A concise guide to scheduling with learning and deteriorating effects

In practical manufacturing systems, the job processing time usually varies with the performance change of manufacturing resources, among which the learning and deteriorating effects are typical characteristics. Due to the interests from both academic exploration and industrial innovation, the research on scheduling problems with these effects is abundant and diverse. However, some studied problems need to be strengthened in combination with realistic production scenarios. This paper provides a concise guide to scheduling problems with these effects, giving a comprehensive review and critical hints for future research. A novel classification scheme is designed based on four levels of different domains, i.e. effects, processing ways, processing time functions, and manufacturing environments. Based on this scheme, the scheduling problems are first distinguished into three categories: learning effects, deteriorating effects, and combined effects. In each category, models are then refined along three lines: general processing way, batch scheduling, and group scheduling. Combined with the attributes of actual processing time functions and manufacturing environments, the evolvement of related scheduling models and a critical analysis on the proposed algorithms are well analysed. Afterwards, the research gaps are revealed and the research directions are indicated from the perspectives of practical applications, time functions, and designed algorithms.


Introduction
In numerous studies of scheduling problems, the processing time of a job is assumed to be constant. However, the job processing time actually changes in various practical manufacturing systems. In some common cases, with repeated work, workers can learn and perform more efficiently in production processes. Through the improvement of workers' skills and experience, the processing time of jobs will gradually decrease (Mosheiov 2001). Such a phenomenon is called as 'learning effect'. Interestingly, there is an opposite phenomenon where the jobs processed later take more time. With the increment of output, machines start to wear out and the conditions become much stricter, the processing time of a job may increase with the delay of starting time (Mosheiov 1994). This is known as 'deteriorating effect'. These two nonignorable phenomena can have significant impacts on the model outcomes, and a growing number of studies have been conducted on the relevant scheduling problems in recent decades.
The phenomenon of the learning effect was first discovered in the aircraft industry (Wright 1936). It was shown that the unit processing time drops by 20% with every redoubling in production. After that, the learning effect has been found in other industries. In the integrated circuit production, manufacturing efficiency can be increased fivefold with the improvement of workers' experience (Webb 1994). On the other hand, deteriorating effect exists widely in steel rolling mills, fire-fighting, hospital emergency ward, and crime scene response scheduling (Kunnathur and Gupta 1990). For example, in steel rolling mills, as the corresponding ingots' current temperature drops with the delay of waiting time, the heating time of ingots will increase on account of the decreasing gas flow intensity (Gupta and Gupta 1988). Hence, based on these practical situa-tions, a variety of learning and deteriorating effects were proposed, which associate with various practical influence factors, such as truncated effect, incompressibility factor, resource consumption, etc. Most of the proposed learning functions are derived from these basic functions including p [jr] = p j r a with learning indicator a < 0 (Biskup 1999), p [jr] = p j (1 + r−1 l=1 p [l] ) a with a ≤ 0 (Kuo and Yang 2006), and p [jr] = p j (1 − r−1 l=1 p [l] n l=1 p l ) a with a ≥ 1 (Koulamas and Kyparisis 2007). Regarding deterioration, most scholars mainly studied time-dependent deterioration, including the starting time-dependent function p [j] = α j t with deteriorating indicator α j ≥ 0 (Mosheiov 1994) and the sum-of-processing-time-based function p [jr] = p j (1 + r−1 l=1 p [l] ) (Liu et al. 2013). In addition to the pure learning effects or deteriorating effects, both effects can co-exist in some practical production processes. For example, in the production of porcelain craftworks, the processing time of a craftwork will be longer as the raw material hardens. However, sometimes the processing time could be shorter due to the effort of a more experienced craftsman (Wang 2007). Similar phenomena also emerge in glass crafts production. The heating time of raw material increases but the shaping time decreases due to deteriorating and learning effects, respectively (Cheng, Wu, and Lee 2008). Motivated by these practical manufacturing processes, Lee (2004) first proposed the learningdeterioration function p [jr] = α j tr a , which combines deterioration p [j] = α j t with learning effect p [jr] = p j r a . In fact, most hybrid functions of subsequent studies are derived from this basic learning-deterioration function.
According to statistics, some studies investigate learning and deteriorating effects from diverse perspectives. Biskup (2008) and Azzouz, Ennigrou, and Ben Said (2018) summarised some well-known learning functions. Gordon, Strusevich, and Dolgui (2012) reviewed some positional polynomial and general linear timedependent deterioration or learning effects, and gave a comprehensive summary and analysis of the scheduling with due date assignment under some special conditions on job processing. Janiak, Krysiak, and Trela (2011) proposed some learning and aging functions through a summary of 33 papers. Additionally, Alidaee and Womer (1999) and Cheng, Ding, and Lin (2004) reviewed the scheduling problems considering the starting time-dependent processing time. To the best of our knowledge, this paper is the first attempt to provide a comprehensive introduction to scheduling problems associated with not only pure learning and deteriorating effects, but also the combination of the two effects.
To give a comprehensive review of scheduling problems with learning and deteriorating effects, we build a specific classification scheme based on two main attributes: production resource restriction and job processing efficiency. Figure 1 shows a four-level framework of the classification scheme, including effects, job processing ways, actual processing time functions, and manufacturing environments. Further, in-depth research is conducted based on the actual processing time functions. Figure 2 shows the common general formulas emerged in scheduling problems with different effects.
In this paper, the main contribution can be summarised as follows: (1) A concise guide to scheduling problems considering learning and deteriorating effects is given according to the papers published from 1990 to now. In this guide, the relative literature is chosen by the representative of actual processing time functions and the innovative models.  (2) A novel classification scheme is designed based on job processing ways and actual processing time functions, i.e. position-based functions, time-dependent functions, and hybrid functions. The characteristics of scheduling problems with diverse functions are described and compared in different manufacturing environments.
(3) The evolvement of related scheduling models is well analysed, and the relationships between the proposed algorithms in various scheduling problems are adequately investigated. (4) The gaps of existing research are revealed and the potential future research directions are proposed by analysing the practical applications, proposed functions, and designed algorithms.
The remainder of this paper is organised as follows: Section 2 describes basic notations, complexity of the problems, and practical applications; Section 3 describes different scheduling problems with learning effects; Section 4 gives an overview of scheduling problems with deteriorating effects; Section 5 reviews papers on scheduling problems with combined effects. Finally, we conclude this paper in Section 6.

