An integrated model of production scheduling and inspection planning for resumable jobs

Journal Pre-proof An integrated model of production scheduling and inspection planning for resumable jobs Wilfrido Quiñones Sinisterra, Cristiano Alexandre Virgínio Cavalcante PII:

S0925-5273(20)30062-1

DOI:

https://doi.org/10.1016/j.ijpe.2020.107668

Reference:

PROECO 107668

To appear in:

International Journal of Production Economics

Received Date: 30 July 2019 Revised Date:

3 February 2020

Accepted Date: 4 February 2020

Please cite this article as: Sinisterra, Wilfrido.Quiñ., Cavalcante, Cristiano.Alexandre.Virgí., An integrated model of production scheduling and inspection planning for resumable jobs, International Journal of Production Economics (2020), doi: https://doi.org/10.1016/j.ijpe.2020.107668. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

Cristiano Cavalcante: Supervision, Methodology, Conceptualization, Investigation, Writing- Reviewing and Editing. Wilfrido Sinisterra: Data curation, Writing- Original draft preparation, Conceptualization, Visualization, Investigation, Software, Validation.

An integrated model of production scheduling and inspection planning for resumable Jobs Wilfrido Quiñones Sinisterraa, Cristiano Alexandre Virgínio Cavalcante a a

RANDOM - Research Group on Risk and Decision Analysis in Operations and Maintenance;

Department of Engineering Management, Universidade Federal de Pernambuco, Recife PE, CEP: 50740-550, Brazil

E-mails: [email protected], [email protected], [email protected]

CORRESPONDING AUTHOR

Cristiano Alexandre Virgínio Cavalcante e-mail: [email protected], [email protected]

ABSTRACT We developed a model that integrates the schedule of a sequence of resumable jobs and inspection policy for the processing of n jobs with different processing times in a single-component system, whose failures follow a two-stage process. The assignment of inspections is performed at the beginning of the job; if an inspection detects the system to be in a defective state, preventive replacement is performed, whereas if the system fails, minimal repair is carried out to restore the system to an operational condition in the defective state. Analytical and simulation models are proposed to establish the best sequence of jobs and inspections that minimize the expected total cost. The expected total cost is composed of expected maintenance cost and expected tardiness cost. We present a numerical study that demonstrates the importance of integrating inspection policy with the sequencing of jobs, as such integration can lead to a significant cost reduction. The most interesting insights brought by the model highlight the importance of the effectiveness of maintenance actions (perform actions in smaller time and less cost), as well as the importance of deciding the correct moment to do them, considering the sequence of jobs.

KEYWORDS: Production; inspection policy; delay time; schedule; integrated maintenance and production planning; job sequence

1 INTRODUCTION AND LITERATURE REVIEW 1.1 Introduction The sequencing of jobs and maintenance planning play a key role in the performance of manufacturing systems, even more so with the current criteria of customization, and this challenge requires these processes are increasingly integrated (Xia et al., 2018). In maintenance planning, the inspection policies have presented dramatic improvements in organizations, however, in the existing literature, only a few studies have considered the joint optimization of the inspection and production schedule (Taghipour and Azimpoor, 2018). Therefore, in order to obtain improvements through the integration of inspection policies and job sequencing, we consider the study of a set of characteristics that are common and widely used in manufacturing systems but have not yet been treated together. We consider a critical single-component system prone to failure which must process resumable jobs with different processing times, whose failure is a two-stage process. The system can visit three different states: good, defective and failed. Whereas it is operational in the good and defective states, failures prevent the system to work and can bring serious consequences related with planning disruption. In fact, some authors have considered interesting approaches, in which, if a failure occurs during the processing of a batch, minimal repair actions are carried out in order to resume the manufacture of the batch and, if required, to perform rework operations (AlSalamah, 2018; Gouiaa-Mtibaa et al., 2018; Xia et al., 2017a). However, these researches do not consider the occurrence of failures that follow a two-phase failure process and neither how jobs sequencing affects the productivity of the production system. Regards, resumable jobs, we observe that they are present in some different production processes, such as the steelmaking process (Ye et al., 2014), milling machine (Seiti and Hafezalkotob, 2019), paper industry (Johnsen, 2017), semiconductor manufacturing (Rose, 2005), among others. Resumability of the jobs is an important feature for an increasing number of manufacturing systems, and is aligned with the current tendency of flexibility imposed by customer satisfaction. So, considering the current customization characteristics, it becomes increasingly necessary that the production systems process small batches in order to satisfy the requirements of each client (Jarebrant et al., 2018). Batch production corresponds to the logical execution of a set of jobs (Pinedo, 2016). For this kind of job processing, to consider that an occurrence of failure during a job forces the job to be restarted from the beginning is quite unrealistic. This becomes even more critical if we consider a job as a batch of small size (Baptista and Antune Simões, 2000; Pham et al., 2007; Pinedo, 2016; Shneor, 2018). In addition, there are some manufacturing systems which, due to their technical aspects, are not acceptable to perform the replacement in the middle of the job processing, which makes it necessary for the execution of minimal repair and the continuity of the job from the point that it stopped.

The main contribution of this article is the development of a model to support decisions associated with scheduling a joint maintenance and production sequence of resumable jobs based on the delay-time concept. A mathematical framework is developed to consider the minimal repair concept in a delay-time model (DTM) for a single-component system. The results are obtained in two different ways: the optimization of analytical expressions regards to total cost and by a simulation process. The remaining paper is structured as follows. In the next sub-section, we discuss some previous works important for the development of the proposed model. Section 2 presents a description of the problem. Section 3 describes the proposed model used to establish the total expected cost under different scenarios. Section 4 defines a numerical study. Section 5 provides details concerning the sensitivity analysis of the model. Finally, section 6 presents the conclusions and recommendations.

1.2 Literature review To address the diversity of the productive processes considering its complexity involved in the establishment of an integrated approach to define a precise production schedule, while taking into account the main elements to avoid the production disruption is a very interesting problem. It demands the use of mathematical models with combinatorial features in order to support the definition of the best job and the inspection sequence (Pan et al., 2010). Some authors defend that this kind of problem demands models that determine the optimal production plan and instances of preventive maintenance actions with the objective of minimizing the sum of preventive and corrective maintenance costs, setup costs, holding costs, backorder costs and production costs, while satisfying the demand for all products over the entire horizon (Fitouhi and Nourelfath, 2012). In the context of a single machine, there are some contributions related to the joint optimization of job sequence and periodic maintenance (Liao and Chen, 2003; Wang et al., 2018). The optimization of job sequencing and preventive maintenance for single machines has been discussed by some authors. A relevant approach was developed by Cassady and Kutanoglu (2005) in which the problem involves
jobs with processing time  and weight  , available at the

beginning of the time horizon. The scheduling aims to jointly determine the optimal job sequence and the optimal decision of whether to perform PM prior to each job, to minimize the total expected weighted completion time of jobs. Due to the large-size configuration of the solution space for this problem, some authors suggest the use of efficient heuristics to minimize the makespan (Hsu et al., 2010). Other authors propose a hybrid approach that combines a mixed integer program and the well-known longest

processing time (LPT)-first heuristic (Souissi et al., 2016). In the same manner, a heuristic method based on variable neighborhood search (VNS) was developed (Pacheco et al., 2018). These authors addressed the production scheduling problem on a single machine by employing flexible periodic preventive maintenances, in which the release dates of the jobs were also considered. Both restartable and non-restartable cases were studied. When failures occur during the processing of jobs, an option is to perform minimal repair if the characteristics of the process and the product allow it. Minimal repair is performed to restore the machine to be “as bad as old” (i.e., in the operating condition immediately before the occurrence of failure) (Feng et al., 2018; Hadidi et al., 2015; Mamabolo and Beichelt, 2004; Pan et al., 2010). Alternatively, Badía et al., (2018) analyzed the optimal replacement policy for a system subject to a general failure and repair model. Failures can be of one of two types: catastrophic or minor, which leads to a replacement or a repair, respectively, after repair, the system recovers the operational state, however, its condition is worse than that just prior to failure. The increasing tendency of customization products has intensified the importance of production systems capable of providing a broader variety and range of the final format of the products, such as batch production. Therefore, the batch production has brought new challenges for maintenance decision-making. The maintenance scheduling should not only consider intervention demand from individual machine deterioration, but also handle additional intervention required for each batch setup with variable batch size (Xia et al., 2015). Xia et al., (2015) proposed two kinds of maintenance activities to reduce unplanned downtime in each batch cycle. For each machine, PM is scheduled during setup times between two batches, while minimal repair is used if it fails during a batch production. Lu et al., (2018) studied an unrelated parallel machine scheduling issue with deteriorating maintenance activities, parallel-batching processing and deteriorating jobs. The objective is to make the joint decisions on job assignments, the maintenance arrangements, jobs batching and batches sequencing on each machine to minimize the makespan. Zahedi et al., (2017) elaborates on an integrated model of batch production scheduling and maintenance scheduling on flow shop where the objective function of the model is to minimize total cost consisting of in process and completed part inventory costs, setup costs, preventive and corrective maintenance costs and rework costs. The importance of this kind of model comes with increase of customization that forces the use of the machine in different production cycles with a small duration and more frequent setups in order to satisfy the requirements of each client (Jarebrant et al., 2018). In the customization processes, it is important to keep the manufacturing system reliable, therefore, a prognostic method is essential (Xia and Xi, 2019). The information from these prognostic methods are useful not only to model the degradation signals, but are also very important to update the time-

to-failure distributions. Xia et al., (2019) proposed an efficient opportunistic maintenance policy that considers relevant information from a prognostic method. In turn, in order to satisfy the rapid market changes,

