A methodology for joint optimization for maintenance planning, process quality and production scheduling

Computers & Industrial Engineering 61 (2011) 1098–1106

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Computers & Industrial Engineering

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A methodology for joint optimization for maintenance planning, process qualityand production scheduling

Divya Pandey a,⇑, Makarand S. Kulkarni a, Prem Vrat b

a Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, Indiab Management Development Institute, Mehrauli Road, Sukhrali, Gurgoan 122 007, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 June 2010Received in revised form 31 May 2011Accepted 29 June 2011Available online 13 July 2011

Keywords:Preventive maintenanceProcess qualityProduction schedulingControl chartIntegration

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.06.023

⇑ Corresponding author. Tel.: +91 26591137; fax: +E-mail addresses: [email protected] (D

tearth.iitd.ernet.in (M.S. Kulkarni), [email protected]

Performance of a manufacturing system depends significantly on the shop floor performance. Tradition-ally, shop floor operational policies concerning maintenance scheduling, quality control and productionscheduling have been considered and optimized independently. However, these three aspects of opera-tions planning do have an interaction effect on each other and hence need to be considered jointly forimproving the system performance. In this paper, a model is developed for joint optimization of thesethree aspects in a manufacturing system. First, a model has been developed for integrating maintenancescheduling and process quality control policy decisions. It provided an optimal preventive maintenanceinterval and control chart parameters that minimize expected cost per unit time. Subsequently, the opti-mal preventive maintenance interval is integrated with the production schedule in order to determinethe optimal batch sequence that will minimize penalty-cost incurred due to schedule delay. An exampleis presented to illustrate the proposed model. It also compares the system performance employing theproposed integrated approach with that obtained by considering maintenance, quality and productionscheduling independently. Substantial economic benefits are seen in the joint optimization.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Production scheduling, maintenance scheduling and processquality is some of the key operational policies, affecting the perfor-mance of any manufacturing system. Despite the possible interac-tion effect between these decisions, not many scientific, model-based approaches are available to optimize them simultaneously.For instance, most production scheduling models, do not considerthe effect of machine unavailability due to failure or maintenanceactivity. Similarly, maintenance planning models seldom considerthe impact of maintenance on due dates to meet customer require-ments. However, maintenance effectiveness cannot be measured ina meaningful way without taking into account whether the mainte-nance function is meeting the production requirements (Paz & Leigh,1994). On the other hand, delaying the maintenance to meet produc-tion requirement may increase the process variability and risk ofmachine failure, which in turn may cause higher rejections or down-time looses. Ollila and Malmipuro (1999) observed that mainte-nance has a major impact on efficiency and quality along-withequipment availability. In a case study carried out in five Finnish in-dustries, they showed that well functioning machinery is a prerequi-site to quality products. They also showed that lack of proper

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91 1126582053.. Pandey), mskulkarni@ne-n (P. Vrat).

maintenance is usually among the three most important causes ofquality deficiencies. Thus these three shop floor level operationalpolicies also have some interaction effect on each other and hencea joint consideration of various policy options pertaining to quality,maintenance, and scheduling provides an important research areafor investigation. Seen in this light, the present paper provides amethodology for joint consideration of these three shop floor poli-cies. It first develops an approach to integrate maintenance planningand process quality policy. Specifically, the aim is to determine opti-mal preventive maintenance interval and control chart parametersthat minimize the expected cost per unit time of joint policy. Subse-quently, an approach is proposed in which the optimal preventivemaintenance interval obtained from above mentioned joint mainte-nance and quality model is integrated with production schedule de-cisions. It provides the optimal batch sequence that will minimizethe penalty-cost associated with schedule delay. Finally, it comparesthe performance of the proposed integrated approach with themethodology that treats these issues independently.

The paper is organized as follows: in Section 2, a brief review ofrelevant literature is presented. In Section 3, the problem state-ment is discussed in detail. Section 4 provides a solution methodol-ogy. It is divided into two parts. First part provides a mathematicalmodel for optimization of integrated maintenance scheduling andquality control chart policy and a numerical example is presentedfor illustration. Second part provides an integrated model of main-tenance scheduling and production scheduling, also illustrated

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Nomenclature

E½CCM �FM1expected cost of corrective maintenance (CM) due tofailure mode I

E[CPM] expected cost of preventive maintenance (PM)E[TCQ]process-failure expected total cost of quality due to process

failureMTCM mean time for CMPR production rateClp cost of lost productionLC labor costCFCPCM fixed cost per CMPFM1 probability of occurrence of failure due to failure mode INf number of failuresMTPM mean time for PMCFCPPM fixed cost per PME[TI] expected in-control periodARL1 average run length during in-control periodT0 expected time spent searching for a false alarmPFM2 probability of occurrence of failure due to failure mode

IITeval evaluation periodARL2M/c average run length during an out-of-control period due

to machine failureARL2E average run length during an out-of-control period due

to external reasons

s mean elapse time from the last sample before theassignable cause to the occurrence of assignable causewhen the maintenance and quality policies are inte-grated

Ts time to sample and chart one itemT1 expected time to determine occurrence of assignable

causeTreset time to perform theresetting of theprocess which moves

to out-of-control condition due to external reasona Type I error probabilityCRej cost of rejection(Rd)M/C probability of non-conforming items produced due to

machine failure mode IIbM/C Type II error probability due to machine failure mode II(Rd)E probability of non-conforming items produced due to

external causebE Type II error probability due to external reasonsCreset cost of resettingE½ðCCMÞFM2

� expected cost corrective maintenance due to failuremode II

[ECPUT]M �Q expected cost per unit time of integrated mainte-nance and quality policy

D. Pandey et al. / Computers & Industrial Engineering 61 (2011) 1098–1106 1099

through an example. Comparison of the result obtained from inte-grated model with those obtained from the independent models isalso given. In the concluding section some possible extensions ofthe proposed approach are given.

