a novel approach for production scheduling of a high...

ORIGINAL ARTICLE

A novel approach for production scheduling of a high pressuredie casting machine subjected to selective maintenanceand a sampling procedure for quality control

Pravin P. Tambe • Makarand S. Kulkarni

Received: 21 May 2013 / Revised: 30 June 2013

� The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and

Maintenance, Lulea University of Technology, Sweden 2013

Abstract Effective maintenance keeps machines in good

working condition, improving the machine availability during

production. It also minimizes the process failure rate due to

component failure, resulting into improved product quality. In

this paper the interrelationship between maintenance, pro-

duction scheduling and quality control is captured using an

integrated approach. A mathematical model comprising of

total cost of selective maintenance, process quality control

using a sampling procedure and production scheduling is

developed for a single machine manufacturing system.

Simultaneous optimization using the proposed integrated

model results into the decision on maintenance actions namely

repair, replace and do-nothing for each component, along with

the values of parameters for the sampling procedure and the

optimal production schedule. A numerical study is presented

to demonstrate the applicability of the proposed model. A

simulated annealing algorithm is used for obtaining the near

optimal solution to the decision parameters. The effectiveness

of the proposed approach is compared with the conventional

approach of decision making for maintenance, quality control

and production scheduling. The results indicate that integrated

approach is better as compared to the conventional approach.

Keywords Production scheduling � Selective

maintenance � Sampling procedure � Integrated

approach � Simulated annealing � Maintenance

optimization

List of symbols

AReq Required system availability

a Age reduction factor for component

b Weibull shape parameter for component

g Weibull scale parameter for component (h)

CC Cost of component (Rs.)

Cf Failure cost for component (Rs.)

CLP Cost of lost production (Rs.)

CLM Labour cost of maintenance (Rs./h)

Csp Cost of sub-components and consumables

(Rs.)

CR Replacement cost for component (Rs.)

Cr Repair cost for component (Rs.)

CRL Cost of loss of residual life (Rs.)

E[Nf]FC2 Expected number of failure of the machine

during the operating period leading to FC2

E(DT) Expected downtime (h)

E[TC]M Expected total cost of selective maintenance

(Rs.)

E[TC]M/Q Expected total cost of maintenance with

quality control(M/Q) decision (Rs.)

ML Mean life (h)

MRL Mean residual life (h)

MTTrA Mean time to repair for component (h)

MTTRA Mean time to replacement for component (h)

MTTCA Mean time to corrective action for component (h)

PR Production rate (units/h)

<i Index of i-th component undergoing

replacement work

ri Index of i-th component undergoing repair

work

Ri Reliability of i-th component

Ri(t/T) Conditional reliability of i-th component

having survived upto time T

P. P. Tambe (&) � M. S. Kulkarni

Department of Mechanical Engineering, Indian Institute of

Technology Delhi, Hauz Khas, New Delhi 110016, India

e-mail: [email protected]

M. S. Kulkarni

e-mail: [email protected]

123

Int J Syst Assur Eng Manag

DOI 10.1007/s13198-013-0183-4

RF Restoration factor for component

TPMS Time between the current maintenance and

next expected opportunity (h)

TAvl Time available to carry out maintenance work (h)

TLM Time elapsed between the last maintenance

and opportunity (h)

TCf Total cost of random failures (Rs.)

TCR Total cost of replacement (Rs.)

TCr Total cost of repair (Rs.)

vi Effective age of i-th component at the end of

any period (h)

(vi)o Effective age of i-th component at the

opportunity (h)

(v’i)o Effective age of i-th component after

maintenance at the opportunity (h)

For process quality control

as Type 1 error of sampling procedure

bs Type 2 error of sampling procedure

s Expected time of occurrence of

assignable cause (h)

ARLIn Average run length in in-control state

ARLOut Average run length in out-of-control state

ATS Average time to signal

Cs Acceptance number

Cins Cost of inspection (Rs.)

Cac Cost of investigating the assignable cause

(Rs./h)

CF Cost of investigating the false alarm (Rs./h)

CRej Cost of rejection per piece (Rs.)

CRes Cost of repairing/restoring the process

(Rs.)

CRew Cost of rework (Rs. per unit)

E[Csampling] Expected total cost of sampling per cycle

(Rs.)

E[CFalse Alarm] Expected total cost of false alarms per

cycle (Rs.)

E[CRejections] Expected total cost of rejections (Rs.)

E[CACD] Expected total cost of assignable cause

detection (Rs.)

E[CRestore] Expected total cost to restore the process

(Rs.)

E[CPQC]cycle Expected cost of process quality control

per cycle (Rs.)

E[CRework] Expected total cost of rework. (Rs.)

E[N]cycles Expected number of cycles

E[T]cycle Expected cycle length of process quality

control

E[T]false Expected total time for investigation of

false alarm (h)

E[TC]PQC Expected total cost of process quality

control (Rs.)

Hs Time between samples (h)

Ns Sample Size

P1 Average proportion of defectives during

in-control

P2 Average proportion of defectives during

out-of-control

Sin Expected number of samples in in-control

state

TF Time required to investigate false alarm

(h)

TS Time required for sampling (h)

T1 Expected time to search the assignable

cause (h)

T2 Expected time to restore the process (h)

k Overall process failure rate

kE Process failure rate due to external causes

kM Process failure rate due to machine

component failure

For production scheduling

bk k-th batch in production schedule

Ch Inventory holding cost per item per unit

time (Rs./item/h)

CT Completion time of a batch (h)

DD Due date for a batch (h)

E[TC]PS Expected total schedule penalty cost (Rs.)

E[TC]Integrated Expected total cost of integrated model

(Rs.)

F(bk)i Probability of failure of i-th component

during k-th batch processing

IHC Inventory holding cost (Rs.)

LT Lateness of a batch (h)

LTF Lateness of a batch due to component

failure (h)

P Processing time of a batch (h)

ST Start time of the batch

Tk Tardiness penalty of k-th batch

Td Delay time due to component failure (h)

W Penalty cost for the batch (Rs./h)

1 Introduction

Over the last few years, competition for business excel-

lence in the global market is increasing very fast and

companies are forced to manage their resources efficiently

to meet several customer demands like quick response,

high product quality, low manufacturing costs, timely

deliveries and better customer service. In the current

competitive industrial environment, the costs of operations

and maintenance, as well as compliance with respect to

production performance requirements, will be a decisive

factor for success (Gao et al. 2010). This can be achieved

through effective utilization of the shop floor resources and


123

efficient collaboration between the shop floor level deci-

sion related to maintenance, production scheduling and

quality control. However, in many situations, possibility of

machine breakdown and machine unavailability is not

considered while planning the production schedules. Sud-

den machine failure during production will affect the

completion time of the jobs, which may increase the

schedule penalty cost and ultimately the production cost.

Hence, to achieve the production volume capacity, main-

tenance becomes an important improvement tool (Gustaf-

son et al. 2013). It has been observed by researchers (Ollila

and Malmipuro 1999) that maintenance is usually not

integrated with quality planning as a practice. Similar

observations have been made by the authors through a

survey conducted in the Indian automotive sector.

The purpose of this paper is to capture interdependency

between maintenance, production scheduling and quality

control by developing an integrated model for the joint

analysis and optimization of maintenance (M), process

quality control (Q) and production scheduling (S) decision

for a multi component, multiple product manufacturing

system. A cost model for selective maintenance, produc-

tion scheduling and process quality control using a sam-

pling based procedure, is developed for a single machine.