Xia et al., (2017b) proposed a dynamic interactive bilevel maintenance

methodology based on predictive maintenance and opportunistic maintenance. The integration of job sequencing and maintenance planning, as treated here, generates a cost corresponding to tardiness penalty cost. This cost refers to the sum of costs incurred in the time that the makespan exceeds a certain due date, as well as costs such as reprogramming of production, inventory, energy, etc. Some researchers consider similar definitions of cost incorporated to the decision criterion in order to handle different problems in manufacturing systems. For example, Sarkar and Moon (2011) proposed a production inventory model, which considers demand with inflation effects, and rework costs due to real-life problems (labor problems, machine breakdown, etc.). To highlight the relationship between quality improvement, reorder point and lead time, as affected by backorder rate (Sarkar and Moon, 2014), consider an embracing cost function (total system cost) associated with an imperfect production process. Other works focus on how the characteristics of random defect rates, inherent to an imperfect production system, affect the decisions regarding to inventories (Sarkar et al., 2014; Sett et al., 2017; Taleizadeh et al., 2017; Tayyab and Sarkar, 2016) . For these articles, the main cost elements are the inventory costs. Besides the impact on the inventory, due to the production of defective items, other important analysis is related to the strategies to reduce the number of defective items. In this way, Kim and Sarkar (2017) investigated a stochastic inventory model with a budget constraint and cost minimization due to imperfect manufacturing process. Another tendency is to incorporate sustainability aspects in the models involving imperfect production system, for example Tiwari et al., (2018) studied a single system with imperfect production, with the purpose of establishing a green production, in which it considers costs for rework holding cost, cost of disposal per scrap, backorder cost, among others. We can observe that despite the differences presented in these articles, a common element is the concerns related with the formulation of the cost criterion. In fact, for all previous work that was mentioned, this criterion is aligned with the essential aspect being treated by the model. In this work, this is not different. The cost is the fundamental aspect of the model and it enables us to integrate production and maintenance planning in order to provide better results in the long term. In order to provide a better understanding for the contributions of this article, we briefly present some prior contributions that were important to the design of the proposed model and address at least one of the fundamental concepts discussed in this article. Table 1 presents the distributions of the articles by the main contribution area.

Some authors developed interesting models that consider optimization of the production schedule and maintenance for resumable jobs in an industrial context. Gouiaa-Mtibaa et al., (2018) studied cases of manufacturing system of the textile industry in order to reduce the impact on the quality of the products, performing a given number of imperfect PM actions before undertaking a perfect one. If a failure occurs, a minimal repair is performed and the production batch must be resumed and at the end rework is performed to establish the best quality of the batch. (Al-Salamah, 2018) studied a manufacturing system composed of a single unreliable machine, where if the machine fails before the batch is completed, the production is interrupted until the machine is repaired and resumes after the machine is operational again. The objective is to find the batch size that minimizes costs. Xia et al., (2017a) used a case study associated with manufacturing of hydraulic steering, using a reconfigurable manufacturing system (RMS). This feature allows the execution of jobs with different processing times where, if failures occur during the production cycle, minimal repairs must be executed. Therefore, the objective is to establish an opportunistic maintenance policy to reduce downtime and costs. Delay time is a concept that divides a system failure process into two stages: from new until the point of an identifiable defect (initial point) and, next, from this point until failure (delay time) (Wang, 2012). The delay time is a window of opportunity to prevent the system to fail as long as an inspection has taken place during the delay time (Ferreira et al., 2009). The delay time concept has been used to support the development of many mathematical maintenance models. Especially due to the possibility to avoid failure, by identification of defects and its strong practical appeal, the number of delay time models in the literature is rapidly increasing (Alberti et al., 2018; Berrade et al., 2018, 2017, 2015, Cavalcante et al., 2015, 2019; Wang, 2012; Yang et al., 2019). Here follows a brief summary of important delay time based models for our proposed model and future extensions. Scarf et al., (2019) developed a delay-time model of inspection-maintenance of a critical system in which the execution of inspections is random. The authors advocate several circumstances where, despite a periodic inspection plan, the exact time of this action being performed becomes random. In our proposed model we consider that inspection might be conducted after a job is finished and before the subsequent job is started. In this way we did not consider periodic inspections, since the jobs have different durations and would not be necessary to inspect at the beginning of every job process. Alternatively, Cavalcante et al., (2018) studied an interesting approach where the influence of opportunities in a hybrid inspection and replacement policy for one-component systems of variable quality where the efficacy of inspection is modeled using the delay time concept. Once there exists some difficulties to follow a schedule plan of inspection to detect the state of the system, do replacements opportunistically provide not only some safes in terms of cost but it also

simplifies the implementation and control of the maintenance activities. In our proposed model we can consider that at the beginning of a resumable job processing that was preceded by another job is an opportunity to conduct an inspection. Thus, the decision is to determine the best sequence of jobs and inspections. Wang et al., (2017) developed a two-phase inspection schedule and an age-based replacement policy for a single plant item subject to a three-stage degradation process. For situations where degradation can be modulated in different levels, especially if the degraded condition can affect the capacity of the production of the system, considering more than one defective state would be interesting. This is not treated in this paper; however, it is already under study for future contributions. For the question related with the modelling of sequences of jobs, Taghipour and Azimpoor (2018) brought important elements regarding the essential aspect considered in a joint preventive maintenance and production scheduling for a single system in a manufacturing plant, which is required to process n jobs. In our work, unlike the proposed model by these last authors, we focus on resumable jobs where a huge class of different kind of processable works, when suffering any interruption, resume from the point that have stopped. Thus, to develop an integrated maintenance and production plan based on the delay-time concept, we propose a mathematical framework to consider the minimal repair concept in a delay-time model (DTM) for a single-component system. In addition to filling a gap in the literature, this feature allows the modelling to be executed in an integrated manner.

Table 1 Contribution of different authors Authors Cassady and Kutanoglu (2005) Pan et al., (2010) Fitouhi and Nourelfath, (2012) Souissi et al., (2016) Wang et al., (2017) Wang et al., (2018) Pacheco et al., (2018) Feng et al., (2018) Lu et al., (2018) Gouiaa-Mtibaa et al., (2018) Cavalcante et al., 2018 Taghipour and Azimpoor, (2018) Scarf et al., (2019) Yang et al., (2019)

Single System

Failure delay-time

Minimal repair

Resumable jobs


























Job sequencing

Inspection policy





























































This paper



















2 PROBLEM DESCRIPTION

We considered a single-component system that is required to process
jobs, each with a

processing time of  ,  = 1, 2, … ,
. All jobs have a common due date . The system can be in one

of three states: good, defective or failed. The system starts operating in the good state, that is, in the absence of any characteristic that is representative of the defective state of the system (noisy but

functioning bearing). The arrival of the defective state is denoted by X, and  denotes its

probability density function. If the defect is left unattended, it will eventually result in failure. The

delay time is denoted by the random variable , which represents the time from the defect arrival until eventual failure, with the probability density function and cumulative density function being

 and  , respectively.

If the system fails while processing a job, minimal repair must be performed as many times

as necessary until the job is finished. A repair takes  time units and incurs cost  . The system is

restored to an operational condition; however, the system age is not altered. This implies that upon system failure, the system operator performs just enough maintenance to resume system function (Cassady and Kutanoglu, 2005). When the system fails, minimal repair is performed to restore the system to be “as bad as old” (i.e., the operating condition just before failure) (Pan et al., 2010). We adapt this concept to be used in a DTM and consider that after a failure, by performing a minimal repair, the system is restored to the defective state, as seen in Fig. 1.

Fig. 1. Graphical representation of delay time and minimal repair

According to the characteristics of the system, inspection can be performed only before

starting a job. , and , represent the downtime and cost per inspection, respectively. If a defect is

detected, the system is replaced preventively; we assume that preventive replacement restores the system to a “good as new” condition. On average, preventive replacement of the system takes ,

time units and incurs cost  .