2. Literature review

Since the 1980s, research on production scheduling with ma-chine failures or varying machine capacity started appearing in lit-erature. Most of the research has taken (Pinedo & Rammouz, 1988;Adiri, Frostig, & Rinnooy Kan, 1991; Hirayma & Kijima, 1992; Fed-ergruen & Mosheiov, 1997; Leung & Pinedo, 2004) a passive ap-proach toward machine unavailability and focused on how toadjust the production schedule to account for the time when ma-chines are unavailable. It is well known that carrying out preven-tive maintenance on a machine with increasing failure rate cansignificantly reduce the occurrence of machine failures. If opti-mally scheduled, this can also increase the machine availability.Grave and Lee (1999) and Lee and Chen (2000) developed ap-proaches that schedule jobs simultaneously with a single preven-tive maintenance. They assume that each machine is maintainedonly once during the planning horizon. Their approaches for pro-duction scheduling and preventive maintenance scheduling consistof two stages:

(1) Determine the interval during which a machine needs to bemaintained only once to increase its availability.

(2) Within the interval found in (1), simultaneously schedulethe jobs and the single preventive maintenance action.

Cassady and Kutanoglu (2003) compared the optimal values oftotal weighted tardiness under production scheduling integratedwith preventive maintenance planning with that obtained underseparate production scheduling and preventive maintenance plan-ning. They assumed that the uptime of a machine follows a Weibulldistribution; the machine is minimally repaired when it fails; and

the preventive maintenance restores the machine to a state asgood as new. Their results indicate that there is an average of30% reduction in the expected total weighted tardiness when theproduction scheduling and preventive maintenance planning areintegrated. Leng, Ren, and Gao (2006) and Sortrakul and Cassady(2007) extended the work of Cassady and Kutanoglu (2003) andproposed a Chaotic Partial Swarm Optimization (CPSO) heuristicand GA-based heuristics respectively to solve the integrated math-ematical model for single machine production scheduling and PMplanning as a multi-objective optimization problem.

Similarly, an increasing number of practitioners and researchershave recognized that there is a strong relationship between prod-uct quality, process quality and equipment maintenance (Ben-Daya & Duffuaa, 1995), and integration of these may be beneficialto the organization. But research in this field is still quite limited.Rahim (1993) jointly determined the optimal design parameterson an x-bar control chart and preventive maintenance (PM) timefor a production system with an increasing failure rate. Ben-Daya(1999), Ben-Daya and Rahim (2000) and Rahim (1994) investigatedthe integration of x-bar chart and PM, when the deterioration pro-cess during in-control period follows a general probability distri-bution with increasing hazard rate. Cassady, Bowden, Liew, andPohl (2000) studied an x-bar chart in conjunction with an age basedpreventive maintenance policy. Rahim and Ben-Daya (2001) pro-vided an overview of the literature dealing with integrated modelsfor production scheduling, quality control and maintenance policy.Recently, Linderman, McKone-Sweet, and Anderson (2005) devel-oped a generalized analytic model to determine the optimal policyto coordinate Statistical Process Control (SPC) and planned mainte-nance to minimize the expected total cost. Panagiotidou and Tag-aras (2007) have proposed an economic model for theoptimization of preventive maintenance interval for a productionprocess with two quality states.

While some literature is available for integrating maintenancewith scheduling and maintenance with quality, integration of allthe three areas i.e. production scheduling, maintenance and qualitycontrol has started getting attention of the research community only

1100 D. Pandey et al. / Computers & Industrial Engineering 61 (2011) 1098–1106

recently (Rahim & Ben-Daya, 2001). Hence it presents a good oppor-tunity for further research. This paper is an attempt in that direction.

2.1. Gaps from the related research

Following are the gaps observed in the literature on thissubject:

� Most of the integrated models focus on process quality pro-blems and completely ignore the possibility of an equipmentfailure in terms of immediate stoppage of machine or improperfunctioning of the equipment that result in a poor product qual-ity and calls for a maintenance action.� Majority of the integrated models focus on the investigation of

preventive replacement or a perfect PM policy that restores theequipment to an as-good-as-new state. Relatively very fewpapers have appeared that incorporate the aspect of imperfectPM.� Most of the maintenance optimization models available in lit-

erature are, in general, considers a fixed value of the cost of cor-rective maintenance. However, as mentioned earlier, the failureof machine also includes performance degradation in terms ofpoor quality leading to rejection of product being manufacturedby the machine. Thus the cost of corrective maintenance notonly includes down time losses and repair/replacement cost,but may also include cost of rejection. The cost of rejectionmay vary depending upon the type of the production system.If the cost of rejection is very high in a particular productionsystem, then the cost of corrective maintenance is higher com-pared to preventive maintenance, which will reduce the pre-ventive maintenance period.

3. Statement of the problem

Consider a production system consisting of a single machine pro-ducing products of the same type with constant production rate (PR,items per hour) on a continuous basis (three shifts of 7 h each,6 days-a-week). Further, consider a single component operating asa part of machine with time-to-failure following a two parameterWeibull distribution. Let the shape and scale parameters of the dis-tribution be B and g respectively. In this paper, machine failure isconsidered in terms of failure mode (FM). It is assumed that when-ever machine fails it leads to one of the following two consequences.