The availability of the machine for maintenance is con-

sidered as an opportunity, which may be either the planned

preventive maintenance (PM) or an unplanned opportunity

due to sudden breakdown or any other reason. For a multi

component system, maintenance of all the system compo-

nents might not be possible during an opportunity because

of limited available time. Therefore, the decision maker has

to select from the possible maintenance actions for each of

the components, so as to complete the maintenance within

the allowable time and meet the availability requirement

till the next expected opportunity. The critical to quality

(CTQ) characteristics for the product under consideration

are attribute type; hence, a sampling procedure based on

the economic design principle is used for process quality

control. In this paper, it is assumed that, whenever a pro-

cess shift occurs, all the lots from the point of occurrence

till its detection are rejected. The rejected lots due to pro-

cess shift need to be manufactured again, which results into

delays in the completion of batches, which may further

increase the schedule penalty cost. Both, earliness and

tardiness penalties are considered for the batches. The

model helps in arriving at optimal maintenance decision by

recommending one of the three maintenance actions

namely, repair, replace or do-nothing for each of the sys-

tem components along with the optimal values of param-

eters for process quality control and the optimal production

sequence. The proposed integrated approach is denoted as

S 9 M/Q which represents full integration, where the

effect of the maintenance decision on quality and the

production schedule will be evaluated to arrive at optimal

values of S, M and Q decisions. Also, a comparison of the

integrated approach with the conventional industrial prac-

tice of independent maintenance, quality control and pro-

duction scheduling decisions is presented.

The rest of the paper is organized as follows:

Section 2 presents a brief literature review on the inte-

grated approaches for production scheduling, maintenance

and quality control. The research gaps based on the

observations from the literature are also given in this sec-

tion. In Sect. 3, the system description for developing the

proposed integrated model is given and Sect. 4 presents the

integration approach. The mathematical models for the

integrated approach are presented in Sects. 5 and 6. In Sect.

7, the solution approach for the proposed integrated model

is presented. Details of the case study for demonstrating the

applicability of the integrated model are given in Sect. 8

and Sect. 9 presents the results and analysis. Finally, the

observations and conclusion are presented in Sect. 10.

2 Literature review

Maintenance has an essential role in ensuring that engineering

systems perform at desired levels of reliability, availability

and safety in a cost effective manner and in keeping with the

corporate demands (Verma et al. 2010). Higher equipment

efficiency results in improved productivity and profitability,

which relies heavily on a reliable maintenance planning.

Realizing the impact of maintenance on the equipment reli-

ability and subsequently the manufacturing productivity,

maintenance planning studies have been combined with reli-

ability optimization. Marais and Saleh (2009) observed that

while most of the maintenance models deal with the cost of

maintenance (as an objective function or a constraint), only a

handful address the notion of value of maintenance, and sel-

dom in an analytical or quantitative way. They proposed that

maintenance has intrinsic value and argued that most of the

existing cost-centric models ignore value of maintenance,

which is one of its important dimensions. They developed a

framework for capturing and quantifying the value of main-

tenance activities by considering systems that deteriorate

stochastically and exhibit multi-state failures. They assumed

that the system under consideration provides a flow of service

per unit time, which has a price for which a discounted cash

flow is calculated resulting in a present value (PV). Tam et al.

(2007) considered maintenance scheduling of a multi-com-

ponent system that optimizes both cost and reliability simul-

taneously. The model is based on the concept of imperfect

maintenance which includes factors such as, ageing due to the

operation rate of the system, maintenance downtime and lead

time for spare parts. The model recommends maintenance

decision for each component such that the PV of the cost for


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the entire planning period is minimized with a constraint on

reliability. Zhang and Nakamura (2005) presented a reliabil-

ity-based optimal maintenance scheduling method based on

the ordering list of element maintenance effects. In this

approach, an ordering list of element maintenance effects with

various maintenance-interval types is constructed and then by

means of the ordering list, reliability-based optimal mainte-

nance scheduling is carried out that reduces the cost. Hoseinie

et al. (2011) presented a reliability based model for PM

scheduling where the model results into PM intervals at dif-

ferent reliability threshold values. Hamadani and Khorshidi

(2013) developed a mathematical model for the system reli-

ability optimization problems during functioning period. The

proposed model considers the time value of money in the

evaluation of system cost and generated income in calculation

of system’s availability which will help the decision makers to

explore more strategies in different periods. A genetic algo-

rithm is used for the optimization problem.

The combined approach of maintenance and production

scheduling has attracted the attention of researcher’s since

the 1990s. In the literature related to joint consideration of

maintenance and production scheduling, mostly two

approaches are followed. In one approach, the maintenance

interval is fixed and the jobs are scheduled accordingly

(Sanlaville and Schmidt 1998; Ji et al. 2007; Kacem et al.

2008) and the approach is termed as ‘‘scheduling with

availability constraint’’. Among the early attempts, Qi et al.

(1999) studied a single machine scheduling problem with

periodic maintenance planning. They considered several

intervals as decision variables in which the machine is

unavailable in order to minimize the total completion time.

Wang et al. (2005) studied the problem of scheduling ‘n’ jobs

on a single machine with availability constraints to minimize

the total weighted job completion time and a single avail-

ability constraint. In the second approach, the maintenance

interval is determined along with the production schedule

which means, the maintenance interval is flexible (Lee and

Yu 2007; Low et al. 2010). The approach in the second

category may be termed as ‘‘scheduling with flexible con-

straint’’. Pan et al. (2010) studied an integrated scheduling

model by incorporating both production scheduling and PM

planning for a single-machine. A variable maintenance time

subjected to machine degradation is considered with an

objective to minimize the maximum weighted tardiness.

Among other approaches, the stochastic behavior of the

machine breakdown has been used in the integrated approach

of production scheduling and maintenance (Cassady and

Kutanoglu 2005; Yulan et al. 2008). A study by Sortrakul and

Cassady (2007) provided a genetic algorithm approach to

improve the solution procedure to solve larger instances of

the problem developed in Cassady and Kutanoglu (2003). In

another study, Mosheiov and Sarig (2009) considered the

problem of scheduling a maintenance activity in a single

machine with the objective of minimizing the total weighted

completion time. Most of the integrated approach problems

are proved to be NP-hard (Lee and Liman 1992) and the

solution methodologies include approaches like the branch

and bound algorithm, (Batun and Azizoglu 2009; Wang and

Liu 2013) and metaheuristic approaches like the genetic

algorithm (Sortrakul and Cassady 2007; Yulan et al. 2008;

Wong et al. 2013). Also, the development of integrated

approaches for multi component systems considering

opportunistic maintenance has started gaining attention of

the research community. Zhou et al. (2012) developed a

dynamic opportunistic PM model for a multi-component

system considering changes in job shop schedule. Job com-

pletion on a machine is considered as a PM opportunity for

the components of the system.

Proper maintenance of production equipments prevents

system failures and thus, improves the quality of the

products being manufactured on the machine. This close

relationship between maintenance and quality has been

considered by many researchers and resulted in the

development of the integrated models (Rahim and Banerjee

1993; Ben-Daya and Duffuaa 1995). Statistical Process

Control (SPC) using control charts is popularly used for

process quality control and the models of PM combined

with SPC have been developed (Cassady et al. 2000; Ben-

Daya and Rahim 2000; Lee and Rahim 2001). Linderman

et al. (2005) developed an integrated model of SPC and

maintenance to minimize the total costs associated with

quality, maintenance and inspection. The SPC is used to

monitor the condition of the production equipment and

provide a signal of its deterioration. The model is based on

perfect maintenance and three scenarios where mainte-

nance brings the process to an in-control state while the

process is out of control or retains it in an in-control state.

Zhou and Zhu (2008) extended their model to include four

scenarios by adding a scenario which incorporates false

alarms of the control chart. Mehrafrooz and Noorossana

(2011) extended the work of Linderman et al. (2005) and

included two more scenarios. These scenarios are equip-

ment failure in an in-control and out-of-control conditions.

Panagiotidou and Tagaras (2012) developed an integrated

SPC and PM model for three-state processes; two opera-

tional states and one non-operational failure state. Also, an

approximate Markovian model which takes both, quality

shifts and failures into account is presented. The SPC–PM

model and the Markovian model showed significant

improvement over isolated SPC and PM. Ho and Quinino

(2012) proposed an integrated model for on-line process

control and corrective maintenance using a variable sam-

pling interval for the process control. They used a Markov

chain approach to describe the system of inspection and the

level of corrective adjustment was based on the location of

the measured quality characteristics on the control chart.


