Joint optimization for production operations considering reworking

Current research in this field can be broadly categorized into two areas. The first category analyzes both production scheduling and reliability. Notable studies include Aggoune (2004), who explored scheduling complexities in a process flow shop considering machine reliability, and uit het Broek et al. (2021), who investigated machine reliability control through production rate adjustments. Basciftci et al. (2020) proposed a data-driven degradation model that incorporates maintenance and production scheduling while accounting for load dependence. It should be emphasized that in this model, production scheduling is regarded as independent of the machine’s current operational status. A more recent line of research explores the interrelationship between a pair of the factors. Kim and Kim (2022) conducted a study on the interrelationship between machine degradation rate and job processing time in a single-machine scheduling problem. Briskorn et al. (2024) studied job-dependent and stochastic machine deterioration affected by processing jobs under a fixed job sequence.

The second category of research involves an analysis of both the quality as well as either job scheduling or reliability within a production system. Chen and Jin (2005) established the foundational coupling relationship between quality and reliability. Lu and Zhou (2017) proposed a novel maintenance strategy for multi-stage manufacturing systems that considered the interaction between machine failure rate and quality. Furthermore, various studies have analyzed the relationship between quality and production scheduling. Eben-Chaime (2022) developed a model that establishes the relationship between production scheduling and quality, proposing an integrated production and quality inspection system. Azimpoor and Taghipour (2021) proposed an integrated approach for production, inspection, and quality planning, taking into account the delay time failure process. These studies primarily focus on monitoring and proposing robust planning strategies to address uncertain non-conforming products. For instance, Hao et al. (2015) devised a soft-decision-based approach to handle the uncertainty caused by quality considerations. Sabri-Laghaie et al. (2022) developed a method to monitor the quality error rate and its impact on product reliability. In sophisticated manufacturing systems, the attainment of high-quality production necessitates simultaneous consideration of job scheduling, machine reliability maintenance, and product quality checks throughout the entire production process. Due to their interdependence and interaction, it is imperative to analyze these three key elements concurrently.

In response to the escalating demand for production systems, recent research has incorporated a synthesis of three foundational elements, including quality, production scheduling, and machine reliability. For instance, Cheng and Li (2020) conducted a comprehensive study encompassing these three elements within a manufacturing system but simplified the job scheduling aspect by setting an index based on machine production efficiency and assuming a predefined deterioration path for the machine. Tasias (2022) proposed a Bayesian model that ingeniously merged quality and machine reliability in a single-machine system, where the production process was presumed to exhibit a constant production efficiency. Similarly, Rivera-Gómez et al. (2021) treated production scheduling as a control variable in a manufacturing system, and Shi et al. (2024) assumed that products are produced at a constant rate in a imperfect manufacturing system. Fakher et al. (2018) integrated these three pivotal elements into a comprehensive model, wherein production scheduling predominantly contributes to the overall product quality, while disregarding the impact of job scheduling on production decisions within a manufacturing system. Sinisterra et al. (2023) developed a model that incorporated the scheduling of a sequence of resumable jobs and inspection policies in a critical single-component system, where the occurrence of defects was contingent upon the hypothesized deterioration process. Reflecting on the aforementioned literature, it is evident that research encompassing quality, job scheduling, production, and reliability tends to simplify the production scheduling aspect by representing it solely as the machine production rate. Consequently, there is currently a lack of a more comprehensive analysis for coordinating job scheduling, quality inspection, and machine maintenance within the manufacturing system.

In order to achieve a high-quality output in terms of both quantity and quality, various research efforts have incorporated rework activities into the production process (Sonntag and Kiesmüller, 2018; Berling and Sonntag, 2022). These studies have considered the existence of random defect rates within the production process. For instance, Gouiaa-Mtibaa et al. (2018) developed an analytical model for maintenance decision-making, where non-conforming products undergo rework and the rate of quality deterioration is pre-assumed. In a similar vein, Gautam et al. (2021) assumed a uniform distribution for the proportion of non-conforming products and established a green supply chain model that integrates product recovery management. Some other researchers have examined the impact of machine deterioration on products, where defective items are either added to the regular production queue at the end of the processing time or reworked promptly (Nobil et al., 2020).

