Reliability allocation method based on maximum entropy ordered weighted average and hesitant fuzzy Linguistic term set

Abstract

Reliability allocation is one of the most important factors to consider when determining the reliability and competitiveness of a product. The feasibility-of-objectives (FOO) technique has become the current standard for assessing reliability designs for military mechanical–electrical systems. However, the FOO method has several drawbacks: For instance, it requires that the value of reliability allocation factors is single linguistic variables, and it does not consider the ordered weight of reliability allocation factors, but simply multiplies the ISPE (Complexity (I), State of Art (S), Performance Time (P), and Environment (E)) values one by one. This can lead to erroneous results. To address these issues, this paper combines the fuzzy allocation method with the maximum entropy ordered weighted averaging method (ME-OWA) to achieve a flexible allocation of system reliability. To verify the effectiveness of the proposed method, the CNC machine tool is taken as an example. The FOO method and the fuzzy allocation method and the proposed method were used to assign reliability to the eight subsystems of a CNC machine tool, and the results were compared to draw conclusions: The proposed method is more flexible and accurate for reliability allocation.

Keywords

Reliability allocation feasibility-of-objectives technique (FOO)the maximum entropy ordered weighted averaging method (ME-OWA)CNC machine tools

1 Introduction

Reliability allocation is one of the most important steps to consider when determining the reliability and competitiveness of a product. The purpose is to convert the reliability index of an entire system into a subsystem or unit reliability index to make it consistent. It is a whole-to-local, top-down decomposition process. The result of reliability allocation directly affects the product quality and market competitiveness.

The Electronic Equipment Reliability Advisory Group [1] developed a reliability allocation method based on the complexity and criticality of a unit or subsystem rather than the failure rate. In contrast to this approach, Aeronautical Radio Inc. [2] released ARINC allocation technology based on failure rate units or subsystems. In addition to these methods, Bracha [3] also introduced a method for assigning reliability using four factors: prior art, subsystem complexity, estimation by number of parts, environmental conditions, and relative runtime, while Karmiol [4] evaluated the complexity of system’s mission objectives, the latest technology, operational overview, and criticality to assign subsystem reliability. More recently, the engineering design guide, Reliability Design Handbook [5], featured the feasibility-of-objectives (FOO) technique, which was incorporated into the Mil-hdbk-338B handbook [6], an established standard for military reliability design. The FOO technique specifically provides a detailed reliability allocation procedure for mechanical–electrical systems; Smedley [7] employed this procedure to perform reliability analysis among low-energy booster (LEB) ring magnet power systems in a superconducting supercollider.

In resolving the FOO technique and average weighting allocation problems, the proposed approach is based on the traditional reliability allocation method, which uses Yager’s OWA [8] and the ME-OWA (Fuller & Majlender, 2001 [9]; Cheng & Chang, 2006 [10]; Chang, Cheng, & Chang,2008 [11]) operators. Yager [8] first introduced the concept of OWA operators to solve the problems described here by using the FOO technique. Additionally, Fuller and Majlender [9] used Lagrange multipliers on Yager’s OWA equation to derive a polynomial equation, which determines the optimal weighting vector under maximal entropy (ME-OWA operator). Thus, the proposed approach thus determines the optimal weighting vector under maximal entropy, and the OWA operator ascertains the optimal reliability allocation rating after an aggregation process. This method is both a simple and effective approach that can efficiently overcome the shortcomings of FOO technique and average weighting allocation.

Hesitant fuzzy sets were introduced by Torra [12] as an extension of fuzzy sets. This method accurately reflects situations in which an expert is hesitant in providing his/her preferences for objects in a decision-making process. The hesitant fuzzy sets are defined as quantitative situations and use a set of values that are possible in the definition process of the membership of an Element [13]. Therefore, this method is unsuitable for modeling qualitative settings in reliability allocation. To this end, Rodriguez et al. [13] proposed a concept of hesitant fuzzy linguistic term sets to address qualitative decision-making problems, combining the fuzzy linguistic approach and hesitant fuzzy sets to reflect situations in which experts equivocate between several possible linguistic values. Thus, hesitant fuzzy linguistic term sets are more appropriate than hesitant fuzzy sets for qualitative settings in reliability allocation. Many studies [14 –18] have been performed on hesitant fuzzy linguistic term set-based methods. Q. Cheng et al. [26] conducted in-depth research in the field of machine tool precision, and the developed method can be used for reference in the field of reliability. L. Wang et al. [27] proposed a prospect theory-based multi-attributive border approximation area comparison (MABAC) method to rank the risks and identify the priority of risks by reflecting the decision-maker’s bounded rationality and behavior psychology. S.Q. Zhang et al. [28] combined the traditional EDAS model for multiple criteria group decision making (MCGDM) with P2TLNs. This was more accurate and effective for considering the conflicting attributes in reliability allocation. R. Wang et al. [29] studied multiple attribute decision-making (MADM) problems with picture fuzzy numbers (PFNs) information. S.M. Peng et al. [32] utilized induced OWG (IOWG) operator to develop picture fuzzy induced OWG (PFIOWG) operators to solve the picture fuzzy multiple attribute decision making problems, leading to enhance data quality and reliability. Z. Zhang et al. [36 –38] proposed a general framework for group decision making with IMPRs. Based on this general framework, they proposed an approach to linguistic large-scale multi attribute group decision making and proposed a new procedure for group analytic hierarchy process to deal with multi criteria group decision making problems. The developed method can be used as a reference in the field of reliability.

Studies on reliability allocation problems using fuzzy-related mathematical models are growing in number. For example, Ebrahimipour et al. [19] used an emotional learning-based fuzzy inference system to improve system reliability evaluation in a redundancy allocation problem. Garg and Sharma [20] implemented fuzzy numbers and particle swarm optimization to solve a fuzzy multi-objective optimization problem under several constraints. Z.S. Chen et al. [33] integrated the possibility allocation into hesitant fuzzy linguistic term set (HFLTS), adding an extra dimension to individual opinion approximation process and significantly leading to enhanced data quality and reliability. G.W. Wei et al. [30, 31] quoted the Maclaurin symmetric mean (MSM) operator and discussed some desirable properties and special cases of these operators in detail. Then, they illustrated the feasibility of the proposed methods by using a numerical example. This is a new direction to study fuzzy allocation. W.Y. Yu et al. [34] developed a new method to deal with MCGDM problems with unbalanced byconsidering the psychological behavior of decision makers. Zhen Zhang et al. [35] defined the best additive consistency index, the worst additive consistency index, and the average additive consistency index to measure the consistency level of a hesitant fuzzy preference relation (HFPR). Sriramdas et al. [21] used trapezoidal fuzzy numbers to express allocation factors during the early design and development stages. In their method, experts can use linguistic terms rather than crisp values to represent their decisions realistically.

However, these methods cannot simultaneously address an expert’s equivocal preferences and the ordered weight of reliability allocation factors, which can bias the conclusions. To overcome the shortcomings of the above mentioned methods, this paper proposes a more general reliability allocation method, which integrates hesitant fuzzy linguistic term set and maximum entropy OWA weights to achieve a flexible allocation of system reliability. The method of hesitant fuzzy linguistic term set can handle the problem of experts’ equivocal preference and the method of maximum entropy OWA weights can address the problem of the ordered weight of reliability allocation factors. By combining these two methods, the problem of experts‘ equivocal preference and the problem of the ordered weight of reliability allocation factors can be solved simultaneously, so that the accuracy of reliability allocation can be improved greatly.

The rest of the paper is organized as follows. Chapter 2 proposes a reliability allocation method that combines hesitant fuzzy linguistic sets and maximum entropy OWA weights to achieve flexible allocation of system reliability. In Chapter 3, the FOO and fuzzy allocation methods are reviewed. Then, using the CNC machine system as an example, these two methods and the newly proposed method were used to distribute the reliability. The reliability allocation results of the three methods were compared, and the effectiveness and superiority of the proposed method were proven. The final chapter provides the conclusions of the study.