123

In developing the integrated models, the process failure

mechanism is assumed to follow a specific distribution. Most

common are exponential distribution (Tagaras 1988;

Panagiotidou and Nenes 2009) and Weibull distribution

(Linderman et al. 2005; Zhou and Zhu 2008). The models of

Chiu and Huang (1996), Ben-Daya (1999), Ben-Daya and

Rahim (2000) and Panagiotidou and Tagaras (2010), con-

sider a general probability function with an increasing hazard

rate. Most of the models on integrating maintenance with

process quality depend on the control charts to declare the

process to be out of control and then initiate maintenance.

Taguchi loss function is also considered in the integrated

model (Pandey et al. 2012; Chen et al. 2011; Ben-Daya and

Duffuaa 2003). These approaches resulted in optimal values

of control chart decision variables like sample size, sample

frequency, control limit coefficient along with PM interval.

The condition of a machine affects both, the production

schedule as well as the quality and integrated approaches

have been developed considering the two functions. How-

ever, the integrated approach considering the three aspects in

a single model is not observed in the literature (Hadidi et al.

2012). To the best of our knowledge, Pandey et al. (2010,

2011) have first hypothesized the interaction effect between

the three shop floor level operational policies and developed

a framework of their joint consideration. In their approach,

first the integrated model of PM and quality control policy is

optimized to obtain the PM interval which is then superim-

posed on the production schedule. The objective of the

integrated model is to obtain the optimal sequence of batches

that minimizes the schedule penalty cost. However, their

work was limited to a single machine scheduling, a block

replacement PM and a x-control chart policy for developing

the integrated approach.

2.1 Research gaps

Two important research gaps related to integrated approa-

ches are observed in the literature

1. It is observed that, most of the models on integrating

production scheduling and maintenance either assume

a fixed maintenance interval which is an independently

optimized interval or consider as a decision variable

(flexible) for optimization. However, in many real life

situations, PMs are likely to be skipped and machines

are run at their full capacity in order to meet customer

demands on time. Approaches for such situations

are available in literature. However, the combined

approach of selective maintenance with production

scheduling is not observed in the literature.

2. Researchers have enriched the literature on integrated

planning; but the approaches are mostly limited to two

functions. Even though, a machine has a direct impact

on production as well as quality, very few approaches

exists in the literature that simultaneously consider the

three functions; maintenance, quality and production

scheduling, in a single integrated model.

This paper is an attempt to address the above mentioned

gaps identified from the literature.

3 System description

The machine under consideration operates on a continuous

basis in three shifts and a set of jobs in batches, (b1, b2,…,

bm) are processed on the machine. It is assumed that all the

batches to be processed on the machine are available at the

start of the production schedule. The machine will operate

for TPMS duration, during which the batches needs to be

scheduled such that the total penalty cost should be mini-

mum. The system considered in this paper is shown in

Fig. 1.

Further, the system is having ‘n’ components, where

each component is subjected to degradation due to con-

tinuous operation over a period of time and has an

increasing failure rate (IFR). The time to failure distribu-

tion of each component is assumed to follow a two

parameter Weibull distribution. The machine is available

for maintenance after a time of TLM from the last main-

tenance action with each component in the system is hav-

ing a certain age which depends on the last failure time.

The next expected planned opportunity duration TPMS is

known, which is the operating period for which the main-

tenance decision needs to be optimized. However, not all

the system components can be/need to be considered for

possible maintenance because of the limited resources and

hence, the decision is selective. Only few components can

be repaired or replaced during the maintenance and for few

components, maintenance has to be skipped (do nothing)

and to be considered during next opportunities. However,

the maintenance actions should meet the target availability

requirement in the next operating period and to be com-

pleted within the available time for maintenance. Both, the

time till next expected opportunity and the available time

are the user inputs, which may be decided mutually by

the production and the maintenance function. Even though

the maintenance actions will improve the condition of the

system, random failures may occur. In this paper, we have

considered two failure consequences associated with the

component failures. The two failure consequences are:

1) Failure will lead to conditions where, the detection is

immediate and the machine needs to be stopped. This

failure consequence is termed as FC1.

2) The failure is in terms of a degraded state where

the machine will run, but lead to deterioration of the


123

product quality being manufactured on the machine.

This failure consequence is termed as FC2.

The failure consequence FC1 is immediately detectable

as it brings the machine under breakdown state; however,

the detection of FC2 is not immediate and occurs after a

time lag using a quality control scheme. The magnitude of

the time lag depends on the sensitivity of the quality con-

trol scheme. With the occurrence of both the consequences,

machine will stop; hence, the effect of both will result into

the delay in completion time of the batches, which may

increase the penalty cost of the schedule.

The process is assumed to operate in two states; in-control

and out of control. During the ‘in-control’ state, the average

proportion defective units produced are P1. However, with

the occurrence of FC2, the process enters in an ‘out-of-

control’ state, where the proportion defectives will increase

to P2. It is assumed that the process shift occurs instanta-

neously in time and that the system continues to produce with

proportion defectives P2, until a repair action is taken. The

process quality is monitored using a sampling procedure.

The quality state of the process is inferred based on the

sample statistics falling in the in-control or out-of-control

state; and accordingly the decision to continue or stop the

process is taken. The sampling procedure will have an as

error (type 1 error) and a bs error (type 2 error). These errors

will depend on the sampling parameters and the values of P1

and P2 respectively. The procedure considered in this paper

is to reject all the units produced between the point of

detection of a valid assignable cause and the expected time of

its occurrence. This time span corresponds to the average

time to signal (ATS) for the sampling procedure.

Thus, it is clear from the above discussion that the

effectiveness of the maintenance decision will affect the

decision on quality as well as the production schedule.

Hence, the objective of this paper is to develop an inte-

grated model that will help in minimizing the expected

total cost of maintenance decision, schedule delays and

poor quality.

In the next section, the approach for integrating the three

functions is presented.

4 Approach for integration

The proposed integrated approach is shown in Fig. 2,

which is represented as S 9 M/Q. During optimization, for

each maintenance decision (M1, M2 …, Mr), the effect of

components failure leading to FC2 on the product quality is

evaluated. The effect in terms of the increase in rejection

cost and thus the overall process quality cost is considered

for each quality control decision i.e. the parameters of the

sampling procedure over the range selected for optimiza-

tion. The combined maintenance decision with quality

control is regarded as the M/Q decision. The M/Q decision

is then superimposed on all the possible schedules (S1,

S2,…,SN) generated by the scheduling algorithm. When the

maintenance and quality decision is implemented, the

effect of component’s random failure will be in terms of

downtime and for the components leading to FC2; it will

result into rejections and additional quantity to be manu-

factured. This will increase the completion time of the

batches. The associated schedule penalty cost is calculated

for each of the generated schedules. The optimal mainte-

nance decision (M*), quality control parameters (Q*) and

the optimal schedule (S*) will be based on the optimization

of total cost for all such decisions. The S 9 M/Q, in this

way, is the full integration of the maintenance decision

with quality and production schedule.

The following section present the details of the models

and the methodology used in this paper.

5 Development of integrated model

In this section, first the individual models for selec-

tive maintenance, process quality control procedure are

b1 b2 bm

12

n

TPMSTLM

FailureRandom

Δ −× -

ComponentSystem

Last maintenance

Current maintenance opportunity

Next expected maintenance opportunity

Productionschedule

Fig. 1 Problem structure of the

integrated approach


123

developed. Then, an integrated model of maintenance

decision with quality (M/Q model) is developed by jointly

considering the total cost of process quality control plan

and the total maintenance cost. In the following section, the

total cost model of selective maintenance is developed.

5.1 Model for expected total cost of selective

maintenance

In this section, we develop an expected total cost model for

selective maintenance. Various aspects associated with the

model and different cost components of the total cost are

discussed in the following subsections.

