Stress-Strength Weibull Analysis Applied to Estimate Reliability Index in Industry 4.0

With technological advances, companies are allowed to integrate digital data, physical supplies, and human resources, and all this integration capability can be done thanks to Industry 4.0. This concept, also called the fourth industrial revolution, refers to smart companies that work with intelligent cyber-physical systems. Industry 4.0enables automation, data interchange, and big data processing, among others. Then, the process decision-making, efficiency, and productivity improvement for companies will become faster and more accurate, thanks to real-time data processes and all supply chain integration allowed by Industry 4.0. However, the implementation of Industry 4.0 carries several challenges for companies to have success in the transformation of a normal industry into an Industry 4.0, like the necessity of adding new hardware, software, and other technologic devices. Because of this, the implementation and control of Industry 4.0 come with new issues to handle and new failure modes for both hardware and electronic devices. These problems can be faced using reliability engineering tools. Then the object of this research is the use of reliability engineering methodology stress-strength Weibull analysis, highlighting that the behavior of frequency emitted by electronics devices follows a Weibull distribution most of the time. Also, a stress-strength Weibull with a different shape parameter close solution is presented to increase the efficiency and productivity in Industry 4.0 electronic devices.


Introduction
Currently, the increment in technology is very fast, even faster than the level at which companies could understand the advance in this technology [1]. One of the most significant advances in technology is the Internet and the Internet of Things, according to different authors [2]- [5]. Such is the impact in the form of communication, the data interchange, and the data process, among others, that now is possible to make it in real-time. In this sense, because these advances are now possible, all types of devices are connected; this technological advance is called the Internet of Things [6]. Then, the implementation of the Internet of Things in companies to allow the transformation into smart industries is called Industry [7], [8].
Industry 4.0 makes companies create new systems and innovate into smart industries, electronic devices, and instruments [9]. These innovations facilitate infrastructures for industry development but also generate new and unknown failure mechanisms, technical and structural dependencies among system components, and eventually, new and unknown hazards and risks [10].
Thanks to technological advancement allowed by Industry 4.0, new industries or companies can get more control over products, processes, human resources, supply chain, and transportation, including the final disposition of products [11]. Since the Internet of Things allows a connection between different devices, such as can be computers, cars, refrigerators, smart phones, industrial machines, tools, etc., then now they can share information, data analysis, and decision-making in realtime [12]. However, nowadays, the main issue is the development of infrastructure with physical systems and models to manage Industry 4.0 facilities efficiently [11].
Another issue to be solved is the period of this Industry 4.0, also called the fourth industrial revolution.
Hence it should be solved in a short time-lapse because the emergence of new industrial areas is inescapable [13]. This puts huge pressure on organizations to learn, understand, and implement this technology and to be prepared for the progressive technology advance and process transformation [14]. This research presents the technology key to Industry 4.0, such as the Industrial Internet of Things, big data analysis, horizontal system integration, autonomous robots, and cyber security [15]. This aims to explain how reliability and stress-strength analysis can be used in these fields. Finally, it is important to mention that the connectivity devices depend on the reliability of the hardware and technological devices level [16], [17].
On the other hand, Reliability engineering, as a sub-discipline of system engineering, includes the systematic application of the engineering principles and techniques throughout a product lifecycle, so reliability should be considered from the concept plan to wear out of the device's lifetime [18]. Reliability Engineering defines reliability requirements and analyzes, assesses, and optimizes system performance via reliability techniques [19].
In that sense, the use of engineering reliability methodologies presents a powerful option for companies to implement in Industry 4.0 with the object of measuring accurately both efficiency and productivity [20]. Whereby the stress-strength analysis in reliability engineering is a useful method to measure the performance of hardware and electronic devices that are employed in Industry 4.0 [14]. Stress-strength analysis determines the stress level at which the element fails. Because both the stress and the strength are random variables, the analysis of these variables is made using probability density functions [21]. Industry 4.0 creates a new opportunity for thought leaders and engineers to create new systems and innovate smart devices and instruments. These innovations facilitate infrastructures for industry development but also generate new and unknown failure mechanisms, new and unknown economic, functional, technical, and structural dependencies among system components, and eventually new and unknown hazards and risks. On the other hand, with complexity and dependence increasing, the implementation of new concepts, and the advancements in knowledge, methods, and techniques, such as big data, the Internet of Things, and quick response to changes, make new opportunities for developing reliability engineering techniques and improve reliability prediction capability.
In addition, it should be considered that electronic elements like hardware or any electronic devices almost always present a Weibull distribution behavior [22]- [24]; in such a way, this manuscript also presents a stress-strength Weibull analysis close solution for the case of the different shape parameter ≠ .
The structure of the paper is as follows. Section 2 presents the Internet of Things; section 3 shows Industry 4.0; in section 4, reliability engineering in Industry 4.0 is presented; section 5 presents the generalities of Weibull distribution: in section 6, Weibull Distribution and Industry 4.0 is shown; section 7 presents the proposed method and the numerical application; finally, section 8 presents the conclusion and references.