2 Proposed integration of hesitant fuzzy linguistic term set and the maximum entropy OWA weights

Reliability allocation is one of the most important issues in the product design and development phase, and the results directly affect the quality of product and market competitiveness. Traditionally, the industry and military have used the FOO method to distribute the reliability of electromechanical systems, requiring Complexity, State of Art, Performance Time, and Environment (ISPE) to achieve clear linguistic values. However, experts have hesitant linguistic values in the product design and development phase to represent the corresponding I, S, P, and E factors, increasing the difficulty of reliability allocation, and the traditional FOO method cannot completely solve these problems. In addition, the FOO method does not consider the ordered weight of reliability allocation factor, which may cause bias in the conclusion. Therefore, to solve these two problems, this paper combines the hesitant fuzzy linguistic term set and the maximum entropy OWA weight and flexibly allocates the reliability of the system, improves the accuracy of allocation result, and reflects the reality. In this chapter, we will briefly introduce the hesitant vague linguistic and the maximum entropy OWA, and then discuss the proposed method.

2.1 Hesitant fuzzy linguistic term set

Torra [12] introduced a hesitant fuzzy set to manage the hesitant management of several values between experts when evaluating indicators.

Rodriguez et al. [13] proposed a hesitant fuzzy linguistic term set based on hesitant fuzzy sets and fuzzy linguistic methods.

Definition 1. [13, 23]: Let S = {s₀, s₁, . . . , s_g} be a set of linguistic term. H_s, $H_{s}^{1}$ , and $H_{s}^{2}$ are three arbitrary hesitant fuzzy linguistic model sets on S, and their complements, unions, and intersections can be defined as follows: $\begin{matrix} H_{s}^{c} = S - H_{s} = {s_{i} | s_{i} \in S and s_{i} \notin H_{s}} \\ H_{s}^{1} \cup H_{s}^{2} = {s_{i} | s_{i} \in H_{s}^{1} or s_{i} \in H_{s}^{2}} \\ H_{s}^{1} \cap H_{s}^{2} = {s_{i} | s_{i} \in H_{s}^{1} and s_{i} \in H_{s}^{2}} \end{matrix}$

Definition 2. [13, 14]: Let S = {s₀, s₁, . . . , s_g} be a set of linguistic term, and H_s is an arbitrary hesitant fuzzy linguistic term set. The lower limit H_s-and the upper limit H_s+ of hesitant fuzzy linguistic term set H_s can be defined as follows: $\begin{matrix} H_{s -} = min (s_{i}) = s_{j}, s_{i} \in H_{s} and s_{i} ⩾ s_{j} \forall i \\ H_{s +} = max (s_{i}) = s_{j}, s_{i} \in H_{s} and s_{i} ⩽ s_{j} \forall i \end{matrix}$

2.2 Maximum entropy ordered weighted averaging method (ME-OWA)

2.2.1 ME-OWA operator

Yager (1988) first introduced the concept of OWA operator, an important aggregation operator in the weighted aggregation method [8]. The optimal weight of an attribute can be derived based on the rating of weight vector after the aggregation process (see Definition 5).

Definition 3. [8]. The mapping of an OWA operator of dimension n is F : Rⁿ → R, and its associated n-dimensional weight vector W = [w₁, w₂, . . . , w_n] ^T, where $\sum_{i = 1}^{n} w_{i} = 1, \forall w_{i} \in [0, 1]$ This operator can be defined as follows: $f (a_{1}, a_{2}, . . ., a_{n}) = \sum_{i = 1}^{n} w_{i} b_{i}$ (1) where b_i is the ith largest element in the aggregate object set (a₁, a₂, . . . , a_n), and b₁ ⩾ b₂ ⩾ . . . ⩾ b_n

Yager [8] also introduced two important feature measurements of weight vector W of OWA operator. One of these two metrics is the orness of the aggregate, which is defined in Definition 4.

Definition 4. [8]. Suppose F is an OWA aggregation operator with a function of W = [w₁, w₂, . . . , w_n] The degree of orness associated with it can be defined as follows: $Orness (W) = \frac{1}{n - 1} \sum_{i = 1}^{n} (n - i) w_{i}$ (2) where orness (w) = α is a situation parameter.

Obviously orness (w) ∈ [0, 1] is true for any weighting vector

The second feature measurement introduced by Yager [8] is a measure of aggregate dispersion, as defined in Definition 5.

Definition 5. Suppose W is a weight vector with the element w₁, . . . , w_n; then, the discrete metric of W can be defined as follows: $Dispersion (W) = - \sum_{i = 1}^{n} w_{i} ln w_{i}$ (3)

O’Hagan [24] combines the principle of maximum entropy with the OWA operator to propose a specific OWA weight with a maximum entropy at a given orness level. This approach is based on the following solutions to mathematical programming problems: $Maximize : - \sum_{i = 1}^{n} w_{i} ln w_{i}$ (4) $Subject to : \frac{1}{n - 1} \sum_{i = 1}^{n} (n - i) w_{i} = α, 0 ⩽ α ⩽ 1,$ (5) $\sum_{i = 1}^{n} w_{i} = 1, 0 ⩽ w_{i} ⩽ 1$ (6)

2.2.2 Calculation of ME-OWA weights

Fuller and Majlender [9] used the Lagrangian multiplier method on the Yager OWA equation to derive a polynomial equation that determines the optimal weight vector at the maximum entropy. By using their method, the relevant weight can be easily obtained from Equations (7)–(9).

$\begin{matrix} ln w_{i} = \frac{i - 1}{n - 1} ln w_{n} + \frac{n - i}{n - 1} ln w_{1} \Rightarrow w_{i} \\ = \sqrt[n - 1]{w_{1}^{n - i} w_{n}^{i - 1}} \end{matrix}$ (7) $w_{n} = \frac{((n - 1) α - n) w_{1} + 1}{(n - 1) α + 1 - {nw}_{1}}$ (8)

$\begin{matrix} w_{1} [(n - 1) α + 1 - {nw}_{1}]^{n} \\ = ((n - 1) α)^{n - 1} [((n - 1) α - 1 n) w_{1} + 1] \end{matrix}$ (9) where ω is the weighting vector, n is the number of attributes, α is the value of situation parameter.

2.3 Proposed method

The above two sections are proposed as the theoretical basis for the proposed method in this paper. The proposed method can be mainly divided into the following nine steps:

Step 1. List the configuration of system.

Step 2. Determine the system reliability requirements and mission runtime.

Step 3. Determine the complexity (k), maintenance (M), cost (Co), criticality (Cr), prior art (S), and operating time (T) scale. Then, each reliability allocation team member and the corresponding linguistic values for K, M, Co and Cr, S, and T are determined for each subsystem evaluation.

Step 4. Suppose there are n subsystems U_i = (i = 1, . . . , n) to be evaluated and allocated by a system design and analysis team consisting of m members, TM_j (j = 1, . . . , m).