1. Failure Mode I (FM1): leads to immediate breakdown of themachine.

2. Failure Mode II (FM2): leads to reduction in process quality byshifting the process mean.

Failure Mode I (FM1) is detected immediately as it brings themachine instantly to breakdown state. However, Failure Mode II(FM2) is detected after a time lag through control chart mechanism.It is assumed that whenever the failure is detected, corrective ac-tions are taken to restore the machine back to the operating condi-tions. Thus, (FM1) results in an expected corrective maintenancecost ðE½CCM�FM1

Þ comprising of cost of down time, cost of mainte-nance labor, and fixed cost of repair/restoration. However, FailureMode II (FM2) affects the functionality of the machine and causesthe process to shift, resulting in an increase in the rejection rate, tillit is detected. Thus (FM2) in addition; also incurs the cost of lostquality. Preventive maintenance (PM) is carried out to reduce theunplanned down time cost. However, PM also consumes timeand resources, which could otherwise be used for production. Pre-ventive maintenance optimization is therefore done to strike a bal-ance between cost of failure and cost of preventive maintenance.

Let us suppose the process quality can be evaluated by measur-ing one key quality characteristic of the finished product. Let x de-note the measurement of this characteristic for a given product. Itis assumed that is a normal random variable having mean l and astandard deviation r. The value of l is referred to as the processmean, and the value of r is referred to as the process standarddeviation. When the process is in-control, the process mean is atits target value. The process mean can instantaneously shift, dueto machine failure or due to some external causes ‘E’ like environ-mental affects, operators’ mistake, use of wrong tool, etc. The pro-cess is also restored if an external cause ‘E’ is detected. After a shiftthe process is said to be out-of-control and the new process meanis given by: l = l0 + dr0, where d is some non-zero real number.Usually, the failure which causes this shift is relatively subtle.Therefore, the cause of failure cannot be identified without shut-ting down the process and performing a close inspection of theequipment. The process also has to be restored similarly if anexternal cause ‘E’ is detected.

Since (FM2) and E cannot be directly detected, a control chart isused to monitor the quality characteristic ‘x’. Hence, the time to de-tect (FM2) and E depends on the power of the control chart. Theparameters of the chart are: (h) the time (in hours) between sam-ples, (n) the sample size, and (k) the number of standard deviationsof the sample distribution between the center line of the controlchart and the control limits. The resulting upper and lower controllimits for the �x-chart are given by:

UCL ¼ l0 þ krffiffiffinp ; LCL ¼ l0 � k

rffiffiffinp

There are costs involved in control chart design. These includecost of sampling, cost of false alarm, cost of process shift due tomachine failure or external events, etc. Thus an economic designof control chart is done to obtain the optimal values of controlchart design parameters (n, h, and k).

However, it is clear from the above description that, machinefailure and maintenance may affect the process quality. Thereforepreventive maintenance optimization and economic design of thecontrol chart must be done simultaneously.

The first problem addressed in this paper is to jointly optimizethe preventive maintenance schedule and control chart policy.Specifically, the objective is to obtain optimal values of h, n, kand tPM (preventive maintenance interval) that minimize the totalexpected cost associated with poor quality, inspection/sampling,corrective/preventive maintenance and process downtime.

The second problem is to integrate the optimal preventivemaintenance interval obtained from the joint maintenance andquality model with the production-schedule to determine the opti-mal job-sequence that will minimize the penalty-cost due to sche-dule delay.

4. Methodology for joint optimization of maintenance andquality policy

In order to demonstrate the benefits of combining preventivemaintenance and Statistical Process Control, a cost model has beendeveloped that captures the costs associated with the manufactur-ing process which are affected by quality control policies andmaintenance planning. These costs comprise of cost of poor qual-ity, cost of sampling/inspection, cost of preventive maintenance,cost of downtime and fixed cost of repair/restoration.

A block replacement policy is considered in this paper. Preven-tive maintenance is treated as imperfect. Imperfect maintenancemakes the system better than what it was before maintenancebut not as good as new. In this paper the Duncan’s model (Duncan,1956) of economic design of control chart is modified to capture


the cost of process shift incurred due to external reasons ‘E’ andFM2.

Due to the stochastic nature of the quality characteristic andmachine failures, the actual cost of implementing a specificblock-replacement imperfect preventive maintenance policy and�x chart is a random variable. Therefore, the performance of themanufacturing process is measured by C(h, n, k, tPM), the expectedcost per unit time. These four variables are therefore the decisionvariables. All other parameters (P, r0, l0, dE, dM/C, T0, Treset, TS, MTCM,l0, dE, dM/C, T0, Treset, TS, MTCM, MTPM, B, g) are assumed to be knownand treated as input constants.

4.1. Development of integrated cost model

In this section we develop an integrated cost model for jointoptimization of optimal preventive maintenance interval and de-sign parameters of control chart.

The total expected cost in the model includes:

1. The expected cost for minimal corrective maintenance due toFM1, (E½CCM �FM1

).2. Expected cost per preventive maintenance, (E[CPM]).3. The expected total cost of quality loss due to process failure,

(E[TCQ]process-failure).