Internet of Things
The concept of the Internet of Things has become relevant in the world because of the advance in technology and mobile devices. The Internet of Things is about billions of products or objects that can sense, communicate, and share data, and all these are interconnected by the Internet protocol network. The objective of this interconnection of objects is the capacity for planning, management, and decisionmaking; in that sense, the definition of the Internet of Things is a physical object communicated by a network. It is important to remember that the Internet of Things is not only about computer networks, is about all types of devices and all sizes (see fig.1) [25].

Industry 4.0
Industry 4.0 can be defined as a new organization and control level over the value chain if the lifecycle of devices or products is geared to individualizing customer requirements [7]. This control in an organization begins with the raw material and ends with the recycling of the product, including customer service during the lifetime of the product [26]. The relevance of Industry 4.0 is the availability of information in realtime by connecting the instances of the value chain, which results in vital for industries to optimize each link of the product in the value chain [13]. On the other side, the connection of systems, things, and people creates a dynamic value-added organization [27]. i.e., the customers are closer to the industries, and the information is dual between the company and the client (see Table 1) [28].
Industry 4.0 encourages manufacturing efficiency by collecting data smartly, making correct decisions, and executing decisions without any doubts. By using the most advanced technologies, the procedures of collecting and interpreting data will be easier. The operating interoperability ability acts as a `connecting bridge´ to provide a reliable manufacturing environment in Industry 4.0 makes full use of emerging technologies and rapid development of machines and tools to cope with global challenges to improve industry levels. The main concept of Industry 4.0 is to utilize advanced information technology to deploy IoT services. Production can run faster and smoothly with minimum downtime by integrating engineering knowledge. Therefore, the product built will be of better quality, production systems are more efficient, easier to maintain, and achieve cost savings.

Magdalena (2017)
Modern and more sophisticated machines and tools with advanced software and networked sensors can be used to plan, predict, adjust and control the societal outcome and business models to create another phase of value chain organization. It can be managed throughout the whole cycle of a product. Thus, Industry 4.0 is an advantage for staying competitive in any industry. To create a more dynamic flow of production, optimization of the value chain has to be autonomously controlled.

Factors that Affect Industry 4.0
The introduction of Industry 4.0 in organizations represents a huge challenge that is for all the facts that are needed in the implantation of Industry 4.0, a relevant factor is the hardware technology [29]. This is because to support all data, processes, and interconnectivity, hardware must provide the requirements to allow Industry 4.0, i.e., the hardware used to process data and connection needs. The correct hardware is not only what the industries need, but it also is very important how long the hardware installed will be efficiently performed through time that is going to support optimal condition communication and data in the industry [30]. Another important aspect in the implementation of Industry 4.0 is the software used in the connection and data process. As is well known, some corporations provide software in the implementation of (ERP) Enterprise Resources Planning SAP and Oracle. But now, Industry 4.0 is not only about communication with all the branches that integrate the whole enterprise, the suppliers, and finally, the customer. This new software needs to support quantities of data that a decade before was only impossible even to imagine. For now, the data represents quantities like terabytes and even petabytes to support data interchange. Moreover, the industries must not deal only with the correct software but with the time that software will be performed in an efficient way [31].
Another relevant factor that can be part of the success or loss of a company is the connection; this can be the most important factor in dealing with Industry 4.0. the transmission of big data (see fig. 2) between every node in the Internet protocol is a very important problem to deal with, i.e., Industry 4.0 makes those industries must evolve into virtual global companies. We are not just talking of suppliers connection, the necessary connection in Industry 4.0 include multiple countries to interchange data with companies, suppliers, clients, transport, and customer, and at the same time, each one is also connected [32]. On the other hand, it is also important to consider the connection time and the speed of data interchange, it is to say, not only to get the connections but the time of optimal and efficient connection support [33]. Finally, reliability is necessary for optimal performance in Industry 4.0 [14].

Reliability Engineering in Industry 4.0
It is important to mention that the efficiency of the interconnection, the hardware performance in optimal conditions, and software support over time depends on the reliability of the user device to perform the interconnection; moreover, wireless transmission of big data depends on a frequency, it is to say, the used signal to connect devices like, smart phones, computers, smart , is possible th , which means eless technolo ensity functio ribution is w using this d it to real-world stribution is al hen, a metho devices is pres 0 approach [36 y predicts ments, or sys n Industry 4.0 ecause reliabil of devices lik Without any of d. Consequent in any compan , challenges n will be i is unknown i devices and con t will be sign ain challenges uncertainty can lack and unco ed using reliabi y methods imp ant to deal with ical and virtua cation and da col. In the next ation are presen r, to say tha afe for use an dentify signifi device's perfo his failure mod ed with a highs important to failure and im rate or decelera predict life a ilures; in that h possesses an time probabili all the pos ll be used [39] bility Engine the best quanti systems design ality over tim out failure in ]. In other wor under the c stablished for a 21 hrough the use , that connectio gy are repres on [35]. Mor widely used istribution pro d data, and as lso widely used odology to d sented in the 6].
analysis an stems. This r to be efficien lity methods re ke hardware, f this, the fram tly, Industry ny or organiza s, and inv irrelevant if i.e. if we don nnectivity, the nificant [14]. U in both virtual nnot be elimin ontrollable pro ility methodolo plementation in h uncertainty a l fabric in a g ata interchang t section, reliab nted. at an electron nd qualified a icant factors a rmance and fi de, the lifetime -reliability leve understand the mportant is to ate the failure accurately and sense, the me n advantage ity because ssible stresses ].

