Evaluation analysis for sortie generation of carrier aircrafts based on nonlinear fuzzy matter-element method

Abstract

This paper proposes a new, comprehensive evaluation method of a principal, nonlinear fuzzy matter-element theory, which considers the complexity, hierarchy, contradiction and relevance of the factors in the sortie generation of carrier aircrafts. First of all, the index system of the sortie generation capacity is ascertained. The importance of the index is reflected in the degree of difference between the observed values of the index. Then the entropy value method is applied to determine the indexes’ weight. Secondly, in view of the situation that one index value is high and other index values are relatively low, the index with a high value is classified as good or bad in the actual situation. However, the prominent effect of this index cannot be reflected. This is due to the shortage of weight and the weight average method. Thus, the evaluation result will not consort with the actual situation. Therefore, this paper uses the nonlinear fuzzy matter-element method to evaluate the first class indexes and the second class indexes. The problem that highlight influences of some indexes are difficult to deal with can be solved by using the nonlinear evaluation method. Finally, the Surge operation of the aircraft carrier “Nimitz” is taken as an example to evaluate the sortie generation capacity. The results verify the usefulness and reliability.

Keywords

Fuzzy matter-element evaluation method nonlinear fuzzy operator entropy value method sortie generation capacity of carrier aircrafts comprehensive evaluation

1 Introduction

The aircraft carrier is an important part of modern, naval warfare. The research on the warfare capacity of the aircraft carrier has become a hot issue [2 , 9]. The study of the sortie generation capacity of the aircraft carrier in different operational schemes is helpful to determine the final plan [7 , 15]. Therefore, the evaluation for the sortie generation capacity of the aircraft carrier has important, theoretical significance.

The linear, fuzzy matter-element method has been adapted to evaluate in a wide field. Xin et al. [11] proposed an optimization model for locating city fire stations based on the fuzzy matter-element analysis. Wu et al. [12] solved the selection of atmospheric, environmental monitoring sites based on the geographic parameters’ extraction of GIS and the fuzzy matter-element analysis. Cheng et al. [14] considered the geographical profile establishment based on the ANFIS and the fuzzy matter-element analysis. Gong et al. [20] applied fuzzy matter-element evaluation method for a reliability analysis of an existing highway tunnel. The fuzzy matter-element evaluation model was effectively used to evaluate the reliability of one highway tunnel structure. Tan et al. [5] proposed the introduction of the concept of the information entropy in order to establish a fuzzy matter-element evaluation method based on the principle of the fuzzy matter-element analysis. Bai et al. [18] integrated engineering economics and a reliability theory; the fuzzy matter element and entropy theory presented a new optimization model about an enterprise-equipment investment project. It also presented detailed methods and steps of application for the integrated model in a concrete example. This can offer a reference for the project investment activity of the practical enterprise equipment. Yiannis et al. [4] investigated the indirect, adaptive regulation problem of the unknown affine in the control, nonlinear systems. Salam et al. [3] considered the utilization of the reclaimed wastewater for olive irrigation, as well as the effect on soil properties, tree growth, yield and oil content. Ebrahimabadi et al. [1] proposed fuzzy multi-criteria decision-making methods. Nagpal et al. [17] illustrated that the usability evaluation of the website using a combined, weighted method: the fuzzy AHP and entropy approach. The fuzzy matter-element analysis has been used in many other areas [8]. However, most of these evaluation methods used the linear weighted evaluation method. Its defect is that the prominent influences of some indexes cannot be reflected.

A new evaluation method of a nonlinear fuzzy matter-element evaluation method (NFMEEM) is proposed in this paper to evaluate the sortie generation capacity of carrier aircrafts. This can avoid the problem that the prominent influences of some indexes cannot be reflected using the linear, fuzzy matter-element evaluation method (LFMEEM). The main contents are as follows: In Section 2, the hierarchy structure of the index system for the sortie generation capacity is determined. The entropy value method is applied to determine the indexes weight in Section 3. In Section 4, the nonlinear fuzzy operator is introduced. NFMEEM is introduced in Section 5. Finally, the sortie generation capacity of the carrier aircrafts is evaluated with NFMEEM and LFMEEM, and the differences between the two methods are discussed.