The expected total cost per unit time of integrated preventivemaintenance and process quality control chart policy [ECPUT]M�Qis the ratio of the sum of the expected total cost of the processquality control (E[TCQ]process-failure), expected total cost of thepreventive maintenance (E[CPM]) and expected total cost of ma-chine failure E½CCM�FM1

to the evaluation time. Therefore the ex-pected total cost per unit time for the integrated model can bewritten as:

½ECPUT�M�Q ¼E½CCM�FM1

þ E½CPM� þ E½TCQ �process�failure

Tevalð1Þ

Optimal preventive maintenance interval (tPM) and process controlchart design parameters (n, h, k) are obtained by minimizing[ECPUT]M�Q.

Models for each of the ingredient costs in [ECPUT]M�Q model aredeveloped in the following sub-sections.

4.1.1. Expected cost model for corrective maintenance due to FM1

Failure Mode 1 (FM1) is detected immediately, hence it incursonly the down time cost during repairing/restoring the machineplus the labor and material cost. Thus corrective maintenance costdue to FM1 during the given period of time can be expressed as.

E½CCM�FM1¼ fMTCM � ½PR � Clp þ LC� þ CFCPCMg � PFM1 � Nf ð2Þ

where PR = production rate (jobs/h), Clp = cost of lost production(Rs/job), LC = labor cost (Rs/h of preventing machine), MTCM = meantime to corrective maintenance (h) and CFCPCM is the fixed cost of thecorrective maintenance. It includes the cost of material, lubricant,maintenance equipment, etc. For example, if maintenance involvesonly the replacement of the component, then in such cases it maybe taken as equal to the cost of the component replaced.

In the present study, a simulation based approach (using ReliaSoft’s BlockSim (Reliasoft, 2009)) is used for obtaining the expectednumber of corrective maintenance as a function of PM interval andrestoration factors for a given g and B during a given evaluationperiod.

4.1.2. Cost per preventive maintenanceCost per imperfect preventive maintenance action of compo-

nent incurs only the down time cost during repairing/restoring

the machine plus the labor and material cost and it can be ex-pressed as:

E½CPM� ¼ fMTPM � ½PR � Clp þ LC� þ CFCPPMg �Teval

tPMð3Þ

where CFCPPM = fixed cost per preventive maintenance (Rs/prevent-ing component), MTPM = mean time to preventive maintenance(h), and Teval

tPM¼ NPM; the number of preventive maintenances (NPM)

is rounded off to the lower whole value.

4.1.3. Model for expected total cost of quality loss due to processfailure, (E[TCQ]process-failure)4.1.3.1. Process cycle length. The cycle length consists of the in-con-trol time, the out-of-control time and process resetting or machinerestoration time. Assume that the in-control time is a negativeexponential distribution with mean 1/k. The expected in-controltime consists of the mean time to failure and the expected amountof time for investigating false alarms (Lorenzen & Vance, 1986),

E½TI� ¼ 1=kþ T0 �S

ARL1ð4Þ

where T0 is the expected search time for a false alarm and If thesampled statistics is independent, then ARL1 = 1/a, and the symbola is used to represent the probability of a Type I error and givenas: a = 2F( � k) .

The expected number of samples while the process is in-control(S) with a process failure rate of (k) can be calculated as (Lorenzen& Vance, 1986):

S ¼X1i¼0

i Pr½assignable cause occurs between the ith and

ðiþ 1Þst samples�

¼X1i¼0

i½e�khi � e�khðiþ1Þ�

¼ e�kh=ð1� e�khÞ

In this paper we consider machine failure in terms of machineoperating with degraded functionality (increased rejection rate)and the sudden breakdown which ceases the machine operation.The probability of occurrence of machine failures can be capturedfrom past failures data. The process may fail because of machinedegradation or due to some external reasons as mentioned above.Let the failure rate due to machine degradation (FM2) be k2 and dueto external reason ‘E’ be k1. Thus the overall process failure rate kdue to (FM2) and ‘E’ is k = k1 + k2.

Where k2¼Nf�PFM2

Tevaland k1¼ 1

MeanTimebetweenprocessfailure.Where (Nf) is the number of failures during (Teval).The out-of-control time consists of the expected time of the fol-

lowing events:

1. The time between occurrence of an assignable cause and thenext sample.

2. The expected time to trigger an out-of-control signal.3. The expected time to plot and chart a sample.4. The expected time to validate the assignable cause.5. The expected time to reset the process if failure is due to exter-

nal reasons or the expected time to restore the machine if fail-ure is due to (FM2).

This can be mathematically expressed using the followingmathematical form.

Out-of -control time ¼ h� ARL2M=Ck2

kþ ARL2E

k1

k

� �� s

�

þn� Ts þ T1 þ Treset �k1

kþMTCM �

k2

k

� ��ð5Þ


where ARL2 is the average run length when the process has shiftedto an out-of-control state. The expected process cycle length is thesum of the Eqs. (4) and (5) and given as:

E½TCycle� ¼1kþ T0 �

SARL1

þ h� ARL2M=Ck2

kþ ARL2E

k1

k

� ��

� sþ nTs þ T1 þ Treset �k1

kþMTCM �

k2

k

� �ð6Þ

where Treset is the time to perform the resetting of the process whichmoves to out-of-control condition due to external reason and MTCM

is mean time for CM, when process moves out-of-control due to ma-chine degradation and T1 is the expected time to determine occur-rence of assignable cause.

4.1.3.2. Process quality cost. The process quality cost consists ofthree main components: the cost of rejection incurred while oper-ating the process in ‘in-control’ (CI) and ‘out-of-control’ (Co), thecost of sampling, the cost of evaluating the alarms-both false andassignable and the cost of resetting or restoring (through correctivemaintenance) the process.

