Keywords

1 Introduction

The increasing intensity of global competition to develop new products in shorter times, with higher reliability and overall quality, as well as a significant reduction of products life cycles, has amplified the need for new and efficient methods for quality engineering and reliability estimation and prediction [6, 17] and is visible in various fields of application (see e.g. [7, 10, 11, 21]).

The reliability issues in manufacturing mostly refer to the product reliability evaluation mostly connected with quality ensuring or manufacturing system reliability modelling being related to functions of the components [15]. Taking one step further, in order to provide a complex assessment analyses, decision-makers should focus on manufacturing process reliability perceived in a broader way relating to machines, materials, operators, methods, measurement and inspection, as well as other process elements. Such an approach makes that traditional manufacturing system reliability modelling methods are not suitable to solve this problem.

The model that integrates product quality with manufacturing process reliability is presented in e.g. in works [5, 15]. In [5], authors present the solution that is based on the analytic network process (ANP) use and process knowledge network approach implementation. In the second work, authors focus on mission reliability modelling method for manufacturing systems with low degree of automation. The third interesting approach regards to quality-reliability chain modelling for complex manufacturing process, where authors investigate the QR-co-effect between product quality, system reliability and component reliability and is presented in [2].

However, the known approaches mostly focus on the two problems of product quality and system reliability. Other important process elements that influence the reliability characteristics are omitted in the analyses. Following this, the paper is focused on the development of the seven-stated Markov model for production process reliability assessment based on multidimensional production process reliability definition, given e.g. in [4]. The developed model is also analysed with taking into account the expert opinions in the area of model parameters estimation. The presented article is the continuation of authors research work connected with development and investigation of production process reliability assessment method that helps decision managers in their every-day work.

Following the above considerations, in the next section, there is provided a short literature review in the investigated research area. Based on this research, there is presented a Markov model for production process performance. Later, the sensitive analysis of the given model is developed based on the obtained expert opinions in the area of model parameters estimation. The work ends up with summary and directions for further research.

2 Production Process Reliability Assessment with Markov Approach Use – Literature Review

General review of the literature concerning the reliability of assessment of production process is presented in publication [3], where the main classification of reliability assessment models and methods is investigated.

In this article authors focus on the application of a Markov process approach in the evaluation of production processes performance. In this area there are a few articles worth mentioning. In one of them [20], the authors use a discrete time Markov chain model of a multistage manufacturing process with inspection and reworking. They focus on influence on interferences at the input (such as the effectiveness of inspections at the input to the process), load requirements and manhourly requirements, as well as total cost in order to achieve the process without distortions.

It is also worth to refer to another article [12], where the authors propose a new technique to reduce the number of Markov states. It will allow to adopt Markov models with continuous and discrete Time Markov Chain in more complex processes or systems. Other authors [2] reject the application of Markov model as a tool valuable to solve problems in production systems be-cause of too complicated calculation and construction. On the other hand, the authors in [14] apply Markov model of Quick Response Manufacturing System to evaluate the reliability of the production system. They identify weaknesses of their system (like e.g. the need of rework of defective product) and give advices how to improve those weaknesses. Moreover, among the many articles in this area, the publication [13] presents the advantages and disadvantages of Markov processes implementation in the field of machinery enterprises and possibilities of application are presented.

The presented short literature review in the analysed research area indicates that there are investigated various aspects of manufacturing systems reliability. The authors of this paper focus on the more comprehensive view on the reliability issues of production processes performance taking into account five main aspects of manufacturing system operation and maintenance.

3 Markov Model for Production Process Performance

Based on the presented literature review, one of the possible modelling approach for system/process reliability estimation is a Markov model.

3.1 Production Process Reliability Definition

In order to develop a Markov model for production process performance, the main definitions should be provided. Based on the literature review given e.g. in [4], authors propose the production process reliability definition as an ability of a production system to completely fulfil the production plan of fully valuable final products in a specified period of time under stated conditions. The given process conditions regard to the five main areas [4]:

  • machines and equipment performance and their failures occurrence possibility,

  • maintenance and logistic support infrastructure performance and their failures occurrence possibility,

  • information flows and information reliability,

  • possibility of the occurrence of unwanted random hazards/threats (internal and external type),

  • processes of decisions making by policy makers and human factor reliability.

The complex reliability assessment analysis should cover the investigation all of the mentioned areas providing a valuable contribution to processes performance improvement. To satisfy these requirements, the multi-state reliability theory should be implemented, where the process may assume many states ranging from perfect functioning to complete failure. Taking one step further, the Markov processes approach may be used as a valuable tool to model the main relations between the defined production process performance conditions.

Because a Markov model is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states [1, 19], the main production process states should be defined.

Let’s consider a manufacturing system under continuous monitoring. Taking into account the presented reliability definition, production process may be in one of seven reliability states being defined in accordance with the production plan and process conditions performance (Table 1). The production process state is given as a pair: the first give the possibility of production plan performance, while the second one the process condition state. The two states marked at grey in the table are not possible to occur in the real-life systems, thus they are excluded from the further reliability analysis.

