Mathematical Modeling of Manufacturing Lines with Distribution by Process: A Markov Chain Approach

Abstract

Today, there are a wide variety of ways to produce goods in a manufacturing company. Among the most common are mass or line production and process production, both of which are antagonists. In an online production system, materials move from station to station, receiving added value on a well-defined layout. In a production line by process, the materials randomly visit a set of machines strategically located in order to receive a treatment, almost always through metalwork machines, according to the final product of which they will be part. In this case, there is not a predefined layout, as the incoming materials are sectioned and each piece forms a continuous flow through different workstations to receive some process. This activity depends on the function of the product and its final destination as a component of a finished product. In this proposal, Markov chain theory is used to model a manufacturing system by process in order to obtain the expected values of the average production per machine, the total expected production in all the facilities, the leisure per machine and the total productive efficiency of the system, among other indicators. In this research, we assume the existence of historical information about the use of the equipment, its failures, the causes of failure and their repair times; in any factory, this information is available in the area of manufacturing engineering and plant engineering. From this information, statistical frequency indicators are constructed to estimate transition probabilities, from which the results presented here are derived. The proposal is complemented with a numerical example of a real case obtained from a refrigerator factory established in Mexico in order to illustrate the results derived from this research. The results obtained show their feasibility when successfully implemented in the company.

1. Introduction

Manufacturing is one of the most important activities in the world economy because the production of consumer goods directly influences a country’s wealth, contributing to around 70% [1]. To this day, the ways of producing consumer goods have evolved in an impressive way along with the technology. Without a doubt, 5G and 4.0 technologies applied to manufacturing systems, in conjunction with hybrid optimization and artificial intelligence, make this environment a highly significant medium in modern global industry [2].

A Flexible Manufacturing System (FMS) is viewed as the integration of assembly processes, material flow, computer communications and control processes [3]. This type of structure represents the central axis around which the activities of a production line revolve. Its importance lies in exchanging machinery and processes immediately in order to manufacture a large number of different products. Usually, the topology (layout) of the plant distribution is defined through the nature of the product to be manufactured, the used materials, the machining operations, the physical arrangement of the equipment, the location of the raw material and the finished product warehouses, among other issues. The literature in this regard is quite abundant in information, although most of the authors agree on some general aspects in the ways of producing.

More specifically, the following ways of producing are common in today’s manufacturing engineering:

-

Unit production: It is characterized by manufacturing a single product that is made at a predetermined time according to its demand. Usually, they are unique products tailored to the client. From the perspective of large-scale manufacturing, these types of production models are of little interest.

-

Batch production: It is described by the means of producing a predetermined quantity of items in a single production run. The quantity manufactured and the run times are programmed in a Master Production Schedule (MPS) according to the demand of the product. Usually, the quantity to be produced is forecast considering the available inventory using a policy $\left(S,s\right)$. This production system becomes interesting when several batches of varied products must be manufactured. This raises the question in what order should the products be produced? From a quantitative point of view, the modeling, simulation and/or optimization of this type of system are more interesting since it involves computational problems of the NP-hard type, especially when calculating the order of production, which translates into the classic problem of sequencing operations. This scheme is especially useful in the production of seasonal goods.

-

Mass production: In this system the manufacture of the product is performed under a strict order in line. At the beginning of the system is the raw material that feeds the first workstation. Product flows downstream through $\rho$ specialized workstations increasing their added value with each visit. At the end of the line, there is a finished product warehouse where the production generated during a period of time is kept. Each workstation contributes with an incremental improvement in the functionality of the product until the final result constitutes a good that goes directly to the consumer or is a sub-assembly that will later be incorporated into another line to form part of the finished product, which can also be considered a spare part. These types of designs usually include a set of $\rho$ (one for each machine) intermediate buffers between each workstation in order to temporarily decouple the operation of the system and thereby avoid bottlenecks or leisure times, see [3,4]. These systems are highly efficient as they generate large amounts of fully standardized product. The control of the operation can be conducted under the classic PUSH type production scheme (under a demand forecast concept) or the PULL type system using the KANBAN philosophy [5].

-

Continuous production: This production system generates one or more products that generally cannot be measured in discrete units. Examples of this are the production of gasoline, milk, gas, liquors, etc. Here, production never stops and only stops when corrective and/or preventive maintenance is required on the equipment. An important characteristic of this type of production is that most of the work is performed automatically by industrial equipment with almost no human intervention.

-

Production by process: This design, which is of interest in this document, is characterized by manufacturing a variety of products in quantities that can be large depending on the demand for the multiple goods to be produced. The design topology originates when there is a series of machining equipment installed in the shop in various positions (not necessarily ordered). These distributions are achieved when the machines to be used are highly automated, heavy or highly specialized, which prevents their movement (for example, electric furnaces, punching machines, cutters, shears, paint booths, etc.) and makes their movement prohibitive. Here, the product must visit the machines that will carry out the operations on the products, not necessarily in an ordered sequence.

The difficulty in mathematically modeling the latter type of process is the inherent randomness in the trajectories that the component materials (or by-products) follow. After leaving the raw material warehouse, the sub-products can follow any path required by manufacturing engineering, except the finished product warehouse. Such trajectory is determined by the design conditions required based on the model of the product to be manufactured, the quantity requested, the actual production capacity, the machining operations required, the availability of equipment and machinery, as well as the availability of tooling and labor. The uncertainty of the path that each component will follow depends on the previous state and, therefore, in this proposal such a sequence is modeled through a matrix of transition probabilities of visiting state j when it comes from state i. Hence, this model can be represented by means of a Markov chain.

Figure 1 shows two typical trajectories that could be followed by two different products within a manufacturing by process system formed by 12 workstations (placed without a specific order), a raw material warehouse and a temporary buffer or warehouse of finished products in a system in mass. That is, the finished product of this process is part of the raw material that enters another production system that is after the current process.

The objective of this research is to model this type of manufacturing process through a Markov chain in order to calculate the main efficiency and productivity indicators required by manufacturing engineering. This work extends in several aspects [6] and is distinguished with respect to the specialized literature in the following: (1) it uses Markov chain theory along with its properties and results; (2) it obtains efficiency indicators based on equipment availability; and (3) it applies the proposal to a refrigerator factory in Mexico.

The rest of the document is organized as follows. Section 2 provides a short literature review in order to establish a reference framework. Section 3 models a production line by process with a Markov chain. Section 4 illustrates the results obtained through an application developed with real information from a refrigerator manufacturing system in Mexico. Section 5 discusses the empirical results obtained. Finally, Section 6 provides the conclusions.

2. A Short Literature Review

The literature is abundant on the analysis and mathematical modeling of mass production systems since the assumptions used to build such models are easy to verify in real cases. The most representative literature of the topic of interest is mentioned below.

There are classical books that address the problem fundamentally from a stochastic perspective. Among the most popular are [7,8,9,10,11].

Regarding the publication of papers related to the subject, it is worth mentioning, for example, [12], which develops a model based on multi-server tandem queues with buffers and blocking, after service is developed. The model focuses on performance characteristics such as throughput and mean sojourn times. The main idea developed is the decomposition of the system into two-station subsystems using a spectral expansion method.

Similarly, Ref. [13] proposes a set of statistical indicators to evaluate the efficiency of the production line as well as the expected value of the manufactured products per unit of time, using the reliability of the installed equipment as the main element. On the other hand, Ref. [13] presents interesting results related to the pharmaceutical industry referencing the system over time. The resulting model has NP-hard characteristics, and the authors decompose the problem into two sub-problems that are easier to solve using discrete and continuous representations.

