {\text{ if and only if }}\mu_{{P_{1} }} \ge \mu_{{P_{2} }} {\text{ and }}v_{{P_{1} }} \le v_{{P_{2} }}$$

A score function is developed by Zhang and Xu [16] to compare two Pythagorean fuzzy numbers of magnitude given as follows:

$$s(P_{1} ) = (\mu_{{P_{1} }} )^{2} - (v_{{P_{1} }} )^{2}$$

(4.7)

Definition 4

For the Pythagorean fuzzy numbers which are given above according to the proposed score functions, to compare two Pythagorean fuzzy numbers, the following laws are defined [16]:

• (1) ${\text{ If }}s(P_{1} ) < s(P_{2} ),{\text{ then }}P_{1} \prec P_{2}$

• (2) ${\text{ If }}s(P_{1} ) > s(P_{2} ),{\text{ then }}P_{1} \succ P_{2}$

• (3) ${\text{ If }}s(P_{1} ) = s(P_{2} ),{\text{ then }}P_{1} \sim P_{2}$

4.1.2 VIKOR Method

The term “VIKOR” is originally in Serbian (VlseKriterijumska Optimizacija I Kompromisno Resenje) and means multi-criteria optimization and a compromise solution. It has been initially proposed by Opricovic [20]. It needs a criteria weight matrix and a decision matrix that cover alternatives, criteria, and their respective performance measures (values of alternatives with respect to the criteria). The procedural steps of VIKOR are specified as follows [21]:

• 1. Identify the problem and build a payoff (decision) matrix,

• 2. Define the best and the worst values of all criterion functions,

• 3. Compute S and R values that are specific for VIKOR,

• 4. Compute Q values according to the computed S and R values from Step 2,

• 5. Sort alternatives in descending order by S, R, and Q values, and

• 6. Propose a compromise solution provided two conditions (acceptable stability and acceptable advantage) are fulfilled.

4.2 Proposed Fine–Kinney-Based Approach Using IVPFVIKOR

In this chapter, the problem has t OHS experts $E_{m} \left( {m = 1\, {\text{to}}\, t} \right)$, f hazards $H_{a} \left( {a = 1\, {\text{to}} \,f} \right)$, and s Fine–Kinney risk parameters $RP_{z} \left( {z = 1\, {\text{to}}\, s} \right)$. Each OHS expert $E_{m}$ has a weight value $(w_{m} > 0 \,{\text{and}}\, \sum w_{m} = 1)$. In the lights of this initial notations and indices, the application phases of IVPFVIKOR are provided as follows:

Step 1: In the first phase, the Pythagorean fuzzy decision matrix with respect to the OHS experts’ subjective judgments is constructed. In evaluating the hazards by the OHS experts, the seven-point Pythagorean fuzzy linguistic scale of Yazdi [22] is used. Each OHS expert’s judgment is combined into a group consensus to set up the decision matrix in Pythagorean fuzzy numbers.

Let $\widetilde{r}_{az}^{k} = \left( {\mu_{az}^{k} ,v_{az}^{k} } \right)$ be the Pythagorean fuzzy values provided by $E_{m}$ on the assessment of $H_{a}$ with respect to $RP_{z}$.

After that, the Pythagorean fuzzy hazard ratings $\left( {\widetilde{r}_{az}^{k} } \right)$ according to each of the risk parameters are computed with a Pythagorean fuzzy weighted averaging (PFWA) operator of Yazdi [22].

$$\begin{aligned} \widetilde{r}_{az} = & \,{\text{PFWA}}\left( {\widetilde{r}_{az}^{1} ,\widetilde{r}_{az}^{2} , \ldots ,\widetilde{r}_{az}^{t} } \right) = \oplus_{m = 1}^{t} \lambda_{m} \widetilde{r}_{az}^{m} \\ = & \left( {\sqrt {1 - \mathop \prod \limits_{m = 1}^{t} \left( {1 - \left( {\mu_{az}^{m} } \right)^{2} } \right)^{{w_{m} }} , \mathop \prod \limits_{m = 1}^{t} } \left( {v_{az}^{m} } \right) ^{{w_{m} }} } \right) a = 1, 2, \ldots , f, z = 1, 2, \ldots , s \\ \end{aligned}$$

(4.8)

Then, the problem is formed into a matrix form as in Eq. (9):

$$\tilde{R} = \left[ {\begin{array}{*{20}c} {\widetilde{r}_{11} } & \cdots & {\widetilde{r}_{1s} } \\ \vdots & \ddots & \vdots \\ {\widetilde{r}_{f1} } & \cdots & {\widetilde{r}_{fs} } \\ \end{array} } \right]$$

(4.9)

where $\widetilde{r}_{az} = \left( {\mu_{az} ,v_{az} } \right)$. is an element of the aggregated Pythagorean fuzzy decision matrix $\tilde{R}$.

