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Developing a Flexible Manufacturing Control System Considering Mixed Uncertain Predictive Maintenance Model: a Simulation-Based Optimization Approach

Abstract

Nowadays, with the development of information technology infrastructure, most systems are moving towards intelligent models, among which the field of production is no exception. One of the important sketch of production models is in term of predictive maintenance control systems such as reliability centered maintenance (RCM). This research focuses on developing a popular flexible manufacturing system called flexible job shop scheduling problem (FJSP) with predictive maintenance terms of RCM that are able to measure the level of production system reliability and determine the required maintenance activities. Moreover, since in production environments, processing time is an approximate parameter due to the activities in which manpower is involved, processing times of the model are defined with fuzzy functions. Meanwhile, to cope with stochastic nature of RCM and fuzzy nature of the process times, Buckley fuzzy numbers are implemented. In fact, this research introduces a mixed uncertain model that considers mentioned natures of probability and possibility at the same time by means developing fuzzy numbers of process time thorough confidence intervals of them. Then, since developed model is NP-Hard and stochastic, two simulation-based optimization (SBO) approaches are introduced based on two meta-heuristic algorithms called genetic algorithm (GA) and imperialist competition algorithm (ICA). Finally, various creative statistical and qualitative outputs are presented to analysis the performance of the introduced SBOs for solving the developed FJSP integrated with RCM control terms.

Introduction

FJSP, which is the extended version of JSP, consists of two sub-problems called the operation sequence and machine assignment problem, in which each operation is processed by different machines and has its own processing time. In this issue, the goal is to minimize the maximum makespan (Cmax) by optimizing the sequence of operations and allocation of machines [1]. Usually, it is assumed that any operation can be performed on a set of available machines. In other words, each machine has the ability to perform several different operations. Therefore, this issue is much more complex compared to the issue of job shop scheduling problem, because it also includes the issue of assigning operations to machines [2].

Today, production scheduling and maintenance activities are part and parcel, so that maintenance activities affect many issues related to production management such as reliability, lack of access to machinery, and various costs [3]. In most production scheduling issues, it is assumed that machines are always available, whereas in the real world this is not the case; they are often out of reach due to machine failure or the need for periodic services and production stops for a while [4].

In the meantime, one of the FJSP goals is to minimize the maximum makespan to respond to requests from the system, which is of particular importance. As mentioned, part of the available time in the production process is spent on maintenance operations. Therefore, in these cases, due to the unavailability of machinery and production breakdown, mainly supply is not done on time and this issue causes customer dissatisfaction and sometimes leads to late payment fines, which in turn increases the cost of the system.

To overcome these challenges and increase the operational reliability of a system, simply choosing a maintenance strategy cannot be considered a complete success and a different strategy must be introduced for the causes of failure. RCM has been able to be a good answer to this issue as a strategy or process that seeks multiple maintenance strategies.

Therefore, RCM is a process that determines the measures required for the maintenance of machinery and equipment in different working conditions so that each of these equipment perform their main tasks in the best possible way [5].

RCM is an intelligent method that identifies and classifies failure modes and tries to maintain system reliability at a level which prevents these modes from occurring [6]. In other words, this method uses the classic maintenance concepts such as preventive maintenance (PM) or corrective maintenance (CM) in its intelligent system and develops them with new maintenance methods and new reliability.

In the developed RCM of this research, the shocking and tracking process is randomly used to reduce the reliability level of machines [7]. Then, according to the reliability of the system and the machines, the intelligent system executes the necessary instructions in order to perform the required action at any time and on each of the machines in such a way that performance indicators such as cost are optimized and maintain system reliability at the best level [8].

Production scheduling issues are divided into types of single machine, parallel machines, flowshop, job shop, and flexible job shop, which will be followed by a summary of recent research in the field of maintenance in these issues and also a comparative Table 1 between these and previous accomplished researches.

Table 1 Literature review of joint production and maintenance systems

In the field of single-machine scheduling maintenance, Low et al. [16] integrated the single-machine scheduling problem with PM and developed several innovative algorithms. Ozturkoglu and Bulfin [17] studied the single machine workshop schedule and preventive maintenance with two objective functions under the titles of total completion times (TCT) and workshop completion time. Kim and Ozturkoglu [18] also presented a common timetable of the single machine problem with multiple preventive maintenance. Ying et al. [19] considered four single-machine scheduling problems (SMSPs) with a variable machine maintenance in their paper. The objectives of the four SMSPs were to minimize mean lateness, maximum tardiness, total flow time, and mean tardiness, respectively. Cui and Lu [20] addressed the production scheduling problem on a single machine with flexible periodic preventive maintenances (PM), where jobs’ release dates are also considered. Shen and Zhu [21] studied a single machine scheduling problem with periodic maintenance, in which processing time and repair time were nondeterministic. Liu et al. [22] presented an integrated decision model that coordinated predictive maintenance decisions based on prognostics information with a single-machine scheduling decisions so that the total expected cost was minimized.

Regarding maintenance on parallel machines scheduling, Lee and Chen [23] examined the parallel machines scheduling model and assumed that workshops could only be supported and maintained during the planning period. They also hypothesized two strategies under which the machines can be maintained and repaired. The first strategy was at the same time and simultaneously and the second strategy was separate. They also solved medium- and small-size problems with the column generation (CG) algorithm. Rebai et al. [55] also examined the scheduling of workshops and maintenance activities of parallel machines, assuming that each machine on the planning horizon is only allowed to perform maintenance activities once. In that study, the aim was to minimize TWCT and preventive maintenance costs.