Moreover, Ref. [14] establishes a mathematical programming model that describes the behavior of a production line in a discrete way. The model specifies a multi-stage system with intermediate buffers and stochastic production times. Similarly, Ref. [15] develops a model based on the administration of tasks in hierarchical active systems including incentives.

In relation to representative models of continuous flow systems, Ref. [16] develops a mathematical programming model of discrete events in continuous time with a discrete and continuous mixed state space. The research focuses on optimally controlling the flow of the product through the system with high efficiency.

Also, an integrative approach to mathematical modeling is found in [17]. Here, the authors develop optimal manufacturing control policies for tandem production lines including KANBAN and CONWIP.

Alternative models of mathematical programming applied to the production capacity of a manufacturing line, which in their design uses optimal operating parameters of production lines, can be found in [18]. The authors in [18], develop a model to derive a production line capacity assignment problem considering an equilibrium requirement from the production in a cold rolling area. They also provide a structure in which there are several types of materials that are delivered with variable times in order to maximize the efficiency of the installed capacity in the plant.

Another interesting point of view is the optimization of the efficiency of production lines. This approach attempts to conveniently group work centers that have sequential activities into unit centers. A model associated with this approach is found in [19]. Here, a cost-oriented objective function is developed for a multiple-manned assembly line balancing problem using mixed integer mathematical programming as a tool to optimize the balancing of the production line. Finally, other applications associated with the topic can be found in [20,21,22,23].

3. The Mathematical Proposal Using Markov Chains

The use of Markov chains in various fields is very widespread. In this document, we make use of the properties of this tool to model and implement a manufacturing system by process.

3.1. Object of the Investigation

For the development of this research, the following steps are carried out:

• Definition of the problem: The problem proposed here is a real case presented by a specialized engineering firm. The request comes from a metalworking company that manufactures different types of refrigerator models for various products. In particular, the company wants to calculate its promises to the customer based on the installed capacity in the manufacturing area in order to obtain a master production plan and calculate its plans for materials and manufacturing requirements.
• Solution approach: Due to the complexity of calculating a quantitative model to estimate the production (avoiding the digital simulation of discrete systems) of the company in a closed way, it was determined that a Markov chain should be used to represent the dynamics of the production system by virtue of having historical information reliable to obtain relevant indicators.
• Mathematical properties of the proposal: The definition of the matrix of transition probabilities associated with the problem uniquely characterizes the shape of the problem as well as its properties (for example, the division of communicating classes). With this, it is possible to mathematically characterize the structure by process to be used in this proposal.
• Feasibility and relevance of the model: The feasibility and relevance of the chosen model are demonstrated in its application to the case raised by a real firm. For this reason, the set of equations that will govern the model is formally characterized through the use of statistical estimators obtained in situ.

3.2. Formalization of the Model through a Markov Chain

Let $S$ be a finite state space. The elements of S could be vectors. Each $i\in$ $S$is called a state. Hereafter, $\pi$ = {${\pi }_i|i$ $\in S$} is a probability distribution defined in some probability space $\left(\mathsf{\Omega },ℱ,\hspace{0.17em}\mathbb{P}\right)$. Thus, $0\le$ ${\pi }_i\le 1$for all $i\in S,\mathrm{and}$ the total mass satisfies ${\sum }_{i\in S}{\pi }_i=1.$ Let $X\left(t\right)$ be a random variable (vector) for each $t$. From now on $t$ will denote time and it can be discrete or continuous.

The proposal developed here attempts to model a manufacturing line by process, through a stochastic process in which the probability that $X\left(t+\delta t\right)$ (position of a product at time $t+\delta t$) depends only on the previous value of $X\left(t\right)$. That is, $X\left(t\right)$ is a Markov process. Let $X\left(s\right),s\le t,$ where $X\left(s\right)\mathrm{is}$the history of the values of $X$ before time $t$ and $z$ is the possible value of $X\left(t+\delta t\right)$. Then, it is satisfied that

$$\begin{array}{ll}\mathbb{P}\hspace{0.17em}[X\hspace{0.17em}(t+\delta t)=& z\hspace{0.17em}|\hspace{0.17em}X\left(s\right)=x\left(s\right),\hspace{0.17em}\hspace{0.17em}s\hspace{0.17em}\le t]\\ & =\mathbb{P}\hspace{0.17em}\left[X\left(t+\delta t\right)=z\hspace{0.17em}|\hspace{0.17em}X\left(t\right)=x\left(t\right)\right].\end{array}$$

(1)

In a particular case when the process is defined in discrete times, these will be numbered by $0,1,2,\dots ,$ or in a convenient way as the case may be. In the continuous time case, the notation $0,\delta ,2\delta ,\dots ,$will be used, or simply $\delta t$. In the same way, the transition probability from $i$ to $j$ in a unit of time will be denoted by

$$P_{ij}=\mathbb{P}\hspace{0.17em}\left[X\left(t+1\right)=j\hspace{0.17em}|\hspace{0.17em}X\left(t\right)=i\right].$$

(2)

In the case of a Markov chain, $X\left(t\right)=i$ denotes that the random quantity $X\left(t\right)\mathrm{is}\mathrm{in}\mathrm{state}\hspace{0.17em}i.\mathrm{Moreover},{\pi }_i\left(t\right)=\mathbb{P}\hspace{0.17em}\left[X\left(t\right)=i\right]$satisfies the normalization equation given by ${\sum }_i{\pi }_i\hspace{0.17em}\left(t\right)=1$. It is also satisfied the Chapman-Kolmogorov equation in the sense that $P_{kj}\left(s,t\right)={\sum }_i{P}_{ji}\left(s,u\right)P_{kj}\left(u,t\right),$for any $t>u>s\ge 0$where$P_{kj}\left(s,t\right)=\mathbb{P}\hspace{0.17em}\left[X\left(t\right)=k\hspace{0.17em}|\hspace{0.17em}X\left(s\right)=j\right]$. Thus, by the Law of Total Probability and from (2), it is also true that

$${\pi }_i\hspace{0.17em}\left(t+1\right)={\sum }_j{P}_{ij}\hspace{0.17em}{\pi }_j\hspace{0.17em}\left(t\right)\mathrm{with}{\sum }_i{\pi }_{i\hspace{0.17em}}\left(t\right)=1.$$

(3)

with $P_{ij}=\mathbb{P}\hspace{0.17em}\left(X\left(1\right)=j\hspace{0.17em}|\hspace{0.17em}X\left(0\right)=i\right)$. Equivalently, we may write

$${\mathit{\pi }}^{\mathit{t}}=\left(\begin{array}{c}{\pi }_1\hspace{0.17em}\left(t\right)\\ {\pi }_2\hspace{0.17em}\left(t\right)\\ ⋮\\ {\pi }_n\hspace{0.17em}\left(t\right)\end{array}\right),\hspace{0.17em}\hspace{0.17em}\mathit{P}=\left(\begin{array}{ccc}{P}_{11}& \cdots & P_{1n}\\ ⋮& \ddots & ⋮\\ P_{n1}& \cdots & P_{nn}\end{array}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathit{\vartheta }=\left(\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right)$$

where

$${\mathit{\pi }}^{\mathit{t}+\mathbf{1}}=\mathit{P}{\mathit{\pi }}^{\mathit{t}}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathit{\vartheta }}^{\mathit{T}}{\mathit{\pi }}^{\mathit{t}}=1$$