Step 2: In the second phase, the Pythagorean fuzzy positive and negative ideal solutions, which are (PFPIS) $\tilde{p}_{z}^{*} = \left( {\mu_{z}^{*} ,v_{z}^{*} } \right)$. and (PFNIS) $\tilde{p}_{z}^{ - } = \left( {\mu_{z}^{ - } ,v_{z}^{ - } } \right)$, respectively, are determined as follows:

$$\begin{aligned} \tilde{p}_{z}^{*} = & \left\{ {\begin{array}{*{20}c} {\mathop {\hbox{max} }\limits_{a} \widetilde{r}_{az} {\text{for}} \,{\text{benefit}}\, {\text{criteria}}} \\ {\mathop {\hbox{min} }\limits_{a} \widetilde{r}_{az} {\text{for}}\, {\text{cost }}\,{\text{criteria}}} \\ \end{array} } \right. z = 1, 2, \ldots , s \\ \tilde{p}_{z}^{ - } = & \left\{ {\begin{array}{*{20}c} {\mathop {\hbox{min} }\limits_{a} \widetilde{r}_{az} {\text{for}} \,{\text{benefit}}\, {\text{criteria}}} \\ {\mathop {\hbox{max} }\limits_{a} \widetilde{r}_{az} {\text{for}}\, {\text{cost}}\, {\text{criteria}}} \\ \end{array} } \right. z = 1, 2, \ldots , s \\ \end{aligned}$$

(4.11)

Step 3: In the third phase, $S_{a}$ and $R_{a}$ which are both VIKOR-specific scores as formulated and calculated with the aid of generalized Pythagorean fuzzy ordered weighted standardized distance operator (GPFOWSD) of Yazdi [22] are

$$S_{a} = {\text{GPFOWSD}}\left( {\tilde{p}_{1}^{*} ,\tilde{p}_{1}^{ - } ,\tilde{r}_{a1} , \ldots ,\tilde{p}_{1}^{*} ,\tilde{p}_{1}^{ - } ,\tilde{r}_{as} } \right) = \left( {\mathop \sum \limits_{m = 1}^{s} w_{m} \bar{d}_{m}^{\lambda } } \right)^{1/\lambda } ,a = 1, 2, \ldots , f$$

(4.12)

$$R_{a} = \left( {\max_{m} \left( {w_{m} \bar{d}_{m}^{\lambda } } \right)} \right)^{1/\lambda } ,a = 1, 2, \ldots , f$$

(4.13)

where $w_{m}$ are Fine–Kinney risk parameters’ ordered weights.

Step 4: In this fourth phase, the index of “$Q_{a}$” is calculated as follows:

$$Q_{a} = v\frac{{S_{a} - S^{*} }}{{S^{ - } - S^{*} }} + \left( {1 - v} \right)\frac{{R_{a} - R^{*} }}{{R^{ - } - R^{*} }} \;\;\;\;\;\;\;\;\;\;\; a = 1, 2, \ldots , f$$

(4.14)

where $S^{*} = \min_{a} S_{a} , S^{ - } = \max_{a} S_{a} , R^{*} = \min_{a} R_{a} , R^{ - } = \max_{a} R_{a}$.

v is the maximum group utility weight, whereas (1−v) is the individual regret weight. Mostly, v is set to 0.5.

Step 5: In the fifth phase, ranks of hazards are determined considering $S_{a} , R_{a}$ and $Q_{a}$ values in increasing order.

Step 6: The last phase concerns the conditions of the compromise solution of VIKOR. The alternative (A⁽¹⁾) which was best ordered by the measure $Q_{a}$ was recommended if the conditions in [20, 21] were satisfied. The main steps of the proposed approach are graphically demonstrated in Fig. 4.2.

Fig. 4.2

The main steps of the proposed approach

4.3 Case Study

To show the proposed approach applicability, a case study was executed in the chrome plating unit of a gun factory. The hazards and associated risks regarding surface treatment are analyzed. Three experts involved in assessing occupational hazard risks in the study. Different weights are assigned for each OHS expert in risk assessment. These OHS experts are denoted as E-1, E-2, and E-3. The weights of experts are assigned considering their work experience period in the worksite following the computation procedure of [23, 24]. The assigned weights are 0.4, 0.3, and 0.3, respectively. 23 different hazards have an impact on the safety risk of the observed gun production company. The hazard list is demonstrated in Table 4.1.