Table 1. Production process states general characteristic
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According to the defined reliability states, the perfect performance of the production plan means that there are manufactured fully valuable products in the right quantity and without any delays. Partially imperfect plan performance may regard to time delays or problems with products right quality and quantity. Imperfect performance of production plan means that based on the available resources there is no possibility to fulfil the defined plan in the defined period of time. The production process conditions performance perfectly if there is no occurred problems in machines, human factor, and support asset performance, the information is perfect (available and reliable), and there is no additional unwanted hazard events identified. Partially perfect performance of production process conditions regards usually at least to have reliable production machines and available human and information resources to fulfil the production plan. These three resources are usually critical for any manufacturing system flexible and reliable performance.

Let’s also assume that production process experiences random failures in time and each failure entails a random duration of repair before the manufacturing system is put back into service. System after repair is as-good-as-new. Let’s also assume that any information about failures in the system is reliable and comes immediately.

3.2 Seven-State Markov Model

For the analysis of the production process performance, we can associate its performance to a Markov seven-state model, on the basis of the defined set of states and defined assumptions (Fig. 1).

Fig. 1.
figure 1

Seven-state Markov model

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The graph describes behaviour of the production process with two successful (0, 1) and five mutually exclusive failure states (3, 4, 5, 7, 8) as shown in the Fig. 1 and Table 1. The modelled process is repaired from states: 3, 4. Worth taking a consideration is the state 4 that defines partially imperfect plan and process conditions performance. From this state, a process may pass to UP state (if the plan will be properly realized) or to another DOWN states, depending on the resources availability and plan performance level.

The detailed interpretation of the given theoretical model is presented in work [18].

Following [8, 16] we can formulate the probability expression for the model:

$$ \left\{ {\begin{array}{*{20}l} {P^{\prime}_{0} (t) = - (\lambda_{01} + \lambda_{03} )P_{0} (t) + \mu_{10} P_{1} (t) + \mu_{30} P_{3} (t)} \hfill \\ {P^{\prime}_{1} (t) = \lambda_{01} P_{0} (t) - (\mu_{10} + \lambda_{14} )P_{1} (t) + \mu_{41} P_{4} (t)} \hfill \\ {P^{\prime}_{3} (t) = \lambda_{03} P_{0} (t) + (\mu_{30} + \lambda_{34} )P_{3} (t) + \mu_{43} P_{4} (t)} \hfill \\ {P^{\prime}_{4} (t) = \lambda_{14} P_{1} (t) + \lambda_{34} P_{3} (t) - (\mu_{43} + \mu_{41} + \lambda_{45} + \lambda_{47} )P_{4} (t) + \mu_{54} P_{5} (t) + \mu_{74} P_{7} (t)} \hfill \\ {P^{\prime}_{5} (t) = \lambda_{45} P_{4} (t) - (\mu_{54} + \lambda_{58} )P_{5} (t)} \hfill \\ {P^{\prime}_{7} (t) = \lambda_{47} P_{4} (t) - (\mu_{74} + \lambda_{78} )P_{7} (t) + \mu_{87} P_{8} (t)} \hfill \\ {P^{\prime}_{8} (t) = \lambda_{58} P_{5} (t) + \lambda_{78} P_{7} (t) - \mu_{87} P_{8} (t)} \hfill \\ \end{array} } \right. $$
(1)

Where the following notations are used:

λ ij :

– failure rate associated with states i and j

μ ij :

– repair rate associated with states i and j

P i (t):

– probability of being in state i at time t

P’ i (t):

– first derivative of P i (t) function with respect to t

Taking into account steady-state solutions, there is possible to estimate the steady-state availability ratio K:

$$ K = \mathop {\lim }\limits_{t \to \infty } K(t) = \mathop {\lim }\limits_{t \to \infty } [P_{0} (t) + P_{1} (t)] $$
(2)

The availability is a calculation of various operational functions at the system and process level, and is interpreted as the percentage of time that the system will be ready to perform satisfactory in its intended operational environment. For the presented model, the level of production process availability depends upon the particular level of parameters λ ij and μ ij . This confirms the necessity of the developed model detailed analysis performance in addition to the real-life manufacturing systems operation.

4 Sensitivity Analysis of the Developed Markov Model

The preliminary sensitivity analysis of the developed model was presented in work [18]. The analysis was based on the assumptions that a parameter ρ = µ/λ obtained the following values: ρ = 1, ρ = 0.1, and ρ = 0.01. The performed analysis mainly focused on the calculation of availability function K(t) of the system and probability of being in the state 4 – P 4(t). The presented analysis results were obtained when all λ ij parameters and all μ ij parameters are the same.