5.1.1 Maintenance actions at the opportunity

For each opportunity, the model considers one of the fol-

lowing three types of maintenance actions:

5.1.1.1 Repair action In this category, during an opportu-

nity, repair work is carried out for a component. The main-

tenance action improves the condition of the component with

an improvement factor and effectively its age is reduced. The

degree of restoration can be defined using a restoration factor

(RF). Kijima et al. (1988) first used the concept of RF for

imperfect repair to suit the real life situations where the failed

unit could be repaired or replaced. The RF is usually between 0

and 1 for repair action. In case of subsystems with a large

number of components, repair usually results in replacing only

a few of these. For such situations, it may be reasonable to

assume minimal repair at subsystem level i.e. RF = 0.

The effective age of a component at any instance or an

opportunity is the sum of effective age of the component

after the last maintenance and the operating time elapsed

thereafter. This can be expressed as:

ðviÞO ¼ vi þ TLM ð1Þ

In present work, the type-II RF is assumed for repair

action. According to this, repairs can only fix the wear out

and damage incurred during the last period of operation.

Thus, the effective age of a component after maintenance

at an opportunity, will be given by,

ðv0iÞO ¼ vi þ aiTLM ð2Þ

Where, a is the age reduction factor and is given by,

a ¼ 1� RF ð0� a� 1Þ ð3Þ

If a component is replaced during a maintenance activity by

a new one, then the component becomes ‘‘as good as new’’ and

the effective age of the component after maintenance becomes

zero i.e. ðv0iÞO ¼ 0:The corresponding maintenance activity

SN

M1

bmb3 b1

Mr

M2 bmb1 b5

bmb2 b4

bmb6 b3

S1

S2

S3

M*/Q*bmb2 b4 S

*

Q1

Q2

Q3

Qg

Fig. 2 Integrated approach

S 9 M/Q


123

is known as Perfect Repair. On the other hand if, RF = 0 or

a = 1, the maintenance has no effect on the component age

and the component remains in the ‘‘as bad as old’’ state and the

corresponding maintenance activity is known as minimal

repair. However, many maintenance actions will lead to a

component state between these extremes.

5.1.1.2 Replacement action In this category, during an

opportunity, a component is replaced with a new one and

the component starts its working with an effective age zero

i.e. the RF is ‘1’. When a component is replaced at an

opportunity, the effect of loss of residual life is considered.

Residual life is the remaining lifetime of an item which has

survived up to certain duration of time (t0).Mean residual life (MRL) is the expected additional

lifetime given that a component has survived until time t0.The mean residual life of any component at a given time

is obtained from, (Ebeling 2010).

MRLiðt0Þ ¼1

Riðt0Þ

Z1

ðt0Þ

RiðtÞdt ð4Þ

where, t0 = (vi)O

5.1.1.3 Do-nothing In this category, no maintenance

actions are taken on components and the components are

left as they are, and to be considered during the next

opportunity. For a given component, this could be because

of the maintenance time constraint or it may be more cost

effective if the maintenance action is postponed to a future

opportunity.

In the next section, the cost models for these mainte-

nance actions are developed.

5.2 Total replacement cost

When a component is replaced at an opportunity, the effect

of loss of residual life is considered. The cost of residual

life (CRL) will be proportional to mean life cost of an item.

If we assume that component cost is uniformly distributed

over the lifetime of the component, the cost of loss of

residual life will be given by,

CRLi ¼CCi

MLi

�MRLi ð5Þ

where, ML is the mean life or mean time to failure (MTTF)

of an item and is obtained from,

MLi ¼Z1

0

RiðtÞ dt ð6Þ

The cost of MRL will be incurred only if; the component

is replaced preventively at an opportunity.

The cost of replacement at an opportunity will be given

by,

ðCRÞi ¼ ½MTTRA i � ðPR� CLP þ CLMÞ þ CC i þ CRL i�ð7Þ

Therefore, the total cost of replacement at an opportunity

considering all the candidate components is,

TCR ¼Xn

i¼1

<i � CRð Þi� �

ð8Þ

<i ¼1; if i-th component is replaced:

0; otherwise:

(

5.2.1 Total repair cost

If a component is repaired at an opportunity, the cost of

repair for the i-th component will be given by:

Crð Þi¼ MTTrA i � PR� CLP þ CLMð Þ þ Csp

� �i

� �ð9Þ

where, (Csp)i is the cost of consumables during the repair of

i-th component.

Therefore, the expected total cost of repair (TCr) at an

opportunity considering all the candidate components is

given by,

TCr ¼Xn

i¼1

ri � ðCrÞi� �

ð10Þ

where ri ¼1 ; if i-th component is repaired

0 ; otherwise

(

5.2.2 Total cost of failure

The cost of failure is the future cost consequence of

maintenance actions performed at an opportunity i.e. the

cost consequence of likely failures till the next scheduled

maintenance. The future corrective action for component

can be repair or replacement depending on the type of

component and the available maintenance time. The cor-

responding corrective action time is MTTCA.

TCf ¼Xn

i¼1

Cf

� �i� 1� Ri t=Tð Þ½ � ð11Þ

where, T is the time up to which the component has

survived, which in the present case is (v0i)O and (Cf)i is the

cost of failure of the i-th component which is given by,

Cf

� �i¼ MTTCAi � PR� CLP � CLMð Þ½þ<i � CCi þ ri � Csp

� �i

� ð12Þ

where,

MTTCAi ¼MTTRAi; if i-th component is replaced:

MTTrAi; if i-th component is repaired:

(


123

5.2.3 Total maintenance cost

The total cost incurred due to a maintenance decision is the

sum of the maintenance and the total failure cost, which is

given as:

E TC½ �M¼Xn

i¼1

<i � ðCRÞi� �

þXn

i¼1

ri � ðCrÞi

þXn

i¼1

Cf

� �i� 1� Ri t=Tð Þ½ �

( )ð13Þ

In the next section, the model for total cost of process

quality control plan is developed.

5.2.4 Model for total cost of process quality

In any process quality control procedure, the state of the

process, whether in-control or out-of-control is decided by

taking a sample from the process output at predefined

intervals. A quality cycle is defined as the time between

start of successive in-control periods (Lorenzen and Vance

1986). In this paper, the process is assumed to start in an in-

control state and produce products with proportion defec-

tives equal to P1. A sample of size Ns is taken from the

process with an interval of Hs hours. If the number of

defectives in the sample is less than or equal to a predefined

number Cs, the process is allowed to continue. If the

number of defectives is greater than Cs, it is assumed that

an assignable cause has occurred. The process is stopped

and a search for the cause is initiated. In this paper, we

consider two sources of assignable causes responsible for

shifting the process to an out-of-control state. It may be

either due to the failure of some of the machine compo-

nents or due to variation in process parameters, raw

material, operator faults, etc., which are considered as

external causes. Occurrence of these assignable causes will

increase the expected proportion of defective units to P2,

where P2 is assumed to be known and is greater than P1. In

actual practice, the shop floor data can help in arriving at

the values of P1 and P2. In case an assignable cause is not

detected, it is an indication of a false alarm and the process

resumes. If an assignable cause (either an external cause or

component failure) is detected, a corrective action is ini-

tiated and the process is restored to its in-control state. The

process cycle as discussed above is schematically shown in

Fig. 3. During the operation, the length of the in-control

period of the process depends upon the occurrence of

assignable causes. In this paper, the in control period is

assumed to have a negative exponential distribution with

mean 1/k. This assumption is based on a simulation study

conducted, where failures of the components affecting

product quality were simulated using Monte Carlo simu-

lation technique. The time-to-failure distribution parame-

ters were used for the generation of failure times. The

failure times are generated by considering the current age

of the component and then using the conditional reliability

function to generate the next failure time. The current age

is derived by using the virtual age after the previous

maintenance action and adding to it the time elapsed till the

current event/opportunity. The virtual age is calculated

using the Kijima model (Kijima et al. 1988). The compo-

nents for which a maintenance action is taken have a new

age, which is zero for replacement or some reduced value

(compared to the current age at the opportunity) according

to a RF as suggested by the Kijima model. The system

failure times were recorded for multiple simulations and a

distribution was fitted. It was observed that the exponential

distribution satisfactorily modeled the time between system

failures and hence the assumption was verified. The results

are also intuitive keeping in mind that the simulation per-

iod is small (the time duration till next planned opportu-

nity) compared to the characteristic life of the components.