(4)

and

$${\mathit{\pi }}^{\mathit{t}}={\mathit{P}}^{\mathit{t}}{\mathit{\pi }}^{\mathbf{0}},\hspace{0.17em}\hspace{0.17em}\mathit{P}\mathit{\vartheta }=\mathit{\vartheta }\hspace{0.17em}\hspace{0.17em}\left(\mathrm{each}\mathrm{row}\mathrm{of}\mathit{P}\mathrm{sums}1\right).$$

(5)

In addition, the following conditions are satisfied in steady state

$${\pi }_i={\lim }_{t\to \infty }{\pi }_i\left(t\right)$$

(6)

if it exists, and the transition equations

$${\pi }_i={\sum }_j{P}_{ij}{\pi }_j,$$

(7)

as well as the balance equations in steady state hold

$${\pi }_m{{\displaystyle \sum }}_{m,m\ne i}{P}_{mi}={{\displaystyle \sum }}_{i,i\ne j}{P}_{ij}{\pi }_i$$

(8)

The corresponding normalization equation is given by

$${{\displaystyle \sum }}_i{\pi }_i\left(t\right)=1$$

(9)

The steady-state balance equations intuitively mean that the average number of transitions into some state per unit of time must be equal to the average number of transitions from the same state to other states [8].

In the matrix-vector form, it is written that in the steady state $\mathit{\pi }$ = ${\lim }_{t\to \infty }{\mathit{\pi }}^{\mathit{t}}$ if the limit exists. Moreover, $\mathit{P}=\mathit{\pi }\hspace{0.17em}\mathit{P},$ and the normalization equation is ${\mathit{\vartheta }}^{\mathit{T}}\mathit{\pi }=1.$Analogously, $\mathit{\pi }$ = ${\lim }_{t\to \infty }{\mathit{P}}^{\mathit{t}}\hspace{0.17em}{\mathit{\pi }}^{\mathbf{0}}$provided the limit exists.

Therefore, the characteristics of a production system by process allow it to be seen as a Markov chain since the passage of the material towards the state $X\left(t+1\right),$ given that it is currently at state $X\left(t\right),$only depends on the previous state. Here, the process is defined through the path that materials follow in the form of semi-finished products to be machined once they leave the raw material warehouse. The process ends when the finished product is in a temporary buffer (generally as a partial element of other products or as a spare part) to be assigned to a new assembly line or stored for sale.

Thus, at each step, the material is necessarily in some state $X\left(t\right)$, and will evolve into the state $X\left(t+1\right),$ according to the machining requirements demanded by it.

Once the above is formalized, the modeling of this type of production line is performed under the following considerations:

• All the material entering the system comes from a single source called the raw material warehouse and will be denoted as the initial state $s_1$.
• The set $S^{\prime }\subseteq S$ defined as S′ = {$s_2,\dots ,s_{\rho -1}\}$represents the intermediate stages of the manufacturing process. That is, where the material is machined to add value to become a by-product of the process. In principle, any $s_j\in S^{\prime }$ is accessed from $s_i$ $\in S^{\prime }$.
• For all $s_i\in S^{\prime }$, there is a $s_j\in S^{\prime }$ with $i\ne j$, such that $s_i⟷s_j.$ This means that any material in the manufacturing process can return with positive probability to any of the previous states. For practical purposes this is called material reprocessing.
• The state $s_{\rho }$ is the only one that allows material to exit the system. This represents the finished product warehouse and can be accessed from any $s\in S^{\prime }$. From here, the materials move onto a general assembly line in series and never return to any of the previous nodes.
• The material from the state $s_1$is sent to any $s\in Z^{\prime }$. This constitutes the product or productive unit of the system on which operations will be carried out in the rest of the states $s\in S^{\prime }$.
• The transition matrix $P$ will be assumed to be known. Transition probabilities are easily retrievable from the plant engineering and manufacturing department historical files.
• The reliability of an equipment (the probability that it will function when required) located in the state $s\in Z^{\prime }$ in the instant $t$, is given by the function

  $$R\left(t\right)=\mathbb{P}\hspace{0.17em}\left(\mathcal{T}\hspace{0.17em}\ge t\right)={\int }_t^{\infty }{f}_{\mathcal{T}}\left(t\right)dt,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathcal{T}\ge 0$$

  (10)

  where $f_{\mathcal{T}}\left(t\right)$ is the failure density function of the equipment at the states $s\in Z^{\prime }$, and $\mathcal{T}$ denotes the random variable that represents the instant where a failure occurs. Here, it will be assumed that

  $$\mathrm{E}(\mathcal{T})={\int }_0^{\infty }td{F}_{\mathcal{T}}\left(t\right)<\infty .$$

  (11)

Hereinafter, $\mathrm{E}$ denotes the mathematical expectation.

8.

Historical information is available on the machining equipment and the number of operations carried out on the materials that visit them.

3.3. Characterization of the Model

During our analysis, the state space $S\subset {ℝ}^{\rho }$ is given by the diverse workstations to which the product can arrive during its trajectory. The analyzed system is made up of the states $s_1,s_2,\dots s_{\rho },$where $s_1$ represents the raw material warehouse and the state $s_{\rho }$ represents the finished product warehouse. So, a typical transition matrix would be given by

Table 1. Typical matrix of transition probabilities used in this process. Source: Authors’ own elaboration.

	$s_1$	$s_2$	$\cdots$	$s_{\rho -1}$	$s_{\rho }$
$s_1$	$0$		$\cdots$	$p_{1,\rho -1}$	$0$
$s_2$	$0$	$0$	$\cdots$	$p_{2,\rho -1}$	$p_{2,\rho }$
$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$	$⋮$
$s_{\rho -1}$	$0$	$p_{\rho -1,2}$	$\cdots$	$0$	$p_{\rho -1,\hspace{0.17em}\hspace{0.17em}\rho }$
$s_{\rho }$		$0$	$\cdots$		$1$

.

According to Table 1, $P_{ij}\ge 0$ and ${\sum }_{j=1}^{\rho }{P}_{ij}=1,\forall i$. This research uses the notation $s_i\to s_j$ to express that a state $s_j$ is accessible from a state $s_i$. In our case, the initial state $s_1$ represents the raw material warehouse and, therefore, it is a non-return state. In turn, the final state $s_{\rho }$ represents the finished product warehouse and once the product reaches it, it remains there until it is required later on in another production line. $s_{\rho }$ is a buffer between the first manufacturing process and the main assembly line. This is an absorbing and therefore recurrent state. The intermediate states $s_2,\dots ,s_{\rho -1}$, represent the existing machining teams between the raw material warehouse and the finished product warehouse. From the characteristics of the transition matrix in Table 1, the following results from [23,24,25] hold.