Wang and Tasi [28] established a bi-objective imperfect preventive maintenance (BOIPM) model of a series–parallel system that the total maintenance cost and mean system reliability were optimized simultaneously through determining the most appropriate maintenance alternative. A bi-objective hybrid genetic algorithm (BOHGA) was established to optimize the BOIPM model too. Ebrahimipour et al. [29] considered a multi-objective PM scheduling problem in a multiple production line. They measured reliability of production lines, costs of maintaining, failure and downtime of system, as multiple objectives in their problem and applied different thresholds for available manpower, spare part inventory, and periods under maintenance. Production system in this paper consists of serial and parallel machines. Yoo and Lee [30] aimed to find a coordinated schedule for the workshop and maintenance activities to minimize the cost of scheduling provided by one of the factors of order completion time, total weight of completion times, maximum delay and total delay, have addressed the issue of parallel machines. Shen and Zhu [31] studied a parallel-machine scheduling problem with preventive maintenance which processing and maintenance time are assumed as uncertain variables.

In the field of maintenance in the flow shop scheduling problem, Safari et al. [39] with the development of condition-based maintenance (CBM) for the flow shop scheduling problem, conducted more relevant studies for this field of research and following previous research in the flow shop maintenance domain, with the aim of calculating the order and job completion time, applied the condition-based maintenance constraints and solved the proposed model with simulated annealing (SA) and forbidden search algorithms.

Naderi et al. [38] also developed a flexible flow shop by considering preventive maintenance, implementation of genetic algorithm (GA), and artificial immune system (AIS). Also, Safari and Sadjadi [40] added the failure hypothesis to the previously mentioned research. Their research did not develop a mathematical model and only simulated the concept that this simulation does not assume the possibility of breakdown between inspection times. Huang and Yu [41] proposed a two-step multiprocessor workshop scheduling environment in line with a cleaner production and minimize the completion time by formulating an integer programming. The algorithm used to solve this problem is the particle swarm optimization (PSO) method.

Khatami and Zegordi [42] investigated the simultaneous scheduling of production and planning of maintenance activities in the flow shop scheduling environment. Their problem had been considered in a bi-objective form, minimizing the makespan as the production scheduling criterion and minimizing the system unavailability as the maintenance planning criterion. Finally, they presented a bi-objective ant colony system algorithm in ordering for solving the problem. Ye et al. [43] focused on a joint scheduling problem considering the corrective maintenance (CM) due to unexpected breakdowns and the scheduled preventive maintenance (PM) in a generic M-machine flow shop.

In order to investigate maintenance in the workshop production schedule, Mati [45] also focused on minimizing the makespan in the non-preemptive job-shop scheduling environment where the machines are not available during the whole planning horizon.

As we know, on the planning horizon at some point in time, machines become inaccessible due to breakdown or the need for periodic services. Therefore, Mati in his study considered this important point as a limitation. Ben Ali et al. [46] presented a multi-objective job shop problem that optimizes maintenance costs in addition to the completion time of the workshop and finally solved it with the multi-objective genetic algorithm (MOEA). Zhou et al. [47] presented optimization of preventive maintenance for a multi-component system under job shop changes, which addresses optimization of workshop completion time and maintenance costs and uses a dynamic scheduling solution to solve it accurately.

Regarding maintenance in flexible workshop production scheduling, Wang and Yu [50] developed the issue of flexible job-shop production scheduling by taking into account maintenance measures in a flexible or pre-fixed time window and they also considered limiting maintenance resources in their model. Moradi et al. [51] integrated the FJSP and preventive maintenance problems into a dual-purpose problem that optimizes access and objective end-time functions, and finally, they used the NSGA-II genetic algorithm to solve the model. Dalfard and Mohammadi [56] also used annealing and genetic simulation algorithms to optimize the total weight of their flexible multi-objective workshop production scheduling problem, which they integrated with maintenance policies and compared the obtained results with the LINGO software results in terms of time and optimality for small-, medium-, and large-scale problems. The results showed that the use of annealing and genetic simulation algorithms is much more effective than the solutions obtained from LINGO software. Finally, he compared the results obtained from the two algorithms.

Li and Pan [57] proposed an effective Discrete Chemical Reaction Optimization (DCRO) algorithm to solve a flexible workshop production scheduling problem with limited maintenance activities. This problem is multi-objective and has three minimization objective functions to describe the maximum completion time, total machine workload (TWL), and critical machine workload (CWL), which are considered simultaneously.

Li et al. [58] proposed a discrete artificial bee colony (DABC) algorithm to solve the problem of flexible multi-objective job shop scheduling (Cmax, TWL, and CWL) and integrate it with preventive maintenance. It is obvious that the difference between the two recent studies was in the choice of algorithm used to solve the problem.

Demir and Isleyen [59] provided a comprehensive evaluation of the various mathematical models in the field of FJSP and Mokhtari and Dadgar [52] provided a common model for PM and FJSP, assuming that the failure rate is a time variable. In their proposed model, the PM activity time is fixed and the simulated annealing algorithm is used to solve the problem. Palacios et al. [53] tackled the flexible job-shop scheduling problem with uncertain processing times. The uncertainty in processing times represented by means of fuzzy numbers, hence the name fuzzy flexible job-shop scheduling. Ahmadi et al. [54] studied random machine failure in the workshop production scheduling problem using simulation. Ahmadi et al. [54] also addressed the stable scheduling of multi-objective problem in flexible job shop scheduling with random machine breakdown. Khoukhi et al. [60] investigated the issue of flexible job shop scheduling with limited machine access due to preventive maintenance activities and with the aim of minimizing workshop completion time and used ant colony algorithm to solve and optimize the model.

