The process monitoring described in the previous section can beconsidered as a renewal reward process consisting of the stochastic andindependent identical cycles. ETand EC are defined as the expected time length of a production cycleand the expected cost of a production cycle, respectively. Hence, the expectedcost of the process per time unite (ECT) can be obtained using thefollowing formula: (1) Now, consider a single arbitrary inspection interval such as (ti-1,ti). Given the state of the process immediately after the inspection at ti-1, three different cases can be considered for the evolution of the system in this interval. Table 1 shows these cases along with the occurrence probability of the cases.
Also, for these three cases, duration of time that the process operates in each state, i.e., in state 0 or in state 1, is shown by Table 1. Please insert table 1 near here.The cases are elaborated as follows:Case a: In this case, the process is in state 0 at ti-1 andremains in this state until ti. The occurrence probability of thiscase is as follows:The sum of the two terms inside the square brackets is theprobability of the process operation in state 1 just before the inspection at ti. Also, ifthe process is in state 1 before the inspection at ti ,withthe probability of , the control chart cannot detect the out-of-controlstate, and the process will continue its operation in state 1 after theinspection performed at ti.As, it is assumed that all the maintenance actions are perfect, thus,the following equation holds true at the start of each production cycle:Note that, the sum ofthe two terms in the right side of Equation 21 is not necessarily equal one,because it is possible that a production cycle is terminated before reaching totime point ti-1 due to performance of CM or RM.
Now, each term inequation 8 and 9 is computed and derivation of the integrated model iscompleted.4. Derivation of the stand-alonemodelIn this model, it isassumed that the process starts its operation in the in-control state and attime point tm , the maintenance inspection is conducted onthe process to clarify the true state of the process. In the stand-alone model,only maintenance planning is considered and tm is the onlydecision variable. If the maintenance inspection indicates that the processstate is 1 at tm, then RM is applied; otherwise PM is implemented onthe process. Hence, in this model, there is no sampling inspection.
Similar tothe integrated model, the production cycles can be considered as a renewalreward process consisting of stochastic and independent identical cycles. Thus,ECT can be obtained based on equation 1, while ET and EC arecomputed as follows:Manford 17 presented differenttypes of inspection policies to monitor processes. From a theoretical point ofview, the inspection times, ti (i=1,2,…,m), can be any arbitraryvalues; however, in practice, the inspection frequency should be designed basedon a simple rule such that it can be applied in practice. “Constant hazardpolicy” is one of the commonly applied rules in practice to determineinspection times when the time of quality shift does not follow the exponentialdistribution5. Based on this rule,the probability of quality shift remains constant in each inspection interval, giventhat at the start of that interval the process still operates in the in – controlstate.
According to this rule, the inspection times are obtained based on thisformula:By assigning anarbitrarily value to the first inspection time point ,t1, the other inspection time points can be determinedusing Equation (24). In this formula, h(t)is the hazard rate function of theprocess failure mechanism, and it is obtained as follows: . It is worth noting that if the process failuremechanism obeys the exponential distribution, Equation (24) leads to the fixedsampling frequency. Another simple rule to determine the inspection time pointsis a constant inspection periods rule that conducts the inspection in theequidistance interval.
6. Optimization of themodels The aim of developmentof the integrated model is to optimally determine the parameters associatedwith the SPC and MM so that ECT of the process is minimized. Thus, inthe integrated model, Equation 1 should be optimized, while EC and ETare computed based on equation 8 and 9, respectively.
Optimization of theintegrated model determines the parameters of the used control chart (i.e., thesample size, the control limit parameter, and the inspection time points) alongwith the maximum duration of the production cycle, tm, and themaximum numbers of inspection periods, m. On the other hand, to optimizethe ECT of the stand-alone maintenance model, Equation 1 should beoptimized, while ET and EC are computed based on Equation 22and 23, respectively. The optimization of ECT in the maintenance modeldetermines the optimal value of tm . It is used a grid searchalgorithm to optimize the models. In the algorithm, the continues variables(i.
e., tm, t1 and k) are discretizedin reasonable ranges. The algorithm are coded in Matlab software and it isavailable upon request from the first author of the paper. 7. An illustrativeexample and sensitivity analysisIn this section, a realexample is presented to clarify the performances of the models. This example isselected from Zhou and Zhu article 4.