The total cost of process quality i.e. the cost due to the

sampling procedure is calculated considering the economic

design principle (Lorenzen and Vance 1986).

In the next subsections, the calculations for expected

total cost of process sampling plan are presented.

5.2.5 Calculation of expected process cycle length

The process cycle is as shown in Fig. 3. The cycle consists

of the following individual times:

i) Time until the assignable cause occurs (in-control

period)

ii) Time until the sampling inspection indicates the

process to be out of control

iii) Time to discover and analyze the assignable cause (T1)

iv) Time to restore the process (T2)

The length of in-control period depends on the process

failure rate. In this case, the process failure rate is governed

by the external causes or failure of those components that

lead to FC2. Let the process failure rate due to external

causes be ‘kE’ and the process failure rate due to machine

component failure be ‘kM’. The process failure rate over

the next evaluation period, i.e. till the next planned

opportunity TPMS, can be determined as,

k ¼ kE þ kM ð14Þ

The process failure rate due to machine component can

be calculated as,


123

kM ¼TPMS

E½Nf �FC2

� ��1

ð15Þ

where, E[Nf]FC2 is the expected number of failures of the

machine during the operating period leading to FC2.

In the present study, the expected number of failures

E[Nf] and the expected downtime, E(DT) over a given

period are determined through a failure simulation

approach using the time to failure distribution of the

components (Yanez et al. 2002) and the corresponding

corrective action time.

Let the assignable cause occur between j-th and (j ? 1)-th

sample. The expected time of occurrence ‘s’, within this

interval is calculated as (Ben-Daya and Duffuaa 2003).

s ffi HS

2� k� H2

S

12ð16Þ

In this paper, the probability distribution of the number

of defective units (d) found in a sample, is determined

using a binomial distribution.

The probability that the number of defectives in a

sample will be greater than the acceptance number, when

the process is in-control (false alarm), is the type 1 error

(as) and is given by,

as ¼ 1�XCS

d¼0

NS!

d! � NS � dð Þ !� Pd1 � 1� P1ð ÞNS�d ð17Þ

where, P1 = average proportion defectives being produced

during the in-control state.

Let ARLIn be the average number of samples taken

before the process control plan gives a false alarm. It can

be determined as follows (Montgomery 2002).

ARLIn ¼ 1=as ð18Þ

If TF is the expected time required to investigate a

false alarm, the expected total time spent in investigation

due to false alarms is given by, (Lorenzen and Vance

1986)

E½T �false ¼ TF �Sin

ARLIn

ð19Þ

where, Sin is the expected number of samples taken while

the process is in control which is given by,

Sin ¼e�kHS

1� e�kHSð20Þ

The probability of type 2 error (probability of not

detecting a process shift), bS is given by,

bS ¼XCS

d¼0

Ns !

d! � jNS � dð Þ !� Pd2 � 1� P2ð ÞNS�d ð21Þ

where, P2 = Average proportion defectives produced when

the process is out-of-control.

The occurrence of the assignable cause; the machine

component failure leading to FC2 or the external causes

may have different effect on the process quality. The

proportion of defectives produced when the process shift

occurs due to external causes or machine component

Last sample before assignable cause

First sample after assignable cause

Occurrence of assignable cause

In-control period Out-of- control period

Lack of control detected

Assignable cause detected

Assignable cause removed

T1 T2

Investigation time

Restoration time

τ

sHjth sample

(j+1)th sample

False alarm

TF

Fig. 3 Process quality cycle


123

failure may have different values. Therefore, the aver-

age proportion of defectives produced when the pro-

cess shifts in an out of control state can be calculated

as,

P2 ¼ P2ð ÞE�kE

k

� �þ P2ð ÞM�

kM

k

� �ð22Þ

where, (P2)E and (P2)M are the average proportion of

defectives produced when the process is in out of control

state due external cause and machine component failure

respectively.

Let ARLOut be the average number of samples required

to detect a given magnitude of process shift. It can be

determined as,

ARLOut ¼1

1� bS

ð23Þ

The expected process cycle length is the sum of the

length of the in-control period and the length of out of

control period, and is given by,

E T½ �cycle¼1

kþ E½T�false þ HS � ARLOut � sð Þþ Sin � NS � TS � osð Þ þ T1þ T2 ð24Þ

where; os

¼1; if process stops during sample inspection:

0; if process continues during sample inspection:

(

The expected cycle length in Eq. (24) is for one cycle.

During the process execution, the cycle will be repeated

whenever the process goes out of control. In this paper,

the evaluation period is taken to be the time till next

expected maintenance opportunity (TPMS). Therefore, the

expected number of cycles in the evaluation period (TPMS)

will be,

E N½ �cycles¼TPMS

E T½ �cycle

ð25Þ

5.2.6 Model for the expected total cost of process quality

control

In this section, a model for the expected cost of the process

quality control plan is presented. The expected total cost of

the process quality control plan includes the following

costs:

i) Expected cost of sampling and inspection.

ii) Expected cost of false alarms.

iii) Expected cost of investigation of assignable cause.

iv) Expected cost of rejections/rework.

v) Expected cost of restoring the process.

The calculations of the above costs are given below:

1. Expected cost of sampling and inspection.

Let Cins be the inspection cost per item. The expected

cost of sampling and inspection is,

E Csampling

� �¼ Sin þ

HS � ARLOut � s½ �HS

� � NS

� Cins þ TS � PR� CLP � os½ � ð26Þ

2. Expected cost of false alarm.

Let CF be the cost per hour to investigate the false alarm.

The cost of false alarm is given by,

E CFalse Alarm½ � ¼ TF �Sin

ARLIn

� CF þ PR� CLP½ � ð27Þ

3. Expected cost of investigation of assignable cause.

Let Cac be the cost per hour to investigate the assignable

cause. The expected cost associated with the detection of

assignable cause can be expressed as,

E CACD½ � ¼ T1� Cac þ PR� CLPð Þ ð28Þ

4. Expected cost of rejection/rework.

The proportion of defectives increases significantly,

when the process shifts to an out-of-control state. Hence,

we assume that all the lots produced from the occurrence of

assignable cause till the point of detection, which is the

ATS time span, are isolated. If it is possible to do 100 %

inspection of the isolated units, the defective pieces will be

segregated for possible rework; otherwise all the units will

be scrapped. This will minimize the consumer risk in

receiving the lots with larger proportion of defective units.

Let ‘p’ be the probability, that a defective piece can be

reworked and ‘Tlag’ be the time lag between taking a

sample and generating the sample inspection report.

The cost of rejection and the cost of rework can be

expressed as,

E½CRejection� ¼ CRej � Hs � ARLOutð Þ � sþ Tlag

� �� PR

ð29Þ

E CRework½ � ¼ Hs � ARLOutð Þ � sþ Tlag

� �� PR

� Cins þ p� P2 � CRew½þð1� pÞ � P2 � CRej

� ð30Þ

5. Expected cost of restoring the process.

The expected cost of restoration action associated with

the assignable cause be expressed as,

E CRestore½ � ¼ CRes þ T2� PR� CLPð Þ ð31Þ

where, CRes is the expected cost associated with the restore

action of the process and T2 is the expected time required

for restoration.