However, uncertainties arise in real production systems where rework is influenced by the QRP-effect. These uncertainties relate to the status of pending jobs and the degradation path of machines, as well as the initial product quality. Proactive measures are necessary to address uncertainties in manufacturing systems. Researchers have developed algorithms employing fuzzy methods to proactively mitigate uncertainties in advance (Gao et al., 2020). In addition, robust measures such as surrogate measures (Yang et al., 2020) and slack-based measures (Hazır et al., 2010) are commonly used to plan for uncertain production environments. Recently, flexible strategies have been introduced to dynamically schedule production and enhance efficiency. These flexible strategies utilize conditional information to adapt plans to uncertain production environments, outperforming static algorithms (Shi et al., 2020). For example, a two-stage framework comprising offline and online stages, which incorporate in-situ information, is frequently utilized in production problems. Online information including the work in progress and the buffer levels is employed to update the initial solution and improve its performance (Hoffman et al., 2021). Conditional job information is utilized to develop rescheduling strategies for updating the job arrangement (Wang et al., 2019). However, a limitation of these solution frameworks is the lack of communication between the two stages, which hampers the enhancement of the initial solution’s quality and restricts the performance of the updated solution by not leveraging valuable information from the digital platform.

In conclusion, when dealing with production systems that involve rework influenced by the QRP co-effect, uncertainties permeate the entire production process. Therefore, it is crucial to develop an adaptive approach that jointly determines maintenance strategies, production scheduling, and rework activities in order to effectively navigate and optimize the complex dynamics of such production systems.

We consider an unrelated parallel system with rework consideration under the QRP-co-effect, where non-conforming products are reprocessed to meet quantity demands. The general problem can be formally described as follows: There are $n$ independent jobs, denoted as $N=\left\{1,2,...,n\right\}$, with different qualities, which need to be processed on $m$ distinct parallel machines, denoted as $M=\left\{1,2,...,m\right\}$. The manufacturing system aims to output $N$ conforming products. Due to machine degradation and variation in input job qualities, some non-conforming products require rework during the production process. These non-conforming products and their corresponding jobs are added to the list of pending jobs for reprocessing. To mitigate production losses caused by machine degradation, appropriate maintenance strategies, including corrective maintenance (CM) with replacement capability and imperfect preventive maintenance (PM), are implemented to maintain system reliability. The entire manufacturing system exhibits uncertainty and dynamic characteristics due to factors such as machine degradation paths, job processing times, pending job sequences (including rework jobs). Consequently, the joint optimization problem involves determining the optimal job scheduling, maintenance strategy, and quality control concurrently.

The joint optimization problem is based on the following assumptions:

1. 1.

   Initially, both jobs and machines are assumed to be in an available state.

2. 2.

   CM activity involves restoring a machine to its original operational status when it exceeds the failure threshold.

3. 3.

   Maintenance activity is not allowed while a machine is processing a job.

Based on the analysis of the interrelationship between quality $Q_{i,k}$, machine reliability $R_{k}$, and scheduling $P_{i}$ in the production process, three intermediate variables with uncertainties need to be introduced: machine degradation, actual job processing time, and product quality. Considering the influence of the working environment on machine degradation state $W_{i,k}$ and the QRP-co-effect, the degradation path of the machine can be represented by:

where $\Delta W_{i,k}$ denotes the change in machine degradation resulting from processing a job, while $U_{i,k}$ and $V_{i,k}$ represent the influence induced by the job and the environment, respectively. Specifically, $\Delta U_{i,k}^{(-)}$ represents the wear of machine $k$ after processing job $i$ whose initial quality is ineligible, following a Gaussian distribution (Ye et al., 2019): $\Delta U_{i,k}^{(-)}\sim Normal(\delta_{i}\mu_{k}^{(-)},\sigma_{k}^{(-)})$, where the product quality deviation $\delta_{i}=|D_{i}-SL_{i}^{(+)}|$, where $D_{i}$ and $SL_{i}^{(+)}$ represent the quality characteristic and the specification limit of the product corresponding to job $i$, respectively. $\Delta U_{i,k}^{(+)}$ represents the effect of job $i$ , and $\Delta U_{i,k}^{(+)}(p_{i})\sim Normal(p_{i}\mu_{k}^{(+)},\sigma_{k}^{(+)})$, where $p_{i}$ represents the actual processing time of job $i$. $\Delta V_{i,k}$ illustrates the effect caused by the external environment, which is characterized by a Gamma distribution: $V_{i,k}(\Delta t_{i,k})\sim Gamma(\alpha_{k}\Delta t_{i,k},\beta_{k})$, where $\Delta t_{i,k}$ denotes the time interval between the start time of job $i$ and the next job on machine $k$, expecting the maintenance time.

The actual processing time of jobs is determined by the degradation state of the assigned machine and the nominal processing time, obtained as follows:

where $p_{i,k}$ and $O_{i,k}$ represent the actual and nominal processing time of job $i$ processed on machine $k$, respectively. Job $i^{{}^{\prime}}$ is the job processed prior to job $i$ on machine $k$. The actual processing time of the job increases with the deterioration of the assigned machine. The decline in machine condition leads to a decrease in the quality of processed jobs, resulting in an increased number of non-conforming products and the need for reworking activities. To ensure high sustainability and efficient operation of the system, an appropriate maintenance strategy is necessary to maintain machine’s condition. Production scheduling, which determines job sequencing, serves as the link between product quality and machine reliability. The impact of machine degradation on product quality is expressed as follows:

where $\boldsymbol{D}$ is a dimensional vector representing the quality characteristics of products under the degradation states of machines $\boldsymbol{W}$ and the initial job quality $\boldsymbol{\upsilon}$, with the characteristics of machines being independent of each other, as well as the initial jobs. In this production system, the output of non-conforming products from the manufacturing system can lead to delivery delays and necessitate rework, which also affects the assigned machine’ condition.

The QRP-co-effect of this problem can be addressed by formulating constraints within three main parts: job scheduling, maintenance, and quality considerations. In the first part, decisions concerning the assignment of machines to jobs and the sequencing of jobs on the machines need to be made. Furthermore, the impact of machine reliability on production scheduling is also considered. The second part encompasses constraints related to maintenance activities. This includes characterizing the constraints among different maintenance actions. Additionally, the machine’s degradation path, changes in product quality, and the actual processing time of jobs under the QRP-co-effect during production are formulated. Lastly, in the third part, constraints regarding the relationship between the quality states of products and the rescheduling point of the production system are provided.

The objective of this study is to maximize overall productivity while minimizing maintenance costs. This is determined by two key factors: the makespan $C_{max}$ of jobs and the overall maintenance cost $C^{m}$. The objective function can be expressed as follows:

where the setup cost of preventive maintenance (PM) is distributed evenly among the machines within the joint maintenance group $g$.

In the planning module, our objective is to generate a resilient plan for job scheduling that includes idle spaces, without the need for rescheduling. This framework is adaptable and can accommodate various optimization algorithms. We have chosen to incorporate an Enhanced version of the Multi-Objective Differential Evolution algorithm (EMODE), due to its flexible search capability. To address the impact of uncertain rework activities on the schedule, idle spaces are introduced as a mitigation strategy. The fitness function used in the static optimization module can be defined as follows:

This evaluation is conducted by minimizing both the time cost and maintenance cost related to processing unit jobs.

In this module, the quantity of idle space $n_{e}$ is obtained using formula (34-35), given that the impact of idle space on machine degradation is equivalent to that of qualified jobs on the machine’s degradation path. To accommodate the differing processing capabilities of machines for various types of jobs, the jobs are divided based on their types. Specifically, jobs of the same type are assigned to machines with corresponding processing capabilities. All machines are initially set to their initial states, denoted as $\boldsymbol{W}^{0}$. The total number of non-conforming products $n^{0}_{e}=\sum_{i\in N}(1-q_{i})$, considering the machine’s degradation state $W_{i,k}^{0}$.