Rahmati et al. [3] also presented a new stochastic RCM mechanism in a common multi-objective maintenance and production planning model. In that proposed model, RCM represents and manages maintenance operations of an advanced random production scheduling problem called FJSP and is in fact a common combination of RCM and FJSP. This research designs and presents an independent and intelligent RCM that consistently displays the level of reliability that can determine which maintenance activities should be performed. In this study, four optimization algorithms based on multi-objective simulation called MOBBO, PESA, NSGAIII, and MOEAD are used to solve the problem. Also, Rahmati et al. [4] in another study have examined together the maintenance schedule and production planning. In this research, the advantage of RCM has been used to display and manage the maintenance performance of a complex random production planning problem called the workshop production scheduling problem. In this RCM research, a condition-based maintenance (CBM) approach is implemented, which is based on the concept of reliability. In this research, due to the high complexity in the joint flexible workshop production scheduling system and maintenance, the

problem is solved with a simulation-based optimization approach (SBO). The SBO searches the justified area of the problem through the genetic algorithm and the biographical-based optimization (BBO) algorithm.

Moreover, RCM parameters have both stochastic and vague features since they follow probabilistic conditional-based actions, and they include activities that are engaged with manpower. Therefore, to cope with stochastic nature of RCM and fuzzy nature of the process times, Buckley fuzzy numbers are implemented [61].

According to the information provided as well as the information expressed in Table 1, among the literature of this research, none of the references has focused on mixed considering of stochastic and approximate or fuzzy process time simultaneously. To fill this gap, this research implements Buckley fuzzy numbers which work based on the confidence intervals of the parameters. After this developing all process time-related variables, such start times and finish times of the operations, are impacted. Another affecting fact on the process times in this research is reliability level of the machines that can stop the operations if their level get lower than two controlling levels of PM and CM. In these cases, the crashed operation is postponed after PM or CM operation and the obtained Cmax is shifted too.

Therefore, the chief contribution of this research is considering stochastic and vague assumption features of the RCM and production systems simultaneously. In fact, this research introduces a mixed uncertain model that considers mentioned natures of probability and possibility at the same time by means developing fuzzy numbers of process time. To do so, first develops point estimation of vague and stochastic parameters. Then, place these confidence intervals, one on top of the other, to produce a triangular-shaped Buckley fuzzy number. In this step, because of the high level of complexity of the RCM process, our paper introduces a condition-based maintenance control process based on mentioned fuzzy numbers. This control process conducts the optimization process of our adapted meta-heuristic algorithms, named GA and BBO, by simulating the objective functions of the generated optimization algorithms. Several novel fuzzy and maintenance figures are also sub-innovation of our work. Moreover, our optimization process never violates mixed accepted uncertainty assumptions (stochastic and vague parameter and calculation) during evolution process like some research that implements this simplifier fuzzy technique and exclude parts of their process out of uncertainty assumption to have simpler calculations.

Problem Definition

In this section, our problem which integrates fuzzy stochastic FJSP with RCM controlling modules is developed.

The Concept of RCM

RCM is an intelligent method that identifies and classifies failure modes and tries to maintain system reliability at a level that prevents these modes from occurring [6]. In other words, this method uses the classic maintenance concepts such as PM or CM in its intelligent system and develops them with new maintenance and reliability methods. In the developed RCM of this thesis, the shocking and tracking process is randomly used to reduce the level of reliability of machines [7]. Then, according to the reliability of the system and the machines, the intelligent system executes the necessary instructions in order to perform the required action at any time and on each of the machines in such a way that performance indicators such as cost are optimized and maintain system reliability at the best level [8].

In this RCM, the process of degradation and shocking is responsible for the occurrence of shock and degradation of machines according to the load assigned to them at any time. In addition, the proposed degradation based on the system reliability level that is a function of arisen random shocks, predicts and determines the necessary measures. It should be noted, however, that in this study, the reliability function is merely a function of shocks and time, but in the extended versions of RCM this can also be a function of other factors and variables.

In addition, in this research, two general types of CM and PM measures have been considered, according to the reliability level in RCM. In this proposed RCM model, if the reliability decreased from the first critical L threshold, the process proposes PM and if it is lower than the LL failure rate, a correction or replacement occurs.

Figure 1 shows the fluctuations of the reliability level, failure modes, and random variables of the problem along with the maintenance activities performed in accordance with the reliability status. Also, the descriptions of the symbols and numbers introduced in Fig. 1 are given in Table 2.

Fig. 1
figure1

The maintenance activities according to the reliability level

Table 2 The descriptions of the symbols and numbers introduced in Fig. 1

It is important to note that the letter R in RLPM and RLCM in Fig. 1 has a new reliability centered definition, not level of improvement. S values in this figure also indicate the time of shocks, which reduces the reliability of the machine in the simulation process. This example includes the six shocks S1 to S6, presented on the horizontal axis. M values, such as M1 and M2, also indicate the jth maintenance time of the machine.

After shocks S1 to S3, the reliability of the machine is still higher than L and in the green zone. Therefore, the device does not require any maintenance activities. The fourth accidental shock (S4) then reduces the machine reliability toward the L preventive maintenance (yellow zone). Therefore, during 2 T inspection, PM maintenance activity can be detected. PM maintenance activities are responsible for the reconstruction and improvement of the degradation level in M1 and the return of reliability to green area. At this level of reliability, the machine works until the S5 happens. Since the machine reliability level after the S5 shock is lower than the LL and in the red zone, CM maintenance must be performed.