Let$i=1,\dots ,\rho$ be the set of states in which a by-product can be found within the production line. If the process $X(·)$ begins with a known initial distribution $\pi =\left({\pi }_1,\dots ,{\pi }_{\rho }\right)$, i.e.,

$$\mathbb{P}\hspace{0.17em}\left(X\left(1\right)=s_j\right)={\sum }_{i=1}^{\rho }\mathbb{P}\hspace{0.17em}\left(X\left(1\right)=s_j\hspace{0.17em}|\hspace{0.17em}X\left(0\right)=s_i\right)\hspace{0.17em}\hspace{0.17em}\mathbb{P}\hspace{0.17em}\left(X\left(0\right)=s_i\right)={\sum }_{i=1}^s\hspace{0.17em}{\pi }_i\hspace{0.17em}{P}_{ij},$$

then the distribution of $X\left(1\right)$ is defined through $\mathit{\pi }\mathit{P}$ and, in general, the distribution of $X\left(t\right)$ is defined through $\mathit{\pi }{\mathit{P}}^{\mathit{t}}$. That is,

$${\lim }_{t\hspace{0.17em}\to \infty }\hspace{0.17em}\mathit{\pi }\hspace{0.17em}{\mathit{P}}^{\mathit{t}}\hspace{0.17em}\Rightarrow \hspace{0.17em}{\mathit{\pi }}^{\mathit{t}}$$

where ${\mathit{P}}^{\mathbf{0}}=\mathit{I}$ is the identity matrix. Here, the symbol $\Rightarrow$ means weak convergence and equality ${\mathit{\pi }}^{\mathit{t}}=\mathit{\pi }\hspace{0.17em}{\mathit{P}}^{\mathit{t}}$is equivalent to ${\mathit{\pi }}^{\mathit{t}}={\mathit{P}}^{\mathit{T}}\hspace{0.17em}{\mathit{\pi }}^{\mathit{t}-\mathbf{1}}$, where ${\mathit{P}}^{\mathit{T}}$ means $\mathit{P}$ transposed.

During the manufacturing process, each state commutes with the rest. That is, the state $s_j$ is reachable from the state $s_i$ and vice versa (denoted as $s_i⟷s_j$). Therefore, this relation dissects the state space into a family of classes $G$ that commute mutually. In practice, a commutative relationship between two states, $s_i$ and $s_j$, means that a material in the state $s_j$ can go back to be reprocessed to the previous station $s_i$ except with the raw material warehouse $s_1$ with positive probability. Similarly, a material in the state $s_{\rho }$ cannot return to any other state $s_1,s_2,\dots ,s_{\rho -1}$.

In decomposing the Markov chain into classes $G_{s_i}$, with $s_i\in S$, the following classes are clearly distinguished:

• $G_{s_1}$ is a class with only one non-return state $s_1$or transcendent state since $\mathbb{P}\left(X\left(t\right)=s_{\rho }\right)=0,$for $t>1$ (assumption 1).
• $G_{s_j}$ is a recurrent communicating class since $s_i⟷$ $s_j,$ $\forall s_i,s_j\in S^{\prime }$. This means that $\mathbb{P}\left(X\left(t\right)=s_j\right)=1,\mathrm{for}t>1\hspace{0.17em}$(assumption 2).
• $G_{s_{\rho }}$ is a communicating class with a single absorbing state (assumption 3).

As $G_{s_{\rho }}$ is a closed communicating class with a single state $s_{\rho }$, then for $P_{\rho \rho }^t=1$, for every $t=1,2,\dots .$ Hence $s_{\rho }$ is an absorbing state and, therefore, it is recurrent.

Classes satisfy the relation $G_{s_1}\prec G_{s_j}\prec G_{s_{\rho }}$. That is, given $s_1\in G_{s_1}$ and $s_j\in G_{s_j,}$ $s_1\to s_j,\mathrm{also}$ $s_j\to s_{\rho }.$ Therefore, $G_{s_1}$ and $G_{s_j}$ constitute a transcendent class while $G_{s_{\rho }}$ is a maximal class since the state $s_{\rho }$ is absorbent.

Due to the assumption that a material processed in the state $s\in S^{\prime }$ can return to be reprocessed on previous workstations or be transferred to higher states, it is satisfied that $\mathrm{for}\mathrm{all}{s}_i\in \{S^{\prime }\cup s_{\rho }\}$.

$$\mathbb{P}\hspace{0.17em}\left[X\left(t+\delta t\right)=\{s_i\cup s_{\rho }\}\hspace{0.17em}|\hspace{0.17em}X\left(t\right)=s_1\right]=1,$$

(12)

Thus, the states $s\in S^{\prime }$ are recurrent and, therefore, the class $G_{s_j}$ is recurrent. Consequently, $\mathrm{for}\mathrm{all}{s}_i\in S^{\prime },\mathrm{and}{s}_{\rho },\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{matrix}{v}_{ij}$ such that

$$v_{ij}={\sum }_{t=0}^{\infty }\hspace{0.17em}\hspace{0.17em}P\hspace{0.17em}\left[X\left(t+\delta t\right)=s_j\hspace{0.17em}|\hspace{0.17em}X\left(t\right)=s_i\right]={\sum }_{t=0}^{\infty }{P}_{ij}^t.$$

(13)

Equation (13) defines the expected number of visits of the process to state $s_j$ when it starts in $s_i$. In practical terms, this represents the expected number of demands per unit of time that each workstation on the manufacturing line has during the planning horizon.

The duration of a visit in each state $i$ in the instant $t$ is defined by the random variable $L_i^t$ and this is formally assigned an exponential distribution. Its meaning can be explained as the time required by each workstation to complete an operation on the visiting product. In order to calculate the duration of the visit to each state in the chain, it is defined

$$\tau =\inf \hspace{0.17em}\left\{t\hspace{0.17em}\ge 1\hspace{0.17em}|\hspace{0.17em}X\left(t\right)=s\right\}$$

(14)

where ${\tau }_i^0=0,$ ${\tau }_i^1={\tau }_i,$ and, in general, ${\tau }_i^{t+1}=\inf \left\{t\ge {\tau }_i^t+1|X\left(t\right)=i\right\}.$ Then, the duration of the $t$-th visit is given by

$$L_i^t=\{\begin{array}{c}{\tau }_i^t-{\tau }_i^{t-1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}\hspace{0.17em}{\tau }_i^{t-1}<\infty ,\\ 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{otherwise}.\end{array}$$

In this proposal, the random variable $L_i^t~ℇ\left({\tau }^{-1}\right)$ where for a set of $\gamma$ samples an unbiased estimator for $\tau$ is given by

$$\widehat{\tau }={\sum }_{t=1}^{\gamma }\hspace{0.17em}\left({\tau }_i^t-{\tau }_i^{t-1}\right){\gamma }^{-1}.$$

The following result is relevant for the analysis of this proposal [14]. For $t=2,\hspace{0.17em}3,\hspace{0.17em}\dots \hspace{0.17em}$, conditional on ${\tau }_i^{t-1}<\infty ,\hspace{0.17em}\hspace{0.17em}{L}_i^t$ is independent of $\left\{X\left(t\right)\hspace{0.17em}|\hspace{0.17em}t\hspace{0.17em}\le \hspace{0.17em}{L}_i^{t-1}\right\}$ and

$$\mathbb{P}(L_i^t=k\hspace{0.17em}|\hspace{0.17em}{\tau }_i^{n-1}\hspace{0.17em}\hspace{0.17em}<\infty )=P\hspace{0.17em}\left[\mathrm{X}\left(t\right)\right]=k.$$

(15)

Regarding the conditional probabilities of ever visiting the state $k$, given that the Markov chain was initially at state $j$, and the conditional probability of an infinite number of visits to the state $k$ given that the Markov chain was originally in the state $j$ will be denoted according to conventional literature as $f_{jk}$ and $g_{jk}$. The value of $f_{jk}$ is important as it represents the conditional probability of ever visiting the state $k$, given that the Markov chain was initially at state $j$. In this proposal this indicator will be used as an estimator of the productivity of the corresponding workstation.