Basic notations and complexity of the problems
To describe various scheduling problems clearly, some basic notations of the learning and deteriorating functions are given. For all types of scheduling problems, n and m denote the total number of jobs and machines, respectively. Job j can be processed on batch machines or be grouped into different groups, where the total number of batches and groups is defined as N and Q, respectively. The number of jobs in batch b is denoted as n b , while the number of jobs in group g is represented as n g . Hence, there is N b=1 n b = Q g=1 n g = n. Given the basic settings of the number of jobs, machines, batches, and groups, a list of the commonly used notations is shown in Appendix A of Supplemental Material, including the definitions of job processing time, learning indicators, deteriorating indicators, starting time, truncation parameter, and some other practical influence parameters.
The mathematical expressions for the general objective functions are also listed in Appendix A. Although the research on scheduling problems with learning and deteriorating effects is abundant and heterogeneous, their objectives in various problems are similar, divided into single-objective functions and multi-objective functions. The common single-objective functions consist of makespan C max , maximum tardiness T max , maximum earliness E max , maximum lateness L max , total completion time C j , total tardiness T j , total earliness E j , etc. In terms of multi-objective functions, most of them are the weighted summation of some single objectives and the others are abstracted from the practical manufacturing scenarios.
In addition, a detailed list of complexity of problems proposed in this paper is given in Appendix C of Supplemental Material, which provides a convenient reference for researchers to find new areas to study.

Practical scenarios of the problems
Over the last 20 years, a variety of learning and deteriorating functions were observed through the practical phenomena in different production processes affected by worker efficiency, machine performance, and job property. One basic learning function p [jr] = p j r a was proposed based on the processing characteristics of the automobile assembly line and the processing of memory chips (Biskup 2008;Vahedi Nouri, Fattahi, and Ramezanian 2013b). The other fundamental learning function p [jr] = p j (1 + r−1 l=1 p [l] ) a relies on practical manufacturing scenarios, such as the inspection for product surface defects in the steel plate or bar industries (Lin 2018). Basic deteriorating function p [j] = αt was abstracted from the increased processing time during aircraft maintenance and steel ingot preheating (Bachman and Janiak 2000).
With the going deep of relevant research work, scholars have expanded practical functions of learning and deteriorating. For example, the deteriorating function p [ij] = p j α i t was considered according to the changeable meta-cutting time caused by the difference of cutting tool quality in a sawmill (Hsu et al. 2013). Learning functions p [jr] = p j r a − κ j u j and p [jr] = ( p j r a u j ) σ were observed from the catalytic process of chemical compounds in the chemical industry . Learning function p [jr] = p j max{r a , ρ} was proposed based on the impact of new workers' learning rule on the processing efficiency of painting tile ). In Fu et al. (2018), learning-deteriorating function p [ijr] = (p ij + α ij t ij )r a ij was proposed based on the frequent changing production demand in Industry 4.0-based manufacturing systems. In this situation, smart machines need to learn quickly to produce new products efficiently, while machines also need to bear the loss of efficiency caused by constant wear.
Recently, some practical scheduling models validated by actual data were proposed. Przybylski (2018) proposed a new integral-based learning function according to the continuous change of the worker's experience in auto components production. Asadayoobi, Jaber, and Taghipour (2021) proposed a new learning curve validated by the data from an order-picking task.
However, from the holistic perspective, the relevant studies need to strengthen the elucidation of the research problems and the formation mechanism of the learning and deteriorating functions. On the one hand, the specific objectives and constraints of scheduling models should be explained, combing with the realistic production processes. On the other hand, it is necessary for scholars to explore more innovative processing time functions based on industrial data and advanced smart production processes. With the application of the new generation of information technology in industry, the current learning and deteriorating functions are no longer sufficient to describe the changes in job processing time. This paper further proposed a detailed introduction to the previous studies in this field, facilitating researchers to identify the new and valuable research directions.