The number of idle spaces $n_{e^{th}}$ caused by jobs of the $e^{th}$ type can be expressed as:

where $M_{e^{th}}$ denotes the number of machines capable of processing jobs of the $e^{th}$ type. The efficiency of this algorithm can be further enhanced based on the following proposition.

Proposition 1: For the proposed problem, If two consecutive jobs processed on the same machine meet any of the following conditions, their processing sequences should be interchanged:

(1) Both output products are non-conforming, and the nominal processing time of the first job $i^{{}^{\prime}}$, exceeds that of the second job $i$.

(2) Both output products belong to the same type and are conforming. The nominal processing time of the first job $i^{{}^{\prime}}$ exceeds that of the second job $i$ and satisfies the following condition:

Here, $\Delta W_{i,k}$ represents the difference of machine’s states between two solution after finishing these two jobs. Furthermore, $\Delta D_{i^{{}^{\prime}}}$ represents the discrepancy between the quality feature of job $i^{{}^{\prime}}$ and the specification $SL_{i}^{(+)}$. The proof is represented in Appendix B.

In the production process, job processing plays a pivotal role in the machine’s degradation process, encompassing the impact of factors such as job sequence, quality, and quantity on the machine’s degradation trajectory. The EMODE algorithm incorporates two crucial differential evolution operators: the decoding-based Differential Evolution (DE) operator (Zhou et al., 2019) and the similarity-oriented Ranking Evolution (RE) operator (Hou et al., 2023). The DE operator introduces a significant amount of randomness, contributing to the enhancement of solution diversity. The RE operator takes advantage of the inherent data associations among different jobs, refining solutions to optimize the process. The control parameter $\nu$ in the EMODE algorithm is used to allocate computational resources for algorithm exploration and exploitation. In the early stages of the iterative process, the algorithm focuses on exploration to avoid getting trapped in local optima. As the process continues, the algorithm gradually shifts its emphasis to exploitation, leading to faster convergence. The algorithm performs well, particularly when the control parameter $\nu$ decreases over the iterations. This relationship is mathematically expressed as follows:

where $\varrho^{0}$ represents the total number of iterations, and the current iteration number $\varrho\in[0,\varrho^{0}]$. In each iteration, the control parameter $\nu$ is randomly selected within the range $[0,2]$ using this function and $rand$ is utilized to generate a random number within the range (0,1). In this algorithm, when $\nu>1$ and $rand<0.7$, the similarity-oriented RE operator is used to balance the number of pending jobs on each machine. Otherwise, the DE operator is utilized to generate a new solution. To enhance the ability of local search, two jobs are randomly selected from an individual. If one of the conditions in Proposition 1 is satisfied, the processing sequence of the jobs will be swapped. Otherwise, a job will be randomly selected from the machine with the longest total work completion time and moved to the most idle machine. This algorithm is detailed in Algorithm 1(presented in Appendix G), where the idle spaces are randomly assigned to the machine at the initial stage. A planning solution can be obtained through the EMODE algorithm to provide a job allocation strategy.

In the context of an online improvement module that incorporates rescheduling, each elite individual undergoes a reevaluation by interacting with the actual production environment. A partial rescheduling strategy is employed, which relies on conditional information to determine the rescheduling point. This point is determined based on the rate of non-conforming products, denoted as $Thr^{r}$ in formula (27). The determination of the decision rescheduling scope relies on the rescheduling strategy as presented in the referenced work (Wang et al., 2019), involving the aggregation of available idle spaces on machines within the current planning module to accommodate the inclusion of a new job. In this module, the evaluation function is designed as follows:

where $\sum_{l\in L}C_{l}^{m,o}$ and $\sum_{l\in L}T_{l}^{o}$ represent the time and maintenance cost incurred for scheduling and maintenance in the production process, respectively, and $L$ represents the number of rounds re-scheduled during the actual production. Variable $d^{o}$ represents the deviation between the planned and actual scheduling in terms of job start times and assignments, with further details available in reference (An et al., 2023).