Formally, the conditional probability $f_{jk}$ that the first passage from $j$ to $k$ occurs in exactly $t$ steps is given by

$$f_{jk}=\mathbb{P}\hspace{0.17em}[N_k(\infty )-N_k\hspace{0.17em}\left(t\right)>0\hspace{0.17em}|\hspace{0.17em}\hspace{0.17em}X\left(t\right)=j]={\displaystyle \sum }_{t=1}^{\infty }{f}_{jk}\hspace{0.17em}\left(t\right)$$

(16)

where the random variable $N_k(\infty )$ = ${\lim }_{t\to \infty }{N}_k\left(t\right)$ represents the total occupation time of the state $k$. The conditional probability of infinitely many visits to the state $s$, given that the Markov chain was at state $j$ initially, satisfies

$$g_{jk}=\hspace{0.17em}\mathbb{P}\hspace{0.17em}\left[N_k\hspace{0.17em}(\infty )-N_k\hspace{0.17em}\left(t\right)=\infty \hspace{0.17em}|\hspace{0.17em}X\hspace{0.17em}\left(t\right)=j\right]\hspace{0.17em}=f_{jk}\hspace{0.17em}{g}_{kk}$$

(17)

where

$$g_{kk}={\lim }_{\hspace{0.17em}t\hspace{0.17em}\to \hspace{0.17em}\infty \hspace{0.17em}}{\left(f_{kk}\right)}^t$$

(18)

and

$$N_k(\infty )={\sum }_{m=1}^{\infty }{Z}_k\hspace{0.17em}\left(m\right),N_k\left(n\right)={\sum }_{m=1}^n{Z}_k\hspace{0.17em}\left(m\right),$$

(19)

where

$$Z_k\hspace{0.17em}\left(n\right)=\{\begin{array}{c}1,\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}X\left(t\right)=k,\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}X\left(t\right)\hspace{0.17em}\hspace{0.17em}\ne \hspace{0.17em}k.\end{array}$$

(20)

Similarly, we denote by $f_{jk}\left(t\right)$ the conditional probability that the first passage from $j$ to $k$ occurs in exactly $t$ steps, i.e.,

$$f_{jk}\hspace{0.17em}\left(t\right)=\mathbb{P}\hspace{0.17em}\left[V_k\left(t\right)\hspace{0.17em}|\hspace{0.17em}X\left(0\right)=j)\right].$$

(21)

This magnitude makes sense in this analysis as it represents the time spent in a by-product visiting a workstation for the first time

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{V}_k\left(t\right)=\left[X\left(t\right)=k,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}X\left(m\right)\ne k\right]\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}t-1.$$

(22)

As a consequence, from (21), the probability of ever visiting state $k$ when coming from state $j$ is given by

$$f_{jk}={\displaystyle \sum }_{t=1}^{\infty }{f}_{jk}\hspace{0.17em}\left(t\right).$$

(23)

According to (22), $V_k\left(t\right)$ is the event of visiting for the first time among the times 1, 2 $\dots$ , at which state $k$ is visited at time $t$, see [24]. Two simple extensions of Equation (21) are the probability of never visiting the state $k$ when the process is in $j$, $1-f_{jk}$, as well as the probability of never returning to the state $k$ after leaving it, $1-f_{kk}$.

Notice now that, from Equations (17) and (18), it can be readily seen

$$g_{kk}=\{\begin{array}{c}1,\hspace{0.17em}\hspace{0.17em}\mathrm{iff}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{kk}=1\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}\mathrm{iff}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{kk}<1\end{array}$$

(24)

and

$$f_{kk}=\{\begin{array}{c}<1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{iff}\hspace{0.17em}\hspace{0.17em}{\displaystyle {\displaystyle \sum }_{t=1}^{\infty }}{p}_{kk}\hspace{0.17em}\left(t\right)<\infty ,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{iff}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle {\displaystyle \sum }_{t=1}^{\infty }}{p}_{kk}\hspace{0.17em}\left(t\right)=\infty .\hspace{0.17em}\end{array}$$

(25)

In particular, for the matrix shown in Table 1, $f_{jk}=1$ for $j,\hspace{0.17em}k\hspace{0.17em}\in Z^{\prime }$ if $j\hspace{0.17em}⟷k$. Otherwise, $f_{jk}=0.$ Similarly, for $j\in Z^{\prime }$ and $k=s_1$, $\hspace{0.17em}{f}_{jk}=0.$

From the above, given $Z$ and the absorbing state $s$, and for recurrent state $j$, the absorption probabilities satisfy the following system of equations.

$$\begin{gathered} f_{js}={\displaystyle \sum }_{i\hspace{0.17em}\in Z}{p}_{ji}\hspace{0.17em}{f}_{is},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j\in Z \\ {f}_{ss}=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{js}=0,j\ne s. \end{gathered}$$

Let the matrix of vectors be ${\mathit{Q}}_{\alpha }$ defined as

$${\mathit{Q}}_{\alpha }=\left({\mathit{P}}^1,\hspace{0.17em}{\mathit{P}}^2,\hspace{0.17em}\dots ,\hspace{0.17em}{\mathit{P}}^{\alpha },\hspace{0.17em}{\mathit{P}}^{\rho }\right),$$

(26)

where ${\mathit{P}}^{\alpha }=\mathit{\theta }\hspace{0.17em}\in \hspace{0.17em}{ℝ}^{\mathit{\rho }}$, that is, the matrix ${\mathit{Q}}_{\alpha }$ results from substituting the $\alpha$-th position of the matrix $\mathit{P}$ by the zero vector $\mathit{\theta }$. Then, by Equation (16), it is satisfied

$$f_{jk}\hspace{0.17em}\left(t\right)={\mathit{Q}}_{\alpha }{f}_{jk}\hspace{0.17em}\left(t-1\right),\hspace{0.17em}\hspace{0.17em}t=2,\hspace{0.17em}3,\hspace{0.17em}\dots$$

(27)

Equation (27) represents the probability of visiting the state $k$ in $t$ steps given that, initially, the process was at the state $j$.

Finally, an important equation for the limiting probabilities of the process is given by the expected number of steps $t$ required to return to state $k$. Formally,

$$m_{kk}=E\left(t\right)={\displaystyle \sum }_{t=1}^{\infty }t\hspace{0.17em}\hspace{0.17em}{f}_{kk}\hspace{0.17em}\left(t\right).$$

(28)

Let $\mathsf{\Psi }$ be the set of non-recurring states in $Z$, and let $Q=\left\{P_{jk}\hspace{0.17em}:j,\hspace{0.17em}k\hspace{0.17em}\in \mathsf{\Psi }\right\}$ be the matrix of transition probabilities of the states that map from $\mathsf{\Psi }$ to $\mathsf{\Psi }$. Then, the vector matrix of transition probabilities of the states that map from $\mathsf{\Psi }$ to $\mathsf{\Psi }$. Then, the vector $\mathit{m}$ of mean absorption times of the chain associated with the manufacturing process can be obtained from the equality

$$\hspace{0.17em}\mathit{m}={\left(I-Q\right)}^{-1}\hspace{0.17em}\mathbf{1}$$

(29)

where $\mathbf{1}$ is the column vector whose components are ones.