Learning effects
Given the first effort in the scheduling literature with learning effects (Wright 1936), the boom in research did not really start until more than sixty years later. In 1999, Biskup (1999) considered the learning effect p [jr] = p j r a in two single-machine scheduling problems. After that, the number of papers in this field is gradually growing. Among their studies, some problems were abstracted from practical manufacturing systems, such as semiconductor industry (Webb 1994), steel rolling systems (Ma, Shao, and Wang 2014), electronic guidance systems (Nadler and Smith 1963), etc.
In this section, we classify the previous relevant papers on basis of job processing ways, i.e.scheduling with general processing way, batch scheduling, and group scheduling. In each category, a variety of position-based learning functions, time-dependent learning functions, and hybrid time-dependent and position-based learning functions are summarised, respectively.

Scheduling with general processing way
The studies of learning effects initially focused on general scheduling problems in various shop environments, such as single-machine, parallel-machine, and flowshop environments. The proposed learning effects are usually derived from the basic functions p [jr] n l=1 p l ) a with a ≥ 1, in which the parameters a and a j are learning indicators.

Position-based learning effects
In an early study, scholars capture that learning effects occur when workers repeat the processing-timeindependent jobs, such as controlling and operating machines, reading data, etc. (Biskup 1999). Since then, position-based learning effects, including linear functions and non-linear functions, have been concerned incessantly. This subsection reviews scheduling models and algorithms based on linear and non-linear functions.
It is shown that compared with the poor applications of linear functions, non-linear functions are often analysed in scheduling problems.