Let us consider that  = 1 indicates that an inspection is performed before job , and

 = 0 indicates otherwise. Our aim is to establish a sequence of jobs ȷ = 1, 2, … , 
 as well

as to allocate inspections that can be performed before each job  =  ,   , … , ! ". The

objective is to jointly determine the optimal job sequence →∗ →∗ using (1), which results in the minimum

expected

total

cost

'()* +, 

#

composed

&

of

the

expected

cost

of

maintenance '-* +,  and expected cost of tardiness in the makespan '(. ,/, . In (2), 0

corresponds to the penalty cost per unit of time in which the makespan exceeds the due date. Equation (2) indicates the formula of '()* +, . The system can process only one job at a time. 1 →∗→∗2 = #

&

-
4'()* +, 5 1 3 ,  ,,,

'()* +,  = 0 67890,  '(* +, :" −  + '-* +,  2

Let us consider  = 0. Since the system starts as good as new, it is not necessary to

perform an inspection before the first job. In this manner, with n jobs to be processed, there exists


! × 2!? possible policies, among which the optimal policy must be obtained. The computational

complexity of obtaining the optimal solution is thus @
! × 2!? . 2.1. Notation

AB DEB F_HB z 7 C

J, K, L N T

N
OB PB

 QR . /,

The probability that event C occurs during processing of job  The probability that event C occurs during processing of jobs in z The probability that event C occurs during processing of 7, as the system begins its life just before processing job  Set of jobs without any inspection performed between the jobs A job or set of jobs without any inspection performed between the jobs Events that can occur on a job or in a set of jobs without any inspection performed between the jobs (1: no defects 2: one defect, 3: one minimal repair, 4: two minimal repairs, I: 6 minimal repairs) Indexes that represent the same meanings as C Number of inspections of policy [3  The number of scenarios of policy [3 . ) = 2M Common due date for all jobs Index that represents the scenarios Number of jobs to be processed Makespan corresponding to events C that occur in a job or a set of jobs Cost of maintenance corresponding to events C that occur in a job or a set of job scenarios Processing time of job  Expected value of scenario N corresponding to policy 3 

6

 ,  ,   , ,  , S T    ,  F '(. ,/,  '-* +,  '()* +, 

Estimated value that represents the number of minimal repairs that may occur in a job or a set of jobs Preventive replacement, inspection and repair downtime Replacement, inspection and repair cost Tardiness penalty cost per unit time Random variable for defect arrival Random variable for delay time Probability density function for X Probability density, distribution and reliability functions for H Expected makespan for policy [3  Expected maintenance cost for policy [3  Expected total cost

2.2 Assumptions 1. After performing minimal repair, the system goes from the failed state to a defective state. 2. When two or more jobs are processed without any inspection performed between the jobs, the occurrence of an event b is studied jointly (i.e., it is studied as a job with processing time equal to that of the sum of the processed jobs). 3. The inspections are perfect and reveal the true state of the system. 4. The time to defect arrival X and delay time H are statistically independent. 5. The sequence of jobs and the assignment of inspections must be established before starting production. 6. When a failure occurs, minimal repairs must be made and, therefore, the system goes to the defective state. Thus, even in case there is a subsequent inspection planned, it must be performed. This has the objective to generate a record that allows to know the state of degradation of the system and the correlation with production indicators, such as reduction of both the production rate and percentage of defective items produced.

3 MODEL DEVELOPMENT To establish the model, we begin by describing the expected makespan. Next, we present the corresponding procedure for the calculation of the probabilities used in the model. This information is used to calculate the expected cost of maintenance. Finally, we present the characteristics of the expected total cost. 3.1 Expected Makespan This section defines the procedure to establish the probabilities of the occurrence of events that can be present in a job. Next, we describe the calculation of the makespan, and the procedure for 2, 3 and
jobs is defined. It is important to note that the jobs are resumable, thus, if the system

fails during the processing of a job, minimal repair is performed (i.e., if the system fails 6 times

during the processing of a job, 6 minimal repairs are performed). This mimics the reality for a

variety of manufacturing systems, in which the most frequent failure of a component does not require complex assembly or restarting of the job. •

For one job:

1. No defect takes place while the job is being processed (Fig. 2a). The probability can be defined as in (3). Note that this case does not depend on the delay time h. V

A = U  8 8 3 W

2. A defect takes place while the job is being processed and the system survives beyond  (Fig. 2b). The probability can be given as follows: W

A = U  8F  − 8 8 4 Y

3. A defect and the subsequent failure take place while the job is being processed, therefore, minimal repair is performed, and the system survives beyond  (Fig. 2c). The probability can

be given as follows: W

A[ = U U Y

W ?]

Y

U

V

 ℎ  ℎ  ] 8 ℎ ℎ 8 5

W ?]^_W 

4. A defect and two subsequent failures take place while the job is being processed, therefore, two corresponding minimal repairs are performed, and the system survives beyond  (Fig. 2d). The

probability can be given as follows: W

Aa = U U Y

W ?]

Y

U

W ?]^_b 

Y

U

V

 ℎ  ℎ   ℎ   8 ℎ ℎ ℎ 8 6

W ?]^_b ^_W 

5. A defect and 6 subsequent failures take place while the job is being processed, therefore, 6

corresponding minimal repairs are performed, and the system survives beyond  . The probability can be given as follows: W

Ad = U U Y

W ?]

Y

,...,U

W ?]^_e ^_efW ^...^_b 

Y

 ℎg  8 ℎ ℎ , . . . , ℎg 8

U

V

 ℎ ℎ  , . . . ,

W ?]^_e ^_efW ^...^_b ^_W 

In general, the expected makespan for one job with processing time  is

'(,Y  =  A +  A +  +  A[ +  + 2  Aa + ⋯ +  + 6  Ag^ '(,Y  = j

d

Bk

7 

OB AB dkg^ 8

where 6 represents the number of minimal repairs that may occur in a job or a set of jobs.

The value 6 must be sufficiently large so that the sum of probabilities established in the calculation of the expected values is equal to or very close to one, for example, for one job, ∑dBk AB ≅ 1. In

this way, 6 represents the maximum number of minimal repairs that can be performed in a job or in a set of jobs.

Fig. 2. Graphical representation of occurrence of events in one job: (a) no defect during the job (b) defect (c) one minimal repair (d) two minimal repairs The following section describes the development of the model when it is necessary to process more than one job. Moreover, the influence of performing inspections is described. We assume that the inspections are perfect and reveal the true state of the system. •

For two jobs with inspection To process two jobs with an inspection performed at the start of job 2 '(,

,Y, ,

two scenarios are presented: The first scenario (scenario 1) occurs when the defect arises before the inspection. Scenario 1 contains all the events that may arise during job 1 after the defect arrival, i.e.,

from 1 to 6 minimal repairs, as the state of the system will be defective when performing the

inspection regardless of the number of minimal repairs that have been carried out. Therefore, the

system is renewed at the beginning of job 2. Further, scenario 1 contains all the events that may arise in job 2 (i.e., from b:1 to I events). To illustrate this more clearly, two cases contained in scenario 1 are described: 1. A defect occurs while job 1 is being processed and the system survives to the end of job 1. Next, the defect is detected, and the system is preventively replaced (i.e., the probability that C = 2 occurs during job 1 is A, ) (Fig. 3a). During the processing of job 2, a defect and its

subsequent failure occur, therefore, minimal repair is performed and the system survives to the

end of job 2 (i.e., the probability that C = 3 occurs during job 2 is A ,[ ) (Fig. 3a). The probability that this case occurs is the product of A, and A W

b

A A [  = qU  8F_  − 8 8 r qU U Y

2.

Y

b ?]

Y

U

V

,[ .

 ℎ ℎ   8 ℎ ℎ 8r

b ?]^_W 

9

During the processing of job 1, a defect and subsequent failure occur. Therefore, minimal repair is performed, and the system survives to the end of job 1. Next, the system is

preventively replaced (i.e., the probability that C = 3 occurs during job 1 is A,[ ) (Fig. 3b).

During the processing of job 2, a defect and the two subsequent failures occur, therefore, two

corresponding minimal repairs are performed, and the system survives to the end of the job 2

(i.e., the probability that C = 4 occurs during job 2 is A ,a ) (Fig. 3b). The probability is the

product of A,[ and A W

A[ A a  = qU U b

qU U Y

b ?]

Y

U

Y

W ?]

Y

b ?]^_b 

Y

,a .

U

U

V

V

W ?]^_W 

 ℎ ℎ   8 ℎ ℎ 8r

 ℎ ℎ   ℎ  8 ℎ ℎ ℎ 8r

b ?]^_b ^_W 

(10)

Fig. 3. Graphical representation of cases contained in scenario 1 for two jobs: (a) one defect in job 1 and one minimal repair in job 2; (b) one minimal repair in job 1 and two minimal repairs in job 2 In this manner, scenario 1 contains all the combinations that occur between C: 2 to I that

occur in job 1 and the events C: 1 to I that occur in job 2 (Fig. 4).

Fig. 4. Simplified graphical representation of scenario 1 for two jobs Scenario 2 takes place when no defect occurs during job 1, i.e., before the inspection.

However, a defect can occur during job 2, which can lead to 6 minimal repairs, i.e., from the

occurrence of C: 1 to I. To illustrate, two cases contained in scenario 2 are described:

A defect and the subsequent failure take place while job 2 is being processed. Therefore,

minimal repair is performed and the system survives beyond job 2. Its probability corresponds to F_

[ —the

F_

[

probability that an event C = 3 occurs during the processing of job 2—as the system

begins its life just before processing job 1 (Fig. 5a). This occurs with probability W ^b 

=U

W

U

W ^b ?]