2 Index system for sortie generation capacity of carrier aircrafts

The index system for the sortie generation capacity of carrier aircrafts is established with related research results. A three-level index system with complexity, hierarchy, contradiction and relevance is established by the recursive hierarchy method. The index system for the sortie generation capacity of carrier aircrafts is shown in Fig. 1.

These indexes are defined as follows:

Emergency sortie generation rate (ESGR): maximum number of ready aircrafts taking off in a few minutes.

Surge sortie generation rate (SSGR): average number of aircrafts per day in Surge operation.

Last sortie generation rate (LSGR): average number of aircrafts per day in a continuous operation.

Performing tasks proportion (PTP): time proportion that aircrafts can carry out at least one task under a certain flight plan.

Missing tasks proportion waiting for parts (MTPWP): proportion of aircrafts missing tasks due to waiting for parts.

Missing tasks proportion waiting for repair (MTPWR): proportion of aircrafts missing tasks due to waiting for repair.

Scheduled completion proportion (SCP): the proportion of the completed number in the planned number of aircrafts.

Pilot utilization rate (PUR): average utilization rate of pilots per day.

Plan implementation probability per aircraft (PIPA): plan implementation probability per aircraft under certain constraints in a given period of time.

Sortie generation rate per aircraft (SGRA): sortie generation rate per aircraft under certain constraints.

Preparation time for next sortie (PTNS): preparation time for the next sortie under the condition of a certain resource allocation.

Ejection interval (EI): average time for ejecting a single aircraft per catapult.

Take-off outage proportion (TOOP): proportion of the canceled number in the ready number of aircrafts.

Recovery interval (RI): average time for recovering a single aircraft.

Overshoot proportion (OP): proportion of the number of aircrafts that failed to recover in the number of aircrafts ready to recover.

3 Determining the indexes weight with entropy value method

The concept of “entropy” is from thermodynamics. Assuming that there are m scenarios and n indexes, the evaluation matrix of scenarios can be obtained. If the entropy value is greater, the differences of the index are greater in different scenarios, and it will provide more information for the final decision. Thus, the entropy value reflects the information of objective data.

The entropy value method is to use entropy to determine the weight of the index. Generally, the evaluation object set is {x_i } (i = 1, 2, ⋯ , m). The number of indexes is n. Let j = 1, 2, ⋯ , n. j denotes the jth index. f is the value of theindex.

The entropy of index is expressed in Equation (1). $e (f_{j}) = - K \sum_{i = 1}^{m} \frac{f_{j} (x_{i})}{E_{j}} lg \frac{f_{j} (x_{i})}{E_{j}}$ (1) where, $E_{j} = \sum_{i = 1}^{m} f_{j} (x_{i})$ , f_j (x_i) is the jth criterion value of the ith scenario x_i, K > 0 and K = 1/ - ln(m).

If the values of f_j (x_i) in all of the scenarios are equal, the relative strength is $f_{j} (x_{i}) / - \sum_{i = 1}^{m} f_{j} (x_{i}) = 1 / - m$ . In this situation, e (f_j) isthe maximum (the amount of information is the minimum) and e (f_j) _max = K ln(m). If K = 1/ - ln(m), the total entropy of criterion set F is defined as Equation (2): $E = \sum_{j = 1}^{n} e (f_{j})$ (2)

The amount of information and entropy is the inverse ratio. Therefore, the weight of information is represented as Equation (3): $w_{j}^{2} = \frac{[1 - e (f_{j})]}{[n - E]}, j = 1, 2, \dots, n$ (3)

4 Nonlinear fuzzy operator

In the fuzzy comprehensive evaluation model, the fuzzy operator is defined in Equation (4) [19]:

$\begin{matrix} B & = & A \circ R = (b_{1}, b_{2}, \dots, b_{m}) \\ = & (a_{1}, a_{2}, \dots, a_{n}) \circ [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 m} \\ r_{21} & r_{22} & \dots & r_{2 m} \\ ⋮ \\ r_{n 1} & r_{2} & \dots & r_{nm} \end{matrix}] \end{matrix}$ (4)

The degree coefficient vector of the highlight influence is defined as Λ = (λ₁, λ₂, ⋯ , λ_n), where λ_i > 1. If the highlight influence of the index is greater, the λ_i is greater. If the highlight influence of the index is small, λ_i is 1. Thus. λ_i is an integer, which is greater than 1 or equal to 1. Let λ = max{ λ₁, λ₂, ⋯ , λ_n }. The nonlinear fuzzy operator is defined in Equation (5).