However, in order to analyse the possibility of such model implementation for real-life complex manufacturing systems performance, the more detailed information about the possible values of parameters λ ij and μ ij should be provided. Following this, in the next subsections authors focus on Markov model use for the automotive industry sector.

4.1 Assumptions for Detailed Sensitivity Analysis

The analysis is performed for the Polish automotive industry sector, taking into account the serial production type.

In the first step of the conducted analysis, the probability of occurrence of given production process states (failure rates and repair rates values) is estimated based on the interviews with experts having long experience in manufacturing systems and processes analysing and managing/improving.

The conducted surveys also revealed important information that the perfect production plan usually takes into account some disturbances being possible to occur during normal business operations. A good example here is the scheduling process of production plan that is prepared with taking into account some level of production staff absence (about 6%). As a result, if during the scheduled time period the assumed level of employee absence is not exceeded, the plan is executed perfectly. Following this, we may state that the production plan is resilient for some disruptions occurrence.

The results of experts’ opinions are summarized in the Table 2. The fifteen possible transitions between process reliability states are described providing the short characteristic of possible scenarios occurrence, the descriptive probability of occurrence (according to a scale: very likely to occur, medium possibility, low possibility, almost unlikely to occur), and estimated by experts values of failure/repair ratios.

Table 2. The estimated values of model parameters λij and μij based on expert opinions
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After the analysis of the above presented Table supported by the experts from the area of production processes management, it was established that the most likely to occur are the states no. 0, 1, and 3. Following the current trends in reliability of production systems, usually enterprises cannot allow for failure of production process occurrence. Therefore, the states no. 5, 7, and 8 occur extremely rare in practice. This also implies that they may be omitted in the further research analysis due to their negligible influence on process reliability level.

Moreover, all the reliability states that are greater than state no. 4, are connected with critical situation occurrence from the company’s point of view and cause a significant increase in the maintenance costs or even may lead to production system total uselessness. Following this, in the next Sections the sensitivity analysis of the developed model is provided.

4.2 Preliminary Sensitivity Analysis for Availability Ratio

The sensitivity analysis of the developed Markov model was made using Markov Graph module of GRIF 2011 Software [9]. It was divided into two main steps. In the first step, the seven-state Markov model is analysed (Fig. 2). Later, the simplified Markov model (four-state model) is analysed. The performed analysis mainly focuses on the calculation of availability function K(t) of the system and probability of being in the state no. 4 - P 4(t).

Fig. 2.
figure 2

Markov seven-state model analysed in GRIF Markov software

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Obtained, chosen results of sensitivity analysis for availability ratio K, given by the Eq. (2), and system probability of being in the state no. 4 are presented in the Figs. 3 and 4.

Fig. 3.
figure 3

System availability ratio.

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Fig. 4.
figure 4

Probability P 4(t) of the system.

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The presented analysis results are obtained for transition rates given in the Table 2. For the presented assumptions, the system availability ratio is equal to 0.9657, and the probability P 4(t) equals 0.0155.

4.3 New Markov Model and Its Sensitivity Analysis

The simplified Markov model includes only four main states that may occur in the manufacturing system according to the obtained experts’ opinions. The model is presented in the Fig. 5 and the obtained results of sensitivity analysis are presented in the Figs. 6 and 7.

Fig. 5.
figure 5

Markov four-state model analysed in GRIF Markov software

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Fig. 6.
figure 6

System availability ratio based on simplified Markov model implementation.

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Fig. 7.
figure 7

Probability P 4(t) of the system based on simplified Markov model implementation.

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For the presented new simplified model, the system availability ratio does not change its level, and the probability P 4(t) equals 0.0153. This confirms the expert opinions that the transition states no. 5, 7, and 8 have a negligible influence on manufacturing process performance due to their very limited possibility of occurrence.

Moreover, it’s worth to investigate what is the influence of transition rate level change on system reliability characteristics. Thus, below there is presented the availability ratio in the situation when the failure rates increase two times or repair rates decrease two times (Table 3).

Table 3. The results of sensitivity analysis
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The results of the analysis are indicative. They point to logically justified influence of various defined scenarios on manufacturing system reliability characteristic. The greatest influence on the availability ratio of the system has the transition from the state 1 to the state 4 (perfect production plan into imperfect production plan performance with satisfying imperfect conditions). The influence of other transitions between the states (0–3, 4–1 and 3–4) is negligible.

5 Conclusions

In the presented paper, authors present the results of their research work connected with development and investigation of production process reliability assessment method that helps decision managers in their every-day work.

In the presented work, authors develop a Markov model for production process reliability assessment and present the subjective estimation of transitions probabilities. Based on the expert opinions, authors conclude that the developed seven-state Markov model may be simplified to the version of four-state Markov model providing the similar results of reliability analysis (the same availability ratio level and very similar P 4(t) probability level for both the models).

Following this, the authors’ further research work will be focused on real manufacturing systems performance analysis and more detailed sensitivity analysis for the given Markov model development.