Let $\begin{matrix} {\tilde{k}}_{ij} = ({\tilde{k}}_{ijL}, {\tilde{k}}_{ijM 1}, {\tilde{k}}_{ijM 2}, {\tilde{k}}_{ijU}), \\ {\tilde{C} r}_{ij} = ({\tilde{C} r}_{ijL}, {\tilde{C} r}_{ijM 1}, {\tilde{C} r}_{ijM 2}, {\tilde{C} r}_{ijU}), \\ {\tilde{C} o}_{ij} = ({\tilde{C} o}_{ijL}, {\tilde{C} o}_{ijM 1}, {\tilde{C} o}_{ijM 2}, {\tilde{C} o}_{ijU}), \\ {\tilde{T}}_{ij} = ({\tilde{T}}_{ijL}, {\tilde{T}}_{ijM 1}, {\tilde{T}}_{ijM 2}, {\tilde{T}}_{ijU}), \\ {\tilde{M}}_{ij} = ({\tilde{M}}_{ijL}, {\tilde{M}}_{ijM 1}, {\tilde{M}}_{ijM 2}, {\tilde{M}}_{ijU}), and \end{matrix}$

${\tilde{S}}_{ij} = ({\tilde{S}}_{ijL}, {\tilde{S}}_{ijM 1}, {\tilde{S}}_{ijM 2}, {\tilde{S}}_{ijU})$ be the fuzzy ratings of ith sybsystem by an expert on allocation factors, and h_j (j = 1, . . . , m) be the relative importance weights of the m team members,satisfying $\sum_{j = 1}^{m} h_{j} = 1$ and h_j > 0 for j = 1, . . . , m. Based on these assumptions, we can calculate the subjective opinions of team members on the subsystems K, M, C o, Cr, S, and T by calculating the aggregated fuzzy evolution information of subsystem using Equations (10–15) [21]. $\begin{matrix} {\tilde{k}}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{k}}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} k_{ijL}, \sum_{j = 1}^{m} h_{j} k_{ijM 1}, \sum_{j = 1}^{m} h_{j} k_{ijM 2}, \sum_{j = 1}^{m} h_{j} k_{ijU}) \end{matrix}$ (10) $\begin{matrix} {\tilde{C} r}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{C} r}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} {Cr}_{ijL}, \sum_{j = 1}^{m} h_{j} {Cr}_{ijM 1}, \sum_{j = 1}^{m} h_{j} {Cr}_{ijM 2}, \sum_{j = 1}^{m} h_{j} {Cr}_{ijU}) \end{matrix}$ (11) $\begin{matrix} {\tilde{C} o}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{C} o}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} {Co}_{ijL}, \sum_{j = 1}^{m} h_{j} {Co}_{ijM 1}, \sum_{j = 1}^{m} h_{j} {Co}_{ijM 2}, \sum_{j = 1}^{m} h_{j} {Co}_{ijU}) \end{matrix}$ (12) $\begin{matrix} {\tilde{M}}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{M}}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} M_{ijL}, \sum_{j = 1}^{m} h_{j} M_{ijM 1}, \sum_{j = 1}^{m} h_{j} M_{ijM 2}, \sum_{j = 1}^{m} h_{j} M_{ijU}) \end{matrix}$ (13) $\begin{matrix} {\tilde{S}}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{S}}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} S_{ijL}, \sum_{j = 1}^{m} h_{j} S_{ijM 1}, \sum_{j = 1}^{m} h_{j} S_{ijM 2}, \sum_{j = 1}^{m} h_{j} S_{ijU}) \end{matrix}$ (14) $\begin{matrix} {\tilde{T}}_{i} = \sum_{j = 1}^{m} h_{j} {\tilde{T}}_{ijL} \\ = (\sum_{j = 1}^{m} h_{j} T_{ijL}, \sum_{j = 1}^{m} h_{j} T_{ijM 1}, \sum_{j = 1}^{m} h_{j} T_{ijM 2}, \sum_{j = 1}^{m} h_{j} T_{ijU}) \end{matrix}$ (15)

Step 5. Hesitating defuzzification of fuzzy evolutionary information of aggregate. Let $\tilde{A}$ and $\tilde{B}$ be two trapezoidal fuzzy numbers, which are parameterized by (a₁, a₂, a₃, a₄) and (b₁, b₂, b₃, b₄), respectively. Then, by using the hesitation defuzzification method of parameter mean, $\tilde{X}$ can be calculated as follows: $\tilde{X} = (\frac{a_{1} + b_{1}}{2}, \frac{a_{2} + b_{2}}{2}, \frac{a_{3} + b_{3}}{2}, \frac{a_{4} + b_{4}}{2})$ (16)

We will give an example to explain the hesitant defuzzification method for the above solution. If $\tilde{A}$ and $\tilde{B}$ are two trapezoidal fuzzy numbers, they are parameterized by (1, 2, 3, 4) and (3, 4, 5, 6) respectively. According to Equation (16), the result of aggregated fuzzy evolutionary information hesitating to defuzzify is $\tilde{X} = (\frac{1 + 3}{2}, \frac{2 + 4}{2}, \frac{3 + 5}{2}, \frac{4 + 6}{2}) = (2, 3, 4, 5)$

Step 6. Determine the dimension n of maximum entropy OWA weight vector according to the requirements of specific reliability allocation, and then calculate the vector of the required maximum entropy OWA weight. From Section 2.2 of this paper, the maximum entropy OWA weight can be calculated using Equations (7)–(9).

Step 7. The evaluation result of fuzzy scale factor of calculation subsystem can be obtained as follows. The fuzzy allocation scale factor (FZ_i) of each subsystem can be defined using Equation (17) [21]. Following Steps 4–6, use the various values of situation parameter to calculate the evaluation results of fuzzy proportionality factor of subsystems. ${FZ}_{i} = \frac{{\tilde{k}}_{i} \times {\tilde{C} o}_{i} \times {\tilde{M}}_{i}}{{\tilde{C} r}_{i} \times {\tilde{S}}_{i} \times {\tilde{T}}_{i}}, i = 1, 2, . . ., n .$ (17)

Step 8. Defuzzify and flexibly assign subsystem reliability. In this step, the center of mass is used to deblur to obtain the centroid of fuzzy number. The deblurred centroid of trapezoidal fuzzy number $\tilde{A} = (a, b, c, d)$ can be determined using Equation (18) [21, 25]: $C_{\tilde{A}} = \frac{c^{2} + d^{2} + cd - a^{2} - b^{2} - ab}{3 (c + d - a - b)}$ (18)

Then, according to the different values of situation parameters, the reliability of subsystem can be flexibly allocated.

Step 9. Analyze the above results and select the optimal reliability allocation decision.

3 Reliability allocation of CNC machine tools

In this chapter, by taking a numerical control machine tool system as an example, the FOO, fuzzy allocation method, and proposed method were used to assign reliability to the CNC machine tool to illustrate the effectiveness of the proposed reliability allocation method. The CNC machine tool system consists of eight subsystems: CNC system, spindle system, feed system, cooling system, servo system, lubrication system, hydraulic system, and tool change system. There are six reliability allocation factors: complexity (K), maintenance (M), cost (Co), criticality (Cr), State of art(S), and operating time (T). According to the design requirements and operating environment of system, the system reliability of CNC machine system was set to 0.875, and the task time was set to 1000 h.

Assume that the reliability allocation team consists of four experts to reflect their various expertise; the relative weights of four team members TM1, TM2, TM3, and TM4 are 0.35, 0.15, 0.20, and 0.30 respectively (when using FOO and fuzzy allocation method) At that time, because the fourth expert provided some hesitant information, only three experts could be considered. At the same time, the relative weights of TM1, TM2, and TM3 became 0.45, 0.25, and 0.3, respectively). Each expert uses the linguistic variables and equivalent trapezoidal fuzzy numbers defined in Table 2 for the eight subsystems in Table 7 to evaluate the reliability factor.

3.1 Reliability allocation method based on FOO

The FOO method was released in the military standard MIL-HDBK-338B [6] and mainly used to assign reliability to mechanical and electrical systems without repairing to satisfy military requirements. In this method, the subsystem allocation factor is calculated based on system complexity (I), State of Art (S), performance time (P), and environment (E) numerical levels. According to the design engineering and expert opinions, I, S, P, and E are respectively determined to establish the corresponding linguistic values. The values of I, S, P, and E are based on a rating of 1 to 10. The four standard levels (I, S, P, and E) for each subsystem are used together to produce an overall rating; therefore, ISPE = I × S × P × E.