• Linear functions
Some scholars simplified the expression of learning effects in realistic manufacturing situations, such as handling raw materials, assembling components, etc., proposing the linear position-based functions p [jr] = p j − a j r and p [ijr] = p ij (μ − νr). The factors μ ≥ 0 and ν ≥ 0 are constant numbers, satisfying μ − (n + 1)ν > 0. However, since these linear functions cannot reflect actual learning effects perfectly, the relevant research has not received much attention except for a few papers published between 2000 and 2015. For example, in the single-machine scheduling problems, Bachman and Janiak (2004) studied learning function p [jr] = p j − a j r, and Cheng and Wang (2000) introduced an upper limit of the position to avoid a continuous reduction in job processing time, i.e. p [j] = p j − a j min{n j , n 0j }. The term n j denotes the number of jobs processed before job j, and n 0j is a threshold value. In addition, the function p [ijr] = p ij (μ − νr) is just another expression of p [ijr] = p ij − a ij r, which was applied in flowshop scheduling problems (Sun et al. 2013). In this field, general sequence rules were utilised, such as the shortest processing time first (SPT) algorithm, any busy schedule (ARB) algorithm, the earliest due date (EDD) algorithm, the earliest ready date (ERD) algorithm, etc. These rules can be modified according to the characteristics of the studied problems, such as the weighted shortest processing time first (WSPT) algorithm and the weighted discounted shortest processing time first (WDSPT) algorithm. All the abbreviations throughout this paper are listed in Appendix B of the Supplemental Material.
• Non-linear functions based on p [jr] = p j r a In many practical scenarios such as high-technology manufacturing processes, learning effects can become stronger with the increase of the jobs already processed. Thereby, the pure general non-linear learning functions p [jr] = p j r a and p [ijr] = p ij r a were proposed. Furthermore, it is noted that learning indicator a can be substituted for job-dependent learning indicator a j , machinedependent learning indicator a i , and hybrid learning indicator a ij , where a, a j , a i , a ij ≤ 0. Figure 3 shows the distinctions between these scheduling models in single-machine, parallel-machine, and flowshop cases. In these studies, some practical influence factors were investigated, including due date, release time, machine preemption, idle time, and maintenance activities. The formulations for existing single-machine cases are the extensions of Model 1.1 proposed by Biskup (1999), such as Model 1.2 (Eren and Güner 2007). In Model 1.2, the due date D r = n j=1 x jr d j and the job tardiness T r ≥ C r − D r were analysed, where d j is the due date of job j and C r is the completion time of the rth job. Moreover, the equation x jr = 1 indicates that job j is scheduled in position r and x jr = 0 otherwise. The detailed explanation of scheduling models proposed in this paper is shown in Appendix D of Supplemental Material. Eren and Güner (2007) designed a random search (RS) method and four tabu search (TS) methods. In the cases of parallel-machine scheduling, the learning function p [ijr] = p ij r a was considered. Due to the influence of machine quantity, Mosheiov (2001) built Model 1.3 based on Model 1.1. Furthermore, Xu, Yin, and Li (2010) added the constraints of lateness L max ≥ C ir − D ir and due date D ir = n j=1 x ijr d ij , where the equation x ijr = 1 shows that job j is scheduled on machine i in position r and x ijr = 0 otherwise, see Model 1.4. In these papers, heuristic algorithms were designed to solve parallel-machine problems. It is found that most papers in this area were published between 2000 and 2010.
There were two papers studying flowshop cases with p [ijr] = p ij r a and extending Model 1.1, see Model 1.5 (Eren and Güner 2008) and Model 1.6 (Bai et al. 2018). Eren and Güner (2008) analysed starting time t r ≥ t r−1 + A r−1 and idle time X r = t r + A r + Y r − C r−1 . Thereinto, A r is the actual processing time of the rth job at the first machine, Y r is the duration between the completion time of the job in position r at the first machine and its starting time at the second machine. In addition, motivated by the assembling process of aircraft components, release date R j was introduced by Bai et al. (2018), affecting the constraint of completion time R j + n r=1 x jr p [1jr] ≤ C 1j . They gave the function p [ijr] In terms of the proposed algorithms, the former utilised meta-heuristic algorithms including RS and TS, while the latter proposed branch-and bound (B&B) algorithms and SPT-based heuristics. On the other hand, flowshop scheduling problems with p [ijr] = p ij r a i were also studied on account of Model 1.3. Vahedi Nouri, Fattahi, and Ramezanian (2013a) introduced flexible maintenance activities and tardiness, i.e. constraints (1.7c) -(1.7e) and (1.7i) of Model 1.7. A hybrid meta-heuristic algorithm based on simulated annealing (SA) and firefly algorithms was proposed due to the higher complexity of the problem. The similar function p [ijr] = p ij r a was also investigated in the context of green scheduling, with the objective to minimise the makespan and total energy consumption (Xin et al., forthcoming). With the development of the related studies, there have been a number of extensions proposed on basis of the function p [jr] = p j r a . In the chemical industry, the job processing time can be compressed if extra costs are paid to increase catalysts (Wang and Cheng 2005). Then, scheduling problems with position-and resource-based learning effects were investigated, including a linear function p [jr] = p j r a − κ j u j with 0 ≤ u j < p j n a κ j and a convex function p [jr] = ( p j r a u j ) σ with u j > 0 ( Gao et al. 2018). Apart from the impacts of resource allocation, there were some other assumptions considered. In real-world situations, job processing time cannot drop to zero steeply with the increase of already processed jobs. Hence, a truncation parameter ρ ∈ (0, 1) was introduced into p [ijr] = p ij r a , that is, p [ijr] = p ij max{r a , ρ} (Li, Hsu, et al. 2011). On the other hand, since the part of job processing time is limited by some conditions and it cannot be shortened, DeJong's learning function p [jr] = p j (M + (1 − M)r a ) and some improved functions were proposed, where M denotes the incompressibility of job processing time, M ∈ [0, 1], and a ≤ 0 (Okołowski and Gawiejnowicz 2010). The detailed analysis of these studies can be found in Appendix E.1 of Supplemental Material.

Time-dependent learning effects
In some practices, the actual job processing time depends not on the number of the jobs processed but on the total normal processing time of the jobs processed. For instance, as employees spend more time on practicing, their skills will improve more. Hence, there is concern about scheduling problems with time-dependent learning effects. According to relevant research, it is found that there is research potential in more complicated manufacturing environments, such as multi-machine, flowshop, and jobshop.
• General time-dependent learning functionsp [jr] Based on the sum of processing time of the jobs already processed, there are two time-dependent learning effects observed in the single-machine and flowshop settings, namely, p [jr] Extending single-machine total completion time minimisation problems with learning function p [jr] (Kuo and Yang 2006 were considered with a normalising constant γ ≥ 0 (Wang 2008). In order to solve five regular performance criteria, general sequence rules were used, such as SPT, WSPT, EDD, etc. There are some papers on more practical flowshop scheduling problems with learning effect Lin et al. (2017) investigated a reentrant flowshop scheduling problem abstracted from semiconductor wafer and printed circuit board production. They introduced h re-entrant times into learning Wu et al. (2018) addressed a two-stage three-machine assembly flowshop scheduling problem, which was abstracted from fire engine assembly, personal computer manufacturing, and other realistic situations. To solve these flowshop scheduling problems, Lin et al. (2017) presented genetic algorithms (GAs) accompanied with EDD rule to minimise the total tardiness, while Wu et al. (2018) proposed a B&B algorithm and six hybrids of particle swarm optimisation (PSO) algorithms.