Let Cf be the cost of false alarm. This includes the cost of search-ing and testing for the cause. Then the expected cost for false alarmis given as:

E½Cf � ¼ Cf � ðS=ARL1Þ � T0 ð7Þ

Let CF be the fixed cost per sample and CV be the variable cost perjob. Thus the expected cost per cycle for sampling is the sum ofthe fixed cost per sample and variable cost per job and is given as:

E½CS� ¼ðCF þ CV � nÞ � 1=kþ T0 � S

ARL1þ h� ðARL2M=c

k2k þ ARL2E

k1k Þ

� �� sþ nTs

h

ð8Þ

The expected cost of non-conforming units when the process is ‘in-control’ is as follows:

E½CI� ¼ ðR0 � PR� CFrejÞ � 1=kþ T0 �S

ARL1

� �ð9Þ

Similarly, the cost of rejections when the process is in out-of- con-trol state due to machine failure is given as:

E½CO�M=C ¼ PR�ðRdÞM=C

1� bM=C� CFrej

!� h� ARL2M=c

k2

kþ ARL2E

k1

k

� ��

�sþ T1 þ n� TsÞ �k2

k

� �ð10Þ

where Ts be the expected time to sample and measure one unit.And, the cost of rejection when the process is in out-of- control

state due to external reason is given as:

E½CO�E ¼ PR� ðRdÞE1� bE

� CFrej

� �� h� ARL2M=c

k2

kþ ARL2E

k1

k

� ��

�sþ T1 þ n� TsÞ �k1

k

� �ð11Þ

If the sampled statistics is independent then, ARL2 = 1/(1 � b)where, b = Pr(in-control signal|process is out-of-control).

b ¼ PfLCL 6 �x 6 UCLjl ¼ l1 ¼ l0 þ drpg ð12Þ

Since �x � Nðl;r2p=nÞ, and the upper and lower control limits are

UCL ¼ l0 þ krp=ffiffiffinp

and LCL ¼ l0 � krp=ffiffiffinp

, we may write Eq. (12)as

b ¼ FUCL� ðl0 þ drpÞ

rp=ffiffiffinp

� �� F

LCL� ðl0 þ drpÞrp=

ffiffiffinp

� �

where F denotes the standard normal cumulative distribution func-tion. This reduces to b ¼ Fðk� d

ffiffiffinpÞ � Fð�k� d

ffiffiffinpÞ. In the present

study it is assumed that the process capability of the in-control pro-cess is 1 (i.e. the upper and lower specification limits would be at±3rp). Thus the proportion of non-conforming units ’Rd’ due to shiftd will be: Rd = 1 � F(3 � d) � F( � 3 � d) and the proportion of non-conforming units ’R0’ when the process is in control is given as:R0 = 1 � F(k) � F( � k) (Montgomery, 2004).

It is assumed that the process is stopped during search and re-pair. Let Creset be the cost for finding and repairing the assignablecause plus the downtime cost as the process is stopped. Thus, ex-pected cost of finding and repairing for a valid alarm for assignablecause due to external failure is given by:

E½Creset� ¼ ½Creset � Treset� � ðk1=kÞ ð13Þ

The expected cost of finding and taking a corrective action for a va-lid alarm due to failure mode FM2 is given by:

E½ðCCMÞFM2� ¼ fðMTCMÞ � ½PR � Clp þ LC� þ CFCPCMg � ðk2=kÞ ð14Þ

Adding Eqs. 7, 8, 9, 10, 11, (13), and (14) gives the expected cost ofprocess failure per cycle as:

E½Cprocess� ¼ E½Cf � þ E½CI� þ E½CO�E þ E½CO�M=C þ E½CS� þ E½Creset�þ E½ðCCMÞFM2

� ð15Þ

Hence the expected process quality control cost for the evaluationperiod is given as:

E½TCQ �process-failure ¼ E½Cprocess� �M ð16Þ

where M is calculated as: TevalE½Tcycle �

4.2. Development of stand-alone models

To evaluate the effectiveness of the integrated model, it isimportant to compare the results obtained from the integratedmodel with stand alone model for maintenance and quality. Inthe following sub-sections these stand alone models are discussed.

4.2.1. Maintenance modelsIn this model, we assume that only planned maintenance is

considered and ignore the probability of quality degradation dueto machine failure. Therefore, the expected cost per correctivemaintenance action can be expressed as:

E½CCM� ¼ fMTCM � ½PR � Clp þ LC� þ CFCPCMg � Nf ð17Þ

The expected cost per preventive maintenance action can be ex-pressed as:

E½CPM� ¼ fMTPM � ½PR � Clp þ LC� þ CFCPPMg �Teval

tPMð18Þ

Thus expected cost per unit time for the planning period is given as:

½CPUT�M ¼E½CCM� þ E½CPM�

Tevalð19Þ

Optimal preventive maintenance interval is determined by mini-mizing the [CPUT]M (Eq. (19)).


4.2.2. Statistical Process Control (SPC) modelThis model has been investigated a lot in the literature. The ex-

pected cycle length and the expected cost of control chart are givenas (Lorenzen & Vance, 1986):

E½TCycle�SPC ¼ 1=kE þ T0 �S0

ARL1þ fh� ðARL2EÞg � sþ nTs þ T1 þ Treset

ð20Þ

E½C�SPC ¼ Cf �S0

ARL1

� �� T0

þðCF þ CV � nÞ � 1

kEþ T0 � S0

ARL1

þ h � ðARL2ÞE � sþ n � Ts

h

þ ða � PR � CRejÞ �1kEþ T0 �

S0

ARL1

� �

þ PR � ðRdÞE1� bE

� CFrej

� �� ðh � ðARL2ÞE � sþ n � TsÞ

þ ðCreset � TresetÞð21Þ

where S0 is expected number of samples when the process is in-con-trol while using SPC in isolation. Therefore the expected total costper unit time for the SPC model is given as:

E½CPUT�SPC ¼E½C�SPC

E½TCycle�SPCð22Þ

Optimal values of control chart variables (n, h, k) are determined byminimizing the E[CPUT]SPC (Eq. (22)).