$\begin{matrix} f (a_{1}, a_{2}, \dots, a_{n}; x_{1}, x_{2}, \dots, x_{n}; Λ) \\ = {(a_{1} x_{1}^{λ_{1}} + a_{2} x_{2}^{λ_{2}} + \dots + a_{n} x_{n}^{λ_{n}})}^{\frac{1}{λ}}, \\ λ_{i} \geq 1, i = 1, 2, \dots, n . \end{matrix}$ (5)

Let A = (a₁, a₂, ⋯ , a_n), where a_i > 0. And X = (x₁, x₂, ⋯ , x_n), where x_i ≥ 1. The characters of the nonlinear fuzzy operator are as follows:

First of all, when the influence degree of some index is greater than that of other index, and both the index values are same, the evaluation results of two indexes should be different.

Secondly, when all of the index values of one scenario are greater than those of another scenario, the former evaluation result should be greater than the latter evaluation result.

Thirdly, when the highlight influence degree of the index is greater, the influence of the index on the evaluation result is greater.

To sum up, this nonlinear fuzzy operator can meet the practical evaluation and can make up for the shortcomings of the linear weighted operator.

5 Nonlinear fuzzy matter-element comprehensive evaluation method

5.1 Basic concept of fuzzy matter-element method

The matter-element is the basic element that contained “matter, characteristics and value”. If the value is fuzzy, the fuzzy matter-element can be formed with “matter, characteristics and fuzzy value”, expressed in Equation (6).

$\begin{matrix} fuzzy matter - element \\ = [\begin{matrix} matter \\ characteristics & fuzzy value \end{matrix}] \end{matrix}$ (6)

R is the fuzzy matter-element. M is the matter. C is the characteristics of matter M. μ (x) is the fuzzy value, namely the membership degree of the value x in the characteristics C. Equation (6) is arranged in Equation (7): $R = [\begin{matrix} M \\ C & μ (x) \end{matrix}]$ (7)

If m matters are described with n characteristics and fuzzy values, the fuzzy complex matter is obtained from Equation (8): $R = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ C_{1} & μ (x_{11}) & μ (x_{21}) & \dots & μ (x_{m 1}) \\ C_{2} & μ (x_{12}) & μ (x_{22}) & \dots & μ (x_{m 2}) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C_{n} & μ (x_{1 n}) & μ (x_{2 n}) & \dots & μ (x_{mn}) \end{matrix}]$ (8)

5.2 General steps of fuzzy matter-element comprehensive evaluation method

The general steps of the fuzzy matter-element comprehensive evaluation method are shown in Fig. 2.

(1) Establishing matter-elements

1) The fuzzy complex matter-elements are established. When the matter-elements are evaluated, these matter-elements are divided into m levels considering n main factors. C_j is the ith main factor in the jth level, which also contains p secondary factors. Let C_jk be the kth secondary factor in the jth level. Its corresponding value is x_jk. Then, the fuzzy complex matter-element is R_n, as expressed in Equation (9): $R_{n}^{'} = [\begin{matrix} C_{11} & C_{12} & \dots & C_{np} \\ x_{11} & x_{12} & \dots & x_{np} \end{matrix}]$ (9)

Let μ_jik be the membership degree of x_jk in the jth level. Then, R_mn is the fuzzy complex matter with m levels and n dimensions, which is expressed in Equation (10):

$R_{mn} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ C_{11} & μ_{111} & μ_{211} & \dots & μ_{m 11} \\ C_{12} & μ_{112} & μ_{212} & \dots & μ_{m 12} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C_{np} & μ_{1 np} & μ_{2 np} & \dots & μ_{mnp} \end{matrix}]$ (10)

2) Weight matter-elements of factors are established. The membership degrees in each level are dispersed in (9). Therefore, it is necessary to focus for one value. R_wik is the weight matter-element of the secondary factor. Also, R_wi is the weight matter-element of the main factor.