Suppose a system contains N subsystems. λ_S is the failure rate of entire system; ${\bar{λ}}_{k}$ is the failure rate assigned to each subsystem; R is the system reliability target; T is the duration of system operation. In addition, $ω_{k}^{'}$ refers to the rating of Kth subsystem; $r_{Ik}^{'}$ , $r_{Sk}^{'}$ , $r_{Pk}^{'}$ and $r_{Ek}^{'}$ refer to the ISPE values of Kth subsystem. $C_{K}^{'}$ refers to the complexity of Kth subsystem, and ω′refers to the entire system rating. The basic formula of the FOO method is as follows:

$λ_{s} = \frac{- ln R}{T}$ (19) $λ_{S} T = λ_{k}^{-} T$ (20) $ω_{k}^{'} = r_{Ik}^{'} \times r_{Sk}^{'} \times r_{Pk}^{'} \times r_{Ek}^{'}, \forall k$ (21) $ω^{'} = \sum_{k = 1}^{N} ω_{k}^{'}$ (22) $C_{K}^{'} = \frac{ω_{K}^{'}}{ω^{'}}$ (23) ${\bar{λ}}_{k} = C_{K}^{'} λ_{S}$ (24)

The FOO method uses I, S, P, and E (ISPE values) to distribute system reliability. In this example of a CNC machine system, K, M, Co, Cr, S, and T standards were used in this study to assign subsystem reliability (CNC, spindle, feed, cooling, servo, lubrication, hydraulic System, and tool change system). Based on the required CNC machine tool reliability of 0.875 and mission time of 1000 h. Using Equation (19), the system failure rate can be obtained as follows: $λ_{s} = \frac{- ln R}{T} = - ln (0.875) / 1000 h = 1.3353 E - 04$

The FOO method does not consider the differences in the expertise of team members. The reliability allocation factor is required to be a single linguistic variable. Therefore, this method cannot handle hesitant information. This is because team member TM4 provides partially hesitant information. Therefore, the FOO method only considers the information provided by team members TM1, TM2, and TM3. Therefore, first use Equation (18) to defuzzify the information shown in Table 4, then use Equations (20)–(24) to calculate the overall rating $w_{k}^{'}$ , complex degree $C_{k}^{'}$ , ${\bar{λ}}_{k}$ (subsystem allocation failure rate). Finally, the allocation reliability of system can be calculated using Equation (25), and the results are shown in Table 2. (Table 1 shows the abbreviation of each subsystem of CNC machine tools).

Table 1

Subsystem division of the CNC milling machine tool

No.	Subsystem	Abbreviation
1	CNC system	CNC
2	Spindle system	SP
3	Feeding system	FD
4	Cooling system	CO
5	Servo system	SO
6	Lubricant system	LU
7	Hydraulics and pneumatics system	HP
8	Automatic tool changing system	ATC

Table 2

Reliability allocation results of the CNC machine tools(FOO)

Subsystems	K	M	Co	Cr	S	T	Overall rating	complexity	Allocated failure rate	Allocated reliability
CNC	7.900	4.000	6.500	4.400	3.900	8.500	29959.644	0.152	2.025E-05	0.980
SP	7.400	6.000	6.500	5.100	4.500	8.500	56298.645	0.285	3.805E-05	0.963
FD	5.600	6.500	5.600	3.400	6.500	6.400	28831.130	0.146	1.948E-05	0.981
CO	3.600	5.400	3.100	5.000	2.500	6.000	4519.800	0.023	3.055E-06	0.997
SO	6.400	6.500	6.000	4.200	6.500	7.600	51787.008	0.262	3.500E-05	0.966
LU	3.000	5.400	3.400	3.400	2.500	6.500	3043.170	0.015	2.057E-05	0.998
HP	6.900	3.100	4.800	4.900	3.600	5.400	9780.124	0.049	6.609E-06	0.993
ATC	5.900	6.600	4.500	3.400	6.600	3.400	13369.348	0.068	9.035E-06	0.991
Total							197588.869	1.000	1.335E-04

3.2 Fuzzy-based reliability allocation method

The fuzzy allocation method was proposed by Sriramdas et al. [21]. The trapezoidal fuzzy number is used to represent the linguistic information of allocation factor. In the fuzzy allocation method, the value of reliability allocation factor must be a deterministic and precise linguistic variable. In this case, this method cannot handle the hesitant information, because the team member TM4 provides some hesitant information. Therefore, when evaluating the subsystem experts, only the team members TM1, TM2, and TM3 are considered. The opinions are shown in Table 5. According to the information in Tables 3 and 4, summarize the opinions of team members TM1, TM2, and TM3 on K, M, Co, Cr, S, and T using Equations (10)–(15), as shown inTable 4.

Table 3
Fuzzy rating for reliability allocation factors

Factors/Scale Allocation factors Abbreviation

K M Co Cr S T

(7,8,9,10) Very High Very High Innovation very High Very High Very High Very High VH

(5,6,7,8) High High Innovation High High High High H

(3,4,5,6) Medium Medium Innovation Medium Medium Medium Medium M

(1,2,3,4) Low Low Innovation Low Low Low Low L

Factors/Scale	Allocation factors	Abbreviation
(7,8,9,10)	Very High	Very High	Innovation very High	Very High	Very High	Very High	VH
(5,6,7,8)	High	High	Innovation High	High	High	High	H
(3,4,5,6)	Medium	Medium	Innovation Medium	Medium	Medium	Medium	M
(1,2,3,4)	Low	Low	Innovation Low	Low	Low	Low	L

Table 4

Reliability allocation information from eight subsystems of the Reliability Allocation Team (three members)

Subsystems	Team members	Allocation factors						Weights
		K	M	Co	Cr	S	T
CNC	TM1	VH	M	H	M	M	VH	0.45
	TM2	VH	L	H	H	M	VH	0.25
	TM3	H	M	H	L	L	VH	0.30
SP	TM1	VH	H	H	M	M	VH	0.45
	TM2	H	M	H	M	M	VH	0.25
	TM3	H	H	H	H	M	VH	0.30
FD	TM1	M	H	M	M	M	H	0.45
	TM2	H	H	H	L	H	VH	0.25
	TM3	H	H	H	L	H	M	0.30
CO	TM1	L	H	L	M	L	H	0.45
	TM2	M	M	L	H	L	M	0.25
	TM3	M	M	M	M	L	H	0.30
SO	TM1	H	H	H	L	H	H	0.45
	TM2	VH	H	M	M	H	VH	0.25
	TM3	M	H	H	H	H	VH	0.30
LU	TM1	L	H	M	M	L	H	0.45
	TM2	M	M	L	L	L	H	0.25
	TM3	L	M	L	L	L	H	0.30
HP	TM1	VH	L	H	H	L	H	0.45
	TM2	M	L	M	L	M	M	0.25
	TM3	H	M	L	M	M	M	0.30
ATC	TM1	H	H	M	M	H	M	0.45
	TM2	H	M	M	L	L	L	0.25
	TM3	M	VH	M	L	M	L	0.30

Table 5

Eight subsystems aggregate fuzzy evolution information based on fuzzy allocation method