For each rescheduling, the promising region always exists in the neighborhood of the initial plan when new jobs are inserted in the unrelated parallel machines system (Peng et al., 2019). Algorithm 2 outlines a fast local search method, which utilizes the JS (Job Swapping) and JI (Job Insertion) operators to search for high-quality solutions (Ulaga et al., 2022). This method is particularly effective in scenarios where new jobs are inserted into an unrelated parallel machines system. The combination of JS and JI operators has been shown to yield favorable results. In this approach, a job is chosen with a 0.5 probability to be inserted at position $j$ on the machine with the earliest completion time. This condition is met when the nominal processing time of the selected job is shorter than the processing time of the job assigned at the $(j+1)^{th}$ position, but longer than the job assigned at the $(j-1)^{th}$ position. Alternatively, with a probability of 0.5, the positions of randomly selected jobs are swapped. We introduce Lemma 1 to guide the job scheduling decision.

Lemma 1: In a parallel system, there always exists an optimal schedule where no idle time exists between two consecutive jobs on a machine.

The proof of Lemma 1 is provided in Appendix C.

Based on Lemma 1, idle spaces in the scheduling are filled either by rescheduled jobs or pending jobs as per the plan. An example is given to illustrate this decoding process, which is represented in Figure 7, provided in Appendix D.

This section presents numerical experiments to showcase the effectiveness of the proposed approach, encompassing the maintenance strategy and dynamic exchange module within the production process. A heterogeneous parallel production system is considered, with careful consideration given to rework in order to address potential defects and maintain exceptional quality. The experiment focuses on the utilization of four unrelated parallel processing machines with distinct capacities, taking into account different initial statuses $\boldsymbol{W}^{0}$. Additionally, two types of jobs are considered, characterized by different nominal processing times. We follow Ye et al. (2019) to initiate the production conditions as follows: The first type of jobs can be processed on machines 1, 3, and 4, with corresponding nominal processing times following a uniform distribution: $U\left(2.616,0.3\right)$. The second type of jobs can be processed on machines 2 and 4, with nominal processing times following a uniform distribution: $U\left(1.92,0.5\right)$. The quality of input jobs is assumed to follow a truncated normal distribution: $N(\mu_{q},\sigma_{q})$. The experiment explores three instances of job sizes, consisting of 100, 200, and 300 jobs, respectively. Table 9, presented in Appendix I, outlines the parameter settings for this problem, where $SL^{(+)}_{e^{th}}$ and $\xi_{e^{th}}$ denote the specification limit and qualified threshold for the quality characteristic of the $e^{th}$ job.

In order to assess the effectiveness of the proposed method, four comparative algorithms are utilized: EMA (Enhanced Memetic Algorithm) (Afsar et al., 2022), MOEA-D (Multiobjective Evolutionary Algorithm based on Decomposition) (de Moraes and Coelho, 2022), MVO (Multi-Verse Optimizer) (Kumar et al., 2023), and NSGAII (Non-dominated Sorting Genetic Algorithm-II) (López et al., 2021). The maintenance strategy employed by these comparative algorithms is based on the MCIM approach (Chen et al., 2022), where the maximum number of preventive maintenance actions for new machines is obtained through static optimization, and a right-shift rescheduling policy is adopted. In evaluating the relative performance of the algorithms, three metrics are employed. The IGD-metric (Inverted Generational Distance) and hypervolume-metric are used to assess the approximation and distribution quality of the non-dominated solution set. Additionally, the RPD-metric (Relative Percentage Deviation) is utilized to analyze the maximum completion time $C_{max}$ and maintenance cost aspects of the results. Details of the metric calculations and algorithm parameter settings are provided in Appendix I.