Now, we focus on the conditions of availability of the equipment in each workstation when receiving a visit from a by-product. From Equation (10), the Mean Time to Failure (MTTF) of each equipment is defined as the expectation of the random variable $\mathcal{T}$ as follows

$$\mathrm{MTTF}=\mathrm{E}(\mathcal{T})=-{\int }_0^{\infty }t\hspace{0.17em}\frac{dR}{dt}\hspace{0.17em}dt={\int }_0^{\infty }R\hspace{0.17em}\left(\mathcal{T}\right)\hspace{0.17em}dt\approx {\lambda }^{-1}.$$

(30)

In the same way, let $m\left(t\right)\Delta t=P(t\hspace{0.17em}\le \hspace{0.17em}\mathcal{T}$ $\le t+\Delta t)$be the probability that the repair of failed equipment requis a time between $t+\Delta t$ to be repaired. Then, the Mean Time to Repair (MTTR) is given by

$$\mathrm{MTTR}={\int }_0^{\infty }tm\left(t\right)dt\approx {\mu }^{-1}.$$

(31)

Regardless of the Probability Density Function (PDF) of the random variable $\mathcal{T}$, the ${\lambda }^{-1}$ and ${\mu }^{-1}$ values represent the MTTF and the MTTR, respectively. In turn, these are used as estimators of availability and unavailability (due to repairs) of equipment.

Define now the functions

$$P_{i,j}=P_{1,0}=\{\begin{array}{ll}\lambda ,& if\hspace{0.17em}\hspace{0.17em}i\ne j\\ 1-\lambda ,& if\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=j\end{array}$$

(32)

$$R_{j,i}=R_{0,1}=\{\begin{array}{c}\mu ,\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i\ne j\\ 1-\mu ,\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=j\hspace{0.17em}\end{array}$$

where $i$ and $j$ take the value 1 if the machine located at station $s$ is available, and 0 otherwise. Let $\pi \left(\alpha ,t\right)$ be the probability of being in the $\alpha$-state then, under steady state conditions it is satisfied that [3,8]

$$\pi \left(0,t\right)={\lim }_{t\to \infty }\left[\pi \hspace{0.17em}\left(0,0\right)\hspace{0.17em}{\left(1-\lambda -\mu \right)}^t+\frac{\lambda }{\mu +\lambda }\hspace{0.17em}\left[1-{\left(1-\lambda -\mu \right)}^t\right]\hspace{0.17em}\right]\frac{\lambda }{\mu +\lambda }$$

(33)

$$\hspace{0.17em}\pi \left(1,t\right)={\lim }_{t\to \infty }\left[\pi \hspace{0.17em}\left(1,0\right)\hspace{0.17em}{\left(1-\lambda -\mu \right)}^t+\frac{\mu }{\mu +\lambda }\hspace{0.17em}\left[1-{\left(1-\lambda -\mu \right)}^t\right]\hspace{0.17em}\right]\hspace{0.17em}=\frac{\mu }{\mu +\lambda }$$

(34)

Here, ${\eta }_j=\pi {\left(1,t\right)}_j$ represents the efficiency of equipment located at state $s_j$. Similarly, if the nominal capacity of that equipment is $\mathcal{Q}$ pieces per unit of time, then the average production rate is given by:

$${\overline{\mathcal{Q}}}_j=\mathcal{Q}{\eta }_j,\hspace{0.17em}\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}\rho$$

(35)

The expected productivity associated with the model can be estimated by (in pieces per unit of time)

$${\mathcal{P}}_j^{Tot}=\sum _{i=1}^{\rho }{\overline{\mathcal{Q}}}_j{f}_{ij}\hspace{1em}\forall j=1,2,\dots ,\rho$$

(36)

Finally, another relevant measure of the system is given by its leisure (or period of inactivity due to equipment stoppage). Leisure represents the productivity lost by not having the equipment time to do it. This measure can be approximated by the expression

$${ℒ}_j=1-\pi \left(1,t\right)=\frac{\lambda }{\mu +\lambda }.$$

(37)

Below is an application of the previous concepts.

4. An Application to a Refrigerator Factory

Metalwork machines are devices that are driven by electricity to perform different operations on metals such as sanding, knurling, drilling, boring, facing, threading and turning, among others. The most frequently used are: lathe machine, milling machine, grinding machine, drilling machines, punching machines, bending machines, paint booths, welding stations, shaper machines, broaching machines, saw machines, planer machine, shearing machine, hobbing machines, drill press and more.

To illustrate the use of the proposed technique, consider an integrated process production section of 12 metalwork machines, a raw material warehouse (steel or aluminum sheets) and a temporary buffer where finished products are placed before being used in another assembly line as in Figure 1. The matrix of transition probabilities used in this example is provided in

Table 2. Transition matrix $P_{ij}$ used in the numerical example. Source: Authors’ own elaboration.

States	1	2	3	4	5	6	7	8	9	10	11	12
1	0.0000	0.0870	0.0563	0.0599	0.0011	0.0448	0.0876	0.0807	0.0000	0.0138	0.5689	0.0000
2	0.0000	0.0000	0.0708	0.0424	0.0153	0.1176	0.0464	0.0788	0.0153	0.0616	0.1101	0.4418
3	0.0000	0.0619	0.0000	0.1064	0.0000	0.0687	0.0090	0.0000	0.0166	0.0981	0.0000	0.6392
4	0.0000	0.0375	0.1004	0.0000	0.0554	0.0551	0.0764	0.1163	0.0000	0.0656	0.0806	0.4127
5	0.0000	0.0228	0.1070	0.0158	0.0000	0.1167	0.0065	0.0596	0.1088	0.0943	0.0911	0.3775
6	0.0000	0.0912	0.1235	0.0087	0.0255	0.0000	0.0455	0.0908	0.0094	0.0339	0.0773	0.4941
7	0.0000	0.0830	0.0000	0.1126	0.0293	0.0454	0.0000	0.0000	0.0953	0.0251	0.1001	0.5093
8	0.0000	0.0541	0.1195	0.1035	0.0520	0.1244	0.0458	0.0000	0.0644	0.0376	0.0167	0.3819
9	0.0000	0.0739	0.0343	0.1215	0.0466	0.0000	0.1177	0.0999	0.0000	0.0230	0.0228	0.4603
10	0.0000	0.0000	0.0000	0.1130	0.0818	0.0811	0.0878	0.0000	0.0000	0.0000	0.0000	0.6363
11	0.0000	0.0262	0.0115	0.0000	0.0194	0.0724	0.0605	0.1185	0.1000	0.0978	0.0000	0.4492
12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	1.0000

. In this case, state 1 represents the raw material warehouse from where the material that enters the manufacturing process comes, and state 12 is the finished product warehouse where all production is accumulated before entering the assembly line. Transition probabilities were obtained from historical data generated in manufacturing engineering and plant engineering. The first defines the machining operations to be carried out on each product, and the second provides information about the availability, capacity and efficiency of the equipment.

The steady state transition matrix ${\mathit{P}}^{\mathit{t}}$obtained from the numerical example is achieved for $t=30$, that is, ${\mathit{P}}^{30}$.

Table 3. Matrix ${\mathit{v}}_{\mathit{i}\mathit{j}}$ , the expected number of visits to each state of the process. Source: Authors’ own elaboration.