PM and CM have distinct differences as follows [39].

  • ✔ PM includes scheduled actions that are only done on a predetermined metric such time or duration of usage while CM is not usually predetermined and is performed at random intervals.

  • ✔ CM can happen between inspection intervals and causes machine failure.

  • ✔ PM is usually performed before equipment failure while CM is performed after that.

  • ✔ It restores reliability to a reliability level of a new machine.

  • ✔ In fact, PM mainly aims to avoid CM and expensive repairs.

  • ✔ PM cause far fewer system downtime than CM.

Figure 2 shows a random variable that shows the level of reliability of the machine (Rel (m)). The reliability function of the machine at any time (Rm (t)) is based on the performance of the proposed Eq. (1). In this equation, β0 and β1 are the parameters of the reliability reduction rate and the weighted mean of the reliability centered critical level, i.e., DM = (L + 4 * LL) / 5. Machine degradation (DLm) or Dm (t) follows the exponential distribution with parameter \(\left({DL}_{m}\sim Exp\left(\eta \right)\right)\) It should be noted that in this chapter, DLm and Dm (t) indicate machine failure and RLm and Relm (t) indicate machine reliability.

Fig. 2
figure2

Procedure of updating the reliability at any simulated time

$${Rel}_{m}\left(t\right)=\frac{{e}^{-{\beta }_{0}{D}_{m}\left(t\right)}}{1+{e}^{{\beta }_{1}\left({D}_{m}\left(t\right)-DM\right)}}$$
(1)

Classical FJSP Simulation Considering Fuzzy RCM

One of the common methods of control in maintenance systems is failure-based control, but in this study, control is reliability centered. Therefore, the parameters and variables DLm and Dm (t) are defined with RLm and Relm (t). The model assumptions are as follows:

  • Simultaneous processing times have two categories of uncertainty and randomness of uncertainty.

  • The action of each task is fixed and predetermined.

  • Tasks do not take precedence over each other, but the actions of a task take precedence over each other.

  • There are no restrictions between different tasks operations and they do not have priority over each other.

  • Machines are available from zero time.

  • Works start at zero time.

  • Machine startup time ignored.

  • Shift time between operations is negligible.

  • At any given time, each machine can only perform one operation.

  • The time between two shocks is random.

  • The rate of degradation due to shocks is random.

  • During the process, depending on the level of machine failure, the operation may be interrupted but the jobs is not reversible.

  • The PM time of the machines follows the probable distribution. The machines are not necessarily the same in terms of the type of distribution and its parameters.

  • The CM time of the machines follows the probabilistic distribution. The machines are not necessarily the same in terms of the type of distribution and its parameters.

  • The level of failure recovery after PM is random and does not necessarily adjust to the initial value.

  • The level of failure recovery after CM is set to the initial value.

  • The inspection system reports and shows the actual level of damage to the system without any errors.

According to the above hypotheses, the model of the problem is as follows:

$$Min{Z}_{1}={\tilde{C }}_{max}=max\left({\tilde{S }}_{ijk}+{\tilde{P }}_{kij}\right)\quad\quad\quad\quad\forall i,j=1,k\in {M}_{ij}$$
(2)

Subject to:

$${\tilde{c }}_{ij}-{\tilde{c }}_{ij-1}\ge {\tilde{P }}_{kij}\times {v}_{ijk}\quad\quad\quad\forall i,k,\forall j=2,\dots ,{J}_{i}$$
(3)
$${\tilde{c }}_{ij-1}\ge {\tilde{P }}_{kij}\times {v}_{ijk}\quad\quad\quad\forall i,j=1,k\in {M}_{ij}$$
(4)
$$\left({\tilde{c }}_{hg}-{\tilde{c }}_{ij}-{\tilde{P }}_{khg}\right)\times {v}_{hgk}\times {v}_{ijk}\times {z}_{ijhgk}\ge 0\quad\quad\quad\forall i,h,j,g,k\in {M}_{ij}\cap {M}_{hg}$$
(5)
$$\left({\tilde{c }}_{ij}-{\tilde{c }}_{hg}-{\tilde{P }}_{kij}\right)\times {v}_{ijk}\times {v}_{hgk}\times {z}_{hgijk}\ge 0\quad\quad\quad\forall i,h,j,g,k\in {M}_{ij}\cap {M}_{hg}$$
(6)
$$\sum_{k\in {M}_{ij}}{v}_{ijk}=1\quad\quad\quad\forall i,j$$
(7)
$${z}_{ijhgk}+{z}_{ijhgk}={v}_{ijk}\times {v}_{hgk}\quad\quad\quad\forall i,h,j,g,k\in {M}_{ij}\cap {M}_{hg}$$
(8)
$${\tilde{c }}_{ij}\ge 0\quad\quad\quad\forall i,j$$
(9)
$${v}_{ijk}\in \left\{\mathrm{0,1}\right\}\quad\quad\quad\forall i,j,k$$
(10)
$${z}_{ijhgk}\in \left\{\mathrm{0,1}\right\}\quad\quad\quad\forall i,j,h,g,k$$
(11)
$$0\le Rel\left(m\right)\le 1$$
(12)

Equation (2) as the first objective function of the problem deals with minimizing the stochastic makespan. Equation (3) controls the prerequisite for the actions of a task and forces each task to follow a specific operating sequence. Equation (4) controls a task makespan, which is surely larger than the uncertain processing time. As we know, the makespan of an operation due to maintenance operations may not necessarily be equal to the processing time of that operation, so this relationship ensures that the makespan of the first operation is at least equal to the processing time.