4.3. Numerical illustration

Consider a single machine whose failure is assumed to follow atwo parameter Weibull distribution with g = 1000 and B = 2.5 asthe characteristic life and shape parameter respectively. Machineconsidered here is expected to operate for three shifts of sevenhours each for 6 days in a week. Time to carryout preventive main-tenance action (MTPM) = 3 time units with restoration factor(RFPM) = 0.6 (it implies 60% restoration of life and sets the age ofthe block to 40% of the age of the block at the time of the mainte-nance action) and time to corrective maintenance (MTCM) = 12 timeunits with restoration factor (RFCM) = 0 (repair is minimal, i.e., theage of a repaired machine is the same as its age when it failed).The manufacturer has used �x control chart to monitor the manu-facturing process producing that product. Assuming that the pro-cess is characterized by an in-control state with process standarddeviation of r = 0.01 and a single assignable cause due to externalfailure is of magnitude dE = 1 and let deviation due to machine fail-ure be dM/C = 0.8, which occurs randomly and results in a shift ofprocess mean from l0 to (l0 + dr). The initial values of relevantparameters are given in Table 1.

ðCPUTÞs ¼Total Penalty cost due to batch delayþ Total raw material inventory carrying cost

Schedule completion timeð23Þ

It is assume that the cost of per piece of camshaft is Rs. 500 andthe profit margin for each piece is Rs. 40. But if the defect is de-tected in the product at the end of the last process or at the cus-tomer end then it will cost more to the manufacture in terms ofconsequential cost because of loss of goodwill, or returning of com-plete product lot to the manufacturer and that may be 10 timesmore than the cost of good quality product. Thus we assume costof rejection as high as Rs. 5000/piece.

The global optimization tool box of Maple 13 has been used tosolve the optimization problem. The optimal values of decisionvariables (n�;h�; k�; t�PM) are obtained by minimizing the expectedtotal cost of system per unit time [ECPUT]M�Q as shown in Eq. (1).Optimal values of the decision variables are as follows: n� = 27,k� = 3.9, h� = 7.6, t�PM ¼ 211 and the corresponding expected totalcost of system per unit time is [ECPUT]M�Q = 215.56 .

For the two stand-alone models, the same values of the relevantparameters and policy variables are assigned and correspondingcost per unit time are obtained, which is [CPUT]M+SPC = 226. Thisproves that the integrated model results in better economic behav-ior then the model in isolation.

4.4. Sensitivity analysis

An important practical issue in the application of the proposedapproach is the estimation of the required model parameters. Sincethe process and cost parameters cannot always be estimated withaccuracy, it is important to know the effect of inaccuracies on thequality of the solution obtained from the proposed approach. Toinvestigate this issue, a systematic sensitivity analysis was con-ducted using some of the cost and process parameters, In Table2, level 1 is the basic level which was used to solve the examplein Section 4.3. Level 2 and 3 represent values of these parametersat +10 and +20% of the basic level respectively.

Range of optimum values of the decision variables and the cor-responding values of the objective function at different levels ofthe model parameters are shown in Table 3. The results show thatthe solution is most sensitive to errors in estimating the magnitudeof the process shift due to external reasons (dE), the cost of rejec-tion and hence more emphasis should be placed on the accurateestimation of dE and CRej .

5. Batch sequencing model

Consider a single machine that processes three batches havingthe processing time, setup time, penalty cost, due dates, and otherproduction parameter as given in Table 4.

Following assumptions are made to solve the problem:

1. A job cannot be preempted by another job.2. There are no failures of machine during the schedule.3. Raw material for all the batches is released at starting of the

schedule.4. The batch processing time is equal to the sum of the processing

times of its jobs and the setup time.

The objective is to obtain the batch sequence that minimizesthe Cost Per Unit Time of the schedule (CPUT)S. (CPUT)S can be cal-culated as:

Penalty cost is incurred only when a batch is delayed beyond itsdue date. Penalty cost for a batch can be calculated as:

Batch penalty cost ¼ ðBatch completion time

� batch due dateÞ:Pi ð24Þ

Since it is assumed here that the raw materials for all the batchesare released at the starting of the schedule, raw material inventory

Table 1Relevant parameters used in the numerical example.

Parameter value dE dM/C Ts h CF Rs Cv Rs/job Cf Rs/h CRej Rs/job CFCPCM Rs/component

1 0.8 20/60 40 10 1200 5000 10,000

Parameter value Creset Rs/h T0 h T1 h TReset h PR Jobs/h Clp Rs/job Labor cost Rs/h of preventing m/c CFCPPM Rs/component

1500 1 1 2 20 40 500 800

Table 2Experimental data set.

