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.
3.2. Formalization of the Model through a Markov Chain
Let be a finite state space. The elements of S could be vectors. Each is called a state. Hereafter, = { } is a probability distribution defined in some probability space . Thus, for all the total mass satisfies Let be a random variable (vector) for each . From now on 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
(position of a product at time
) depends only on the previous value of
. That is,
is a Markov process. Let
where
the history of the values of
before time
and
is the possible value of
. Then, it is satisfied that
In a particular case when the process is defined in discrete times, these will be numbered by
or in a convenient way as the case may be. In the continuous time case, the notation
will be used, or simply
. In the same way, the transition probability from
to
in a unit of time will be denoted by
In the case of a Markov chain,
denotes that the random quantity
satisfies the normalization equation given by
. It is also satisfied the Chapman-Kolmogorov equation in the sense that
for any
where
. Thus, by the Law of Total Probability and from (2), it is also true that
with
. Equivalently, we may write
where
and
In addition, the following conditions are satisfied in steady state
if it exists, and the transition equations
as well as the balance equations in steady state hold
The corresponding normalization equation is given by
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 = if the limit exists. Moreover, and the normalization equation is Analogously, = 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 given that it is currently at state 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 , and will evolve into the state 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 .
The set defined as S′ = {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 is accessed from .
For all , there is a with , such that 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 is the only one that allows material to exit the system. This represents the finished product warehouse and can be accessed from any . 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 is sent to any . This constitutes the product or productive unit of the system on which operations will be carried out in the rest of the states .
The transition matrix 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
in the instant
, is given by the function
where
is the failure density function of the equipment at the states
, and
denotes the random variable that represents the instant where a failure occurs. Here, it will be assumed that
Hereinafter, 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
is given by the diverse workstations to which the product can arrive during its trajectory. The analyzed system is made up of the states
where
represents the raw material warehouse and the state
represents the finished product warehouse. So, a typical transition matrix would be given by
Table 1.
According to
Table 1,
and
. This research uses the notation
to express that a state
is accessible from a state
. In our case, the initial state
represents the raw material warehouse and, therefore, it is a non-return state. In turn, the final state
represents the finished product warehouse and once the product reaches it, it remains there until it is required later on in another production line.
is a buffer between the first manufacturing process and the main assembly line. This is an absorbing and therefore recurrent state. The intermediate states
, 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
be the set of states in which a by-product can be found within the production line. If the process
begins with a known initial distribution
, i.e.,
then the distribution of
is defined through
and, in general, the distribution of
is defined through
. That is,
where
is the identity matrix. Here, the symbol
means weak convergence and equality
is equivalent to
, where
means
transposed.
During the manufacturing process, each state commutes with the rest. That is, the state is reachable from the state and vice versa (denoted as ). Therefore, this relation dissects the state space into a family of classes that commute mutually. In practice, a commutative relationship between two states, and , means that a material in the state can go back to be reprocessed to the previous station except with the raw material warehouse with positive probability. Similarly, a material in the state cannot return to any other state .
In decomposing the Markov chain into classes , with , the following classes are clearly distinguished:
is a class with only one non-return state or transcendent state since for (assumption 1).
is a recurrent communicating class since . This means that (assumption 2).
is a communicating class with a single absorbing state (assumption 3).
As is a closed communicating class with a single state , then for , for every Hence is an absorbing state and, therefore, it is recurrent.
Classes satisfy the relation . That is, given and Therefore, and constitute a transcendent class while is a maximal class since the state is absorbent.
Due to the assumption that a material processed in the state
can return to be reprocessed on previous workstations or be transferred to higher states, it is satisfied that
.
Thus, the states
are recurrent and, therefore, the class
is recurrent. Consequently,
such that
Equation (13) defines the expected number of visits of the process to state when it starts in . 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
in the instant
is defined by the random variable
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
where
and, in general,
Then, the duration of the
-th visit is given by
In this proposal, the random variable
where for a set of
samples an unbiased estimator for
is given by
The following result is relevant for the analysis of this proposal [
14]. For
, conditional on
is independent of
and
Regarding the conditional probabilities of ever visiting the state , given that the Markov chain was initially at state , and the conditional probability of an infinite number of visits to the state given that the Markov chain was originally in the state will be denoted according to conventional literature as and . The value of is important as it represents the conditional probability of ever visiting the state , given that the Markov chain was initially at state . In this proposal this indicator will be used as an estimator of the productivity of the corresponding workstation.
Formally, the conditional probability
that the first passage from
to
occurs in exactly
steps is given by
where the random variable
=
represents the total occupation time of the state
. The conditional probability of infinitely many visits to the state
, given that the Markov chain was at state
initially, satisfies
where
and
where
Similarly, we denote by
the conditional probability that the first passage from
to
occurs in exactly
steps, i.e.,
This magnitude makes sense in this analysis as it represents the time spent in a by-product visiting a workstation for the first time
As a consequence, from (21), the probability of ever visiting state
when coming from state
is given by
According to (22),
is the event of visiting for the first time among the times 1, 2
, at which state
is visited at time
, see [
24]. Two simple extensions of Equation (21) are the probability of never visiting the state
when the process is in
,
, as well as the probability of never returning to the state
after leaving it,
.
Notice now that, from Equations (17) and (18), it can be readily seen
and
In particular, for the matrix shown in
Table 1,
for
if
. Otherwise,
Similarly, for
and
,
From the above, given
and the absorbing state
, and for recurrent state
, the absorption probabilities satisfy the following system of equations.
Let the matrix of vectors be
defined as
where
, that is, the matrix
results from substituting the
-th position of the matrix
by the zero vector
. Then, by Equation (16), it is satisfied
Equation (27) represents the probability of visiting the state in steps given that, initially, the process was at the state .
Finally, an important equation for the limiting probabilities of the process is given by the expected number of steps
required to return to state
. Formally,
Let
be the set of non-recurring states in
, and let
be the matrix of transition probabilities of the states that map from
to
. Then, the vector matrix of transition probabilities of the states that map from
to
. Then, the vector
of mean absorption times of the chain associated with the manufacturing process can be obtained from the equality
where
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
as follows
In the same way, let
be the probability that the repair of failed equipment requis a time between
to be repaired. Then, the Mean Time to Repair (MTTR) is given by
Regardless of the Probability Density Function (PDF) of the random variable , the and 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
where
and
take the value 1 if the machine located at station
is available, and 0 otherwise. Let
be the probability of being in the
-state then, under steady state conditions it is satisfied that [
3,
8]
Here,
represents the efficiency of equipment located at state
. Similarly, if the nominal capacity of that equipment is
pieces per unit of time, then the average production rate is given by:
The expected productivity associated with the model can be estimated by (in pieces per unit of time)
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
Below is an application of the previous concepts.