Subsystems	K	M	Co	Cr	S	T
CNC	[6.4,7.4,8.4,9.4]	[2.5,3.5,4.5,5.5]	[5.0,6.0,7.0,8.0]	[2.9,3.9,4.9,5.9]	[2.4,3.4,4.4,5.4]	[7.0,8.0,9.0,10.0]
SP	[5.9,6.9,7.9,8.9]	[4.5,5.5,6.5,7.5]	[5.0,6.0,7.0,8.0]	[3.6,4.6,5.6,6.6]	[3.0,4.0,5.0,6.0]	[7.0,8.0,9.0,10.0]
FD	[4.1,5.1,6.1,7.1]	[5.0,6.0,7.0,8.0]	[4.1,5.1,6.1,7.1]	[1.9,2.9,3.9,4.9]	[4.1,5.1,6.1,7.1]	[4.9,5.9,6.9,7.9]
CO	[2.1,3.1,4.1,5.1]	[3.9,4.9,5.9,6.9]	[1.6,2.6,3.6,4.6]	[3.5,4.5,5.5,6.5]	[1.0,2.0,3.0,4.0]	[4.5,5.5,6.5,7.5]
SO	[4.9,5.9,6.9,7.9]	[5.0,6.0,7.0,8.0]	[4.5,5.5,6.5,7.5]	[2.7,3.7,4.7,5.7]	[5.0,6.0,7.0,8.0]	[6.1,7.1,8.1,9.1]
LU	[1.5,2.5,3.5,4.5]	[3.9,4.9,5.9,6.9]	[1.9,2.9,3.9,4.9]	[1.9,2.9,3.9,4.9]	[1.0,2.0,3.0,4.0]	[5.0,6.0,7.0,8.0]
HP	[5.4,6.4,7.4,8.4]	[1.6,2.6,3.6,4.6]	[3.3,4.3,5.3,6.3]	[3.4,4.4,5.4,6.4]	[2.1,3.1,4.1,5.1]	[3.9,4.9,5.9,6.9]
ATC	[4.4,5.4,6.4,7.4]	[5.1,6.1,7.1,8.1]	[3.0,4.0,5.0,6.0]	[1.9,2.9,3.9,4.9]	[5.1,6.1,7.1,8.1]	[1.9,2.9,3.9,4.9]

According to the results shown in Table 5, the fuzzy scale factor of subsystem can be calculated using Equation (17), as shown in Table 6.

Table 6

Evaluation result of the fuzzy proportionality factor for eight subsystems of CNC Machine Tools by fuzzy allocation method

Subsystems	K×M×Co	Cr×S×T	Fuzzy proportionality factor
CNC	[80.000,155.400,264.600,413.600]	[48.720,106.080,194.040,318.60]	[0.150,0.801,2.494,2.560]
SP	[132.750,227.700,359.450,534.000]	[75.600,147.200,252.000,396.000]	[0.221,0.904,2.442,2.639]
FD	[84.050,125.460,260.470,403.280]	[38.171,87.261,164.151,274.841]	[0.594,0.764,2.985,2.877]
CO	[13.104,39.494,87.084,161.874]	[15.750,49.500,107.25,195.000]	[0.016,0.368,1.759,1.664]
SO	[110.250,194.700,313.950,474.000]	[82.350,157.62,266.490,414.960]	[0.113,0.731,1.992,2.256]
LU	[11.115,35.525,80.535,152.145]	[9.500,34.800,81.900,156.800]	[0.056,0.434,2.314,1.939]
HP	[28.512,71.552,141.192,272.412]	[27.846,66.836,130.626,225.216]	[0.127,0.548,2.113,2.468]
ATC	[67.320,131.760,227.200,359.640]	[18.411,51.301,107.991,194.481]	[0.609,.1.220,4.429,3.635]

According to the results shown in Table 6, the centroid deblurring method of Equation (18) can be used to calculate the fuzzy allocation scale factor and the clear number of weights. The results are shown in Table 7. In Table 6, the weights of each subsystem were calculated using Equations (22) and (23), and the allocated reliability of each subsystem was calculated using Equation (25).

Table 7

Reliability allocation results of eight subsystems of CNC machine tools based on fuzzy allocation method

Sub-systems	Defuzzified	Weightage	Allocated reliability
	(FZ_i)	(ω_i)	(R_i)
CNC	1.493	0.124	0.984
SP	1.543	0.128	0.983
FD	1.805	0.150	0.980
CO	0.949	0.079	0.990
SO	1.265	0.105	0.986
LU	1.186	0.099	0.987
HP	1.313	0.109	0.986
ATC	2.477	0.206	0.973

$R_{i} = (R)^{ω_{f}}$ (25)

3.3 Proposed reliability allocation method

The proposed method integrates the hesitant fuzzy linguistic term set and the maximum entropy OWA weight to achieve a flexible allocation of system reliability. The hesitant set of vague linguistic terms can address the ambiguity of experts in providing linguistic assessments. The following procedure describes the steps of this method:

Step 1. List the configuration of system. In this method, analyze the eight subsystems of CNC machine system: CNC system, spindle system, feed system, cooling system, servo system, lubrication system, hydraulic system, and change Knife system.

Step 2. Determine system reliability requirements and error time. Based on expert opinion, the reliability requirement of CNC machine tool system is 0.875, and the mission time is 1000 h.

Step 3. Determine the scale of K, M, Co, Cr, S, and T. Using linguistic variables (Table 3), the reliability allocation team members determine the scales of K, M, Co, Cr, S, and T for the eight subsystems of CNC machine tool, as shown in Table 8.

Table 8
Allocation information from the reliability allocation team (four members) to the eight subsystems of the machine tool

Subsystems Team members Allocation factors Weights

K M Co Cr S T

CNC TM1 VH M H M M VH 0.35

TM2 VH L H H M VH 0.15

TM3 H M H L L VH 0.2

TM4 [M,H] [L,M] [H,VH] L [L,M] [H,VH] 0.3

SP TM1 VH H H M M VH 0.35

TM2 H M H M M VH 0.15

TM3 H H H H M VH 0.2

TM4 H [M,H] M [M,H] [M,H] H 0.3

FD TM1 M H M M M H 0.35

TM2 H H H L H VH 0.15

TM3 H H H L H M 0.2

TM4 [M,H] [M,H] M [L,M] M H 0.3

CO TM1 L H L M L H 0.35

TM2 M M L H L M 0.15

TM3 M M M M L H 0.2

TM4 L L L M [L,M] [M,H] 0.3

SO TM1 H H H L H H 0.35

TM2 VH H M M H VH 0.15

TM3 M H H H H VH 0.2

TM4 H VH VH [M,H] M [H,VH] 0.3

LU TM1 L H M M L H 0.35

TM2 M M L L L H 0.15

TM3 L M L L L H 0.2

TM4 M L [L,M] M M [M,H] 0.3

HP TM1 VH L H H L H 0.35

TM2 M L M L M M 0.15

TM3 H M L M M M 0.2

TM4 H L [M,H] [L,M] L H 0.3

ATC TM1 H H M M H M 0.35

TM2 H M M L L L 0.15

TM3 M VH M L M L 0.2

TM4 [H,VH] M H [M,H] [L,M] [M,H] 0.3

Subsystems	Team members	Allocation factors	Weights
CNC	TM1	VH	M	H	M	M	VH	0.35
	TM2	VH	L	H	H	M	VH	0.15
	TM3	H	M	H	L	L	VH	0.2
	TM4	[M,H]	[L,M]	[H,VH]	L	[L,M]	[H,VH]	0.3
SP	TM1	VH	H	H	M	M	VH	0.35
	TM2	H	M	H	M	M	VH	0.15
	TM3	H	H	H	H	M	VH	0.2
	TM4	H	[M,H]	M	[M,H]	[M,H]	H	0.3
FD	TM1	M	H	M	M	M	H	0.35
	TM2	H	H	H	L	H	VH	0.15
	TM3	H	H	H	L	H	M	0.2
	TM4	[M,H]	[M,H]	M	[L,M]	M	H	0.3
CO	TM1	L	H	L	M	L	H	0.35
	TM2	M	M	L	H	L	M	0.15
	TM3	M	M	M	M	L	H	0.2
	TM4	L	L	L	M	[L,M]	[M,H]	0.3
SO	TM1	H	H	H	L	H	H	0.35
	TM2	VH	H	M	M	H	VH	0.15
	TM3	M	H	H	H	H	VH	0.2
	TM4	H	VH	VH	[M,H]	M	[H,VH]	0.3
LU	TM1	L	H	M	M	L	H	0.35
	TM2	M	M	L	L	L	H	0.15
	TM3	L	M	L	L	L	H	0.2
	TM4	M	L	[L,M]	M	M	[M,H]	0.3
HP	TM1	VH	L	H	H	L	H	0.35
	TM2	M	L	M	L	M	M	0.15
	TM3	H	M	L	M	M	M	0.2
	TM4	H	L	[M,H]	[L,M]	L	H	0.3
ATC	TM1	H	H	M	M	H	M	0.35
	TM2	H	M	M	L	L	L	0.15
	TM3	M	VH	M	L	M	L	0.2
	TM4	[H,VH]	M	H	[M,H]	[L,M]	[M,H]	0.3

Step 4. Calculate the set of fuzzy evolution information for each subsystem and summarize the subjective opinions of team members on K, M, Co, Cr, S, and T of the eight subsystems (TM1, TM2, TM3, and TM4), as shown in Table 9.