Equations (5) and (6) also serve as separator constraints, indicating that the operation Ohg should not begin before uncertain completion time of Oij or that the operation Ohg should end before Oij, if assigned to the same machine. Equation (7) states that a machine must be selected from a set of available machines for each operation and Eq. (8) causes one of the two priority relations to be selected (Table 3).

Table 3 The definition of processing time in developed RCM model

The fuzzy parameters of the model are formulated by Buckley numbers. In fact, Buckley fuzzy numbers consider stochastic and vague assumption simultaneously. These two assumptions are also two dimensions of the uncertainty. In the meantime, most of the popular fuzzy numbers, such as triangular or trapezoidal fuzzy numbers, only consider vagueness feature of some real world parameter. However, Buckley’s method goes forward and makes the classical assumption much closer to the real world environment and mixed these two features of the real world uncertain parameters by means of the confidence intervals. In the fuzzy estimation by Buckley method, we should let X be a random variable with probability density function (or probability mass function) f (x; θ) for single parameter θ. Assume that θ is unknown and it must be estimated from a random sample X1, …, Xn. Let Y = u (X1, …, Xn) be a statistic used to estimate θ. According to the values of these random variables \({X}_{i}\) = \({x}_{i}\), 1 ≤ i ≤ n, we obtain a point estimate \({\theta }^{*}\) = y = u (\({x}_{1, }{\dots ,x}_{n})\) for θ. We would never expect this point estimate to exactly equal h, so we often also compute a (1 − β) 100% confidence interval for θ. We are using β here since α, usually employed for confidence intervals, is reserved for α- cuts of fuzzy numbers. We propose to find the (1 − β) 100% confidence interval for all 0.01 ≤ β < 1. Starting at 0.01 is arbitrary and it can also start with smaller values. These confidence intervals as Eq. (13).

$$\left[{0}_{1}\left(\beta \right),{0}_{2}\left(\beta \right)\right],0.01\le \beta <1$$
(13)

Add to this the interval [\({\theta }^{*}\),\({\theta }^{*}\)] for the 0% confidence interval for θ. Then, we have (1 − β) 100% confidence intervals for θ for 0.01 ≤ β < 1. Now place these confidence intervals, one on top of the other, to produce a triangular-shaped Buckley fuzzy number \(\stackrel{\sim }{\theta }\) whose α- cuts are the confidence intervals. We have \(\stackrel{\sim }{\theta }\)[α] = [\({\theta }_{1}\)(α), \({\theta }_{2}\)(α)], 0.01 ≤ α < 1.

Moreover, our optimization process never violates mixed accepted uncertainty assumptions (stochastic and vague parameter and calculation) during evolution process like some researches that implement this simplifier fuzzy technique and exclude parts of their process out of uncertainty assumption to have simpler calculations.

Besides, RCM parameters have both stochastic and vague features since they follow probabilistic conditional based actions and they include activities that are engaged with manpower. Therefore, Buckley fuzzy is implemented in our process design. More description of the method and difference with classical numbers are referred to the main research to reduce reintroduction [61].

Algorithm Simulation Factor

As mentioned in the developed scheduling model, the multi-objective FJSP includes various random components such as RL, PMD, RLPM, CMD, RLCM, or TBS to provide a near-realistic version of RCM. These variables dynamically change the response modes and are automatically controlled and managed at every instant of random programming under the control of the simulator controller, meta-heuristic algorithms optimizing. Figure 3 presents the general structure of the proposed simulator.

Fig. 3
figure3

General structure of proposed simulated model

The input to Fig. 3 is an answer of optimization process and the output is a simulated version of the objective function. The simulation algorithm implements a simulation loop with the number of iterations of Numsim to obtain the mean value of the objective functions of the answers, to report a robust solution to the optimization algorithm. In this flowchart, dt sets the simulation sampling time.

Figure 4 is the proposed decision-making function of maintenance measures, and its costs are based on instantaneous reliability and perform the instantaneous reliability update operation based on the latest status of the devices. According to Fig. 4, the necessary decisions are determined based on the area in which the reliability is located. These areas are also shown in Figs. 1 and 2. The operator is also responsible for updating the accidental maintenance costs based on the load assigned to the machines and the maintenance operations performed on them and implements paragraph one of the While loop measures of Fig. 3. Due to the fuzzy nature of the problem, all parameters related to time and cost are marked with a “ ~ ” which indicates the fuzzy nature of these items.

Fig. 4
figure4

Proposed reliability updating and maintenance decision function

Figure 5 shows the instantaneous update function of job and machine modes for RCM, which, like Fig. 4, is active with respect to their processing times and maintenance actions. This decision function determines the start and end modes of tasks and machines at each instant of the simulation clock.

Fig. 5
figure5

Proposed machine and job status determination function

Figure 6 also shows the timing function of shocks used by other operators. In the event of a shock, the simulation-based optimization (SBO) updates the level of reliability of the machines during the operation time of each task operation on that machine and provides appropriate maintenances accordingly.

Fig. 6
figure6

Proposed shock generation function for fuzzy RCM

Solving Methods

This section develops two simulation optimization approaches of the research based on the GA and ICA metaheuristic algorithms. Actually, these paper introduces two fuzzy simulation-based optimization (SBO) algorithms named ICA and GA. These general huge SBO algorithms have following sub-parts.