Moreover, another function p
was widely used as well, which relies on the proportional job processing time. In the context of two-machine flowshop scheduling, Koulamas and Kyparisis (2007) utilised the SPT algorithm to minimise C max and C j in consideration of idle time on the second machine. For single-machine cases, two realistic situations have received some attention, i.e. scheduling problems with unequal release time R j of job j abstracted from diffusion and oxidation processes of wafer fabrication (Wu, Hsu, and Lai 2011), and two-agent scheduling problems springing from different demands of telecommunication services (Wu 2014). Both of them proposed B&B and SA algorithms to find optimal and near-optimal solutions.

• Exponential time-dependent functions
To the best of our knowledge, the general timedependent learning functions show that the decline in the job processing time presents an accelerating trend. However, the learning process may be slow under many practical situations, such as memory chips production (Bai, Wang, and Wang 2012). Exponential timedependent learning functions were analysed in the single-machine and energy-aware flowshop scheduling problems. In the single-machine cases, two functions (Wang, Sun, and Sun 2010) and (Ma, Shao, and Wang 2014) were studied. The parameters μ, ν ≥ 0 are obtained empirically, where μ + ν = 1. For regular objectives, they proposed some general sequence rules, i.e. SPT, WSPT, WDSPT, and EDD rules.
was also applied to an energy-aware flowshop scheduling problem (Liu, Shi, and Shi 2018). They combined power consumption rates μ(M i ) with total working time τ c (M i ) on machine M i , aiming to minimise the total energy consumption. A new efficient framework Bounds-based Nested Partition (BBNP) was designed, which is more effective than the conventional heuristic algorithms such as SPT, NEH (Nawaz, Enscore, and Ham 1983), and FL (Framinan and Leisten 2003).

Hybrid time-dependent and position-based learning effects
In some manufacturing processes such as the inspection of steel plates and the cutting operation of papers, diverse raw material, setups, and inspection are required, which can affect actual job processing time (He 2016). Hence, some hybrid time-dependent and position-based functions were proposed to reflect the complicated learning effects.
It is summarised that the hybrid learning effects are mainly obtained from the functions p [jr] (Koulamas and Kyparisis 2007), p [jr] = p j r a (Biskup 1999), and p [jr] = p j a r−1 (Cheng and Sun 2006). These hybrid functions were proposed in some singlemachine and flowshop cases. For example, Wu and Lee (2008) n l=1 p l ) a 1 r a 2 with a 1 , a 2 < 0, and Cheng, Wu, and Lee (2008)  with a 1 ≥ 1 and 0 < a 2 < 1. In these papers, SPT, WSPT, and EDD algorithms can provide the optimal solutions for regular objectives.
In addition, hybrid exponential time-dependent and position-based learning functions p [jr] were applied in single-machine cases (Bai, Wang, and Wang 2012) and flowshop cases (He 2016), respectively, where 0 < a 1 , a 2 ≤ 1. The parameter w l denotes the weight, μ and ν are obtained empirically, where μ, ν ≥ 0 and μ + ν = 1. The former study considered p-s-d setup time s [r] = γ r−1 l=1 p [l] and presented some general sequence rules. The latter study introduced the tail operations for each job π j = d max − d j with due date d. With the objective to minimise the lateness, they proposed a hybrid solution framework made up of two heuristics, a B&B algorithm, and a new Nested-Partitionbased solution approach.

Batch scheduling and group scheduling
Batch scheduling is characterised by the fact that machines can process multiple jobs at the same time.
Most batch scheduling problems with learning effects assume that the actual processing time is a decreasing function of the job's position under the consideration of job and batch learning indicators. Moreover, a number of papers introduced the combined effects of learning and group technology into scheduling problems. In group scheduling problems, job and group learning indicators were also analysed. Compared with scheduling problems within unit-capacity processors, it is more difficult to solve batch and group scheduling problems with learning effects (Geiger and Uzsoy 2008).

Position-based learning effects
This subsection introduces batch and group scheduling problems with position-based learning effects. For most batch scheduling problems, the learning function p [bjr] = p bj r a was often proposed with constant learning indicator a ≤ 0. As for other learning functions, there are relatively few relevant studies on p [bjr] = p bj r a b with batch-dependent job learning indicator a b ≤ 0 and p [bjr] = p bj r B b with batch learning indicator B b ≤ 0 (Yang and Kuo 2009). In group scheduling problems, job and group learning indicators are usually investigated simultaneously. The majority of learning functions are derived from p [gjr 1 r 2 ] = p gj r a 1 1 r a 2 2 . Specifically, the term p [gjr 1 r 2 ] is the actual processing time of job J gj scheduled in the r 1 th position and in the r 2 th group, and the terms a 1 ≤ 0 and a 2 ≤ 0 represent the internal job and group learning effects, respectively.

