Reliability Evolution of Elevators Based on the Rough Set and the Improved TOPSIS Method

Focusing on reliability evaluation of elevator equipment operation, this paper builds a model for reliability evaluation of elevator equipment. First, we analyze the factors influencing the reliability of elevators and extract the initial evaluation indicators from four macro-perspectives: people, management, equipment, and environment, to build the index system for reliability evaluation of elevator equipment. And, we reduce the indicator set to remove the unnecessary indicators by using the rough set and weight the reduced indicators according to expert opinions. Finally, on this basis, we combine the grey number theory with the TOPSIS method to build an improved TOPSIS evaluation model, to evaluate the reliability of elevators and determine the optimal ordering.


Introduction
With the rapidly developing economics and society, the accelerating urbanization process, and the improving living standards in China, the passenger elevator becomes the essential vertical transport in people's daily life and society production, which is the most common special equipment in life.With the increasing number of elevators, elevator safety accidents caused by elevator failure happen frequently.The safety and reliability of elevator equipment are related to the people the safety of life and property, and any elevator accident can cause enormous loss of life and property.The statistic issued by AQSIQ shows that, in 2014, a total of 48 elevator accidents caused by various reasons killed 36 people in China; in 2015, a total of 58 elevator accidents killed 46 people in China.Therefore, the reliability of elevator safety has been concerning by the public, and it is significant to evaluate the reliability of elevators, which can find out problems instantly, get rid of security danger, and avoid the safety accidents.
Relative researchers have studied a lot on the reliability of elevator safety.Zhang applied the fuzzy evaluation method and fuzzy AHP into the evaluation system for risk of elevator system, by using the concrete mathematical model to calculate the evaluation, improving the speed and accuracy of evaluation [1].Gu et al. provided the reliability evaluation method of elevator safety based on the risk principle, with integral various evaluation theory, including safety checklist, fuzzy evaluation, and fault tree analysis, and built the expert system of elevator safety evaluation on this basis simultaneously [2].Liu et al. provided a risk model for using elevators based on inspection data and analyzed it [3].Fan et al.Seung-Tae Park introduced the risk prediction technique into the maintenance inspection of elevators, to analyze the parts that can fail and the damages that can occur, and the maintainer maintained the elevator in terms of the risk analysis [4].
By referring to the relative literature, we find that there have been lots of researches on the safety and reliability evaluation of elevators, which have formed their own system.However, most of the researches focus on a certain system and a certain part of elevators, with few reliability researches on overall equipment.As for the selection of indicators used in reliability evaluation of elevator equipment, it mostly depends on the subjective judgments or experience, with the lack of systematic theory and a single research method, and the objectivity of evaluation results is easily called into question because of these defects.Thus, how to build an evaluation The reliability of elevators Management The setting of management organization  3 The management of maintenance and use  4 The management regulations for elevators  5 The management of safety and accident emergency treatment  6 The reliability of elevators Environment The environment of equipment operation  7 The condition of routine maintenance  8 The reliability of elevators Equipment The evaluation of hardware  9 The evaluation of software  10 The interaction evaluation of hardware and software  11 system of elevator equipment completely and reasonably and explore an effective evaluation method becomes an important research subject.In view of this, this paper studies the reliability of elevator equipment from the perspective of multiobjective.First, we systematically select the indicators influencing elevator operation, from four perspectives: people, management, equipment, and environment.Secondly, we preprocess the reliability indicators of elevators by using the rough set method.Finally, we build a combination weightinggrey TOPSIS model for reliability evaluation of elevators and sort [5], to provide the reference of reliable elevator maintenance and operation plan for enterprises.

The Analysis of Reliability Factor for Elevators and the Construction of Index System
There are many factors influencing the reliability of elevators.
On the basis of consulting data and spot examination, under the condition of fully acquaint with the structure of elevator system, this paper builds the index system for reliability evaluation of elevator equipment by dividing the reliability of elevators in a certain unit into the macro and micro aspects (As shown in Table 1).
From four perspectives, people, management, equipment, and environment, we determine the specific indicators evaluating the reliability of elevator equipment.
People: People, the first-class indicator, can be classified into two parts: (1) maintainers and (2) operators.They consist of the personnel factor influencing the reliability of elevators.
Equipment: The reliability indicators of main elevator equipment are subdivided into (1) safety evaluation of main parts; (2) evaluation of operating performance; (3) evaluation of safety protection devices.
Management: Based on the principle of PDCA Cycle, this paper divided the management into four indicators: (1) the setting of management organization; (2) the management of maintenance and use; (3) the management regulations for elevators; (4) the management of safety and accident emergency treatment.
Environment: The environment is comprises of (1) the environment of equipment operation and (2) the condition of routine maintenance, to build the index system of reliability evaluation.
In particular, these indicators, people, management, equipment and environment, are not independent or separated but interrelated and relying on each other, and changes in any indicator can cause the change of other indicators, to lead the deviation of business decision.Therefore, in the process of evaluating the reliability of elevator equipment, we need to fully consider the interacting factors among indicators, clarify the interrelation among indicators, and build the figure of interrelation among the indicators of the reliability of elevators, which is shown in Figure 1.[6][7][8] Definition 1.Let  = (, , , ) be a knowledge expressing system, where  is the universe,  is the attribute set,  is the attribute-values set, and  is the information function.