Y

U

V

 ℎ  ℎ  ] 8 ℎ ℎ 8 11

W ^b ?]^_W 

A defect and the two subsequent failures take place while job 2 is being processed. Therefore, two corresponding minimal repairs are performed and the system survives beyond job 2.

The probability corresponds to F_

a— probability

that an event C = 4 occurs during the processing

of job 2—as the system begins its life just before processing job 1 (Fig. 5b). This occurs with probability F_

W ^b 

ak U W

U

W ^b ?]

Y

U

W ^b ?]^_b 

Y

U

V

 ℎ ℎ   ℎ  8 ℎ ℎ ℎ 8

W ^b ?]^_b ^_W 

(12)

Thus, scenario 2 contains all the events that could occur in job 2 (i.e. C: 1 vw I), as no

defect occurs during job 1 (Fig. 6)

Fig. 5. Graphical representation of cases contained in scenario 2 for two jobs: (a) one minimal repair in job 2 (b) two minimal repairs in job 2

Fig. 6. Simplified graphical representation of scenario 2 for two jobs To determine the expected total makespan '(,

,Y, ,

it is necessary to obtain the

sum of the expected values of the two scenarios, as shown in (13):

'(,

,Y, 

= Q,

,Y,

+Q

Q11,2,0,1 =

, ,Y,

{ +  +  +  A A   +  +  +  +  A A  + ⋯ +  +  + 6  +  +  A A y ⋮ +  +  + 6  +  +  Ag^  A  " +  +  + 6  +  +  Ag^  A " + ⋯ + z y  +  + 26  +  +  Ag^  A g^  " x

Q, Q

,Y,

d

=j

, ,Y,k

€k

j

d

k

O€ A€ A



g^  " +

y ~ y }



 +  +  F_

Q

, ,Y,



+  +  +  F_  +  +  +  +  F_  +  + 6  +  F_  g^  d

=j

O F_

k

[

+⋯+ ‚



(13) •

For two jobs without inspection When two or more jobs are processed without any inspection between the jobs, we consider

the occurrence of an event C as if it was one job with processing time  +  . The expected total

makespan '(, '(,

,Y,Y 

,Y,Y 

=j

d

is

O D,

k

 14

The number of scenarios represents the number of terms included in the expression for the

expected total makespan for the policy [3 . To determine the number of scenarios or terms, it is necessary to know the number of inspections that will be carried out in the policy, therefore, the number of scenarios is determined using the equation ) = 2M . Thus, when a policy is evaluated

with 1, 3 and N inspections, the number of scenarios is 2, 8 and T, respectively. The content of the

scenarios represents the possibilities of renewal of the system. Fig. 7 illustrates the scenario involving three jobs with two inspections. Therefore, to represent the possibilities of renewal, it is enough to illustrate the corresponding occurrence of the defect. Thus, each term used in the expected total makespan of policy [3  corresponds to a scenario.

Fig. 7. Simplified graphical representation of scenarios for three jobs with two inspections •

For three jobs

For the processing of three jobs with processing times  ,  7
[ , first we consider the

policy that assigns two inspections,   = 1, [ = 1, where four scenarios are generated as a

result. Fig. 7 illustrates in a simplified form that each of the events contained in each of the four scenarios represents the possibilities of renewal. The following is the expected makespan expression of this policy, where each term represents a scenario.

'(, '(, j

d

€k

,[,Y,,  ,[,Y,, 

j

• '(,

'(,

d

k

O€ F

= Q. /, + Q =j

d

j

€k

_[ A€ +

d

j

€k

d

ƒk

, . /

j

j

d

d

+ Q[. /, + Qa. /,

k

A€ A ƒ A[ O€ƒ +

O€ F_

k

€ A[

+ j

d

k

O F_[ (15)

For three jobs with an inspection at the start of job 2 For this case, two scenarios are presented (Fig. 8) ,[,Y,,Y 

= Q. /, + Q d

,[,Y,,Y  = j

€k

j

, . /

d

k

O€ A€ D

,[ + j

d

k

O F_

,[ 16

Fig. 8. Simplified graphical representation of scenarios for three jobs with one inspection at the start of job 2 • '(, '(,

For three jobs with an inspection at the start of job 3 For this case, two scenarios are presented (Fig. 9) ,[,Y,Y,  ,[,Y,Y, 

= Q. /, + Q d

= j

€k

j

d

, . /

k

A[ D,

€ O€ + j

d

F_[ O

k

(17)

Fig. 9. Simplified graphical representation of scenarios for three jobs with one inspection at the start of job 3 •

For three jobs without inspection.

'(, •

For three jobs without inspection there is only one scenario ,[,Y,Y,Y 

d

= j

k

For n jobs without inspection

'(,…,!,Y,Y,…,Y  = j •

D,

d

,[ O 18

O D,

k

,…! 19

for n jobs with one inspection

In this case, it is necessary to divide the jobs into two groups; the jobs that precede the inspection and the jobs that follow the inspection, therefore
=
 +
'(!W ,!b ,Y,  = Q!W ,!b ,Y, + Q Q!W ,!b ,Y, = j

Q

!W ,!b ,Y,

•

=j

d

€k

d

O€ D,

!W ,!b ,Y,

,…,!W € j

d

O D,

k

,…,!b 

O F!W _!b  20

k

For
jobs with two inspections

Due to the two inspections, it is necessary to divide the number of jobs into 3 groups,
=
 +
+
[

'(!W ,!b ,!„ ,Y,,  = Q. /, + Q '(!W ,!b ,!„ ,Y,,  = j j

d

€k

j

d

k

d

€k

j

O€ F!b _!„  A!W € + j

d

€k

d

ƒk

, . /

j

j

d

d

+ Q[. /, + Qa. /,

O€ƒ A!W € A!b ƒ A!„  +

k

k

O€ F!W _!b € A!„  + j

d

O F!W _!„  21

k

•

For
jobs with
− 1 inspections

To establish the expected makespan it is necessary to obtain the sum of the ) scenarios, as

shown in (22)

…

'(,…,!,Y,,…,  = j

Rk

QR. /,

'(,…,!,Y,,…,  =

{ y y y y y y y

j j

d

€k

j

d

€k d

j

j

€k d

j

d

d

k d

j

k

O F_! +

k

O€ F_!?€ A! + O€ A€ F

k

O€ F_!?

_!

+

€ F!?_!

+

 y y y y y y y

⋮ z ~ d d d y y y j€k jƒk jk O€ƒ A€ A ƒ F[_! + y y d y d d yj j j O€ƒ A€ F _[ƒ Fa_! + y ƒk k y €k y ⋮ y y d d d y y j j … j O€ƒ… A€ A ƒ … A! x } €k ƒk k

22

3.2 Expected Maintenance Cost The expected maintenance cost corresponds to the relation between the replacement cost  ,

inspection cost  and repair costs  , in accordance with policy [3 . The description of the

probability of occurrence of the different events is presented in section 3.1. The expected maintenance costs are •

For one job without inspection

'-,Y  = 0A + 0A +  A[ + 2 Aa + ⋯ + 6 Ag^ '-,Y  = j •

d

PB AB dkg^ 23

Bk

For two jobs with inspection

To determine the expected maintenance cost '-,

,Y, ,

sum of the expected values of the two scenarios, as shown in (24): '-, Q,

,Y, 

,Y, =



= Q,

,Y,

+Q

, ,Y,

it is necessary to obtain a

‰

 +  A12 A21  +  +  A12 A22  + ⋯ +  +  + 6K A12 A26+2 " + ⋮

+  +  + 6K A16+2 A21 " +  +  + 6K A16+2 A22 " + ⋯ +  +  + 26K A16+2 A26+2 "

Q,

,Y,

d

=j

j

€k

Q21,2,0,1=

d

k

P€ A€ A



‹ F1_21 +  F1_22 + K +  F1_23 + 2K +  F1_24 + ⋯ + 6K +  F1_26+2 Œ

Q

, ,Y,

•

P F_

k

 24

For 2 jobs without inspection

'-, •

d

=j

,Y,Y 

=j

d

P D,

k

For n jobs without inspection d

'-,…,!,Y,Y,…,Y  = j •

k

 25

P D,

For
jobs with one inspection

,…! 26

It is necessary to establish two groups of jobs that precede and follow the inspection
=
 +


'-!W ,!b ,Y,  = Q!W ,!b ,Y, + Q Q!W ,!b ,Y, = j

Q

!W ,!b ,Y,

•

=j

d

€k

d

P€ D,

!W ,!b ,Y,

,…,!W € j

d

P D,

k

,…,!b 

P F!W _!b  27

k

For
jobs with two inspections

Due to the two inspections, it is necessary to generate 3 groups of jobs
=
 +
+
[

'-!W ,!b ,!„ ,Y,,  = Q. /, + Q '-!W ,!b ,!„ ,Y,,  = j j

d

€k

•

j

d

k

d

€k

d

j

P€ F!b _!„  A!W € + j

d

ƒk

€k

For
jobs with
− 1 inspections

, . / d

j

j

d

+ Q[. /, + Qa. /,

P€ƒ A!W € A!b ƒ A!„  +

k

k

P€ F!W _!b € A!„  + j

d

P F!W _!„  28

k

To establish the expected maintenance cost, it is necessary to obtain the sum of the T scenarios, as shown in (29)

Š

'-,…,!,Y,,…,  = j

…

Rk

QR. /,

'-,…,!,Y,,…,  =

{ y y y y y y y

j j

d

€k

j

d

j

€k d

j

€k d

j

d

d

k

k

P€ F_!?€ A! +

k d

j

P F_! +

k

P€ A€ F

P€ F_!?