➀ The weight matter-element of the second class indexes is expressed in Equation (11): $R_{wik} = [\begin{matrix} C_{11} & C_{12} & \dots & C_{np} \\ w_{ik} & w_{11} & w_{12} & \dots & w_{np} \end{matrix}]$ (11) where, w_ik denotes the weight of kth secondary factor in the ith main factor at each level.

➁ The weight matter-element of first class indexes is expressed in Equation (12): $R_{wi} = [\begin{matrix} C_{1} & C_{2} & \dots & C_{n} \\ w_{i} & w_{1} & w_{2} & \dots & w_{n} \end{matrix}]$ (12) where, w_i denotes the weight of the ith main factor at each level.

3) The concentration fuzzy complex matter is determined. In order to determine the concentration fuzzy complex matter-element, the weight average should be carried out in Equation (13): $R_{b} = R_{wik} * R_{mn}$ (13) where, * is the weighted average operator and the expression of R_b is given in mode M (• , +):

$R_{b} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ b_{1} & b_{11} = \sum_{k = 1}^{p} w_{1 k} u_{11 k} & b_{21} = \sum_{k = 1}^{p} w_{1 k} u_{21 k} & \dots & b_{m 1} = \sum_{k = 1}^{p} w_{1 k} u_{m 1 k} \\ b_{2} & b_{12} = \sum_{k = 1}^{p} w_{2 k} u_{12 k} & b_{22} = \sum_{k = 1}^{p} w_{2 k} u_{22 k} & \dots & b_{m 2} = \sum_{k = 1}^{p} w_{2 k} u_{m 2 k} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ b_{n} & b_{11} = \sum_{k = 1}^{p} w_{nk} u_{n 1 k} & b_{11} = \sum_{k = 1}^{p} w_{nk} u_{2 nk} & \dots & b_{11} = \sum_{k = 1}^{p} w_{nk} u_{mnk} \end{matrix}]$ (14) where, b_j (j = 1, 2, ⋯ , m, i = 1, 2, ⋯ , n) denotes the concentration value of the membership degree of the ith main factor.

(2) Comprehensive evaluation of matter-elements

1) The single evaluation of the fuzzy complex matter-element is determined. R_x is the single evaluation of the fuzzy complex matter. The linear fuzzy matter-element is expressed in Equation (15): $R_{x} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ x_{1} & x_{11} = w_{1} b_{11} & x_{21} = w_{1} b_{21} & \dots & x_{m 1} = w_{1} b_{m 1} \\ x_{2} & x_{12} = w_{2} b_{12} & x_{22} = w_{2} b_{22} & \dots & x_{m 2} = w_{2} b_{m 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n} & x_{1 n} = w_{n} b_{1 n} & x_{2 n} = w_{n} b_{2 n} & \dots & x_{mn} = w_{n} b_{mn} \end{matrix}]$ (15)

At the same time, the nonlinear fuzzy matter-element is expressed in Equation (16): $R_{x} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ x_{1} & x_{11} = w_{1} 2^{b_{11}} & x_{21} = w_{1} 2^{b_{21}} & \dots & x_{m 1} = w_{1} 2^{b_{m 1}} \\ x_{2} & x_{12} = w_{2} 2^{b_{12}} & x_{22} = w_{2} 2^{b_{22}} & \dots & x_{m 2} = w_{2} 2^{b_{m 2}} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n} & x_{1 n} = w_{n} 2^{b_{1 n}} & x_{2 n} = w_{n} 2^{b_{2 n}} & \dots & x_{mn} = w_{n} 2^{b_{mn}} \end{matrix}]$ (16)