The fuzzy process is based on Buckley approach that is addressed in previous comment. The RCM’s sub-part that engaged in the modules of our metaheuristic algorithms are all selected from related professional references and then introduces an integrated process for conditional monitoring of the RCM activities. Various figures are also provided to present this integrated RCM process.

Then, all sub-parts are implemented in two SBO versions of GA and ICA. The reason for choosing these two algorithms is that ICA is designed for random key or continuous solution structure while GA is designed for both continuous and discrete structures. Therefore, the paper presents a better opportunity for future algorithm adaption of the researches. Moreover, these two different optimization processes can be used to validate each other on a problem with such complexity and vast search area.

Genetic Algorithm

Genetic algorithms are random search techniques based on natural selection and natural genealogy. This algorithm is one of the most important meta-heuristic algorithms that is used to optimize various functions. In this algorithm, past information is extracted due to the inheritance of the algorithm and used in the search process [62].

Algorithm Structure

As shown in Fig. 7, in this study, the genetic algorithm developed from the first step, which is the development of the model and example to the end, performs all the calculations in a fuzzy environment. The listed simulation operators are described in Figs. 1, 2, 3, 4, and 5. Up to this point, an iteration or a generation of algorithms has been completed. After several generations, the algorithm gradually converges towards the optimal answer. The condition for stopping the problem is to go through a certain number of iterations, which is determined by the user before the start of the algorithm. The general structure of a genetic algorithm can be expressed in Fig. 7.

Fig. 7
figure7

The structure of the GA used in this research

In this research, the mentioned processing times are fuzzily entered into the algorithm. For this purpose, it is assumed that the processing times follow the normal probability distribution and the Buckley method is used to make fuzzy the data throughout the solution path. The fuzzy inputs listed in Table 4 are shown in Fig. 8 [61].

Table 4 Example of the crisp FJSP with three jobs, four machines, and eight operations
Fig. 8
figure8

Fuzzy input of FJSP

In this diagram, which shows the fuzzy input data matrix in Table 4, if a machine could not do a task, it is not provided a fuzzy number for that as the same. In addition, algorithms never deviate from the intended fuzzy space during the optimization path.

In this research, the answer structure consists of two vectors as shown in Fig. 9. Figure 10 shows how to decode this answer to calculate the objective function.

Fig. 9
figure9

Solution structure including sequencing and assignment vectors

Fig. 10
figure10

Decoding method to calculate the objective function

In Fig. 10, the first vector shows the sequence of operations of different tasks. The second vector shows the machine assigned to each operation among the four available machines. In this regard, the ability of the machine to perform the operation has also been checked.

Crossover

The crossover operator in this research is designed in such a way that the feasible structure of the solutions is not destroyed. This algorithm has three steps. Figure 11 also presents an illustration of this operator.

Fig. 11
figure11

Crossover structure for sequence and assignment vector

Mutation

In this research, location swap is used for the mutation operator. The algorithm of this operator is shown in Fig. 12 in the following three steps. This structure provides two different types of the mutation including the uniform and the random point with same probability during the evolution process.

Fig. 12
figure12

Mutation structure for sequence and assignment vector

Imperialist Competitive Algorithm

The second meta-heuristic algorithm of the paper is ICA. Since our solution structure is real code and continuous (Figs. 9 and 10) the basic platform of the basic reference is implemented and all process are referred to that [63].

Analysis and Numerical Results

Solving the Classical FJSP

Since this study simulates the implementation of the RCM system in the flexible job shop production scheduling problem, we need to solve the classical FJSP to ensure the correctness of the basic problem model. For this purpose, we consider two GA and combined ICA with simulated annealing and solve FJSP by these two algorithms and compare their results with each other. In fact, the validity of the solution results is obtained by comparing the results of solving the basic problem by two mentioned algorithms. We will also compare these results with the literature to ensure the accuracy of the baseline model. It should be noted that the computer system used to solve this research problem was a personal laptop with a CORE i7 CPU and 8 GB RAM.

Design of Experiments and Parameter Adjustment of Hybrid Genetic Algorithm with Simulated Annealing Algorithm by Taguchi Method

The basic parameters of the problem are considered at three levels as described in Table 5. Considering the number of these parameters and the levels intended for each of them, in order to reduce the frequency of problem solving, we use the technique of designing experiments in Minitab software and finally setting the parameters by Taguchi method.

Table 5 The parameters of hybrid Genetic algorithm with Simulated Annealing algorithm

By executing the experiment design in the software, 27 experiments are suggested as described in Table 6 (Plan section). The output of each of the described experiments is then taken on the basis of the Brandimart problem, with information on Cmax, Mean Cmax, and Time being extracted and listed in Table 6 (Output section).

Table 6 Design of experiments and solving the classical FJSP

In this research, the signal to noise ratio (S/N) method is used to analyze the results and outputs of the previous step. Therefore, the S/N diagram and the ranking table of influential factors are as follows. As you know, in the S/N method, the signal to noise ratio is checked and its maximum value is always the most optimal mode. Therefore, in Fig. 13, the maximum values are selected, which are marked with red circles and then, based on each parameter, the optimal counter and its corresponding value are extracted according to Table 5, which are mentioned in Table 7.