• Batch scheduling problems
The research on batch scheduling problems with learning effects has not drawn amounts of attention until now, and most of the related papers mainly took into account a fixed learning indicator. Regarding serialbatching scheduling problems, batch setup time s b = θt with batch deteriorating indicator θ was investigated by Pei et al. (2018) and Pei, Cheng, et al. (2019). In particular, Pei et al. (2018) innovatively introduced the compression of job's processing time τ j into the learning function p [jr] = (p j − τ j )r a , where a < 0. The factor τ j does not exceed the maximum reduction in the processing time of job j. The studied problems were solved by hybrid algorithms, including a novel hybrid GSA-TS algorithm combining the Gravitational Search Algorithm (GSA) and TS, and a hybrid VNS-GSA algorithm combining variable neighbourhood search (VNS) and GSA.

• Group scheduling problems
As seen from Figure 4, the applications of two learning functions in single-machine and flowshop group scheduling are depicted, and resource allocation is introduced into some cases. It is found that almost all position-based learning functions are the extensions of p [jr] = p j r a . Moreover, in contrast to the research on batch scheduling, the studies of group scheduling usually considered not only internal job learning indicators, but also group learning indicators. One learning function p [gjr 1 r 2 ] = p gj r a 1 1 r a 2 2 and group setup time s [gr 2 ] = s g r a 2 2 were presented in the flowshop environment (Qin, Zhang, and Bai 2016). They proposed four heuristic algorithms with worst-case ratios, and two meta-heuristic algorithms based on GA and quantum differential evolutionary (QDE) algorithm. There were some other papers presenting two functions p [gjr 1 r 2 ] = ( p gj r a 1 1 r a 2 2 u gj ) σ with u gj > 0 and p [gjr 1 r 2 ] = p gj r a 1 1 r a 2 2 − κ gj u gj with 0 ≤ u gj < p gj Q a 2 (n Q ) a 1 κ gj in view of resource allocation (Zhu et al. 2011;Lu et al. 2017). The parameter u gj is the amount of resource allocated to job j in group g. In the context of general autonomous and induced learning effects, Huo, Ning, and Sun (2018) set the internal job indicator and the group learning indicator as a 1 log 2 (1 − τ )a and a 2 log 2 (1 − τ )a. They proposed the learning function p [gjr 1 r 2 ] = p gj r a 1 log 2 (1−τ )a 1 r a 2 log 2 (1−τ )a 2 and group setup time s gr 2 = s g r a 2 log 2 (1−τ )a 2 . Thereinto, a 1 , a 2 > 0 are the learning coefficients, 0 < a ≤ 1 is the standard learning indicator, and 0 ≤ τ < 1 is the percentage reduction of a. The studied problems were formulated as assignment models in these three papers, i.e. Models 1.8, 1.9, and 1.10 (see Figure 4).

Time-dependent learning effects and hybrid learning effects
In the existing studies on batch scheduling problems, time-dependent learning effects are rarely applied. Filling this gap, Liu et al. (2020) proposed a truncated linear function p where t b is the starting time of the jobs within batch b. They designed a hybrid CS-SADE algorithm based on an improved cuckoo search algorithm (CS) and self-adaptive DE for the parallel-batching machines scheduling problem.
In group scheduling problems, two time-dependent learning functions were proposed and extended in single-machine cases. Kuo (2012) ) a g with group-dependent job learning indicator a g ≤ 0. Liu, Lee, and Wu (2008)  ) a . Thereinto, a ≤ 0 and a g ≤ 0 are the learning indicators of the group setup times. These proposed problems were usually solved by heuristic algorithms to minimise C max or C j .
In addition, hybrid time-dependent and positionbased learning effects also occur in single-machine group scheduling problems. There exists one paper that investigated a generalised function p [gjr 1 ] = p gj f 1g ( r 1 −1 l=1 p g [l] ) f 2g (r 1 ) and group setup time s [gr 2 ] = s g ϕ( r 2 −1 l=1 s [l] )θ (r 2 ) (Yin et al. 2013). In these functions, f 1g : [0, +∞) → (0, 1] is a differentiable non-increasing function with f 1g (0) = 1, in which f 1g is non-decreasing in [0, +∞), and f 2g : [1, +∞) → (0, 1] is a non-increasing function with f 2g (1) = 1. The constraints of functions ϕ and θ are the same as those of functions f 1g and f 2g , respectively. Similar group scheduling problems can be solved in polynomial time. Furthermore, another learning func- ng l=1 p gl ) a 1 a 2 r 2 −1 was studied by Low and Lin (2012). The parameters a 1 ≥ 1 and 0 < a 2 < 1 are the internal job learning indicator and the group learning indicator, respectively. They presented the SPT algorithm to minimise C max and C j . Gupta and Gupta (1988) and Browne and Yechiali (1990) first introduced deteriorating effects in single-machine makespan minimisation problems. After that, extensive studies were conducted, some of which have been considered in real scenarios, such as steel production, firefighting, and cleaner production (Gupta and Gupta 1988;Kunnathur and Gupta 1990). However, few papers have conducted a specialised survey in this field. In this section, a variety of deteriorating functions are summarised, mainly depending on job starting time or the sum of processing time of jobs already processed, i.e.

