An integrated model of maintenance planning and statistical processcontrol is developed for a production process. The process has two operationalstates including an in-control state and an out-of-control state, where the processfailure mechanism is supposed as a general continuous distribution withnon-decreasing failure rate. Based onthe information obtained from the control chart, three types of maintenance actionsmay be implemented on the process. The integrated model optimally determinesthe parameters of the control chart and maintenance actions so that theexpected cost per time unit is minimized.
To evaluate the performance of the integrated model, a stand-alone model isdeveloped. 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. IntroductionMaintenance management (MM) and statistical process control (SPC)are two key tools for management and control of production processes. Although foryears, from the academic and practical point of view, these two key tools areconsidered and analyzed separately, some integrated models have been recently developedto consider MM and SPC jointly.
It is mentioned by many authors that there are manyinteractions and interrelations between MM and SPC that verify the development ofthe integrated models (1,2,3,4).Integrated models of MM and SPC can be classified based on thedifferent criteria such as: type of the control chart employed for the processmonitoring, process failure mechanism, number of the process states, inspectionpolicy applied for the process monitoring, impact of the maintenance on theprocess, and maintenance policy in the different situations. Different types ofcontrol 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 integratedmodels, it is assumed that the probabilities of process transitions betweendifferent 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 mechanismfollows a general distribution (5,13). In some models, the number of the process states is assumed tobe two states including an in-control state and an out-of-control state (12,4). Some integrated models assume three states for a system includingan in-control state, an out-of-control state and a failure state (5,14).
Also in some studies, a system has several operational statesplus a failure state (3,15). Different inspection policies are applied to monitor processes suchas equidistance interval policy (14,2) and constant hazard policy(16,5). In some integrated models, the effect of maintenance on systemsis supposed to be perfect (12,13,10), while in some models, it is assumed that the maintenance effect isimperfect (5,3,16). While a perfect maintenance restores the system to the best-as-newstate, 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 implementedon the process. A compensatory maintenance is applied when a false alarm isissued from the control chart, a reactive maintenance is implemented when facingthe out-of-control state, and a corrective maintenance is applied in the stateof complete process failure. In this paper, a process that has two operational states (anin-control state and an out-of-control state) is considered. The processfailure mechanism follows a general continues distribution with non-decreasingfailure rate.
Based on the information obtained from the control chart, threetypes of maintenance actions are possible to be conducted on the process, andfour scenarios are possible for the evolution of the process in a productioncycle. An integrated model of MM and SPC is presented for the process. Toevaluate the performance of the integrated model, a stand-alone maintenancemodel is also developed.
The rest of the paper is organized as follows: in section 2, thegeneral structure of the problem is described. Derivation of the integratedmodel is described in section 3. In section 4, a stand-alone maintenance modelis developed. Section 5 elaborates the inspection policy applied in theintegrated model. In section 6, details about the optimization of the modelsare presented. Section 7 presents a reals case study.
Also, some sensitivityanalyses is conducted in section 7, and finally section 8 concludes the paper. 2. Problem description Consider a production process that has twooperational states: an in-control state denoted as state 0 and an out-of-controlstate denoted as state 1. The operationof 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 qualitycosts. The time that theprocess spends in state 0 before transition to state 1, the process failuremechanism, follows a general continues distribution function withnon-decreasing failure rate. The process is monitored as follows: at specific time points suchas (t1,t2,…,tm-1), these time pointes are the decisionvariables of the model, n units of the produced items of the process areselected and a suitable quality characteristic (characteristics) is (are)measured and then a suitable statistic is calculated.
This statistic is plottedon a desired control chart. If the statistic falls within the control limits ofthe control chart, the process will continue its operation without anyinterruption. If the statistic falls outside the control limits, an alarm isissued from the control chart. After that, an investigation is performed on thesystem to verify this alarm. If the investigation concludes that the chartsignal is incorrect (i.e., the process is in state 0), a compensatorymaintenance (CM) is conducted on the process; but if the investigationconcludes that the chart signal is correct, a reactive maintenance (RM) isimplemented on the system.
Henceforth, we call the investigation performedafter releasing the alarm of the control chart as the maintenance inspection todistinguish 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 maintenanceinspection is applied to determine the true state of the process. If the maintenanceinspection indicates that the system is in the in-control state at tmthen a preventive maintenance (PM) is conducted, but if the maintenanceinspection indicates that the system state is out-of-control at tmthen RM is applied. Hence, a production cycle of the process starts in state 0and is terminated due to implement one type of the maintenance actions (RM, PMor CM). Based on the descriptions presented so far, four scenarios arepossible for the evolution of the process in a production cycle.
Thesescenarios are illustrated in figure 1 and elaborated as follows:Please insertfigure 1 near here.Scenario 1: The process remains in state 0 until tm andno 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 falsealarm is released from the control chart. Hence, CM is implemented, and theprocess is renewed.
Scenario 3: The Process shifts to state 1 before tm-1, andan alarm is released from the control chart in one of the remaining inspectionperiods. Thus, RM is implemented and the process is renewed. Scenario 4: The process shifts to state 1 before tm , butthe control chart cannot release this state. In other words, no alarmindicating the out-of-control state of the process is issued by the controlchart in the remaining inspection periods. Hence, at tm, after the maintenanceinspection, the true state of the process is identified, and RM is conducted.