2) The comprehensive evaluation of the fuzzy complex matter-element is established. In order to overcome the one-sidedness, the evaluation index comprises the mean d_j1, the maximum d_j2 and the minimum d_j3 values of the fuzzy value in the main factors. They are expressed in Equation (17): ${\begin{matrix} d_{j 1} = (x_{j 1} + x_{j 2} + \dots + x_{jn}) / n \\ d_{j 2} = max (x_{j 1} + x_{j 2} + \dots + x_{jn}) \\ d_{j 3} = min (x_{j 1} + x_{j 2} + \dots + x_{jn}) \end{matrix}, j = 1, 2, \dots, m$ (17)

R_d denotes the evaluation of the fuzzy complex matter-element, which is expressed in Equation (18): $R_{d} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ d_{j 1} & d_{11} & d_{21} & \dots & d_{m 1} \\ d_{j 2} & d_{12} & d_{22} & \dots & d_{m 2} \\ d_{j 3} & d_{13} & d_{23} & \dots & d_{m 3} \end{matrix}]$ (18)

R_D is the comprehensive evaluation of the fuzzy complex matter-element, which is expressed in Equation (19): $R_{D} = [\begin{matrix} M_{1} & M_{2} & \dots & M_{m} \\ d_{j} & d_{1} & d_{2} & \dots & d_{m} \end{matrix}]$ (19) where, d_j is the comprehensive evaluation of the jth level, which is expressed in Equation (20): $d_{j} = \frac{1}{3} \sum_{i = 1}^{3} d_{ji}, j = 1, 2, \dots, m$ (20) where the corresponding level M of the maximum value d_max in the comprehensive evaluation d_j is the quantization level of the evaluated object.

6 Evaluation for sortie generation capacity of carrier aircrafts

6.1 Evaluation samples

The object of evaluation is the Surge operation of the “Nimitz” carrier in 1997 [2]. Ten scenarios are selected randomly in order to ensure the scientific nature, which are shown in Tables 1–4.

6.2 Fuzzy matter-element comprehensive evaluation of sortie generation of carrier aircrafts

According to the above scenarios, the entropy value method is applied to determine the indexes weight. Then, the fuzzy matter-element comprehensive evaluation method is used to evaluate 10 scenarios. The quantization level is divided five levels: into I (4, 5), II (3, 4], III (2, 3], IV (1, 2], V (0, 1].

The indexes of the sortie generation of the carrier aircrafts are analyzed with the relative weights of the first class indexes, relative weights of the second class indexes, quantization levels of the second class indexes and the absolute weights of the second class indexes. The results of LFMEEM and NFMEEM are compared. Also, the results of NFMEEM are compared with the results of the maximum entropy configuration model of the objective index weight based on the grey-correlation deep coefficient (MECM) [16].

(1) LFMEEM

The results of LFMEEM are shown in Table 5 and Figs. 3–5.

The following facts of LFMEEM can be obtained from Table 5. The relative weight of the availability capacity (R₂) is the greatest one and it has the greatest influence on the evaluation result. The second greatest influences are support, ejection and recovery capacity (R₄). The third is the sortie generation rate capacity (R₁). The smallest one is the tasks completion capacity (R₃). The absolute weights of the emergency sortie generation rate, surge sortie generation rate, last sortie generation rate, performing tasks proportion, missing tasks proportion waiting for parts and missing tasks proportion waiting for repair in the second class indexes are greater than those of the other second class indexes. Thus, their influence on the evaluation result is greater.

(2) NFMEEM

The results of NFMEEM are shown in Table 6 and Figs. 6–7.

Figure 6 illustrates the deviation of the quantization levels of the second class indexes under LFMEEM and NFMEEM. The quantitative levels of 8 second class indexes are the same under LFMEEM and NFMEEM. The quantitative levels of 3 second class indexes under LFMEEM are 1 level higher than those under NFMEEM. The quantitative levels of 2 second class indexes under LFMEEM are 2 levels higher than those under NFMEEM. The quantitative levels of 2 second class indexes under LFMEEM are 3 levels higher than those under NFMEEM. It is not difficult to figure out the better method between LFMEEM and NFMEEM by analyzing the original data.

The original data show that the last sortie generation rate is generally less than the surge sortie generation rate. Therefore, the quantitative level of the last sortie generation rate should be lower than that of the surge sortie generation rate. However, the evaluation result under LFMEEM is that the quantitative level of the last sortie generation rate is higher than that of the surge sortie generation rate, which is not reasonable. The evaluation result under NFMEEM is convincing.