123

Therefore, the total expected cost of process quality

control in the evaluation period is,

E TC½ �PQC¼ E CPQC½ �cycle�E N½ �cycles ð32Þ

where, E[CPQC]cycle is the expected cost per cycle of the

process quality control and is determined as,

E CPQC½ �cycle¼ E Csampling

� �þ E CFalse Alarm½ �

þ E CRejection =Rework

� �þ E CACD½ �

þ E CRestore½ � ð33Þ

5.3 Expected total cost of the integrated model (M/Q)

The expected total cost of the integrated model (M/Q), is

the sum of the expected total cost of the opportunistic

maintenance and the expected total cost of the process

quality control. This cost is calculated using Eqs. (13) and

(32), and is given by,

E TC½ �M=Q¼ E TC½ �MþE TC½ �PQC ð34Þ

In the next section, we develop the integrated model of

the production schedule and the maintenance with quality

(M/Q).

6 Production scheduling model

In this section, a model to find an optimal production

schedule is developed. In this paper, we consider a

single machine which is required to process a number of

jobs in ‘m’ batches. Let us assume for the k-th batch, Pk

denote the processing time, DDk denote the due date, Wk

be the penalty cost of batch delay and Chk be the

inventory holding cost per item per unit time. Also, let

CTk denote the completion time of the batch k, where

k = 1, 2, …, m.

In the present study, we consider both the earliness as

well tardiness penalty cost for the batch. For earliness

penalty, the inventory holding cost for the batch is con-

sidered. The in-process inventory holding cost is calculated

as a continuous function of time. The lateness of the batch

can be defined as,

LTk ¼ CTk � DDk ð35Þ

and the tardiness penalty of the k-th batch is,

Tk ¼ Wk � max 0; LTk½ �f g ð36Þ

Let ‘ST’k denote the start time of the k-th batch in the

production schedule. Then, the inventory holding cost is

calculated as follows:

i) If DDk �CTk, then

IHCk ¼ZCT�STð Þ

0

CT � ST � tð Þ � PR� Ch� dtdt

8><>:þ DD� CTð Þ � PR� Ch� CT � STð Þgk ð37Þ

ii) If DDk\STk\CTk

IHCk ¼ZCT�STð Þ

0

CT � ST � tð Þ � PR� Ch� dt

8><>:

9>=>;

k

ð38Þ

iii) If STk\DDk\CTk

IHCk ¼ZDD�STð Þ

0

DD� ST � tð Þ � PR� Ch� dt

8><>:

þZCT�DDð Þ

0

CT � DD� tð Þ � PR� Ch� dt

9>=>;

k

ð39Þ

The objective of production scheduling is to find an

optimal sequence of batches such that total cost of batch

delay and inventory holding cost for all batches is

minimized. The total schedule penalty cost is given by,

Schedule penalty cost

¼Xm

k¼1

Wk � max 0; LTk½ �ð Þ þXm

k¼1

IHCk

( )ð40Þ

The optimal production sequence can be obtained by

minimizing the total schedule penalty cost using a

scheduling algorithm. In this paper, a backward-forward

heuristic (Sule 2007) will be used to obtain the optimal

batch sequence. Following assumptions are made for the

batch scheduling problem.

i) The machine can process no more than one batch at a

time.

ii) Each batch is independent of each other.

iii) A batch cannot be pre-empted by another batch.

iv) The setup time is included in the processing time for

the batch.

In the next section, an integrated model of production

schedule with M/Q is developed.


123

6.1 Integrated model of production scheduling

and M/Q

In this section, an integrated model for selective mainte-

nance considering quality (M/Q) and production schedul-

ing is developed. The objective of the integrated model is

to find the optimal combination of maintenance actions for

each of the machine component, the parameters of the

quality control procedure along with the optimal produc-

tion schedule. The decision on maintenance actions for

components is based on the condition of the equipment i.e.

the effective age of the component at the opportunity and

the availability requirement in the next operating period

during which the batch processing will take place. If the

length of the operating period is much smaller than the

mean life of a component, it can be assumed that the

component can fail only once during the operating period.

Based on the knowledge of the system being considered in

this paper, it is also assumed that only one component can

fail at a time for which system will be down for corrective

action.

Let a0i denote the age of the i-th component at the

beginning of the production schedule.

The probability of failure of the i-th component during

the processing of the k-th batch is given by,

F bkð Þi¼ F CTk a0ijð Þ � F CTk�1 a0ijð Þ ð41Þ

where,

F CTk a0ijð Þ ¼ZCTk

0

f t j a0ið Þ dt;

f t a0ijð Þ ¼ � d

dtR t j a0ið Þ½ � ;

R t j a0ið Þ ¼ R t þ a0ið ÞR a0ið Þ

Similarly,

F CTk�1 a0ijð Þ ¼ZCTk�1

0

f t j a0ið Þ dt ð42Þ

As already discussed, a component failure can result

into either FC1 and/or FC2, therefore, the amount of time

by which a batch can get delayed due to the failure of i-th

component is,

Td½ �i¼ MTTCAi þ ATS� QFi ð43Þ

where,

QF ¼1; if failure of i-th component affect product quality

0; otherwise

(

Here, it is assumed that the quantity rejected due to FC2

has to be produced again and the corresponding additional

time required will be ATS.

The completion time of the k-th batch, in case the

component fails during the processing of batch, is the sum

of the completion time of the preceding batch, the batch

processing time and the expected delay time.

CTF½ �k¼ CT½ �k�1þPk þ ½Td�i ð44Þ

For the previous and subsequent batches, it will be

CT½ �k�1þPk

The total lateness of k-th batch, in case the component

fails during the processing of the batch is,

LTF½ �k¼ CT½ �k�1þPk þ Td½ �i �

� DDk ð45Þ

For the previous and subsequent batches, it will be

CT½ �k�1þPk � DDk

Then, the expected total schedule penalty cost consid-

ering the failure of i-th component can be expressed as,

E½TC�PS ¼Xm

k¼1

Wk � max 0; LTk½ �f g þXm

k¼1

IHCk

þXn

i¼1

Xm

k¼1

fFðbkÞi � Td½ �i�Wk � Skg ð46Þ

From Eqs. (34) and (46), the resulting final math-

ematical model for the integrated approach can be written

as,

Minimize; E TC½ �Integrated¼ E TC½ �MþE TC½ �PQCþE TC½ �PS

ð47Þ

Subject to

iÞXn

i¼1

<i � ðMTTRAiÞ þXn

i¼1

ri � ðMTTrAiÞ" #

� TAvl

ð48Þ

ii) 1�

Pni¼1

<i � ðMTTRAiÞ½ � þPni¼1

ri � ðMTTrAiÞ½ � þ E DT½ �TPMS

n o� �

TPMS

�AReq ð49Þ


123

iii) ATS � ATSL ð50Þiv) <i þ ri� 1 ð51Þ

It should be noted that, in the integrated model, the

expected cost of repair will not be considered in the

expected total cost of process quality control in Eq. (33) as

it is already accounted for in the maintenance cost

associated with the failed component which results into

FC2.

The constraints of the mathematical model are explained

below:

The first constraint is on the available time for carrying

out the maintenance activities. Every maintenance oppor-

tunity will have a constraint in terms of the allowable time

for carrying out the maintenance activities selected. The

first expressionPni¼1

<i � ðMTTRAiÞ is the total replacement

time for those components which are replaced and the

second partPni¼1

ri � ðMTTrAiÞ is the total repair time for the

components which are repaired during the maintenance

opportunity. The sum of these maintenance activity times

should be less than or equal to the available time for

maintenance. The proposed methodology requires this

allowable time to be given as a user input at an opportunity.

The second constraint is about the system availability.

Random failures lead to unplanned down time and affect

the system availability in the planning horizon. Therefore,

while taking the maintenance decisions at the opportunity,

the probability of random failure should be considered to

ensure the required system availability. The numerator in

the expression is the total downtime which includes the

active maintenance downtime and the expected downtime

due to random failures.

The third constraint is on the ATS. The ATS is the time

between the occurrences of assignable cause till its detec-

tion. The limiting value (ATSL) of the ATS is considered

as a user input.

The last constraint is for the repair or replace decision

on the component. The component can either be repaired or

replaced.