Fig. 13
figure13

The S/N diagram of hybrid Genetic algorithm with Simulated Annealing algorithm

Table 7 The optimal counter and its corresponding value

In Fig. 14, in the Rank row, the most effect of the parameters is observed in optimization, which due to this gist, iteration parameter has the most effect, followed by the parameters Maxit, Pm, Maxsubit, Alpha, Popsize, and Pc, respectively.

Fig. 14
figure14

Ranking of influential factors

Now, based on the results in Table 5, the Brandimart’s Flexible Job shop scheduling problems are solved with Matlab software and the outputs are mentioned in Table 6.

Design of Experiments and Parameter Adjustment of Hybrid ICA-SA Algorithm by Taguchi Method

The basic parameters of the problem are considered in three levels as described in Table 7. In this section, we will use the technique of designing experiments in Minitab software and adjust the parameters by Taguchi method.

By executing the experiment design in the software, 27 experiments are suggested as described in Table 8 (Plan section). The output of each of the experiments described is then taken on the basis of the Brandimart problem, with information on Cmax, Mean Cmax, and Time being extracted, which is listed in Table 8 (Output section).

Table 8 The outputs of classical FJSP according to the tuned parameters

In this section, as in the previous algorithm (genetic algorithm combined with annealing simulation algorithm), the signal-to-noise ratio (S/N) method is used to analyze the results. Therefore, by implementing this method in Minitab software, the relevant chart and ranking table of influential factors are presented as Figs. 15 and 16.

Fig. 15
figure15

The S/N diagram of hybrid Imperialist competition algorithm with Simulated Annealing algorithm

Fig. 16
figure16

Ranking of influential factors

As you know, in the S/N method, the signal to noise ratio is checked and its maximum value is always the optimal. Therefore, in Fig. 15, the maximum values are also indicated by red circles. The ranking table of influential factors is also presented in Fig. 16, which according to the information in the Rank row, it can be seen that the Popsize parameter has had the greatest effect on optimization. The parameters MaxDecades, nImp, Maxsubit, Maxit, and Alpha have the following rankings (Tables 9 and 10).

Table 9 The parameters of hybrid Imperialist competition algorithm with Simulated Annealing algorithm
Table 10 Design of experiments and solving the classical FJSP

By identifying the optimal points (red dots) in Fig. 15 and considering their counter, the corresponding value of each is extracted according to Table 9 and is presented as described in Table 11.

Table 11 The optimal counter and its corresponding value

Based on the information in Table 12, the Brandimart’s Flexible Job shop scheduling problems are solved with Matlab software and the outputs are mentioned in Table 12.

Table 12 The outputs of classical FJSP according to the calculated tuned parameters

Evaluation and Validation of the Results

In order to ensure the accuracy of the outputs obtained in Tables 8 and 12, we compare the outputs with the literature. For this purpose, we review Rahmati and Zandieh [64] research in which a new biography-based optimization algorithm (BBO) for a flexible job shop job scheduling problem is presented. In this research, the results of applying the proposed BBO algorithm have been compared with Ho and Brandimart problems. Table 13 shows the detailed outputs in Tables 8 and 12 related to Cmax along with the basic problems, which is observed that there is no significant difference between the outputs obtained from the problem of this research and the literature. Therefore, it can be concluded that the outputs have the necessary validity.

Table 13 Evaluation and validation the results

After ensuring the accuracy of the obtained results, we will compare the two algorithms used based on the results obtained in Tables 8 and 10. For this purpose, comparative graphs of the results are drawn and also these results are analyzed using the test of basic assumptions (Fig. 17).

Fig. 17
figure17

Comparison on Cmax, Mean, and Time

Statistical Analysis of Results

In this section, in order to analyze the results and reach the conclusion whether there is a significant difference between the results or not, we use the statistical hypothesis testing. Therefore, in order to examine the evidence, whether there is a significant difference between the results of the used algorithms (population of the two communities under study) or not, the hypothesis test is defined as follows, which is performed by Minitab software and t-test method.

$$\left\{\begin{array}{c}{H}_{0}:{\mu }_{GA,SA}={\mu }_{ICA,SA}\\ {H}_{1}:{\mu }_{GA,SA}\ne {\mu }_{ICA,SA}\end{array}\right.$$

By selecting the data contained in the Cmax, Mean, and Time columns related to the problem solving results by the two algorithms and performing the t-test, P-value is obtained. Also by the comparison of P-value and α = 0.05, the results are obtained which are mentioned in Table 13. If P-value is larger than α, hypothesis H0 and otherwise hypothesis H0 is accepted. Hypothesis H0 states that there is no significant difference between the means of the two statistical populations (Table 14). The boxplot of statistical analysis is presented in Fig. 18.

Table 14 The summary result of statistical analysis
Fig. 18
figure18

Boxplot of Cmax, Mean, and Time for two methods

Solving Fuzzy Stochastic RCM Problem in FJSP with GA

In this part, the proposed model for the single-objective maintenance problem based on fuzzy stochastic reliability in FJSP environment is solved by Genetic algorithm in Matlab software. The objective function of this problem addresses the optimization of makespan (Cmax). In the previous sections, for solving the classical FJSP, the parameters of algorithm were considered according to Table 5 and finally by using the Taguchi method, the parameters were adjusted and then according to optimal value, the outputs of problem were received and recorded. Due to the complexity of the problem under investigation and the limitations of the computer system, it is not possible to solve the maintenance problem based on fuzzy stochastic reliability in FJSP with the parameters obtained by Taguchi method, so in order to determine the value of these parameters, we use the literature parameters. Rahmati et al. [3] considered the values in Table 15 for each of the parameters of this algorithm in their paper.