The Rough Set
Definition 2. Let  be a cluster equivalence relation, where  ∈ .When ind() = ind( − {}), then  is not necessary in  and vice versa.For each  ∈  is necessary in ,  is independent and vice versa.Definition 3. The set which comprises all necessary attributes in R is denoted as core() and called the core of the attribute set.When  ∈  and both  and  are independent, i.e., ind() = ind(), then  is the reduction of , which is denoted as red().

Definition 4.
Where  ⊆ , /ind() denotes the relevant knowledge of equivalence relation family  and its trail information is denoted as

The Grey Number
Theory.Grey number generally refers to the number without determined value [9], and the value can only be determined by scope; usually we use "⊗" to The setting of management organization The management regulations for elevators The management of maintenance and use The management of safety and accident emergency treatment The environment of equipment operation The condition of routine maintenance The evaluation of hardware denote the grey number; for instance, the grey number of  is denoted as ⊗ = [⊗, ⊗], where ⊗ and ⊗ are denoted as the upper and lower bounds, respectively.

The Model for Evaluating the Reliability of Elevators
Among the amount of data of the reliability of elevators, not all the data have the influence on the reliability of elevators.To accurately capture the key indicators and effectively evaluate the reliability of elevators, we need to filter the indicators.For this, this paper builds the model for index evaluation based on the rough set preprocessing, which is shown in Figure 2.This model reduces the initial indicators of elevator equipment by using the rough set theory firstly, to get rid of the unnecessary indicators, obtaining the simple and effective index system.And, the grey number theory is applied to the TOPSIS [10] matrix, and the weights of reduced indicators can be determined by using the combination weighting method.In this paper, the objective weights of indicators are determined by using the rough set then combined with the subjective weights of experts to determine the final weights by weighted calculation.Finally, the evaluation result of the reliability of elevators cab be deduced by calculating the paste progress of grey ideal solution.The steps are shown as follows.
Step 1 (build the initial evaluation index system).According to the character of elevator equipment, we select the evaluation indicators and determine the evaluation index system from four aspects: people, equipment, environment, and management.Suppose that there are m elevators, which are denoted as  1 ,  2 ,  3 , . . .  , respectively; there are n indicators, which are denoted as  1 ,  2 ,  3 ⋅ ⋅ ⋅  n , respectively.The weight of each indicator is denoted as   , for  = 1, 2, 3 ⋅ ⋅ ⋅ n.
Step 2 (build the decision matrix based on the grey theory).According to the scoring interval and information identification of the reliability indicators of elevators by experts, we take the corresponding grey number, which is expressed as follows: for  = 1, 2, 3 ⋅ ⋅ ⋅ , m,  = 1, 2, 3 ⋅ ⋅ ⋅ , n, and build the grey decision matrix as Step 3 (reduce the indicators by using the rough set theory).
The reduction of indicators is to get rid of the redundant indicators of the reliability of elevators and form the new evaluation indicators, to simplify the evaluation.The specific steps are as follows: (1) Establish the information system of evaluation indicators; (2) Discretize data; the rough set theory is only able to process the discrete data, so the sample data need to be discretized before reducing the initial data.(3) Reduce indicators.
Step 4 (determine the weights).This paper adopts the combination weighting method combining the subjective weighting method by expert scoring with the objective weighting method by calculating the weights through the rough set, to determine the weights of indicators, finding the weights of indicators as reasonable and formal as possible.The specific steps are as follows: (1)   denotes the weight of the ith attribute judged by experts; (2)   denotes the weight of the ith attribute determined by the rough set; (3) Combine the subjective weights with the objective weights to find the combined weight, which is expressed as where  is the experiential factor. ( Step 5 (build the grey standardized decision matrix).To get rid of the influencing among dimension and make the comparison between different attributes, we standardize the grey matrix by the following: Step 6 (build the grey weighted division matrix).To calculate the weighted matrix ⊗  = [  ,   ], where ⊗  =   × ⊗  , for  = 1, 2, 3 ⋅ ⋅ ⋅ , m,  = 1, 2, 3 ⋅ ⋅ ⋅ , n,   is denoted as the indicator weight of combined weight.
Step 7 (determine the grey ideal solution).Let   = max    ,   = max    ,   = min   ij , and   = min    ; The positive ideal solution is The negative ideal solution is Step 8 (calculate the ideal distance).The ideal distance of the reliability of each elevator equipment and the distance to the positive ideal solution are The distance to the negative ideal solution is The close-degree to grey ideal solution is sorted by the results. (6)