In the next section, the solution approach for optimiza-

tion of the integrated model is presented.

7 Solution approach for the integrated model

In this paper, a Simulated Annealing (SA) algorithm is

used for obtaining the near optimal solution. SA is an

iterative search method proposed by Kirkpatrick et al.

(1983). It is a generalised probabilistic approach for

approximately solving large combinatorial optimization

problems. The physical behaviour of the annealing process

is simulated in the SA technique to find the optimal or near

optimal solutions for complex combinatorial optimization

problems. The general approach of the algorithm is given

in Fig. 4. The SA algorithm starts from an initial solution

which is randomly generated. During the search process,

the algorithm generates a new solution by some perturba-

tion mechanism in the neighbourhood of the current solu-

tion. If the new solution is better than the current solution

then the new generated solution is accepted as the current

solution. If the new solution is inferior to the current

solution, the algorithm will accept the inferior solution

with a certain probability which decreases with time. The

most important characteristic of the algorithm is the pos-

sibility of accepting inferior solutions with certain proba-

bility, which helps the algorithm to avoid being trapped in

a local optimum. The algorithm continues until a stopping

condition is attained.

In the next section, a numerical example in the form of a

real life case study is illustrated.

8 Numerical example: a case study

In this section, we consider a numerical example, which is

a real life case study to evaluate the performance of the

proposed integrated model as discussed in Sect. 3. The case

study focus on one of the high pressure die casting

machines on the shop floor, which is required to process the

jobs in batches. The company supplies castings to leading

automobile manufacturers in India and abroad. The die

casting machine on which the products for a particular

order processed is considered in this paper. The machine

operates for 8 h per shift in three shifts per day. The

machine is required to produce seven different die-cast

products for which the details of the batch sizes, processing

times, due dates, penalty cost and other parameters are

given in Table 1.

The repair/replacement time, failure cost, replacement

cost and the details of other maintenance parameters of the

machine are given in Table 2. (‘NA’ in Table 2 indicates

data value ‘not applicable’). The minimum required

availability in the next operating interval is taken to be

0.92, the time available at an opportunity for maintenance

is 20 h and the limiting value (ATSL) is taken to be 4 h. A

set of maintenance actions needs to be taken for each

component so that these constraints are satisfied.

In the present case, a sampling procedure is used for

monitoring the process quality. The CTQ requirement

can be stated as the absence of casting defects like blow

holes, non-filling, etc. These sub surface defects are

difficult to observe by the naked eyes and hence a

radioscopic inspection (X-ray) of the casting is done.

The defective percentage when the process is in-control


123

is 5 % and increases whenever a process shift occurs.

The occurrence of external causes increases the average

proportion defectives to 10 %, while it increases to 20 %

due to machine component failure. This data was

obtained from the shop floor records. The quality control

policy is to take samples at a regular interval and carry

out the radioscopic inspection. It was observed from the

shop floor records, that there are five machine compo-

nents (Sr. Nos. 22–26 in Table 2) whose failures leads to

the quality defects like blow holes, non-filling, etc. and

increase the defective percentage to 20 %. However, it

should be noted that, the given increase in the defective

rate is specific to the machine under consideration. The

details of the cost and time data for the process quality

control is given in Table 3. The cost of rejection, pro-

duction rate and the cost of lost production values in

Table 3 are the weighted volume values for the products

mentioned in Table 1.

9 Results and discussion

In order to evaluate the performance of the proposed

integrated approach (S 9 M/Q), a planned maintenance

opportunity is considered. The effective age values of the

components at the current opportunity are given in Table 4

which indicates the initial condition of components for

maintenance decision evaluation. The data for the set of

batches to be scheduled in the next operating period is

given in Table 1. It is assumed that the machine will not be

available for maintenance till the processing of all the

batches is over, the time till next expected opportunity is

taken equal to the makespan time plus an additional 20 %

to accommodate for any random failure events. A SA

algorithm as discussed in Sect. 7 is used for optimization of

the model parameters. The parameters selected for the SA

are, an initial temperature (Ts) of 20,00,000, a temperature

reduction factor of 0.95, a loop factor of 10 and a stopping

SelectAn initial temperature, Ts (a large number)An initial solution, So

A cost function, GA temperature reduction factor, θS

A neighborhood structure for the solution spaceRepeat

Repeatnew solution, SNew =perturb(So)G = G(SNew) - G(So)

If G ≤ 0 OR random number Rn[0, 1] < exp( - G/T)then, So= SNew

Until iteration count = Max_number_iterationTs = Ts× θS

Until stopping condition

Fig. 4 Simulated annealing

algorithm

Table 1 Production scheduling parameters

Sr. No. Product Set up

time (h)

Production

rate

(units/h)

Penalty

cost

(Rs./h)

Profit

per unit

(Rs./unit)

Rejection

cost

(Rs./unit)

Batch

size

Processing

time (h)

Inven-tory holding

cost (Rs./unit/h)

Due

date (h)

1 Product 1 1 142 533 15 63 1,510 11.6 0.0014 41

2 Product 2 1 72 648 30 160 23,440 326.6 0.0037 1143

3 Product 3 1 142 568 20 104 5,580 40.3 0.0024 141

4 Product 4 1 142 426 20 95 5,000 36.2 0.0022 127

5 Product 5 1 142 511 18 99 4,860 35.3 0.0023 124

6 Product 6 1 142 320 15 70 7,870 56.4 0.0016 197

7 Product 7 1 142 682 16 72 9,870 70.5 0.0016 247


123

condition of temperature (Ts) [1. Multiple starting solu-

tions were used to address the issue of multiple local

minima and ensure that the final solution is the best one

among all the obtained solutions. The total cost of the

objective function is considered as the fitness function for

the decision parameters. The algorithm is implemented

using a program developed in Matlab R2012a software. For

programming purpose, the maintenance actions repair,

replace and do nothing for the component are represented

by the numbers 1, 2 and 3 respectively. Hence, the main-

tenance decision generated by the program will be a string

of the numbers 1, 2 and 3. However, it should be noted

here, that for the non repairable components, only 2 and 3

are the possible maintenance actions. The objective of the

integrated model S 9 M/Q is to find the optimal combi-

nation of the maintenance actions for the components and

parameters of the quality control along with the optimal

production schedule. The results of the optimization of the

S 9 M/Q model for maintenance, quality control and

production scheduling decision are given in Tables 5, 6 and

7, respectively. The progress in the objective function

value during iterations of the SA algorithm is shown in

Fig. 5. From Fig. 5, it is seen that during initial iterations

when the temperature is high, some large increases in the

objective function value are accepted and some areas far

from the optimum value are explored. As execution con-

tinues and temperature falls, the search converges and the

iterations are spent searching around the optimum.

In order to compare the performance of the integrated

approach with the conventional approach of independent

decision on maintenance, quality control and production

scheduling, the results are obtained as per the stand-alone

models. The same values of the relevant parameters and

policy variables are considered to obtain the model

parameters. The selective maintenance decision (M) and

parameters of the sampling procedure are optimized inde-

pendently based on the models presented in Sects. 5.1 and

5.2, respectively. A backward-forward heuristic is used to

Table 2 Maintenance data for machine components

Sr. No. Component b g (h) Component

cost during

replacement

(Rs.)

Failure

cost (Rs.)

Sub-component/

consumables

cost during

repair (Rs.)