Table 9

Aggregated fuzzy evolution information for the eight subsystems by the proposed method

Subsystems	K	M	Co	Cr	S	T
CNC	[5.4,6.4,7.4,8.4]	[2.1,3.1,4.1,5.1]	[5.0,6.0,7.0,8.0]	[2.3,3.3,4.3,5.3]	[2.0,3.0,4.0,5.0]	[6.4,7.4,8.4,9.4]
	[6.0,7.0,8.0,9.0]	[2.7,3.7,4.7,5.7]	[5.6,6.6,7.6,8.6]		[2.6,3.6,4.6,5.6]	[7.0,8.0,9.0,10.0]
SP	[5.7,6.7,7.7,8.7]	[4.1,5.1,6.1,7.1]	[4.4,5.4,6.4,7.4]	[3.4,4.4,5.4,6.4]	[3.0,4.0,5.0,6.0]	[6.4,7.4,8.4,9.4]
		[4.7,5.7,6.7,7.7]		[4.0,5.0,6.0,7.0]	[3.6,4.6,5.6,6.6]
FD	[3.7,4.7,5.7,6.7]	[4.4,5.4,6.4,7.4]	[3.7,4.7,5.7,6.7]	[1.7,2.7,3.7,4.7]	[3.7,4.7,5.7,6.7]	[4.9,5.9,6.9,7.9]
	[4.3,5.3,6.3,7.3]	[5.0,6.0,7.0,8.0]		[2.3,3.3.4.3.5.3]
CO	[1.7,2.7,3.7,4.7]	[3.1,4.1,5.1,6.1]	[1.4,2.4,3.4,4.4]	[3.3,4.3,5.3,6.3]	[1.0,2.0,3.0,4.0]	[4.1,5.1,6.1,7.1]
				[1.6,2.6,3.6,4.6]	[4.7,5.7,6.7,7.7]
SO	[4.9,5.9,6.9,7.9]	[5.6,6.6,7.6,8.6]	[5.3,6.3,7.3,8.3]	[2.7,3.7,4.7,5.7]	[4.4,5.4,6.4,7.4]	[5.7,.6.7,7.7,8.7]
				[3.3,4.3,5.3,6.3]		[6.3,7.3,8.3,9.3]
LU	[1.9,2.9,3.9,4.9]	[3.1,4.1,5.1,6.1]	[1.7,2.7,3.7,4.7]	[2.3,3.3,4.3,5.3]	[1.6,2.6,3.6,4.6]	[4.4,5.4,6.4,7.4]
			[2.3,3.3,4.3,5.3]			[5.0,6.0,7.0,8.0]
HP	[5.4,6.4,7.4,8.4]	[1.4,2.4,3.4,4.4]	[3.3,4.3,5.3,6.3]	[2.8,3.8,4.8,5.8]	[1.7,2.7,3.7,4.7]	[4.3,5.3,6.3,7.3]
			[3.9,4.9,5.9,6.9]	[3.4,4.4,5.4,6.4]
ATC	[4.6,5.6,6.6,7.6]	[4.5,5.5,6.5,7.5]	[3.6,4.6,5.6,6.6]	[2.3,3.3,4.3,5.3]	[2.8,3.8,4.8,5.8]	[2.3,3.3,4.3,5.3]
	[5.2,6.2,7.2,8.2]			[2.9,3.9,4.9,5.9]	[3.4,4.4,5.4,6.4]	[2.9,3.9,4.9,5.9]

Step 5. Fuzzy defuzzification processing is performed on the aggregated fuzzy information. A clear trapezoidal fuzzy number can be obtained using Equation (16). The results are shown in Table 10.

Table 10

Hesitant defuzzification of aggregated fuzzy evolution information

Subsystems	K	M	Co	Cr	S	T
CNC	[5.7,6.7,7.7,8.7]	[2.4,3.4,4.4,5.4]	[5.3,6.3,7.3,8.3]	[2.3,3.3,4.3,5.3]	[2.3,3.3,4.3,5.3]	[6.7,7.7,8.7,9.7]
SP	[5.7,6.7,7.7,8.7]	[4.4,5.4,6.4,7.4]	[4.4,5.4,6.4,7.4]	[3.7,4.7,5.7,6.7]	[3.3,4.3,5.3,6.3]	[6.4,7.4,8.4,9.4]
FD	[4.0,5.0,6.0,7.0]	[4.7,5.7,6.7,7.7]	[3.7,4.7,5.7,6.7]	[2.0,3.0,4.0,5.0]	[3.7,4.7,5.7,6.7]	[4.9,5.9,6.9,7.9]
CO	[1.7,2.7,3.7,4.7]	[3.1,4.1,5.1,6.1]	[1.4,2.4,3.4,4.4]	[3.3,4.3,5.3,6.3]	[1.3,2.3,3.3,4.3]	[4.4,5.4,6.4,7.4]
SO	[4.9,5.9,6.9,7.9]	[5.6,6.6,7.6,8.6]	[5.3,6.3,7.3,8.3]	[3.0,4.0,5.0,6.0]	[4.4,5.4,6.4,7.4]	[6.0,7.0,8.0,9.0]
LU	[1.9,2.9,3.9,4.9]	[3.1,4.1,5.1,6.1]	[2.0,3.0,4.0,5.0]	[2.3,3.3,4.3,5.3]	[1.6,2.6,3.6,4.6]	[4.7,5.7,6.7,7.7]
HP	[5.4,6.4,7.4,8.4]	[1.4,2.4,3.4,4.4]	[3.6,4.6,5.6,6.6]	[3.1,4.1,5.1,6.1]	[1.7,2.7,3.7,4.7]	[4.3,5.3,6.3,7.3]
ATC	[4.9,5.9,6.9,7.9]	[4.5,5.5,6.5,7.5]	[3.6,4.6,5.6,6.6]	[2.6,3.6,4.6,5.6]	[3.1,4.1,5.1,6.1]	[2.6,3.6,4.6,5.6]

Step 6. Calculate the maximum entropy OWA weights in conjunction with Equations (7)–(9). The results are shown in Table 11.

Table 11

The maximum entropy OWA weights

alpha	Weight
	w1	w2	w3
0.5	0.3333	0.3333	0.3333
0.6	0.4384	0.3232	0.2383
0.7	0.5539	0.2921	0.1541
0.8	0.6819	0.2953	0.1279
0.9	0.8267	0.1434	0.0249
1	1.0000	0.0000	0.0000

For example, when n = 3, α = 0.6. Using Equation (7), $\begin{matrix} ω_{1} [(n - 1) α + 1 - n ω_{1}]^{n} \\ = ((n - 1) α)^{n - 1} \end{matrix}$ and the result is $ω_{1} = 0.4384$

Using Equation (8), $ω_{3} = \frac{((3 - 1) \times 0.7 - 3) 0.4384 + 1}{(3 - 1) \times 0.7 + 1 - 3 \times 0.4384} = 0.3232$

Using Equation (9), $ω_{2} = \sqrt{ω_{1} ω_{3}} = 0.2383$

The value of α in the range 0.5–0.9 represents the optimism of decision makers on the current reliability allocation. When the decision maker faces the reliability allocation, it can take the OWA weight when α = 0.5, α = 0.9. When it is relatively representative of the degree of optimism, α = 1 is used to represent the most optimistic situation of decision makers (pure optimist: Because the reliability of entire system of CNC machine tools is extremely high, we do not need to consider the sentence of α = 1.0).