Result Analysis.
After experts analyze all equipment on spot, the linguistic variables of each indicator from experts should be transformed into grey number, to build the grey decision matrix shown as follows: To use [0, 40] → 1, [40, 80] → 2, [80, 100] → 3 we discretize the information in the sample as shown in Table 3.
According to the theory of rough set and its reduction function in the above, we can elicit that the setting of management organization  3 and the management regulations for elevators  5 have no obvious influence on elevators.So, the simplified index system is  = ( 1 ,  2 ,  4 ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ) .   the importance of each indicator by using the concept and equations of the rough set Find  (−{ 1 }) ( 1 ) = () − ( − { 1 }) = 0.15.Similarly, this is shown in Table 4.The weight of each indicator can be elicited by equations calculating the weights of the rough set, which is shown in Table 5.
The subjective weighted can be given by comprehensive analysis from experts with their experience and existing knowledge.Let   ( = 1, 2, . . .) be the subjective weight of each indicator from the index system  = { 1 ,  2 , ⋅ ⋅ ⋅ ,  n }, respectively, which satisfies ∑ n i=1   = 1, as is shown in Table 6.
The combined weight is   =   + (1 − )  , where  is the experiential factor.According to the actual situation, we set the experiential factor as 0.2 with synthetic consideration, to get the comprehensive weights shown in Table 7.
To standardize the grey matrix, we standardize the reduced grey matrix by using the rough set method.
To weight the grey matrix, get the weighted grey matrix.
To calculate the distance between each evaluation object and the ideal solution, the results are as follows: The distance to the positive ideal solution is The distance to the negative ideal solution is To calculate the close-degree to the ideal solution of each evaluation object, the result is presented in Table 8.
By Table 8, we can intuitively find that the ordering of the reliability of these 5 elevators is  4 >  1 >  3 >  5 >  2 .Therefore, whether regularly maintaining the equipment or lending, the specific and reasonable arrangement can be given in terms of the ordering.

Conclusions
To solve the problem on reliability evaluation of elevator equipment, this paper builds the reliability evaluation index system of elevator equipment with a microscopic perspective and conducts a comprehensive analysis combining with objective data, to reasonably find the actual reliability of evaluated elevators.The conclusions obtained by studying are as follows: (1) According to the rough set theory and the combination weighting approach, this paper builds the multiobjective evaluation model for the reliability of elevator equipment.This model mines the relationship among indicators by using the combination weighting approach, under the condition that the key influencing factors are extracted reasonably, determining the weights of key indicators scientifically and effectively by combining with the objective data of indicators, to reasonably evaluate the reliability of several elevators.
(2) While analyzing the internal logical relationship among indicator data in the multiobjective evaluation model for the reliability of elevator equipment, we combine the qualitative analysis with quantitative calculation of indicators, providing a more reliable reference for determining the plans for elevator equipment of maintenance quantitatively.
This paper extracted the key indicators in a more reasonable way by using the rough set theory and reduced the indicators which influence the reliability of elevators not significantly.Although this method reduces amount of work on calculating to some extent, it is unable to reflect the relationship between results and indicators integrally, with certain error in sorted results.Therefore, under the circumstance of maintaining the integral data of indicators, to reduce amount of work on calculating as much as possible is the focused direction for further study.

Figure 1 :
Figure 1: The figure of interrelation among the indicators of the reliability of elevators.

Figure 2 :
Figure 2: The model for reliability evaluation of elevators.

Table 1 :
The index system of the reliability of elevators.

Table 2 :
The raw data of indicators evaluation.
5.1.Data Sources.This paper takes five elevators from a special equipment corporation in Fujian as an example, begins with the 11 indicators determined by 4 perspectives: people, management, environment, and equipment, and asks experts to carry out actual analysis and evaluation of each equipment.The raw data of indicators evaluation is shown in Table2.

Table 3 :
The discrete initial data.

Table 4 :
The importance of indicators.

Table 5 :
The weights of indicators.

Table 6 :
The subjective weights of indicators.

Table 7 :
The comprehensive weights of indicators.

Table 8 :
The close-degree of evaluated elevators.