MTTCA (h) MTTrA (h) MTTRA (h)

1 Electrodes 1.61 2,388 1,100 2,776 100 1 1 0.33

2 Electrode insulator 1.07 3,458 100 2,776 NA 1 NA 1

3 Electric wire 1.06 7,877 300 2,976 NA 1 NA 1

4 Arm bearings 2.83 1,837 6,200 7,538 200 0.5 1 0.5

5 Limit switch 3.33 14,027 20,000 22,676 NA 1 NA 1

6 Chain 6.04 1,623 5,000 6,352 800 2 2 3

7 Chain lock 4.14 3,368 150 2,826 NA 1 NA 1

8 Bearing(cup side) 2.83 1,837 2,500 11,028 200 3 1 3

9 Pneumatic cylinder 3.33 14,027 30,000 35,402 NA 2 NA 2

10 Seal 4.40 6,680 7,000 12,352 NA 2 NA 2

11 Dia. valve 5.56 3,265 8,000 13,352 NA 2 NA 2

12 Connector 2.49 7,738 250 1,133 NA 0.33 NA 0.33

13 Shock Absorber 3.33 14,027 10,000 12,676 NA 1 NA 1

14 Valve screw 2.59 22,803 100,000 18,028 10,000 3 3 4

15 Gear box 3.93 20,364 100,000 38,760 12,000 10 10 4

16 Servo valve 3.33 14,027 200,000 202,676 200 1 4 1

17 Inj. unit piston 3.93 20,364 160,000 288,448 20,000 48 8 48

18 Shot sensor 3.76 6,840 100,000 106,338 NA 0.5 NA 0.5

19 Teflon seal 5.94 1,624 150 819 NA 0.25 NA 0.25

20 Extractor bearings 2.69 14,027 12,200 33,608 NA 8 NA 8

21 Length adjustor 4.64 9,958 250 2,926 50 1 1 1

22 Acc. piston 2.69 3,744 120,000 63,380 50,000 5 5 6

23 Acc. seal 2.83 1,837 50,000 60,704 NA 4 NA 4

24 Safety valve 3.01 36,023 10,000 3,176 500 1 1 1

25 O’ring set 3.33 14,027 40,000 74,788 NA 13 NA 13

26 Couplings 2.49 7,738 16,200 14,580 1,000 5 5 1


123

obtain the optimal production sequence (S) for the batches.

The optimal maintenance decision is superimposed on the

production schedule and the quality control plan (Q) to take

into consideration the effect of random failures during the

actual operation. In this paper, the conventional approach is

represented as ‘Separate SMQ’ approach. The results for

maintenance, quality control and production scheduling

decision using independent models are given in Tables 8, 9

and 10, respectively, while Fig. 6 shows the cost progress

of the optimization of the maintenance decision for this

approach.

From the results obtained for both the approaches,

S 9 M/Q and Separate SMQ, it is observed that:

1. From Table 10, the schedule penalty cost in case of

Separate SMQ approach is Rs. 55,587, which is higher

than the integrated approach. This is due to the fact

that maintenance decision is not considering the

production schedule. Hence, when the schedule is

implemented on the shop floor, the penalty cost due to

component failure increases the total schedule penalty

cost. However, in the case of integrated approach, the

effect of component failures on the production

schedule is considered during the optimization and

the maintenance decision having minimum effect on

the production schedule is selected.

2. From Tables 5 and 8, it is observed that the availability

achieved for the integrated approach is better than the

independent approach.

3. From Tables 6 and 9, the total cost of process quality

including sampling and rejection cost in the next

operating period is Rs. 319,393 and that for Separate

Table 5 Maintenance decision

using S 9 M/Q approachMaintenance decision Cost of

maintenance (Rs.)

Cost of

failure risk (Rs.)

Availability

Repair Components—NIL 85,326 158,863 0.9438

Replace Components—3, 7, 8, 19, 21, 23

Do nothing Components—1, 2, 4, 5, 6, 9, 10,

11, 12, 13, 14, 15, 16, 17, 18,

20, 22, 24, 25, 26

Table 3 Data for process quality control procedure

Cost of inspection (Rs./unit) Cost of investigating

false alarm (Rs./h)

Time to detect

false alarm (h)

Cost of assignable

cause detection (Rs./h)

Time to detect

assignable cause (h)

150 80 1.0 160 2

Cost of rejection (Rs./unit) Production rate (Units/h) Cost of lost production (Rs./unit) Mean time between

process failures due

to external causes (h)

114.3 115 22.4 500

Table 4 Effective age of components at the opportunity

Component 1 2 3 4 5 6 7

ðviÞO(h) 395 954 6,341 1,963 9,922 67 3,159

Component 8 9 10 11 12 13 14

ðviÞO(h) 1,041 4,209 8,286 3,949 6,086 4,511 27,953

Component 15 16 17 18 19 20 21

ðviÞO(h) 1,906 16,416 17,282 1,041 2,008 14,117 12,821

Component 22 23 24 25 26

ðviÞO(h) 2,982 2,167 26,507 7,420 2,835

Table 6 Quality control decision using S 9 M/Q approach

Process quality control

parameters

Cost of process

quality control (Rs.)

Sample size = 5 319,393

Acceptance number = 0

Sampling interval = 2.5 h


123

SMQ approach is Rs. 377,100. This shows the

maintenance decision’s role in quality. In the case of

Separate SMQ approach, the maintenance decision is

taken considering only the availability requirement

in the next operating period. The maintenance deci-

sion does not consider the possibility of machine

failure and the maintenance actions are not evaluated

for the possible effect on the quality and production

schedule. However, in the integrated approach, each

maintenance decision’s effect in terms of its probabil-

ity of failure is considered on the quality as well as

production schedule to obtain the optimal decision for

S, M and Q based on the minimum total cost. The

direct cost of maintenance decision in case of Separate

SMQ appears to be less as compared to the integrated

approach; however the cost of consequence is higher.

4. The total cost for the integrated approach S 9 M/Q is

Rs. 617,009 and for the Separate SMQ is Rs. 650,572.

Thus, the total cost for S 9 M/Q is less as compared

to the Separate SMQ approach and the integrated

approach has resulted into approximately Rs. 33,563

i.e. 5.44 % saving over the Separate SMQ approach.

Keeping in mind that this saving is for one machine,

the total saving for a large number of machines will be

quite significant.

0 50 100 150 200 250 3002

2.5

3

3.5

4

4.5

5x 10

5

Iterations

Tot

al c

ost(

Rs.

)

Fig. 6 Cost progress of maintenance optimization for separate SMQ

approach

0 50 100 150 200 250 3006

6.5

7

7.5

8

8.5x 10

5

Iterations

Tot

al c

ost (

Rs.

)

Fig. 5 Total cost progress of S 9 M/Q approach

Table 8 Maintenance decision using Separate SMQ approach

Maintenance decision Cost of

maintenance

(Rs.)

Cost of

failure

risk (Rs.)

Availability

Repair Components—4 5,125 212,760 0.9332

Replace Components—7,

8, 19, 21

Do nothing Components—1, 2,

3, 5, 6, 9, 10, 11,

12, 13, 14, 15, 16,

17, 18, 20, 22, 23,

24, 25, 26

Table 9 Quality control decision using separate SMQ approach

Process quality control parameters Cost of process

quality control (Rs.)

Sample size = 5 377,100

Acceptance number = 0

Sampling interval = 2 h

Table 10 Production scheduling decision using separate SMQ

approach

Batch sequence Production schedule

penalty cost (Rs.)

1 5 4 3 6 7 2 55,587

Table 7 Production scheduling decision using S 9 M/Q approach

Batch sequence Production schedule penalty cost (Rs.)

1 5 4 3 6 7 2 53,427


123

10 Conclusion

This paper has presented an approach for integrating

selective maintenance, process quality control and pro-

duction scheduling for a multi component, multiple product

manufacturing system. A cost model for the joint consid-

eration of selective maintenance, quality control and pro-

duction scheduling is developed. The optimization of the

proposed integrated model results into maintenance actions

namely repair, replace and do-nothing for each of the

components, parameters of quality control procedure and

the optimal production schedule. The comparison of the

integrated models with the conventional approach is pre-

sented using the data from a real life case study. The results

of the case study reveal that the integrated approach is

beneficial over the conventional approach and has shown

about 5–6 % saving in the total cost over the conventional

approach. This saving is for one machine in one period;

therefore the approach can result in significant annual

savings, if implemented for all the machines in a plant. The

approach presented in this paper is generic and can be

applied at planned as well as unplanned opportunities.

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