It is about amanufacturer producing nonreturnable glass bottles which are designed topackage a carbonated soft drink beverage. The manufacturer used control chart for the process monitoring. When theprocess is in the in-control state, the quality characteristic follows a normaldistribution with the mean and variance of , respectively. In the out-of-control state, the mean ofthe process shifts to , while the variance of the process remains unchanged. Als, indicates the magnitude of the shift, and it is assumed to beconstant. The thickness of the bottles is an important quality characteristic.Suppose that the thickness of the bottle in the in-control state is 10mm and asingle assignable cause leads to a shift in the mean of the process with the magnitudeof For the chart, the probability of type I error and type II error aregiven by:where, v is the shapeparameter, and ? is the scale parameter. For the considered process in thissection, the value of the shape parameter is 2 and the value of the mean of theWeibull distribution , , is 17.
5 hours. The other parameters of the process areillustrated in Table 2. In this table, Cf and Cv are thefixed and variable sampling cost respectively. Thus, WQC for n unitsis Cf+n×Cv.Please insert table 2 near hereThe results of the optimizationof the two models are elaborated as follows. In the integrated model, the valuesof the decision variables are: ETC=130, t1=2.3, k=3.
5, n=9, m=53 andtm=121.9. These results indicate that the process monitoringshould be started at time point 2.
3. The other time points of inspection arecomputed using Equation 24. At each time point of the inspection, a sample withsize 9 is taken from the bottles produced by the process and the thickness ofthe bottles is measured as a critical quality characteristic. The control limit parameter of the chart is 3.5, and themaximum duration of a production cycle is 121.
9. For the maintenancemodel, the results of the optimization is ECT=157.31, tm=28.5.These results indicate that, based on the value of ECT, the integratedmodel has a better performance. Table 3 shows the results of a sensitivityanalysis for some important parameters of the integrated model. Please insert table 3 near hereAs the results of Table 3denote, increasing the value of ? leads to an increase inthe value of k and a decrease in the value of n.
This trend can be justifiedbased on the fact that it is easier for the control chart to release a biggershift in the mean of the quality characteristic. Also, the increase of ? decreases the value of ETC to the limited extent, while theeffect of this change on the other variables is insignificant. Increasing the value of C0yields to an increase in the value of ETC and t1, while the effectof this change on the other variables is little. Finally, changing the mean ofthe process failure mechanism from 17.5 to 25 leads to a decrease in the valueof ECT and an increase in the value of m and tm. The same analysisis conducted about the stand-alone maintenance model.
The results of theanalyses are shown in Table 4.Please insert table 4 near hereAs the results of Table 4show, the integrated model in all the cases leads to less values of ECT incomparison with the maintenance model. Also, in the maintenance model, the lengthof the production cycle, tm , is much less than the correspondingvalue in the integrated model. Since, there is no sampling inspection in themaintenance model, changing the magnitude of the process shift, ?, has no effect on the decision variable of this model.The effects of change in the value of C0 and in the mean of theprocess failure mechanism are similar to the integrated model. 8.
ConclusionIn this paper, aproduction process that has two operational states, i.e., an in-control stateand an out-of-control state, is studied. Two models are developed for theprocess. The first model is an integrated model of maintenance planning andstatistical process control, while the second model is a stand-alone model thatonly considers maintenance planning. In the integrated model, based on theinformation obtained from the control chart, different types of maintenance actionsare possible to be implemented on the process.
The integrated model determinesthe parameters related to the control chart and the maintenance measures sothat the expected cost per time unit is optimized. A real case study ispresented to clarify the performances of the models. The result of the exampleindicates that the integrated model leads to the less expected cost per unittime in comparison with the stand-alone model. Finally, a sensitivity analysis is conducted for the three keyparameters of the models. The proposed integrated model has a general structurebecause a general form is considered for the process failure mechanism.
Moreover, different types of inspection policies can be applied in the model.