Parameter Basic (1) Level 2 (+10%) Level 3 (+20%) [ECPUT]M�Q

Basic Level Level 2 Level 3 Range

dE 1 1.1 1.2 215.56 226 238 215–238dM/C 0.8 .88 .96 215.56 214 213 215–214T0 1 1.1 1.2 215.56 216 216 215–216T1 1 1.1 1.2 215.56 215 215 215–215.56Treset 2 2.2 2.4 215.56 216 217 215–217CRej 5000 5500 6000 215.56 228 241 215–241Cv 10 11 12 215.56 219 222 215–222Cf 1200 1320 1440 215.56 216 216 215–216Clp 40 44 48 215.56 217 219 215–219CFCPCM 10,000 11,000 12,000 215.56 216 217 215–217CFCPPM 800 880 960 215.56 216 216 215–216CF 40 44 48 215.56 216 217 215–217

Table 3Range of optimal values obtained from sensitivity analysis.

Decision variable n h k tPM

Range 19–27 5–8.3 3.82–3.91 186–219


for a batch is carried until it starts getting processed, i.e. for theduration of the processing and setup time of all the previousbatches (if any) and the setup time of the current batch. Hence

ðCPUTÞS�ðM�QÞ ¼Total penalty cost due to batch and maintenance delayþ Total raw material inventory carrying cost

Schedule completion timeð25Þ

the inventory carrying cost is calculated based on the whole batchsize for this period. Secondly, during the processing of a batch,raw material of the batch depletes at a constant rate and thereforethe inventory carrying cost is calculated for this period based on theaverage inventory (half the batch size).

To obtain the optimal production schedule, a complete enumer-ation method is used. In the present problem for the three batches, atotal of 3! batch sequences are possible. These batch sequences areshown in Table 5, where the (CPUT)S for all the six possible sequencesare presented. It is discuss from Table 5 that sequence [B2-B3-B1]gives the minimum Cost Per Unit Time of the production scheduleand hence the same is selected as the optimal sequence. Since thisanalysis ignores the probability of machine failure, this solution istermed as ‘OS’ solution. ‘OS’ stands for ‘‘only scheduling’’.

5.1. Integration of production and maintenance schedule

When the optimal schedule obtained in Section 5 is imple-mented on shop floor, it may get interrupted due to scheduledoptimal preventive maintenance interval obtained in Section 4.1.In order to implement both the policies, it is required to integratethe maintenance interval on the optimal batch sequence. It is as-

sumed that the machine cannot be stopped for PM until all the jobsin a batch are completed.

The objective of combining these two policies is to determine theoptimal production sequence for which the Cost Per Unit Time ofjoint consideration is minimized. However, the problem of integra-tion is complicated by the fact that tardiness values for the jobs arestochastic, since the machine may or may not fail during processingof a job and PM decisions affect the probability of machine failure.

The CPUT for joint consideration of production and maintenanceschedule can be expressed as:

The suffix S � (M � Q) indicates that the preventive maintenanceschedule obtained through the integrated model of maintenanceand process quality control is integrated on the entire feasible pro-duction schedule obtained independently. The total penalty costdue to batch and maintenance delay can be calculated as follows(details can be seen in Pandey, Kulkarni, & Vrat, 2010):

The probability that the machine fails while kth batch is gettingprocessed can be determined using the Weibull probability distri-bution as follows:

uk ¼ Fðp½k� þ �a½k�1� �a½k�1�� Þ ¼ 1

� exp �p½k� þ �a½k�1�

g

� �b

þ�a½k�1�

g

� �b" #

k ¼ 1;2; . . . ;m ð26Þ

uk ¼ 1�uk ð27Þ

The completion time for a job is a discrete random variable that de-pends on: (i) the age of the machine prior to decision making; (ii)the processing time for batches; (iii) the time to complete PM andthe PM decisions; and (iv) the repair time and the probability of ma-chine failure during batches. Let C[k] denote the completion time forthe kth batch. Then:

Table 4Production parameters for the illustrative example.

Batch(I)

Processing time inminutes (II) per job

Batchsize (III)

Setup timein h (IV)

Total processingtime (h) V = (II � III)/60 + IV

Release time(VI)

Due date (h)(VII)

Penalty cost/h/batch(Pi) (in h) (VIII)

Carrying cost/job/h(IX)

1 6 500 3 53.0 0 100 75 1.712 3 500 1 26.0 0 50 50 1.713 2 500 2 18.66 0 40 45 1.71

Table 5(CPUT)S calculation for all possible sequences.

Batch sequence Completion time Tardiness Penalty Cost Inventory Cost (CPUT)S

Batch 1 Batch 2 Batch 3 Batch 1 Batch 2 Batch 3

[B1-B2-B3] 53 79 98 0 29 58 4045 11,407 158[B1-B3-B2] 53 98 72 0 48 32 3808 12,248 164[B2-B1-B3] 79 26 98 0 0 58 2595 9654 125[B2-B3-B1] 98 26 45 0 0 5 210 8742 92[B3-B1-B2] 72 98 19 0 48 0 2383 11,336 140[B3-B2-B1] 98 45 19 0 0 0 0 9583 98

Table 6(CPUT)S�(M�Q) at different locations of PM schedule superimposed in productionschedule.

Batchsequence

Locationof PM

(CPUT)S�(M�Q)

[B1-B2-B3] PM is performed before first batch(in this case it is batch 1 i.e. B1)

142

[B1-B2-B3] PM is performed before second batch(in this case it is batch 2 i.e. B2)

197

[B1-B2-B3] PM is performed before first batch 348[B1-B2-B3] PM is performed after third batch (No PM) 583

Table 7Optimal solution obtained from integration example result.

