An integrated model of maintenance planning and statistical process
control is developed for a production process. The process has two operational
states including an in-control state and an out-of-control state, where the process
failure mechanism is supposed as a general continuous distribution with
non-decreasing failure rate. Based on
the information obtained from the control chart, three types of maintenance actions
may be implemented on the process. The integrated model optimally determines
the parameters of the control chart and maintenance actions so that the
expected cost per time unit is minimized. To evaluate the performance of the integrated model, a stand-alone model is
developed. In the stand-alone model, only maintenance planning is considered. Finally,
a real case study is presented to clarify the performances of these models.
Key words: maintenance; control chart;
statistical process control; process failure mechanism; integrated model
Maintenance management (MM) and statistical process control (SPC)
are two key tools for management and control of production processes. Although for
years, from the academic and practical point of view, these two key tools are
considered and analyzed separately, some integrated models have been recently developed
to consider MM and SPC jointly. It is mentioned by many authors that there are many
interactions and interrelations between MM and SPC that verify the development of
the integrated models (1,2,3,4).
Integrated models of MM and SPC can be classified based on the
different criteria such as: type of the control chart employed for the process
monitoring, process failure mechanism, number of the process states, inspection
policy applied for the process monitoring, impact of the maintenance on the
process, and maintenance policy in the different situations. Different types of
control charts are employed in the integrated models of MM and SPC such as
control chart (4,5,3), Shewhart chart with variable parameters6, Bayesian control chart7, chi-square chart8, cause- selecting control chart 2 and exponential weighted moving average (EWMA) chart (9,10). From the aspect of process failure mechanism, in some integrated
models, it is assumed that the probabilities of process transitions between
different states are based on an exponential distribution (11,8). Some models are developed based on the Weibull distribution (4,12), and in some researches it is supposed that the failure mechanism
follows a general distribution (5,13). In some models, the number of the process states is assumed to
be two states including an in-control state and an out-of-control state (12,4). Some integrated models assume three states for a system including
an in-control state, an out-of-control state and a failure state (5,14). Also in some studies, a system has several operational states
plus a failure state (3,15).
Different inspection policies are applied to monitor processes such
as equidistance interval policy (14,2) and constant hazard policy(16,5). In some integrated models, the effect of maintenance on systems
is supposed to be perfect (12,13,10), while in some models, it is assumed that the maintenance effect is
imperfect (5,3,16). While a perfect maintenance restores the system to the best-as-new
state, an imperfect maintenance renews the system to the state between “as-good-as-new”
state and the current state (3, 5). Based on the process state, different maintenance policies are implemented
on the process. A compensatory maintenance is applied when a false alarm is
issued from the control chart, a reactive maintenance is implemented when facing
the out-of-control state, and a corrective maintenance is applied in the state
of complete process failure.
In this paper, a process that has two operational states (an
in-control state and an out-of-control state) is considered. The process
failure mechanism follows a general continues distribution with non-decreasing
failure rate. Based on the information obtained from the control chart, three
types of maintenance actions are possible to be conducted on the process, and
four scenarios are possible for the evolution of the process in a production
cycle. An integrated model of MM and SPC is presented for the process. To
evaluate the performance of the integrated model, a stand-alone maintenance
model is also developed.
The rest of the paper is organized as follows: in section 2, the
general structure of the problem is described. Derivation of the integrated
model is described in section 3. In section 4, a stand-alone maintenance model
is developed. Section 5 elaborates the inspection policy applied in the
integrated model. In section 6, details about the optimization of the models
are presented. Section 7 presents a reals case study. Also, some sensitivity
analyses is conducted in section 7, and finally section 8 concludes the paper.
2. Problem description
Consider a production process that has two
operational states: an in-control state denoted as state 0 and an out-of-control
state denoted as state 1. The operation
of the process in state 1 is undesirable, because in comparison with state 0,
it leads to much more operational cost and also yields the higher quality
costs. The time that the
process spends in state 0 before transition to state 1, the process failure
mechanism, follows a general continues distribution function with
non-decreasing failure rate.
The process is monitored as follows: at specific time points such
as (t1,t2,…,tm-1), these time pointes are the decision
variables of the model, n units of the produced items of the process are
selected and a suitable quality characteristic (characteristics) is (are)
measured and then a suitable statistic is calculated. This statistic is plotted
on a desired control chart. If the statistic falls within the control limits of
the control chart, the process will continue its operation without any
interruption. If the statistic falls outside the control limits, an alarm is
issued from the control chart. After that, an investigation is performed on the
system to verify this alarm. If the investigation concludes that the chart
signal is incorrect (i.e., the process is in state 0), a compensatory
maintenance (CM) is conducted on the process; but if the investigation
concludes that the chart signal is correct, a reactive maintenance (RM) is
implemented on the system. Henceforth, we call the investigation performed
after releasing the alarm of the control chart as the maintenance inspection to
distinguish it from the sampling inspection.
At the end of the production cycle (at time point tm),
there is no sampling from the produced items; but only the maintenance
inspection is applied to determine the true state of the process. If the maintenance
inspection indicates that the system is in the in-control state at tm
then a preventive maintenance (PM) is conducted, but if the maintenance
inspection indicates that the system state is out-of-control at tm
then RM is applied. Hence, a production cycle of the process starts in state 0
and is terminated due to implement one type of the maintenance actions (RM, PM
Based on the descriptions presented so far, four scenarios are
possible for the evolution of the process in a production cycle. These
scenarios are illustrated in figure 1 and elaborated as follows:
figure 1 near here.
Scenario 1: The process remains in state 0 until tm and
no alarm is released from the control chart in the previous inspection periods.
Hence, PM is conducted on the process at tm.
Scenario 2: While the process is operating in state 0, a false
alarm is released from the control chart. Hence, CM is implemented, and the
process is renewed.
Scenario 3: The Process shifts to state 1 before tm-1, and
an alarm is released from the control chart in one of the remaining inspection
periods. Thus, RM is implemented and the process is renewed.
Scenario 4: The process shifts to state 1 before tm , but
the control chart cannot release this state. In other words, no alarm
indicating the out-of-control state of the process is issued by the control
chart in the remaining inspection periods. Hence, at tm, after the maintenance
inspection, the true state of the process is identified, and RM is conducted.