Step 7. Calculate the evaluation result of subsystem fuzzy proportionality factor

According to the results shown in Tables 9 and 10, the fuzzy proportionality factor of the subsystem can be calculated using the maximum entropy OWA weights (α = 0.5, 0.6, 0.7, 0.8, 0.9) in combination with Equations (1) and (17). The results are shown in Table 12.

Step 8. Defuzzify the fuzzy proportionality factor and flexibly allocate the reliability of subsystem.

According to the results shown in Table 12, the fuzzy scale factor can be defuzzified using Equation (18), and the weights of each subsystem can be obtained using Equations (22) and (23). Then, according to the different values of scene parameters, the subsystems reliability can be analyzed flexibly. The results are shown in Table 13.

Table 12

Evaluation results of the fuzzy proportionality factor for the eight subsystems by the proposed method

Subsystems	Weights	K×Co×M	Cr×S×T	Fuzzy proportionality factor
CNC	α= 0.5	[4.466,5.466,6.466,7.466]	[3.766,4.766,5.766,6.766]	[0.147,0.948,1.357,2.121]
	α= 0.6	[4.784,5.784,6.784,7.784]	[3.348,4.348,5.348,6.348]	[0.330,1.082,1.560,2.269]
	α= 0.7	[5.075,6.075,7.075,8.075]	[2.978,3.978,4.978,5.978]	[0.527,1.220,1.779.2.421]
	α= 0.8	[5.335,6.335,7.334,8.334]	[2.659,3.659,4.659,5.658]	[0.731,1.360,2.004,2.575]
	α= 0.9	[5.532,6.527,7.522,8.517]	[2.398,3.393,4.388,5.383]	0.924,1.488,2.217,2.714]
SP	α= 0.5	[4.833,5.833,6.833,7.833]	[4.466,5.466,6.466,7.466]	[0.067,0.902,1.250,2.057]
	α= 0.6	[4.970,5.969,6.969,7.969]	[4.214,5.213,6.214,7.214]	[0.145,0.961,1.337,2.122]
	α= 0.7	[5.120,6.121,7.121,8.121]	[3.999,4.999,5.999,6.999]	[0.223,1.020,1.425,2.187]
	α= 0.8	[5.286,6.285,7.285,8.285]	[3.826,4.825,5.825,6.824]	[0.303,1.079,1.510,2.252]
	α= 0.9	[5.453,6.448,7.443,8.438]	[3.691,4.686,5.681,6.676]	[0.376,1.135,1.588,2.310]
FD	α= 0.5	[4.133,5.133,6.133,7.133]	[3.533,4.533,6.533,7.533]	[0.132,0.786,1.353,1.939]
	α= 0.6	[4.069,5.069,6.069,7.069]	[3.241.4.240.5.240.6.240]	[0.194,0.967,1.431,2.158]
	α= 0.7	[4.021,5.021,6.021,7.021]	[2.944.3.944.4.944.5.944]	[0.273,1.016,1.527,2.218]
	α= 0.8	[3.986,4.985,5.985,6.985]	[2.638,3.638,4.638,5.637]	[0.371,1.075,1.645,2.291]
	α= 0.9	[3.954,4.949,5.944,6.939]	[2.306,3.301,4.296,5.291]	[0.499,1.152,1.801,2.384]
CO	α= 0.5	[2.067,3.066,4.066,5.066]	[3.000,4.000,5.000,5.999]	[0.0,0.613,1.017,1.814]
	α= 0.6	[1.937,2.936,3.936,4.936]	[2.915,3.915,4.915,5.915]	[0.0,0.597,1.005,1.801]
	α= 0.7	[1.828,2.828,3.829,4.829]	[2.886,3.886,4.886,5.886]	[0.0,0.579,0.985,1.783]
	α= 0.8	[1.743,2.743,3.743,4.743]	[2.917,3.917,4.916,5.916]	[0.0,0.558,0.956,1.761]
	α= 0.9	[1.683,2.678,3.673,4.668]	[3.024,4.019,5.014,6.009]	[0.0,0.534,0.914,1.733]
SO	α= 0.5	[5.266,6.266,7.266,8.266]	[4.466,5.466,6.466,7.466]	[0.146,0.969,1.329,2.124]
	α= 0.6	[5.196,6.196,7.195,8.195]	[4.167,5.167,6.167,7.167]	[0.199,1.005,1.393,2.167]
	α= 0.7	[5.125,6.125,7.125,8.126]	[3.872,4.872,5.872,6.872]	[0.257,1.043,1.462,2.214]
	α= 0.8	[5.051,6.051,7.050,8.050]	[3.575,4.575,5.575,6.574]	[0.323,1.085,1.541,2.656]
	α= 0.9	[4.950,5.945,6.940,7.935]	[3.261,4.256,5.251,6.246]	[0.397,1.132,1.631,2.322]
LU	α= 0.5	[2.333,3.333,4.333,5.333]	[2.866,3.866,4.866,5.866]	[0.0,0.685,1.121,1.891]
	α= 0.6	[2.218,3.218,4.218,5.218]	[2.646,3.645,4.645,5.645]	[0.0,0.693,1.157,1.908]
	α= 0.7	[2.114,3.114,4.115,5.115]	[2.466,3.466,4.466,5.466]	[0.0,0.697,1.187,1.921]
	α= 0.8	[2.021,3.021,4.021,5.021]	[2.330,3.330,4.330,5.330]	[0.0,0.698,1.208,1.929]
	α= 0.9	[1.935,2.930,3.925,4.920]	[2.248,3.243,4.238,5.233]	[0.0,0.691,1.210,1.926]
HP	α= 0.5	[3.466,4.466,5.466,6.466]	[3.033,4.033,5.033,6.033]	[0.107,0.887,1.355,2.086]
	α= 0.6	[3.865,4.864,5.864,6.864]	[2.933,3.933,4.933,5.933]	[0.237,0.986,1.491,2.189]
	α= 0.7	[4.258,5.259,6.259,7.259]	[2.876,3.876,4.877,5.877]	[0.356,1.078,1.615,2.283]
	α= 0.8	[4.647,5.647,7.647,8.646]	[2.867,3.867,4.866,5.866]	[0.460,1.161,1.978,2.571]
	α= 0.9	[5.015,6.010,7.005,8.000]	[2.914,3.909,4.904,5.899]	[0.537,1.226,1.792,2.428]
ATC	α= 0.5	[4.333,5.333,6.333,7.333]	[2.766,3.766,4.766,5.766]	[0.416,1.119,1.682,2.329]
	α= 0.6	[4.384,5.384,6.384,7.384]	[2.761,3.761,4.761,5.761]	[0.432,1.131,1.697,2.341]
	α= 0.7	[4.459,5.459,6.459,7.459]	[2.746,3.746,4.747,5.747]	[0.457,1.150,1.724,2.361]
	α= 0.8	[4.559,5.559,6.559,7.559]	[2.718,3.717,4.717,5.717]	[0.495,1.178,1.764,2.391]
	α= 0.9	[4.679,5.674,6.669,7.664]	[2.659,3.654,4.649,5.644]	[0.553,1.221,1.825,2.436]

Table 13

Reliability allocation of the eight subsytems of the CNC machine tools by the proposed