Batchsequence

Location of PM (CPUT)S�(M�Q)


142


134


133


108


110


130


C ½k� ¼ MTPM �Xm

k¼1

y½k� þXm

k¼1

p½k� þM½k� k ¼ 1;2; . . . ;m ð28Þ

Let Nk = {1, 2, . . . k}, and let NKq denote a subset of Nk containing qelements. Then, M[k] is a discrete random variable having the fol-lowing probability mass function

p½k;q� ¼ PrfM½i� ¼ q�MTCMg ¼XNk;q

Yl2Nk;q

u½k�Y

lRNk;q

u½k�; q

¼ 0;1; . . . k ð29Þ

For all k = 1, 2, . . ., m, let

C ½k� ¼ ðMTTRÞPM �Xm

k¼1

y½k� þXm

k¼1

p½k� þ q�MTCM q ¼ 1;2; . . . ; k

Let H[k] denote the tardiness of the kth batch, k = 1, 2, . . ., m. Notethat H[k] has k + 1 possible values,

h½k;q� ¼maxð0;C½k;q� � d½k�Þ q ¼ 0;1; . . . ; k ð30Þ

Thus,

EðH½k�Þ ¼Xk

q¼0

h½k;q� � p½k;q� ð31Þ

Thus the total penalty cost incurred due batch tardiness is given as

ðTPCÞbatchtardiness ¼Xm

k¼1

P½k�EðH½k�Þ ð32Þ

where d[k] and P[k] are the due date and penalty cost for the kthbatch respectively. Table 6 shows the calculations of (CPUT)S�(M�Q)

for all the four possible locations of PM for the given batch sequence

[B1-B2-B3]. For example, in the sequence of [B1-B2-B3] if PM is per-formed before the first batch (B1), then the value of (CPUT)S�(M�Q) forthe given sequence is 183. This process is repeated for each set ofPM decisions, and the set with the smallest objective function valueis identified as optimal for that job sequence. The optimal decisionfor this sequence is to perform PM only before the second batch(marked bold in Table 6).

Once this analysis has been performed for all the batch se-quences and the integrated solution for the batch sequence andPM decisions with the overall minimum objective function valueare identified as the optimal solution. The results for all the six fea-sible sequences for this example are presented in Table 7. The glo-bal optimal solution out of all obtained solutions is to use the batchsequence [B2-B3-B1] with PM performed before second batch i.e.B3.

Similarly, the objective function value in the case where oneconsiders scheduling only and no PM is obtained for all the six fea-sible batch sequence as shown in Table 8.

5.2. Analysis of results

The objective function value obtained for the case with schedul-ing only and no PM’’ is 563 (computed from Eq. (25), see Table 8)for batch sequence [B3-B2-B1] which is not the same sequence asobtained from the integrated production scheduling and PM modeli.e. [B2-B3-B1]. The minimum objective function value obtained

Table 8Results for the illustrative example when considering scheduling only and no PM.

Batch sequence Location of PM (CPUT)S�(M�Q)

[B1-B2-B3] No preventive maintenance 583[B1-B3-B2] No preventive maintenance 571[B2-B1-B3] No preventive maintenance 585[B2-B3-B1] No preventive maintenance 571[B3-B2-B1] No preventive maintenance 563[B3-B1-B2] No preventive maintenance 568


from integrated solution is 108 (computed from Eq. (25), see Table7) shows a saving of 80%. Thus, the results indicate that integratingproduction scheduling and PM policies is substantially better thansolving the two problems independently.

6. Conclusions and future extensions

This paper proposes a model for integrating PM and processquality control. The model allows joint optimization of quality con-trol charts and preventive maintenance interval (n, h, k, tPM) tominimize the expected total cost per unit time. To examine theeffectiveness of the integrated model, two stand-alone modelsare also developed. Numerical example indicates that the proposedintegrated model performs better than the stand alone models. Asensitivity analysis of the model is presented to study the effectof various model parameters on the behavior of the system. Thismay help the manufacturer to identify the parameters which aremore sensitive from those which are not.

Further, the optimal PM interval obtained from the joint main-tenance and process quality control model is integrated with theproduction schedule. For the example studied, integrating thetwo decision making problems i.e. production scheduling and PMresults in an average improvement of approximately 80%. Depend-ing on the nature of the manufacturing system, the average savingmay be different but still it can be very substantial. The proposedmodel can lead to a number of potential extensions as follows:

� The work presented in this paper is limited to a single machine,however it will be interesting to apply the proposed methodol-ogy to different shop floor environments, like flow-shop, open-shop, job-shop, etc., which contain multiple machines and dif-ferent flow patterns and sequence dependent/independentsetup times.� This study assumes three batches of jobs. However, this can be

extended to more number of batches, which will increase thecomplexity of the problem but take it closer to reality. To solvesuch problems, different meta-heuristics like Genetic Algorithm(GA), Particle Swarm Optimization (PSO), Simulated Annealing(SA), TABU Search, etc. can be used and their performancemay be compared.� Taguchi quality loss function approach could be employed to

quantify loss due to process shift.� As part of future research, the researchers can try using Cumu-

lative Sum (CUSUM) chart instead of X-bar chart for detectingthe small shift in the process and the results can be compared.� As an extension of this model, a more realistic enrichment can

be attempted by either analytical or simulation model in futureextensions.� In this paper minimization of expected cost per unit time is con-

sidered as an objective function. As an extension other objectivefunctions like machine availability, process capability, etc. couldbe used.

Acknowledgement

The authors would like to take the opportunity to thanks theanonymous referees for their constructive suggestions which hasenhanced the readability of the paper.

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A methodology for joint optimization for maintenance planning, process quality and production scheduling