_!

+

€ F!?_!

+

 y y y y y y y

⋮ z ~ d d d y y y j€k jƒk jk P€ƒ A€ A ƒ F[_! + y y d y d d yj j j P€ƒ A€ F _[ƒ Fa_! + y ƒk k y €k y ⋮ y y d d d y y j j … j P€ƒ… A€ A ƒ … A! x } k €k ƒk

29

The expected total cost corresponds to the costs incurred due to failures in the system; thus, this cost is composed of the expected tardiness cost and expected maintenance cost, as shown in (2) '()* +,  = 0 67890,  '(* +, :" −  + '-* +, 

For the first term, which represents the cost for tardiness, we assume that all jobs have a

common due date, therefore, if the makespan '(. ,/,  is larger than the due date, the difference

corresponds to the time of tardiness, which subsequently corresponds to costs for production replanning and misuse of workforce, as well as increase in inventory costs, energy cost (Masmoudi et al., 2019), among others. These costs are thus referred to as 0 , which is the cost of penalty per unit

of time. The second term corresponds to the expected maintenance cost of the policy 3 , as presented in section 3.2. As shown in (1), the objective is to determine the optimal policy that minimizes the expected total cost 1 →∗→∗2 = #

&

-
4'()* +, 5 3 ,  ,,,

The model corresponds to a problem having significant computational complexity due to

the number of integrals that the objective function contains and the corresponding evaluation of each of the possible policies, however, we assume that the processing time of the jobs that do not contain an inspection between the jobs is equal to the sum of the jobs involved. Next, the probabilities of some events are presented considering different scenarios; these can be recorded

and used when necessary. The abovementioned steps reduce the number of calculations and can, therefore, reduce the computational complexity.

4 NUMERICAL CASE To describe the proposed model we use realistic data, which can be observed in different real contexts, these data are important to evaluate the performance of the model and show the contribution of the model based on the results obtained. Let us consider a system in which the arrival of defects follows a Weibull distribution with shape parameter β and scale parameter η, and the delay time ℎ, ℎ , … , ℎ used in this model follows an exponential distribution with mean µ. The

parameters are defined in Table 2. This system should process three jobs, and the processing times are listed in Table 3. The data related to downtime by preventive replacement, inspection and repair are presented in Table 4, where the incurred costs for inspection, preventive replacement, repair and the tardiness penalty cost are also described. The common due date is 110 hours where the value of 6 for one job is three and, for two and three jobs not intervened by inspection, the values of 6 are 5 and 8, respectively.

According to the model proposed in section 3, we can obtain the optimal policy that minimizes the expected total cost. First, we calculate the expected makespan for each of the policies (Table 5) in order to obtain the expected cost for tardiness. Next, we calculate the cost of expected maintenance (Table 6) and, finally, we establish the expected total cost for each policy (Table 7) and define the optimal policy. Table 2 Parameters of defect arrival and delay time Parameters for defect arrival Weibull distribution β =3 η =140 hours

Delay time Exponential distribution µ= 40 hours

Table 3 Processing times Job 1 2 3

Processing time (hours) 19 48 35

Table 4 Costs and downtimes Description of the parameters Preventive replacement (dp, cp) Inspection (di, ci)

Downtime (hours) 3 2

Cost ($) 150 20

Repair (dr, cr) Penalty for each unit time of job tardiness downtime (ct)

65

60 50

Table 5 Expected makespan for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 116.1626 116.1626 116.1626 116.1626 116.1626 116.1626

0,1,0 117.7972 117.7912 114.9757 114.9757 116.6115 116.6115

0,0,1 112.5725 113.6012 112.5725 113.0185 113.6012 113.0185

0,1,1 114.3040 115.4870 113.4136 112.7845 115.1897 113.6347

0,1,0 33.1581 33.1581 35.9646 35.9646 33.9609 33.9609

0,0,1 43.2334 37.2013 43.2334 55.9965 37.2013 55.9965

0,1,1 62.9624 57.0615 62.1667 74.0140 56.7922 74.7555

0,1,0 423.0181 422.7181 284.7496 284.7496 364.5359 364.5359

0,0,1 171.8584 217.2613 171.8584 206.9215 217.2613 206.9215

0,1,1 278.1624 331.4115 232.8467 213.2390 316.2772 256.4905

Table 6 Expected maintenance cost for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 13.1448 13.1448 13.1448 13.1448 13.1448 13.1448

Table 7 Expected total cost for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 321.2748 321.2748 321.2748 321.2748 321.2748 321.2748

According to the results presented in Table 7, two policies allow the minimization of the

total cost. The first corresponds to 1 →∗→∗2 = Ž1,2,3, 0,0,1 and the other policy is 1 →∗ #

&

#

→∗2 = Ž2,1,3, 0,0,1. Therefore, in the first policy, the job with processing time 19 should be &

processed first, followed by the jobs with processing times 48 and 35. Further, an inspection should be performed before processing the third job. The other optimal solution is to start the job with processing time 48, followed by jobs with processing times 19 and 35. As in the previous policy, it is necessary to perform an inspection before the third job. The expected total cost for the two

optimal policies is $171.8584. The proposed model states that for a set of jobs that do not involve the inspection between the jobs, the processing time should be considered as the sum of the processing times of the jobs. Considering

the

five

policies

that

present

the

lowest

total

expected

costs,

Ž1,2,3, 0,0,1, Ž2,1,3, 0,0,1, Ž2,3,1, 0,0,1, Ž3,2,1, 0,0,1 and Ž2,3,1, 0,1,1,

four of these policies assign an inspection before job 3, which indicates that there is a greater probability of reducing the number of minimal repairs. This is associated with a lower makespan and, in turn, less tardiness time. Consequently, these policies have lower total costs compared to those of the other policies. In the best policies, the job with the longest processing time (i.e., job 2 with processing time 48) should be assigned first or, alternatively, in the set of jobs not intervened by an inspection that initiates the sequence of jobs. By reviewing each of the corresponding columns of the expected total cost, it can be seen that the least-cost policies correspond to those that begin with the job with the longest processing time.

In general, the policies that do not involve any inspection  = 0,0,0 are affected mainly

by the minimal repairs that may occur during the processing of the three jobs, which affects the

makespan. This could, in turn, increase the cost per tardiness, and require increased maintenance costs due to the costs for minimal repairs,  .

For the set of policies that involve performing the inspection at the end of job 1,  = 0,1,0

(Table 7), performing this inspection is premature as there is little probability of the occurrence of defects before job 1. Therefore, there is little possibility of minimizing the number of minimal repairs during the process of the jobs. This incurs a cost per inspection and a downtime per inspection that directly affects the makespan and generates costs for tardiness. However, when the job with the largest processing time in the sequence is assigned to the first position, there is a reduction of at least 22 percent of the total expected cost in relation to the other costs of this policy set (i.e., the policies {(2,3,1), (0,1,0)} and (2,1,3), (0,1,0)} are at least 22% less expensive than the other four policies in this column). By performing two inspections, for the parameters used on this numerical study, it minimizes the probability of carrying out minimal repairs during the process of the jobs, however, it increases the expected maintenance cost, as shown in the policy set with double inspection,

 = 0,1,1 (Table 6). A two-inspection policy does not achieve minimal cost, when compared to other policies. The sequence of jobs with minimal cost, considering two-inspection policy and three jobs corresponds to a sequence of jobs with a decrease ordering of processing times {(2,3,1), (0,1,1)}. In practice, jobs with the largest processing time are assigned first on the sequence of jobs.