States	1	2	3	4	5	6	7	8	9	10	11	12
1	0.0000	0.1741	0.1523	0.1602	0.0639	0.1866	0.1930	0.2215	0.1112	0.1413	0.6468	27.1230
2	0.0000	0.0556	0.1354	0.1088	0.0531	0.1891	0.1013	0.1410	0.0605	0.1216	0.1583	28.6759
3	0.0000	0.0916	0.0456	0.1461	0.0295	0.1156	0.0505	0.0447	0.0348	0.1322	0.0401	29.2213
4	0.0000	0.0929	0.1659	0.0776	0.0931	0.1400	0.1287	0.1720	0.0523	0.1302	0.1333	28.6471
5	0.0000	0.0833	0.1729	0.0975	0.0429	0.1910	0.0741	0.1288	0.1487	0.1571	0.1398	28.5885
6	0.0000	0.1353	0.1761	0.0763	0.0561	0.0767	0.0911	0.1403	0.0511	0.0931	0.1221	28.8286
7	0.0000	0.1274	0.0615	0.1663	0.0647	0.1111	0.0600	0.0746	0.1321	0.0820	0.1523	28.7764
8	0.0000	0.1156	0.1931	0.1763	0.0907	0.2000	0.1060	0.0732	0.1041	0.1063	0.0815	28.6536
9	0.0000	0.1255	0.1035	0.1909	0.0853	0.0830	0.1691	0.1598	0.0488	0.0842	0.0869	28.7563
10	0.0000	0.0395	0.0526	0.1505	0.1061	0.1285	0.1211	0.0479	0.0338	0.0423	0.0498	29.1678
11	0.0000	0.0780	0.0738	0.0767	0.0596	0.1392	0.1167	0.1692	0.1371	0.1424	0.0486	27.5663
12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	30.0000

shows the Matrix ${\mathit{v}}_{\mathit{i}\mathit{j}},$i.e., the expected number of finished products entering the final warehouse, in Equation (13). In this case, state 12 has been selected because it is the most representative of the system, since it is the connection between two different areas in the manufacturing process. In this case, manufacturing by process and mass manufacturing. Note that the last column (state 12) reflects the expected total quantity of finished goods arriving at the finished goods warehouse.

Derived from the results of Table 3 and

Table 4. Probability matrix $f_{jk}\left(t\right)$ for $j=1,\dots ,12,k=12,t=1,2,\dots ,12$. Source: Authors’ own elaboration.

States	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8	t = 9	t = 10	t = 11	t = 12
1	0.4614	0.2611	0.1337	0.0699	0.0359	0.0185	0.0095	0.0049	0.0025	0.0013	0.0007	0.0003
2	0.2760	0.1371	0.0704	0.0363	0.0187	0.0096	0.0049	0.0025	0.0013	0.0007	0.0003	0.0002
3	0.1799	0.0872	0.0457	0.0233	0.0120	0.0062	0.0032	0.0016	0.0008	0.0004	0.0002	0.0001
4	0.2902	0.1449	0.0738	0.0381	0.0196	0.0101	0.0052	0.0027	0.0014	0.0007	0.0004	0.0002
5	0.3197	0.1455	0.0764	0.0393	0.0202	0.0104	0.0054	0.0028	0.0014	0.0007	0.0004	0.0002
6	0.2510	0.1246	0.0631	0.0327	0.0168	0.0086	0.0044	0.0023	0.0012	0.0006	0.0003	0.0002
7	0.2214	0.1304	0.0674	0.0347	0.0179	0.0092	0.0047	0.0024	0.0013	0.0006	0.0003	0.0002
8	0.3085	0.1508	0.0771	0.0396	0.0204	0.0105	0.0054	0.0028	0.0014	0.0007	0.0004	0.0002
9	0.2453	0.1434	0.0733	0.0377	0.0194	0.0100	0.0051	0.0026	0.0014	0.0007	0.0004	0.0002
10	0.1623	0.0987	0.0498	0.0256	0.0132	0.0068	0.0035	0.0018	0.0009	0.0005	0.0002	0.0001
11	0.2662	0.1361	0.0722	0.0370	0.0191	0.0098	0.0051	0.0026	0.0013	0.0007	0.0004	0.0002
12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

, we show the probabilities $f_{jk}\left(t\right)$ for $k=12,t=1,2,\dots ,12$, i.e., the conditional probability $f_{jk}$ that the first passage from $j$ to $k$ occurs in exactly $t$ steps (as in Equation (27)).

Table 5. Probability matrix $f_{jk}$ for the proposed instance. Note that $g_{kk}=0,\forall k.$ Source: Authors’ own elaboration.

States	1	2	3	4	5	6	7	8	9	10	11	12
1	0.0000	0.0872	0.0895	0.0890	0.0603	0.1286	0.0946	0.1257	0.1060	0.1217	0.0479	0.9712
2	0.0000	0.0556	0.0588	0.0586	0.0357	0.0580	0.0493	0.0525	0.0424	0.0550	0.0409	0.5512
3	0.0000	0.0298	0.0436	0.0292	0.0283	0.0386	0.0387	0.0416	0.0165	0.0287	0.0382	0.3590
4	0.0000	0.0554	0.0583	0.0721	0.0339	0.0750	0.0451	0.0439	0.0498	0.0592	0.0465	0.5814
5	0.0000	0.0606	0.0585	0.0748	0.0412	0.0607	0.0634	0.0604	0.0330	0.0564	0.0422	0.6163
6	0.0000	0.0442	0.0450	0.0621	0.0284	0.0713	0.0405	0.0398	0.0394	0.0554	0.0390	0.5005
7	0.0000	0.0444	0.0589	0.0418	0.0329	0.0577	0.0567	0.0695	0.0306	0.0536	0.0451	0.4839
8	0.0000	0.0615	0.0652	0.0602	0.0350	0.0614	0.0543	0.0682	0.0348	0.0644	0.0611	0.6144
9	0.0000	0.0517	0.0647	0.0557	0.0353	0.0771	0.0419	0.0490	0.0465	0.0577	0.0601	0.5358
10	0.0000	0.0395	0.0503	0.0267	0.0199	0.0383	0.0265	0.0446	0.0322	0.0406	0.0474	0.3615
11	0.0000	0.0518	0.0591	0.0713	0.0378	0.0569	0.0497	0.0392	0.0307	0.0388	0.0463	0.5041
12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

shows the results of the probability $f_{jk}$of ever visiting state $k$ when coming from state $i$, as in Equation (23).

Finally, vector $\mathit{m}\hspace{0.17em}$of mean absorption times associated with the system from state 2 to state 11 is shown in

Table 6. Mean absorption time ($\mathit{m})$ , states 2 to 11. Source: Authors’ own elaboration.

2	3	4	5	6	7	8	9	10	11
2.9924	2.1727	1.8129	2.2781	2.3696	2.1348	2.0858	2.3853	2.1720	1.7708

, as in Equation (29). Note that station 1 has a zero-value associated with it because, although its work is important, it does not contribute to the added value of the product since it is only a temporary warehouse for raw materials. Similarly, workstation 12 represents a point of accumulation of material of the entire manufacturing line, thus the quantities obtained represent the average values of accumulation of material per unit of time.

5. Discussion of Empirical Results

In this section, empirical results are presented and discussed.

Table 7. Average production per workstation (states) pieces/unit of time.