Subsystems	Defuzzified (Fzi)					Weightage(Wi)					Allocation reliability(Ri)
	α= 0.5	α= 0.6	α= 0.7	α= 0.8	α= 0.9	α= 0.5	α= 0.6	α= 0.7	α= 0.8	α=0.9	α= 0.5	α=0.6	α=0.7	α=0.8	α=0.9
CNC	1.252	1.450	1.665	1.893	2.112	0.129	0.141	0.153	0.162	0.173	0.983	0.981	0.980	0.979	0.977
SP	1.149	1.232	1.316	1.398	1.473	0.119	0.120	0.121	0.119	0.121	0.984	0.984	0.984	0.984	0.984
FD	1.250	1.327	1.425	1.548	1.714	0.129	0.129	0.131	0.132	0.141	0.983	0.983	0.983	0.983	0.981
CO	0.934	0.925	0.906	0.877	0.835	0.096	0.090	0.083	0.075	0.068	0.987	0.988	0.989	0.990	0.991
SO	1.225	1.285	1.352	1.429	1.518	0.127	0.125	0.124	0.122	0.124	0.983	0.983	0.984	0.984	0.984
LU	1.037	1.078	1.113	1.139	1.146	0.107	0.105	0.102	0.097	0.094	0.986	0.986	0.986	0.987	0.988
HP	1.258	1.390	1.511	1.763	1.679	0.130	0.135	0.139	0.151	0.138	0.983	0.987	0.982	0.980	0.982
ATC	1.578	1.593	1.619	1.659	1.719	0.163	0.155	0.148	0.142	0.141	0.978	0.980	0.980	0.981	0.981

3.4 Comparison and discussion

To verify the effectiveness of the proposed reliability allocation method, the eighth chapter of numerical control machine tool was taken as an example for verification in the fourth chapter. This paper also compares the simulation results with FOO and fuzzy allocation methods. The allocation results of the three methods are shown in Table 14. A comparison of FOO, fuzzy allocation, and suggested reliability allocation methods is shown in Table 15.

Table 14
Comparison of the results of three methods for reliability assignment of CNC machine tools

Subsystems FOO method Fuzzy allocation method Proposed method

Alpha

α= 0.5 α= 0.6 α= 0.7 α= 0.8 α= 0.9

CNC 0.980 0.984 0.983 0.981 0.980 0.979 0.978

SP 0.963 0.983 0.984 0.984 0.984 0.984 0.983

FD 0.981 0.980 0.984 0.983 0.983 0.983 0.982

CO 0.997 0.990 0.987 0.988 0.988 0.989 0.990

SO 0.966 0.986 0.983 0.983 0.983 0.982 0.983

LU 0.998 0.987 0.986 0.986 0.987 0.988 0.988

HP 0.993 0.986 0.983 0.982 0.982 0.981 0.982

ATC 0.991 0.973 0.979 0.980 0.981 0.982 0.982

Consider TM1 TM1 TM1 TM1 TM1 TM1 TM1

Information TM2 TM2 TM2 TM2 TM2 TM2 TM2

TM3 TM3 TM3 TM3 TM3 TM3 TM3

TM4 TM4 TM4 TM4 TM4

Overall Reliability 0.875 0.875 0.875 0.875 0.875 0.875 0.875

Subsystems	FOO method	Fuzzy allocation method	Proposed method
CNC	0.980	0.984	0.983	0.981	0.980	0.979	0.978
SP	0.963	0.983	0.984	0.984	0.984	0.984	0.983
FD	0.981	0.980	0.984	0.983	0.983	0.983	0.982
CO	0.997	0.990	0.987	0.988	0.988	0.989	0.990
SO	0.966	0.986	0.983	0.983	0.983	0.982	0.983
LU	0.998	0.987	0.986	0.986	0.987	0.988	0.988
HP	0.993	0.986	0.983	0.982	0.982	0.981	0.982
ATC	0.991	0.973	0.979	0.980	0.981	0.982	0.982
Consider	TM1	TM1	TM1	TM1	TM1	TM1	TM1
Information	TM2	TM2	TM2	TM2	TM2	TM2	TM2
	TM3	TM3	TM3	TM3	TM3	TM3	TM3
			TM4	TM4	TM4	TM4	TM4
Overall Reliability	0.875	0.875	0.875	0.875	0.875	0.875	0.875

Table 15

The main difference between the three methods

Method selection	Consider factor
	order weight	fuzzy information	hesitant information	Situation parameter
FOO method	NO	NO	NO	NO
Fuzzy allocation method	NO	NO	NO	NO
Proposed method	YES	YES	YES	YES

As shown in Tables 14 and 15, the proposed reliability allocation method has several advantages:

First, the proposed reliability allocation method considers the order weights of K, M, Co, Cr, S, and T. The ordered weights allow more accurate system reliability assignments.As is known to all,the machine tool system is a relatively complicated system. The reliability allocation of the machine tool system by using FOO method or fuzzy allocation method alone will lead to the deviation of the results.The two methods do not consider the order weight,It can be seen from Equation 21 that in the FOO method, simply multiplying the evaluation values of K, M, Co, Cr, S, and T,which leads to deviation!As a result, ME-OWA method is combined in this paper. The method can solve the problem of ordered weights, so that reliability allocation can be performed according to actual requirements when performing reliability allocation.

Second, the method can handle fuzzy and hesitant information. In the reliability allocation, experts will generate hesitant linguistic values in the product design and development stage to represent the corresponding K, M, Co, Cr, S, and T factors, thus increasing the difficulty of reliability allocation. Although the FOO method is simple and widely used in industry, it cannot address ambiguity or hesitant information to reliability allocation.To handle the abovementioned problems, this paper optimizes by adding a hesitant fuzzy linguistic term set based on the fuzzy allocation method, so that the fuzzy and hesitant information can be solved.

Finally, this article uses situation parameters to reflect the current optimism of decision makers (maximum optimism, optimism/pessimistic neutrality, and minimum optimism), which can be combined with K, M, Co, Cr, S, and T,so that the corresponding situation parameters can be selected according tothe current optimism of decision makers to perform reliability allocation, which makes the reliability allocationn more accurate.However, apparently FOO and fuzzy allocation methods do not consider the situation parameters, which can lead to erroneous conclusions.

4 Conclusion and project

4.1 Conclusion

Reliability allocation is a key task in reliability design, including setting the target reliability of components or subsystems to satisfy operational needs. The main purpose is to allocate limited resources while achieving reliability goals. However, traditional FOO and fuzzy allocation methods fail to deal with expert hesitation information, delete such data and cause some useful information to be ignored. In addition, the two methods do not consider the ordered weights of K, M, Co, Cr, S and T, which may lead to wrong conclusions. To solve this problem effectively, this paper integrates the hesitant fuzzy linguistic term set and the maximum entropy OWA weight to achieve a flexible allocation of system reliability.

The main advantages of the proposed reliability allocation method are as follows:

The proposed method not only assigns system reliability according to the ordered weights of K, M, Co, Cr, S, and T, but also flexibly allocates the reliability of system according to the specific situation.

The proposed method can deal with fuzzy or hesitant information, and the information of reliability factor can be described as a linguistic variable, so that the reslt better corresponds to the reality.

The proposed method uses the situation parameter value to flexibly allocate the system reliability,so that the corresponding situation para-meters can be selected according to the specific situation to perform reliability allocation, which makes the reliability allocation more accurate.

The proposed method is applicable to any large-scale complex system that requires reliability allocation and is not limited to only K, M, Co, Cr, S, and T.

4.2 Insufficient and project

In this paper, for the reliability allocation weighting factors (K, M, Co and Cr, S, T) of CNC machine tools, the method of expert scoring in fuzzy allocation was used in a single way. Although the problem of hesitant information allocation can be solved by adding the fourth expert, but overall, it is biased towards theorizing, purely relying on the experience of experts to solve. This can significantly shorten the cost and time of reliability allocation in the field of solving some simple systems, but for heavy-duty CNC machine tools. In the reliability allocation of complex systems, owing to the complexity of system, sometimes the problem of reliability allocation cannot be solved by simply assigning each subsystem. Sometimes it should be specific to the allocation of parts. Therefore, in the future, when facing the reliability allocation of some complex systems, we should consider the fault information and data of some subsystems. Later research will combine some data analysis methods to synthesize expert ratings for reliability allocation.

Footnotes

Acknowledgments

Financial supports from National Natural Science Foundation of China (51575010)and National Science and Technology Major Project (2018ZX04033001-003) are acknowledged.

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