The original data show that the performing tasks proportion is affected by the missing tasks proportion waiting for parts and the missing tasks proportion waiting for repair. Thus, the quantitative level of the performing tasks proportion should be lower than that of the missing tasks proportion waiting for parts and the missing tasks proportion waiting for repair. However, the evaluation result under LFMEEM is that the quantitative levels of the performing tasks proportion, missing tasks proportion waiting for parts and missing tasks proportion waiting for repair are II, IV, III, respectively. This result is opposite to the original data.

In the actual condition, the influence of the preparation time is large for the next sortie and the recovery interval on the sortie generation of the carrier aircrafts. However, the evaluation result under LFMEEM is that the quantitative levels of the preparation time for the next sortie and the recovery interval are II, which is not reasonable.

The reason for this result is that weight-average model cannot reflect the significant impact of some of the indexes on the evaluation result, which leads to the distortion of the evaluation result.

(3) MECM

Wang et al. [16] proposed the characterization of the information size that is contained by the objective index weight by using the grey-correlation deep coefficient. Thus, this paper compared NFMEEM with MECM.

The results of MECM are shown in Table 7 and Figs. 8–9.

The quantization levels of the second class indexes under MECM are analyzed. Table 7 and Figs. 8–9 illustrate the deviation of the quantization levels of the second class indexes under NFMEEM and MECM. The quantitative levels of 2 second class indexes under NFMEEM are 1 level higher than those under MECM. The quantitative level of 1 second class index under NFMEEM is 1 level lower than those under MECM.

The original data show that performing tasks proportion is affected by missing tasks proportion waiting for parts and missing tasks proportion waiting for repair. Thus, the quantitative level of performing tasks proportion should be lower than that of missing tasks proportion waiting for parts and missing tasks proportion waiting for repair. The evaluation results under NFMEEM and MECM are that missing tasks proportion waiting for parts and missing tasks proportion waiting for repair are IV, III, respectively. The evaluation results under NFMEEM and MECM are that performing tasks proportion are IV, V, respectively. Both of these results are reasonable.

In the actual condition, the influence of the scheduled completion proportion is lower than that of the sortie generation rate per aircraft. However, the quantitative level of the scheduled completion proportion under MECM is equal with that of the sortie generation rate per aircraft, which is not reasonable. The quantitative level of the scheduled completion proportion under NFMEEM is 1 level lower than that of the sortie generation rate per aircraft, which is more reasonable.

In the actual condition, the influence of the take-off outage proportion is not lower than that of the overshoot proportion. However, the quantitative level of the take-off outage proportion under MECM is 1 level lower than that of the overshoot proportion, which is not reasonable. The quantitative levels of the take-off outage proportion and the overshoot proportion under NFMEEM are same, which is more reasonable.

The reason for this result is that MECM and NFMEEM can reflect the significant impact of some indexes on the evaluation result. Also, the advantage of NFMEEM in reflecting the different, significant impacts of the indexes is a little larger than that of MECM.

Therefore, in order to improve the sortie generation of carrier aircrafts, the indexes with low a quantitative level and a high weight value should be firstly improved. Then, the sortie generation of carrier aircrafts will be improved in making full use of resources.

7 Conclusion

This paper proposes a new, comprehensive evaluation method based on NFMEEM to deal with complexity, hierarchy, contradiction and relevance of the factors in the sortie generation of carrier aircrafts. This method can make up for the deficiency of the traditional fuzzy comprehensive evaluation method. The importance of the index is reflected with the degree of the difference in the observed values of index. In view of the situation that one index value is high and other index values are relatively low, the index with a high value is classified as good or bad in the actual situation. NFMEEM solves this situation. The evaluation of an example illustrates the feasibility and superiority of the nonlinear fuzzy matter-element evaluation method, which helps the evaluation results to coincide with the actual situation. Finally, the important indexes are discovered.

This evaluation method can also be spread to the general comprehensive evaluation. Also, this method can make up for the shortage of the weight-average method.

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