Row	Station Measure	1	2	3	4	5	6	7	8	9	10	11	12
1	$\mu$	2.0000	3.1824	1.6939	2.1028	3.9182	1.2838	0.8273	0.7283	0.9273	1.4828	1.2303	0.9891
2	$\lambda$	0.0100	0.0300	0.2091	0.1912	0.0245	0.0439	0.0234	0.0129	0.0121	0.3726	0.0123	0.8373
3	$\mathcal{Q}$	14	16	12	15	14	17	16	14	12	16	18	14
4	$\eta$	0.9950	0.9907	0.8901	0.9166	0.9938	0.9669	0.9725	0.9826	0.9871	0.7992	0.9901	0.5416
5	$\overline{\mathcal{Q}}$	13.9303	15.8506	10.6815	13.7496	13.9130	16.4378	15.5599	13.7563	11.8454	12.7869	17.8218	7.5818
	Station	1	2	3	4	5	6	7	8	9	10	11	12
6	1	0.0000	1.3815	0.9555	1.2234	0.8393	2.1139	1.4726	1.7294	1.4892	1.5566	0.8534	7.3634
7	2	0.0000	0.8813	0.6278	0.8053	0.4972	0.9541	0.7666	0.7228	0.6224	0.7037	0.7283	4.1790
8	3	0.0000	0.4716	0.4658	0.4013	0.3937	0.6352	0.6016	0.5726	0.4931	0.3669	0.6808	2.7216
9	4	0.0000	0.8789	0.6229	0.9916	0.4721	1.2321	0.7013	0.6037	0.5198	0.7575	0.8293	4.4080
10	5	0.0000	0.9599	0.6246	1.0288	0.5739	0.9983	0.9872	0.8313	0.7158	0.7215	0.7516	4.6725
11	6	0.0000	0.7006	0.4805	0.8541	0.3945	1.1720	0.6301	0.5480	0.4719	0.7082	0.6959	3.7947
12	7	0.0000	0.7044	0.6291	0.5748	0.4572	0.9488	0.8827	0.9555	0.8228	0.6849	0.8040	3.6690
13	8	0.0000	0.9747	0.6962	0.8274	0.4875	1.0090	0.8445	0.9386	0.8083	0.8237	1.0884	4.6586
14	9	0.0000	0.8191	0.6915	0.7662	0.4910	1.2681	0.6522	0.6739	0.5803	0.7379	1.0704	4.0623
15	10	0.0000	0.6261	0.5374	0.3677	0.2774	0.6293	0.4117	0.6136	0.5284	0.5189	0.8456	2.7405
16	11	0.0000	0.8206	0.6314	0.9797	0.5262	0.9361	0.7732	0.5387	0.4638	0.4964	0.8256	3.8223
17	12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
18	Average produtivity	0.0000	9.2186	6.9628	8.8204	5.4100	11.8970	8.7238	8.7281	7.5157	8.0762	9.1733	46.0917
19	$ℒ$	0.0000	0.0093	0.1099	0.0834	0.0062	0.0331	0.0275	0.0174	0.0129	0.2008	0.0099	0.4584

shows the average production per workstation (states) pieces/unit of time. The estimators in Table 7 are part of other necessary calculations, such as material requirement planning (MRP), and manufacturing requirement planning (MRP II). Likewise, the estimators found allow us to approximate a Master Production Schedule (MPS) in order to create commitments with customers and suppliers.

In this proposal, the probabilities of ever visiting each station have been used as an estimator of the probability of arriving at that place. The assumption of independence between the probabilities of visiting and availability of each station, as well as the previous knowledge of the nominal capacity of the equipment, allows the estimation of production by work area. An important aspect of the proposal is the shape of the transition matrix $\mathit{P}$. This is common in processes of the same type; therefore, it facilitates the characterization of these systems and, therefore, their analysis from a quantitative perspective.

With this proposal, several indicators can be obtained from the basic knowledge of the $\mathit{P}$ matrix and the initial distribution $\mathit{\pi }$. Such statistics are normally available in the area of engineering and maintenance departments of the majority metalworking industries.

The way to obtain them and their practical meaning is as follows (The number assigned to each line corresponds to the explanation given below):

• The 12 stations that make up this study are defined.
• Here, the average rates for the repair $\left(\mu \right)$ of the equipment contained in the referred station are defined, i.e., the Mean Time to Repair (MTTR) for each station, Equation (31).
• The Mean Time to Failure $\left(\lambda \right)$ (MTTF) of each equipment is defined, as in Equation (30).
• The productivity ($\mathcal{Q}$) of each machine assigned to the corresponding workstation. It is defined as the relationship between the volume of output and the volume of inputs. This information is provided by the manufacturing engineering area and is obtained from historical data.
• The efficiency ($\eta$) of the machine installed on the corresponding workstation is defined. Mechanical efficiency is a measure of how well the machine converts the input work or energy into some useful output. In our case, we use Equation (34) to calculate this parameter.
• The analysis provides the expected productivity of each machine ($\overline{\mathcal{Q}}$) associated with a workstation, Equation (35).
• Lines 6 through 12 define the expected productivity per station (${\overline{\mathcal{Q}}}_{kj})=f_{kj}\hspace{0.17em}{\overline{\mathcal{Q}}}_j$.
• Row 18 shows the accumulated values of the expected production at station $j$. That is, the total average parts manufactured at station $j$, i.e.,

  $$\mathrm{Average}\mathrm{productivity}={\sum }_{j=6}^{17}\hspace{0.17em}{\overline{\mathcal{Q}}}_{kj},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\hspace{0.17em}\dots ,\hspace{0.17em}12\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

  (38)
• Finally, the leisure index ($ℒ$) generated in each station, due to its unavailability, is also presented as in Equation (37).

Some Additional Comments on the Results

Table 7 fully summarizes the results in this research. Some important aspects to highlight are the following. This methodology was applied to a real metalworking manufacturing model in a refrigerator manufacturing company in Mexico. The actual results were flattering as they are fairly close to the values obtained onsite. It is important to note that the entire project is based on the knowledge of the matrix of transition probabilities. Likewise, the knowledge of the availability of the equipment is essential for the calculation of various parameters.

It is also important to note that at the start of system operations, it is verified that

$$\hspace{0.17em}{\mathit{P}}^{\mathit{t}}\hspace{0.17em}{\mathit{\pi }}^{\mathbf{0}}\to \mathit{\pi }\hspace{0.17em}\in {ℝ}^{12}$$

as t→ $\infty$.

Under stable state conditions, it can be assumed that each component of $\pi$ is equiprobable, then it is verified that ${\lim }_{t\to \infty }\mathit{\pi }\hspace{0.17em}{\mathit{P}}^{\mathit{t}}\stackrel{\mathfrak{D}}{\overbrace{\to }}{\mathit{\pi }}^{\mathit{t}}$, where the symbol $\stackrel{\mathfrak{D}}{\overbrace{\to }}$ means convergence in distribution.

Finally, the approximation of the duration of visits to each station can be approximated using Equation (15) as long as the system is in steady state conditions.

$$\mathbb{P}(L_i^t=k\hspace{0.17em}|\hspace{0.17em}{\tau }_i^{n-1}\hspace{0.17em}\hspace{0.17em}<\infty )=P\hspace{0.17em}\left[\mathrm{X}\left(t\right)\right]=k.$$

(39)

It is interesting to highlight some efforts made to model mass manufacturing systems with the approach proposed here, see, for example, [26,27,28,29,30]. This proposal is a first attempt to develop a model in process manufacturing systems, and more research is needed in the future to improve the modeling.

6. Conclusions

In this proposal Markov chain theory was used to model the activity of a manufacturing line by process. Due to the frequency with which these types of layouts are found in manufacturing engineering, the problem addressed here acquires special importance in order to determine the average production that can be aspired to when the failure and maintenance rates of the system are known.

Other interesting lines of research to explore, in the future, are the association of costs to the process, the time spent at each station (Equation (14) provides an estimator of this), the rejected materials and the presence of preceding buffers. In practice, manufacturing engineers face these challenges using a Monte Carlo simulation in order to obtain the mentioned estimators. However, the credibility of such models depends largely on the quality of the proposed model and the ability to program it. Therefore, an approximate solution is always preferable, despite its uncertainty.

The real results obtained in this project showed its feasibility and above all, its usefulness in terms of obtaining quantitative indicators of the expected production in a manufacturing line with these characteristics. This offers advantages over simulation since the estimators depend only on the matrix of transition probabilities.

Finally, it is important to point out that the reliability of the chain run can be easily obtained from the product of the reliability of each station.