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

1. Introduction

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

or CM).

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:

Please insert

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.