Table 4.1 Hazards in the observed chrome plating unit of the gun factory

4.3.1 Application Results

In this application, as in other chapters, three parameters of Fine–Kinney method are used in the risk assessment. The weights of these parameters are derived from [25] as $w_{P} = 0.289, \, w_{E} = 0.293, \, w_{C} = 0.418$. During the prioritization procedure by IVPFVIKOR, the linguistic terms given in [22] are used. In the chapter, the OHS experts’ evaluations in Pythagorean linguistic terms for each of the 23 hazards have been received. At the end of this assessment, the linguistic evaluations of the hazards and Pythagorean fuzzy decision matrix are obtained by Eq. (8). The results are shown in Tables 4.2 and 4.3.

Table 4.2 Linguistic assessment of the hazards by OHS experts

Table 4.3 The Pythagorean fuzzy decision matrix (aggregated)

Then, employing Eqs. (12–14), Q values are obtained as given in Table 4.4. Figure 4.3 also indicates the values of S, R, and Q for each hazard. The hazards are ordered considering the Q values. The smallest Q value refers to the highest and most serious risk. Risks with the highest Q value indicate the least important risks. Results show that the most serious risks are stemmed from Hazard-5, Hazard-15, and Hazard-23.

Table 4.4 S, R, and Q values and ranking orders for each hazard

Fig. 4.3

Final IVPFVIKOR-specific S, R, and Q values

The last step of a generic risk assessment work is risk control. For this aim, for hazards with lower IVPFVIKOR-specific Q values, a number of control measures are suggested. For instance, these are some of the measures that can be applied to control the system regarding Hazard-5 (regarding insufficient ventilation), Hazard-15 (Noise), and Hazard-23 (professional competence and experience):

• For Hazard-5: Required improvements should be made by taking service from a professional company for ventilation measurements. Especially systematic suction ventilation should be installed on the benches. It will be provided to enter the zone with a half-face mask.

• For Hazard-15: Appropriate ear protectors should be provided and used depending on the results of the ambient measurements made to determine the noise levels of work equipment during operation. Employees should be trained about the usefulness and requirement of personal protective equipment usage.

• For Hazard-23: Employees should be given vocational training and be certified.

4.3.2 Validation Study on the Results

In this subsection, three validation tests on obtained ranking results are performed. The first validation study concerns a comparison between the results of the existed approach (IVPFVIKOR under Fine–Kinney’s method) and classical Fine–Kinney method. We then observe the variations in hazard rankings. The results are shown in Fig. 4.4.

Fig. 4.4

Comparison of rankings by proposed and classic approaches

It is observed from Fig. 4.4 that by both approaches, Hazard-5 is ranked as the most critical hazard, followed by Hazard-15. It has also seen that the least important three hazards (Hazard-13, Hazard-16, and Hazard-17) have partially the same ranking according to both approaches. When we compare the results obtained by both approaches, we observe that there are very small rank variations between them. The Spearman rank correlation between the two approaches is obtained as 0.928. That means there exists a high correlation between the ranking orders of two approaches so that it can be claimed that this proposed approach is applicable for occupational risk assessment in the Fine–Kinney domain.

As a second validation study, we analyze the difference in the ranking of hazards in times of changing of Fine–Kinney parameters’ weights. Therefore, we apply four different weight vectors. The weight vectors of Fine–Kinney parameters designed for the sensitivity analysis are given in Table 4.5. The ranking of hazards with respect to four different weight vectors is shown in Table 4.6.

Table 4.5 The weight vectors designed for the sensitivity analysis

Table 4.6 Ranking order changes in times of parameters’ weight changes

It can be observed from Table 4.6 that when the weights change, there exist variations in the ranking of hazards. Therefore, our proposed approach is sensitive to Fine–Kinney risk parameters’ weights. Hazard-5 is mostly ranked as the most critical hazard according to all the weight vectors. There is no change in the ranking of Hazard-16 for all combinations. It lies in the 19th place in the ranking. When compared to the results with the ones similar to this study from the literature, we can say that the ranking result obtained by our proposed approach is credible and applicable.

We also calculated the Spearman rank correlation (RHO) between weight vectors by an online calculator. The obtained results are given in Table 4.7. Results show that there exist high correlations between the ranking orders obtained by four different weight vectors. Since all values are close to 1. To this end, it can be claimed that this proposed approach is sensitive to the changing of the weight values. It is an expected output when considering similar attempts from the literature.

Table 4.7 Results of Spearman’s RHO between weight vectors

A third validation study (as a sensitivity analysis) was performed in the results of the proposed approach by varying value of v (maximum group utility) which is set to 0.5. Ten different scenarios (excluding the experiment of v = 0.5) are tried to observe the variability. Results of Q values from this sensitivity analysis are given in Fig. 4.5. It is clearly understood from Fig. 4.4 that Hazard-5 has the best ranking for each case, Hazard-13 has the worst ranking for each v value changing situation.

Fig. 4.5

IVPFVIKOR Q values according to different v values

4.4 Python Implementation of the Proposed Approach