It should be noted that the results obtained using the analytical model represent an approximation as we limited the number of minimal repairs to a maximum of three for one job. For two and three jobs not intervened by inspection, the maximum number of minimal repairs is five and eight, respectively, in order to ensure that the calculations are feasible. However, with the increase of processing time of the jobs, the precision of the results obtained using the analytical model decreases. Thus, in the next sub-section, we present results obtained using a simulation process, in which there is no limitation on the number of minimal repairs. Consequently, we expect that the results obtained using the simulation represent both the expected makespan and expected maintenance costs are slightly larger as the simulation includes more cases of minimal repairs. 4.1 Simulation Results To compare the results obtained from the model defined previously, and to present another method of solution, we developed a simulation model that allows the integration of a sequence of jobs and the inspection policy that minimizes the total expected cost. In sub-sections 4.1.1 and 4.1.2, we present the simulation algorithms for the makespan and maintenance cost, respectively, and, subsequently, the results obtained using the simulation are discussed. The results obtained by means of analytical model (Table 5-7) and those obtained using the simulation (Table 8-10) exhibit remarkably similar values. Like those derived using the analytical model, the simulation also produces an optimal policy as follows: 1 →∗→∗2 : Ž1,2,3, 0,0,1 or Ž2,1,3, 0,0,1 #

&

Table 8 Simulation of makespan for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 116.7063 116.7063 116.7496 116.6900 116.7424 116.6800

0,1,0 117.9441 117.9441 115.0066 115.0102 116.7094 117.0034

0,0,1 112.8653 113.8791 112.8403 113.0613 113.9809 113.1536

0,1,1 114.5542 115.6215 113.5566 112.9539 116.0023 113.9555

Table 9 Simulation of expected maintenance cost for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 13.6440 13.4820 13.3015 13.6514 13.6412 13.6297

0,1,0 33.1790 33.2058 36.1523 36.1909 34.2073 34.1722

0,0,1 43.7270 37.9802 43.8256 56.2103 37.2213 56.1597

0,1,1 63.2501 57.5223 62.8341 74.5012 56.8597 75.8310

Table 10 Simulation of expected total cost for all policies Jobs\inspections 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1

0,0,0 349.0590 348.7970 350.7815 348.1514 350.7612 347.6297

0,1,0 430.3840 430.4108 286.4823 286.7009 369.6773 384.3422

0,0,1 186.9920 231.9352 185.8406 208.2753 236.2663 213.8397

0,1,1 290.9601 338.5973 240.6641 222.1962 356.9747 272.6060

4.1.1 Simulation algorithm for expected makespan

For policy =1:
! × 2!? (refers to the evaluation of each of the policies) do

For=1: number of simulations do X: generate arrival time of the defect; the value of x should be generated for each renewal of the system If x < (the time to process all jobs after the most recent renewal of the system) then If x < (the time used to process the jobs between the most recent renewal and I inspection  , where  corresponds to the closest inspection after x) then Generate h: delay time If x + h > (the time to process the jobs until before inspection , ) then Makespan = (jobs processed between the most recent renewal and inspection , ) + inspection (inspections performed in this period) + preventive replacement else repeat Perform a minimal repair Generate h: delay time Until x + ∑  > (jobs processed between the most recent renewal and inspection ,) Makespan = (jobs processed between the most recent renewal and inspection , ) + (inspections performed in this period) + preventive replacement + minimal repairs made ( ∑ ‘’ ) End if

Else repeat Generate h: delay time Perform a minimal repair Until x + ∑  > jobs processed from the most recent renewal to finish all jobs Makespan= (Processing of all jobs from the most recent renewal) + (inspections performed) + el minimal repairs made ( ∑ ‘’ ) End if Else Makespan = (Processing of all jobs from the most recent renewal) + (inspections performed) End if End for Expected makespan= ∑ “”•–—˜”™ /
›6CœK w L6›J7vw
L End for

4.1.2 Simulation algorithm for expected maintenance cost

For policy =1:
! × 2!? (refers to the evaluation of each of the policies) do For=1: number of simulations do X: generate arrival time of the defect; the value of x should be generated for each renewal of the system If x < (the time to process all jobs after the most recent renewal of the system) then If x < (the time used to process the jobs between the most recent renewal and I inspection  , where  corresponds to the closest inspection after x) then Generate h: delay time

If x + h > (the time to process the jobs until before the inspection , ) then Maintenance cost = (cost of inspections performed from the most recent renewal) + jobs (cost of preventive replacement)( else repeat Perform a minimal repair Generate h: delay time Until x + ∑  > (jobs processed between the most recent renewal and inspection , ) Maintenance cost = cost of inspections performed from the most recent renewal + cost of renewal preventive replacement + cost of minimal repairs made ( ∑ ’ ) End if

Else repeat Generate h: delay time Perform a minimal repair Until x + ∑  > jobs processed from the most recent renewal to finish all jobs Maintenance cost = (cost of inspections performed from the most recent renewal until completion of job all jobs) +cost of minimal repairs made ( ∑ ’ ) End if Else Maintenance cost = cost of all inspections End if End for Expected Maintenance cost = ∑ -7
vœ
7
œ wLv /
›6CœK w L6›J7vw
L End for

Table 8 presents the expected makespan corresponding to each of the policies obtained through the simulation algorithm. These values present a mean relative error of 0.26% in relation to the expected makespan obtained using the analytical model. In this way, the obtained results validate the analytical model due to the similarity between the described methods and the small mean relative error presented. Table 9 presents the expected maintenance costs obtained using the simulation algorithm, which are quite similar to the values obtained using the analytical model. The mean relative error for the simulation compared to that of the analytical model is 1.31%. Due to the existing similarity between the results obtained with the methods and the small mean relative error, it is possible to maintain consistent results despite the variety of processing times of the jobs. The total expected cost obtained using the simulation model (Table 10) presents the same order of policies as obtained using the analytical model. The results obtained are related to the assigned position of the inspection and the position of the jobs; for example, {(1,2,3), (0,1,0)} and {(1,3,2), (0,1,0)} present similar costs, specifically, $430.3840 and $430.4108, respectively. It can be observed that the inspection is scheduled at the beginning of job 2. In both cases, job 1 is processed before the inspection and jobs 2 and 3 are processed after the inspection. Such observations are also noted in the results obtained using the analytical model.

There are small differences when comparing the values of the makespan and expected maintenance cost obtained using the analytical model with the result obtained via simulation. However, a slightly greater difference in the results is observed for the expected total cost. This is due to the multiplicative factor corresponding to the tardiness penalty per unit time, which occurs when the makespan exceeds the due date. Note that despite this difference, the order of the policies is identical for both models to determine even closer values between the results obtained from the models, which is enough to increase the number of minimal repairs in the analytical model. However, this consideration makes the expressions of the analytical model significantly larger and, for this reason, we consider keeping the expressions of a suitable size for an adequate presentation of the article, maintaining a tolerable level of mean relative error of 1.31% for the expected makespan and the expected maintenance cost.

5 SENSITIVITY ANALYSES Modifying the scale parameter η from 140 to 70 (cases 1 and 2 in Table 11) decreases the time of arrival of the defect. However, this increases the number of minimal repairs that must be performed during the processing of the jobs, therefore, it is necessary to evaluate both the sequencing of the jobs and the assignment of inspections. In this case, to obtain the optimal policy,

it is necessary to increase the number of inspections to  = 0,1,1 and assign the first position to

the job with the highest processing time. This policy increases the expected cost of maintenance, however, allows better control over the cost per tardiness of the jobs. Based on this modification, the optimal policy corresponds to Ž2,3,1, 0,1,1. This modification leads to a large increase in

the expected total cost due to the increased number of failures during processing. When modifying the mean µ from 40 to 20 (cases 1 and 5), the delay time decreases, leading to the increase of the minimal repairs during the process of the jobs. Consequently, it is necessary to increase the number of inspections in this way the optimal policy is Ž2,3,1, 0,1,1

When modifying  from 2 to 4, Ž2,1,3, 0,0,1 continues to be the optimal policy,

however, when changing  to values of 5 or higher, the optimal policy suggests that inspections should not be performed (cases 1, 3 and 4) as increasing the downtime due to inspections also

increases the makespan, thereby leading to a significant increase in the expected total cost. When increasing  from 3 to 15, the optimal solution continues being Ž2,1,3, 0,0,1 (case 6).

Therefore, it is possible to observe that  is more sensitive than  in terms of the expected total cost, which indicates that the use of inspection procedures with a smaller downtime decreases the expected total cost.

When  is modified to values smaller than 28, the optimal policy suggests that inspections

should not be carried out. For values between 29 and 70, the optimal policy is considered to be {(2,1,3), (0,0,1)}, and for values higher than 70, the optimal policy is {(2,3,1), (0,1,1)}. This can be

observed in cases 1, 7 and 8. Case 7 uses  = 20. For this value, the makespan ends before the due

date (110 hours) and, thus, there is no cost for tardiness. Therefore, the expected total cost is equal to the expected cost of maintenance, which in this case corresponds only to the costs related to the possible minimal repairs made during the processing of the three jobs, knowing that the optimal policy suggests no inspections. It is possible to observe that the use of minimal repair procedures with a smaller downtime significantly decreases the expected total cost.