Scheduling with general processing way
According to the overview, it is found that deteriorating effects were first considered in the scheduling problems with general processing way, and the number of papers on this subject is far bigger than that of papers on batch scheduling and group scheduling. In contrast to timedependent deterioration, only a few scholars focused on position-based deterioration as well as hybrid timedependent and position-based deterioration.

• Pure linear starting time-dependent functions
Overall, one of the most popular deteriorating functions is p [j] = α j t with deterioration indicator α j > 0 and job starting time t ≥ t 0 > 0. The linear monotone increasing function is a simplified assumption of deterioration, but it can reflect some realistic cases, such as single-server cyclic-queue and external shocks in a production setting.
In the early studies, the pure linear starting timedependent deterioration p [j] = α j t was investigated in the single-machine environment (Mosheiov 1994). With the in-depth research, some practical factors were considered, such as delivery times and release times (Zou 2014) and chain precedence constraints . All these proposed problems can be solved in polynomial time. Besides single-machine cases, there were also two papers on parallel-machine problems. Ji and Cheng (2009)

designed some polynomial time algorithms to solve the problems P|p
which are strongly NP-hard. Li and Yuan (2010) further analysed the total penalty of rejected jobs S e j . They proposed fully polynomial-time approximation schemes (FPTAS) for problems Pm|p [j] = p j + α j t|C max (S) + S e j and Pm|p [j] = α j t, r j = t 0 | S w j C j + S e j , and exploited an optimal O(n 2 )-time dynamic programming (DP) algorithm to solve the problem P|p [j] = p j + αt| S C j + S e j . In these two papers, P and P m denote the parallel-machine scheduling problems with a fixed number of machines and an arbitrary number of machines, respectively, S andS are the set of accepted jobs and rejected jobs, respectively.
The deteriorating function p [ij] = α ij t with α ij > 0 has been widely applied in flowshop scheduling problems, which were usually solved by B&B and heuristic algorithms, see Figure 5. Initially, given deterioration p [ij] = α ij t, the complexity of jobshop, openshop, and flowshop scheduling problems was proved (Mosheiov 2002). However, since then, only flowshop scheduling problems have received much attention. More studies focused on two-machine flowshop scheduling problems rather than three-machine problems. For instance, Zhao and Tang (2012) introduced two machineindependent precedence constraints, and Cheng et al. (2014) built Model 2.1 for bicriteria hierarchical optimisation, as shown in Figure 5. They provided heuristic and B&B algorithms. On the other side, Wu et al. (2019) studied a two-stage three-machine assembly scheduling problem with deterioration p [ij] = p ij + α i t. Three metaheuristic algorithms were proposed including dynamic differential evolution (DDE), SA, and cloud theory-based simulated annealing (CSA) algorithm.

• Linear starting time-dependent functions with fixed processing time
Considering a fixed part of the job processing time, two linear starting time-dependent deteriorating functions are introduced, i.e. p [j] = p j + α j t and p [j] = p j (μ + νt), where α j > 0 and ν > 0 denote the deteriorating indicators. On this basis, some resource consumption functions were also presented. Figure 6 shows the application of p [j] = p j + αt and p [ij] = p ij + α ij t in single-machine, parallel-machine, and flowshop cases. At first, Bachman and Janiak (2000)  and Bachman, Janiak, and Kovalyov (2002) introduced deterioration p [j] = p j + α j t j into single-machine scheduling problems with the objectives to minimise L max and w j C j , respectively. After that, some common problem settings were considered, such as p-sd setup times and maintenance activities. Sun and Geng (2019) proposed a new deteriorating function p [j] = λp j + α(t j − C r − A) by adding a fixed maintenance time A and maintenance rate 0 < λ < 1. For two parallel-machine cases, deteriorating maintenance activities (Hsu et al. 2013) and multiple rate-modifying activities (Woo and Kim 2018) were investigated. Particularly, these two papers built corresponding mathematical models, see Models 2.2 and 2.3 in Figure 6. In Model 2.3, the constraint j∈J x j jk ≤ 1 guarantees that at most only one job can be assigned to the first sequence of a potential bucket, which is different from n j=1 x ijr = 1 of Model 2.2. Thereinto, x j jk = 1 if job j precedes job j in bucket k and x j jk = 0 otherwise. The constraints (2.3f)-(2.3h) show the limitation of the job completion time. In addition, there were a number of papers on m-machine flowshop scheduling problems with p [ij] = p ij + α ij t ij , such as Wang, Huang, and Wang (2019). They proposed a mathematical model and developed a novel algorithm called multi-verse optimiser (MVO) (see Figure 6). In Mode 2.4, the constraints of the starting time of job j on machine i, t ij + p [ij] ≤ t i+1,j and t ij + p [ij ] ≤ t ij + φ × (1 − x j j ) with an infinite number φ, show the constraints of job processing and machine capacity, similar to the constraints (1.3a)-(1.3b) in Model 2.2. Thereinto, x j j = 1 if job j is followed by job j immediately and x j j = 0 otherwise.
Actually, the deteriorating function p [ij] = p ij (μ + νt) with constant value μ, ν ≥ 0 can be regarded as another representation of p [ij] = p ij + α ij t, which is common in the flowshop cases (Bank et al. 2012a(Bank et al. , 2012b. Moreover, considering the compression of job processing time in realistic situations, two resource-dependent deteriorating functions p [j] = p j + α j t − κu j and p [j] = ( κ j u j ) σ + α j t j were presented in single-machine scheduling problems, where α j > 0 and σ > 0 (Wei, Wang, and Ji 2012;Li and Wang 2018). The term u j is the amount of resource allocated to the job j and κ j is a positive parameter that denotes the workload of job j. More details are shown in Appendix E.2 of Supplemental Material.