Table 15 Genetic algorithm parameters

According to the parameters in Table 15 and solving the problem, the outputs are obtained as described in Table 16. Each row of the table shows the obtained fuzzy objective function in two formats of the fuzzy number table and figure. They show how Buckley method put confidence intervals of the vague stochastic parameters to create their fuzzy numbers.

Table 16 The final results of solving the fuzzy stochastic RCM by GA

In Table 16, the responses are presented by a matrix with 10 rows and 3 columns due to the fuzzy logic is followed. In order to observe the optimization trend, the machine assignment to operation diagrams are presented as the Gant chart for the first, third, and 10th problem in Fig. 19. In this figure, PM and CM are drawn by black boxes. According to these diagrams, when a PM is done, the reliability level increases to upper level of L (The Green zone), while when CM is done, the reliability level increases to one.

Fig. 19
figure19

Gant chart of the 1st, 3rd, and 10th fuzzy stochastic RCM problem by GA

Figure 19 illustrates the RCM process and the Gant charts related to each reliability control. In the Gant charts the black boxes include two types of the maintenance activities including the PM and the CM. These boxes force the complementation time of the Gant chart to be increased.

Solving Fuzzy Stochastic RCM Problem in FJSP with ICA

In this part, the proposed model for the single-objective maintenance problem based on fuzzy stochastic reliability in FJSP environment is solved by Imperialist competition algorithm in Matlab software. By the similar explanations in Sect. 4.2, for determination of the parameters, we act according to the literature parameters. So the parameters of this algorithm for this section are selected in the range of literature parameters which are presented in Table 17.

According to the parameters in Table 17 and solving the problem, the outputs are obtained as described in Table 18. Each row of the table shows the obtained fuzzy objective function in two formats of the fuzzy number table and figure.

Table 17 Imperialist competition algorithm parameters
Table 18 The final results of solving the fuzzy stochastic RCM by ICA

Also in this part, in order to observe the optimization trend, the machines assignment to operations diagrams are presented as the Gant chart for the first, third, and 10th problem in Fig. 20.

Fig. 20
figure20

Gant chart of the 1st, 3rd, and 10th fuzzy stochastic RCM problem by ICA

Figure 20 illustrates the RCM process obtained from ICA algorithm. As can be seen, just like GA algorithm the black boxes obtained from ICA are also two types that are determined according to the level of the reliability. For reliability level upper than 0.86 no maintenance activity is done even on the scheduled times of the PM. But, when the reliability level decreases the between PM and CM level, PM actives down the production system when the scheduled times of the PM are met. Besides, as the reliability level goes lower than CM level, CM activities substitute the failed part.

Conclusion

This research focused on the development of FJSP problem by relaxing the assumption of no degradation and implementing RCM controlling modules for determining types of the required maintenance activities according to the real-time condition of the machines. In addition to stochastic nature of the RCM, in our mixed uncertain model, approximate nature of the processing times are also considered by means Buckley fuzzy approach. The developed integrated model creates fuzzy numbers by confidence intervals of the stochastic process times. Then, the whole process of the optimization from the problem generation to objective function calculations is done in mentioned fuzzy environment and never violates the fuzziness to crisp environment. To solve the developed problem, two SBO directed in GA and ICA implemented. Taguchi tuned the parameters of the algorithms. To show the performance of the algorithms, first they solved benchmark problems of the literature in crisp environment and shown non-dominated performance. Then, by means of various creative outputs, the performance of the real-time simulation of RCM depicted in real time controlling Gant charts. Moreover, the final novel fuzzy Cmax of the test problems were also illustrated. Different statistical and graphical outputs also prepared to show the detailed aspects of the developed problem and algorithms. For future work, considering queuing theory, dual resource constraint or redundant inventory control are of interest.

Availability of Data and Material

Not applicable.

Code Availability

Not applicable.

Abbreviations

t:

Time t

m:

Machine

j :

Job j

\({O}_{ij}\) :

Operation jth of job ith

\({\tilde{C }}_{max}\) :

Uncertain makespan

\({\tilde{s }}_{ijk}\) :

Uncertain start time of \({O}_{ij}\)

\({\tilde{p }}_{ijk}\) :

Uncertain processing time of \({O}_{ij}\)

\({\tilde{c }}_{ij}\) :

Uncertain completion time of \({O}_{ij}\)

\({M}_{ij}\) :

Capable machines

\({v}_{ijk}\) :

\({v}_{ijk}\in \left\{0.1\right\}\) Assigning \({v}_{ij}\) to machine k

\({z}_{ijhgk}\) :

\({z}_{ijhgk}\in \left\{0.1\right\}\) Precedence of \({O}_{ij}\) and \({O}_{hg}\) on machine k

RL { m } :

Reliability of machine m

PMD:

Stochastic PM duration

RLPM:

Stochastic recovery level through PM

CMD:

Stochastic CM duration

RLCM:

Stochastic recovery level through CM

TBS:

Stochastic time between two shocks

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Shahbazi, B., Rahmati, S.H.A. Developing a Flexible Manufacturing Control System Considering Mixed Uncertain Predictive Maintenance Model: a Simulation-Based Optimization Approach. Oper. Res. Forum 2, 51 (2021). https://doi.org/10.1007/s43069-021-00098-5

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Keyword

  • Flexible manufacturing model
  • Predictive maintenance system
  • Simulation-based optimization
  • Uncertainty
  • Meta-heuristic algorithms