If we increase the cost of ( from 60 to 100, the optimal policy remains {(2,1,3), (0,0,1)};

the same happens when we increase ( from 150 to 250 and ( from 20 to 60. However, ( is

more sensitive in relation to the expected total cost. For cases 1, 12 and 13, for (S greater than 7, the optimal policy is Ž2,1,3, 0,0,1 and lower costs are incurred due to inspections not being carried

out. In addition, it is observed that the expected total cost is sensitive to the variations in (S , as this

cost is related to the re-planning of production, misuse of workforce and productive inefficiency,

among others factors, and thus, each unit of time lost by tardiness represents significant costs incurred in the production process. By modifying the due date from 10 to 12 (case 14) the tardiness is reduced, allowing a

reduction of the total expected cost, and the optimal policy is Ž2,1,3, 0,0,1. In this case the due

date of 15 is increased to 20, therefore, there are no costs associated with the tardiness thus in this case it is suggested not to carry out inspections. Both the changes that can be generated in the assignment of the inspections and the changes generated in the sequence of jobs produce significant variations in the expected total cost, however, the changes produced in the inspections are more sensitive, as shown in Table 11. The following notation Žȷ, 0,0,0 indicates that the policy can have any sequence of jobs without inspections.

Table 11. Results for different variations of parameters - sensitivity analysis β

µ







(

(

(S

d

60

50

110

112.5725

43.2334

171.8584

60

50

110

133.8171

191.1025

1381.9575

60 60

50 50

110 110

114.5709 116.1626

43.2334 13.1448

271.7784 321.2748

150

60

50

110

119.5138

79.5825

555.2725

20

150

60

50

110

113.8164

43.2334

234.0534

20

20

150

60

50

110

106.3043

13.1448

13.1448

75

20

150

60

50

110

113.7486

43.2334

261.4440

case

η

1

140

3

40

2

3

65

20

150

2

70

3

40

2

3

65

20

150

3 4

140 140

3 3

40 40

4 5

3 3

65 65

20 20

150 150

5

140

3

20

2

3

65

20

6

140

3

40

2

15

65

7

140

3

40

2

3

8

140

3

40

2

3

(

'(. ,/, *

'-* +, *

'()* +, *

→∗→∗ #

&

Ž2,1,3, 0,0,1 Ž2,3,1, 0,1,1 Ž2,3,1, 0,0,1 Žȷ, 0,0,0 Ž2,3,1, 0,1,1 Ž2,1,3, 0,0,1 Žȷ, 0,0,0 Ž2,3,1, 0,1,1

9

140

3

40

2

3

65

60

150

60

50

110

112.5725

83.2035

211.8285

10

140

3

40

2

3

65

20

250

60

50

53.5997

182.2247

140

3

40

2

3

65

20

150

100

50

110 110

112.5725

11

112.5725

48.3664

176.9914

12

140

3

40

2

3

65

20

150

60

7

110

112.5725

43.2334

56.2830

13

140

3

40

2

3

65

20

150

60

200

110

112.5725

43.2334

557.7334

14

140

3

40

2

3

65

20

150

60

50

112

112.5725

43.2334

71.8584

15

140

3

40

2

3

65

20

150

60

50

120

116.1626

13.1448

13.1448

Ž2,1,3, 0,0,1 Ž2,1,3, 0,0,1 Ž2,1,3, 0,0,1 Žȷ, 0,0,0

Ž2,1,3, 0,0,1 Ž2,1,3, 0,0,1 Žȷ, 0,0,0

6 CONCLUSIONS AND RECOMMENDATIONS In this study, we develop a model that allows the integration of the sequence of jobs and inspection policy to process jobs with different processing times in a single-component system. Failures of the system are modelled using the delay-time concept. Therefore, the occurrence of a defect leads to a failure unless an inspection is performed between the jobs. If the system fails, minimal repair is performed to restore the system to be "as bad as old". Thus, immediately after performing the minimal repair, the system is in the defective state. The assignment of inspections must be performed at the beginning of the job. If an inspection detects the system to be in a defective state, preventive replacement is performed. The purpose is to reduce the expected total cost composed of the expected tardiness cost—which represents the costs incurred due to the replanning of production, the misuse of the workforce, the increase in the inventory, among other factors, when the due date is exceeded—and the expected maintenance cost. We illustrate the proposed model using a numerical study which describes that the decisions concerning the assignment of inspections and the sequencing of jobs must be carefully thought out in order to provide good performance in terms of the total expected cost. The sensitivity analysis generates a knowledge base concerning the integration of the job sequence and inspection policy for a single-component system, which can be used for the development of heuristics or meta-heuristics that allow the creation of solution methods that deliver optimal or near-optimal policies for problems involving a greater number of jobs. Although some managerial insights were described in other sections of this paper, below we include other, the batch production corresponds to the logical execution of a set of jobs that increasingly demands greater customization criteria and the proposed model could be used as an aggregate planning tool that can increase the performance of the production process. In turn, the model allows to study manufacturing systems that limit the number of minimal repairs since they can affect the quality of production. In this way, using the procedure proposed in this paper, it is possible to address a wide variety of challenges, such as bottlenecks, quality, transportation, aggregate planning, etc. Future research could consider studying different failure modes that can

affect the system, as well as deepen in the integration of the sequence of jobs and inspection policies in multicomponent systems, in order to study how this integration affects the performance of a production process.

ACKNOWLEDGEMENTS

The work of Cristiano Cavalcante was supported by CNPq (Brazilian Research Council). The work of Wilfrido Quiñones was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

REFERENCES Al-Salamah, M., 2018. Economic production quantity with the presence of imperfect quality and random machine breakdown and repair based on the artificial bee colony heuristic. Appl. Math. Model. 63, 68– 83. https://doi.org/10.1016/j.apm.2018.06.034 Alberti, A.R., Cavalcante, C.A.V., Scarf, P., Silva, A.L.O., 2018. Modelling inspection and replacement quality for a protection system. Reliab. Eng. Syst. Saf. 176, 145–153. https://doi.org/10.1016/j.ress.2018.04.002 Badía, F.G., Berrade, M.D., Cha, J.H., Lee, H., 2018. Optimal replacement policy under a general failure and repair model: Minimal versus worse than old repair. Reliab. Eng. Syst. Saf. 180, 362–372. https://doi.org/10.1016/j.ress.2018.07.032 Baptista, R., Antune Simões, J.F., 2000. Three and five axes milling of sculptured surfaces. J. Mater. Process. Technol. 103, 398–403. https://doi.org/10.1016/S0924-0136(99)00479-3 Berrade, M.D., Scarf, P.A., Cavalcante, C.A.V., 2018. Conditional inspection and maintenance of a system with two interacting components. Eur. J. Oper. Res. 268, 533–544. https://doi.org/10.1016/j.ejor.2018.01.042 Berrade, M.D., Scarf, P.A., Cavalcante, C.A.V., 2017. A study of postponed replacement in a delay time model. Reliab. Eng. Syst. Saf. 168, 70–79. https://doi.org/10.1016/j.ress.2017.04.006 Berrade, M.D., Scarf, P.A., Cavalcante, C.A.V., 2015. Some insights into the effect of maintenance quality for a protection system. IEEE Trans. Reliab. 64, 661–672. https://doi.org/10.1109/TR.2015.2417431 Cassady, C.R., Kutanoglu, E., 2005. Integrating preventive maintenance planning and production scheduling for a single machine. Ieee Trans. Reliab. 54, 304–309. https://doi.org/10.1109/TR.2005.845967 Cavalcante, C., Scarf, P., Almeida, A., Neto, C., 2015. A two-phase inspection policy for a single component preparedness system with a mixed time to failure distribution. Reliab. Risk Saf. Theory Appl. 1, 525– 530. https://doi.org/10.1201/9780203859759.ch72 Cavalcante, C.A.V., Lopes, R.S., Scarf, P.A., 2018. A general inspection and opportunistic replacement