The results of the algorithm comparison experiment based on the IGD-metric and hypervolume-metric are presented in Table 4. Smaller values of the IGD-metric and larger values of the hypervolume-metric indicate more competitive algorithms. It is evident from the results that the DPEIA algorithm consistently achieves smaller objective function values across all instances compared to other algorithms, indicating its superior performance. Additionally, the DPEIA algorithm outperforms other algorithms with the maximum hypervolume value. Table 5 shows the experimental results of the DPEIA algorithm and the four other algorithms in terms of two objectives: maximum completion time and total maintenance cost, under the RPD-metric. Smaller values indicate better algorithm performance. The DPEIA algorithm demonstrates the best performance in both dimensions of maximum completion time and total maintenance cost.

To provide a clearer visualization of the experimental results, box plots of the four other benchmarks, along with three algorithms of ablation experiments, are shown in Figure 4 and Figure 4, representing makespan and cost, respectively. The DPEIA and DPEIA-full methods show notable advantages in terms of makespan compared to the remaining four algorithms, highlighting the effectiveness of the proposed solution framework in uncertain manufacturing systems. Regarding maintenance cost, both DPEIA and DPEIA-full exhibit superior performance with minimal variance fluctuations compared to the other four algorithms, showcasing the advantages of simultaneous maintenance and production scheduling.

Figure 6 and Figure 6 display the final set of non-dominated solutions obtained by DPEIA-full, DPEIA, and the four other algorithms. The solution set generated by the DPEIA method exhibits superior approximation and distribution diversity compared to the other algorithms. Furthermore, the DPEIA method achieves results that are very close to those obtained by the DPEIA-full algorithm while consuming approximately half of the computational resources. Based on the analysis of the experimental results, we conclude that DPEIA demonstrates exceptional performance in addressing the studied problems.

To validate the effectiveness of the proposed DPEIA algorithm, additional numerical experiments were conducted to analyze the sensitivity of the algorithm to different ranges of input job quality. Two qualified deviations, 0.03 and 0.09, were considered in the experiments. The results of these experiments, presented in Table 6, can be observed that the DPEIA algorithm remains competitive compared to the other four algorithms in this experiment.

In addition, ablation experiments were conducted to evaluate the performance of the communication mechanism and the integrated mechanism separately. The DPEIA-full algorithm, where the proposed communication mechanism was replaced by a fully rescheduling strategy, and the DPEIA-single algorithm, which considers scheduling and maintenance decisions separately, were used as comparative benchmarks. Detailed comparison results between the DPEIA algorithm and the DPEIA-full algorithm in terms of IGD and hypervolume indicators are shown in Table 8.

The experimental results show that as the number of jobs or non-conforming input jobs increases, along with an increased frequency of maintenance demands, the disparities in cost and completion time among the algorithms become more pronounced. This emphasizes the advantages of the proposed method in dynamic production environments and its suitability for making maintenance decisions. Based on the IGD and hypervolume indicators, when the job qualified deviation $\sigma_{q}$ is 0.03, the performance of DPEIA is almost the same as that of DPEIA-full, while it is slightly worse when $\sigma_{q}=0.06$. DPEIA-full has advantages when $\sigma_{q}=0.09$, but situations with a high proportion of non-conforming jobs (about 45%) are rare, and using a fully rescheduling strategy consumes a lot of computing resources without meeting practical needs. For example, the computational resources consumed by the DPEIA-full algorithm are almost 1.5 times that of the DPEIA algorithm where $\sigma_{q}=0.09$ and qualified 300 products should be completed. Meanwhile, in all experimental situations, the DPEIA algorithm has significant advantages compared to the other four algorithms, validating the competitiveness of the proposed algorithm. Furthermore, in all experimental situations, the DPEIA algorithm consistently outperforms the other four algorithms, validating its competitiveness. The DPEIA algorithm is superior in approximating the Pareto optimality and addressing joint optimization problems with uncertain factors such as quality, production scheduling, and reliability under the QRP-co-effect.