(6)
From [18], the sample size n is determined as = ( ( )) (7) And the mean μ and the standard deviation σ are given as

Weibull Distribution and Industry 4.0
The information on the mean values of signals in telecommunications like cell phones, Internet protocol, computers, radios, printers, and all devices that use a frequency to communicate, i.e., wireless connectivity, is not sufficient to characterize the performance of communication systems; the variation in time, space and frequency, must also be taken into account [46], [47]. The dynamic behavior of the desired and interfering signals plays a decisive role in the analysis of system reliability and the choice of system parameters; in such a way, the signals must be considered as a random variable, and to model this variable, a probability density function needs it [48].
On the other hand, electronic devices are used physically and virtually everywhere, then; an aspect of Industry 4.0 success is the accuracy of these electronic devices and the communication between all of them; in other words, electronic communication needs to be able to reach the entire world [7]. The wireless that connects the world with information and streaming data into and out of an organization term the Industry 4.0 era [49].
To describe the life of a product or element known as the bathtub curve (see fig. 3). The bathtub curve consists of three periods: an infant mortality period with a decreasing failure rate followed by a normal life period with a low, relatively constant failure rate, concluding with a wear-out period.
The bathtub describes the relative failure rate of an entire population of products over time [50]. The success factor is the choice of the reliability model that captures lifetime variables for parameter estimation and defines the distribution adequately. In that way, an adequate distribution to fit the failure mode in electronic devices and connectivity protocol is the Weibull distribution because the Weibull distribution is based on the weakest link, and this allows electronic and virtual devices to identify the failure mode and calculate with accuracy and precision the reliability of components being physicals or virtual [51].
On the other hand, the Weibull distribution is very useful in modeling Internet traffic data sessions, flows, and packets at different Internet network connections; also, the scale parameter of the Weibull distribution can be used to characterize the Internet traffic and flow [52].

Reliability Design Estimation in Industry 4.0
It is well known that quality in components is very important to ensure the correct function in different aspects of Industry 4.0 and Internet protocol, such as the performance of the routers, the bandwidth, the people knowledge, the data process, etc. [53]. On the other hand, this quality for how long is going to work at optimum conditions, i.e., what is the probability of a component to perform the function in a period without failures [54]. Here is where reliability engineering is very helpful in preventing these failures and knowing the reliability index of any component [55]. In such a way, Industry 4.0 works with very significant factors that can be a key success but also a problem that affects all the chains in the Industry 4.0 performance [56]. Then, using reliability methodologies in Industry 4.0 will help industries to get better quality over time and to have the best knowledge about their supply, helping to prevent possible failures [57]. In the next section, a methodology to estimate the real reliability of different kinds of components is presented and applied to a set of data of routers experiment just like an example of the efficiency of the methodology.

Proposed Method to Estimate the Design Reliability
The proposed method is based on the stress-strength Weibull analysis emphasizing that the Weibull distribution has no additive property, and the stressstrength analysis is an algebraic sum of two probability density functions (see figure 4). Then a method to estimate reliability level in the case of different shape parameters ≠ is performed.  This means that the real reliability of the system speed received with the router is 94.99%, and even more, the probability safety factor can be estimated using equation (21), as shown above.

Conclusion
The technological advances have allowed the growth of the organization to a new level in all the supply chain by using infrastructures or tools like Industry 4.0, which help companies to receive and process data in quantities never seen before and even more this big data process, and transmission is made in real-time. However, Industry 4.0 need the use of both hardware and software that can be very useful, but also the probability of failure is always present because, as it is well known, all products, i.e., software and hardware, are random variables that need to be controlled and analyzed to prevent failures not only in some hardware but in the transmission, received and data processed. In this manner, reliability engineering possesses tools to help with this failure issue. Hereby, in this paper, the reliability stress-strength Weibull distribution with different shape parameters ≠ analysis is presented to estimate the real reliability level of the frequency and durability in internet routers, showing that it is possible to measure the efficiency and productivity of processes in Industry 4.0. On the other hand, since all products or elements are subjected to random stress and in the same way, products possess an inherent random strength to support this stress and both are random variables, a probability density functions need to calculate the reliability level by using the stress-strength Weibull analysis above mentioned. Even more, it is well known that electronic products and devices with additives, i.e., physical phenomena behavior, follow a Weibull distribution; then, the use of a stress-strength Weibull distribution tool to verify the real device reliability is a good option in the decision-making process.