policy for one-component systems of variable quality. Eur. J. Oper. Res. 266, 911–919. https://doi.org/10.1016/j.ejor.2017.10.032 Cavalcante, C.A.V., Scarf, P.A., Berrade, M.D., 2019. Imperfect inspection of a system with unrevealed failure and an unrevealed defective state. IEEE Trans. Reliab. 68, 764–775. https://doi.org/10.1109/TR.2019.2897048 Feng, H., Xi, L., Xiao, L., Xia, T., Pan, E., 2018. Imperfect preventive maintenance optimization for flexible flowshop manufacturing cells considering sequence-dependent group scheduling. Reliab. Eng. Syst. Saf. 176, 218–229. https://doi.org/10.1016/j.ress.2018.04.004 Ferreira, R.J.P., de Almeida, A.T., Cavalcante, C.A.V., 2009. A multi-criteria decision model to determine inspection intervals of condition monitoring based on delay time analysis. Reliab. Eng. Syst. Saf. 94, 905–912. https://doi.org/10.1016/j.ress.2008.10.001 Fitouhi, M.C., Nourelfath, M., 2012. Integrating noncyclical preventive maintenance scheduling and production planning for a single machine. Int. J. Prod. Econ. 136, 344–351. https://doi.org/10.1016/j.ijpe.2011.12.021 Gouiaa-Mtibaa, A., Dellagi, S., Achour, Z., Erray, W., 2018. Integrated Maintenance-Quality policy with rework process under improved imperfect preventive maintenance. Reliab. Eng. Syst. Saf. 173, 1–11. https://doi.org/10.1016/j.ress.2017.12.020 Hadidi, L.A., Al-Turki, U.M., Rahim, M.A., 2015. Practical implications of managerial decisions to integrate production scheduling and maintenance. Int. J. Syst. Assur. Eng. Manag. 6, 224–230. https://doi.org/10.1007/s13198-014-0291-9 Hsu, C.J., Low, C., Su, C.T., 2010. A single-machine scheduling problem with maintenance activities to minimize makespan. Appl. Math. Comput. 215, 3929–3935. https://doi.org/10.1016/j.amc.2009.11.040 Jarebrant, C., Almström, P., Löfving, M., Widfeldt, M., Wadman, B., 2018. Evaluation of flexible automation for small batch production. Procedia Manuf. 25, 177–184. https://doi.org/10.1016/j.promfg.2018.06.072 Johnsen, B., 2017. The Pulp and Paper Industry. Ind. Eng. Chem. 27, 514–518. https://doi.org/10.1021/ie50305a006 Kim, M.S., Sarkar, B., 2017. Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost. J. Clean. Prod. 144, 572–590. https://doi.org/10.1016/j.jclepro.2016.11.052 Liao, C.J., Chen, W.J., 2003. Single-machine scheduling with periodic maintenance and nonresumable jobs. Comput. Oper. Res. 30, 1335–1347. https://doi.org/10.1016/S0305-0548(02)00074-6 Lu, S., Liu, X., Pei, J., T. Thai, M., M. Pardalos, P., 2018. A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity. Appl. Soft Comput. J. 66, 168–182. https://doi.org/10.1016/j.asoc.2018.02.018 Mamabolo, R.M., Beichelt, F.E., 2004. Maintenance Policies with Minimal Repair. Econ. Qual. Control 19, 143–166. https://doi.org/10.1515/EQC.2004.143 Masmoudi, O., Delorme, X., Gianessi, P., 2019. Job-shop scheduling problem with energy consideration. Int. J. Prod. Econ. https://doi.org/10.1016/j.ijpe.2019.03.021 Pacheco, J., Porras, S., Casado, S., Baruque, B., 2018. Variable neighborhood search with memory for a

single-machine scheduling problem with periodic maintenance and sequence-dependent set-up times. Knowledge-Based Syst. 145, 1–14. https://doi.org/10.1016/j.knosys.2018.01.018 Pan, E., Liao, W., Xi, L., 2010. Single-machine-based production scheduling model integrated preventive maintenance planning. Int. J. Adv. Manuf. Technol. 50, 365–375. https://doi.org/10.1007/s00170-0092514-9 Pham, D.T., Dimov, S.S., Petkov, P. V., 2007. Laser milling of ceramic components. Int. J. Mach. Tools Manuf. 47, 618–626. https://doi.org/10.1016/j.ijmachtools.2006.05.002 Pinedo, M., 2016. Scheduling : theory, algorithms, and systems. Rose, O., 2005. Modeling Tool Failures in Semiconductor Fab Simulation. Proc. 2004 Winter Simul. Conf. 2004. 2, 826–830. https://doi.org/10.1109/wsc.2004.1371548 Sarkar, B., Cárdenas-Barrón, L.E., Sarkar, M., Singgih, M.L., 2014. An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. J. Manuf. Syst. 33, 423–435. https://doi.org/10.1016/j.jmsy.2014.02.001 Sarkar, B., Moon, I., 2014. Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process. Int. J. Prod. Econ. 155, 204–213. https://doi.org/10.1016/j.ijpe.2013.11.014 Sarkar, B., Moon, I., 2011. An EPQ model with inflation in an imperfect production system. Appl. Math. Comput. 217, 6159–6167. https://doi.org/10.1016/j.amc.2010.12.098 Scarf, P.A., Cavalcante, C.A.V., Lopes, R.S., 2019. Delay-time modelling of a critical system subject to random inspections. Eur. J. Oper. Res. 278, 772–782. https://doi.org/10.1016/j.ejor.2019.04.042 Seiti, H., Hafezalkotob, A., 2019. Developing the R-TOPSIS methodology for risk-based preventive maintenance planning: A case study in rolling mill company. Comput. Ind. Eng. 128, 622–636. https://doi.org/10.1016/j.cie.2019.01.012 Sett, B.K., Sarkar, S., Sarkar, B., 2017. Optimal buffer inventory and inspection errors in an imperfect production system with preventive maintenance. Int. J. Adv. Manuf. Technol. 90, 545–560. https://doi.org/10.1007/s00170-016-9359-9 Shneor, Y., 2018. Reconfigurable machine tool: CNC machine for milling, grinding and polishing. Procedia Manuf. 21, 221–227. https://doi.org/10.1016/j.promfg.2018.02.114 Souissi, O., Benmansour, R., Artiba, A., 2016. An accelerated MIP model for the single machine scheduling with preventive maintenance. IFAC-PapersOnLine 49, 1945–1949. https://doi.org/10.1016/j.ifacol.2016.07.915 Taghipour, S., Azimpoor, S., 2018. Joint Optimization of Jobs Sequence and Inspection Policy for a Single System with Two-Stage Failure Process. IEEE Trans. Reliab. 67, 156–169. https://doi.org/10.1109/TR.2017.2775958 Taleizadeh, A.A., Samimi, H., Sarkar, B., Mohammadi, B., 2017. Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system. J. Ind. Manag. Optim. 13, 1511–1535. https://doi.org/10.3934/jimo.2017005 Tayyab, M., Sarkar, B., 2016. Optimal batch quantity in a cleaner multi-stage lean production system with

random defective rate. J. Clean. Prod. 139, 922–934. https://doi.org/10.1016/j.jclepro.2016.08.062 Tiwari, S., Ahmed, W., Sarkar, B., 2018. Multi-item sustainable green production system under trade-credit and partial backordering. J. Clean. Prod. 204, 82–95. https://doi.org/10.1016/j.jclepro.2018.08.181 Wang, H., Wang, W., Peng, R., 2017. A two-phase inspection model for a single component system with three-stage degradation. Reliab. Eng. Syst. Saf. 158, 31–40. https://doi.org/10.1016/j.ress.2016.10.005 Wang, T., Baldacci, R., Lim, A., Hu, Q., 2018. A branch-and-price algorithm for scheduling of deteriorating jobs and flexible periodic maintenance on a single machine. Eur. J. Oper. Res. 271, 826–838. https://doi.org/10.1016/j.ejor.2018.05.050 Wang, W., 2012. An overview of the recent advances in delay-time-based maintenance modelling. Reliab. Eng. Syst. Saf. 106, 165–178. https://doi.org/10.1016/j.ress.2012.04.004 Xia, T., Dong, Y., Xiao, L., Du, S., Pan, E., Xi, L., 2018. Recent advances in prognostics and health management for advanced manufacturing paradigms. Reliab. Eng. Syst. Saf. 178, 255–268. https://doi.org/10.1016/j.ress.2018.06.021 Xia, T., Fang, X., Gebraeel, N., Xi, L., Pan, E., 2019. Online Analytics Framework of Sensor-Driven Prognosis and Opportunistic Maintenance for Mass Customization 141, 1–12. https://doi.org/10.1115/1.4043255 Xia, T., Jin, X., Xi, L., Ni, J., 2015. Production-driven opportunistic maintenance for batch production based on MAM-APB scheduling. Eur. J. Oper. Res. 240, 781–790. https://doi.org/10.1016/j.ejor.2014.08.004 Xia, T., Tao, X., Xi, L., 2017. Operation Process Rebuilding ( OPR ) -Oriented Maintenance Policy for Changeable System. IEEE Trans. Autom. Sci. Eng. 14, 139–148. https://doi.org/10.1109/TASE.2016.2618767 Xia, T., Xi, L., 2019. Manufacturing paradigm-oriented PHM methodologies for cyber-physical systems. J. Intell. Manuf. 30, 1659–1672. https://doi.org/10.1007/s10845-017-1342-2 Xia, T., Xi, L., Pan, E., Ni, J., 2017. Recon fi guration-oriented opportunistic maintenance policy for recon fi gurable manufacturing systems 166, 87–98. https://doi.org/10.1016/j.ress.2016.09.001 Yang, L., Ye, Z. sheng, Lee, C.G., Yang, S. fen, Peng, R., 2019. A two-phase preventive maintenance policy considering imperfect repair and postponed replacement. Eur. J. Oper. Res. 274, 966–977. https://doi.org/10.1016/j.ejor.2018.10.049 Ye, Y., Li, J., Li, Z., Tang, Q., Xiao, X., Floudas, C.A., 2014. Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty. Comput. Chem. Eng. 66, 165–185. https://doi.org/10.1016/j.compchemeng.2014.02.028 Zahedi, Rojali, Yusriski, R., 2017. Stepwise Optimization for Model of Integrated Batch Production and Maintenance Scheduling for Single Item Processed on Flow Shop with Two Machines in JIT Environment. Procedia Comput. Sci. 116, 408–420. https://doi.org/10.1016/j.procs.2017.10.081