• Sum-of-processing-time-based functions
Although most deteriorating functions depend on the job starting time, the sum-of-processing-time-based deterioration was also analysed. Two basic deteriorating functions p [jr] = p j (1 + r−1 l=1 p [l] ) and p [jr] = p j (1 + r−1 l=1 p [l] n l=1 p l ) were designed in two-agent single-machine scheduling problems (Liu et al. 2013). Dolgui, Gordon, and Strusevich (2012) considered the deterioration model p [jr] = p j (1 + r−1 l=1 p [l] ) α with α > 0 in singlemachine scheduling with precedence constraints, which can be solved in polynomial time. With the aim to avoid the unrealistic situation of infinite increase in the job processing time, the sum-of-logarithm-processingtime-based function p [jr] = p j (1 + r−1 l=1 log p [l] ) α was designed, where α ≥ 1 (Cheng, Lee, and Wu 2011). They exploited SPT and EDD algorithms to solve singlemachine scheduling problems with several regular performance criteria.

• Position-based functions
Motivated by the characteristics of deteriorating effects, the actual job processing time can also be regarded as an increasing function dependent on the job's position. Gordon et al. (2008) and Gordon and Strusevich (2009) both studied positional deterioration p [jr] = p j f (r) in the single-machine scheduling cases, where f (r) is a non-decreasing function and f (1) = 1. The former focused on scheduling problems under series-parallel precedence constraints with priority-generating objective functions. The latter proposed some polynomial-time DP algorithms in two due date assignment problems. Figure 7 shows the introduction of maintenance and resource allocation into scheduling problems with the position-based deterioration. In the multi-machine environment, Yang (2013) considered a deteriorating maintenance activity and proposed a generalised deteriorating function p [ijr] = p ij f ij (r). Zhang et al. (2018) also studied the impact of deteriorating jobs and rate-modifying activities. The deteriorating function was formulated as p [gjr] = p gj (1 + α) r−1 with 0 < α ≤ 1. Notably, there is concern about resource-dependent deteriorating functions. For instance, Yang, Lee, and Guo (2013) proposed the function p [jr] = p j f (r) − κ j u j with 0 ≤ u j < p j f (n) κ j and the function p [jr] = ( p j f (r) u j ) σ with u j > 0 in single-machine cases. Figure 7 shows that all papers formulated the studied problems as assignment models, see Models 2.5, 2.6, and 2.7.

Batch scheduling and group scheduling
In batch and group scheduling problems, most papers captured starting time-dependent deterioration, while a few papers focused on position-based deterioration. In the context of parallel-batching and serial-batching scheduling, pure linear starting time-dependent deteriorating functions p [j] = αt with α > 0 and p [j] = p j + α j t with α j > 0 were investigated. As for group scheduling problems, most scholars focused on single-machine cases and studied two basic functions p [gj] = α gj t and p [gj] = p gj (μ + νt), where α gj > 0 is the deteriorating indicator of the processing time for job j in group g.

• Batch scheduling with linear starting time-dependent functions
In the studies of batch scheduling problems with the common linear deteriorating functions, various hybrid algorithms are proposed to obtain near-optimal solutions.
For parallel-batch scheduling problems with deterioration p [j] = α j t, some scholars assumed that the maximum number of jobs processed simultaneously as a batch is infinite or bounded (Li, Ng, et al. 2011). Li, Ng, et al. (2011) solved the unbounded problem (b = ∞) by an O(n log n)-time DP algorithm. They also solved the bounded problems with identical job release dates by an FPTAS. Recently, some bounded problems with job sizes and job rejection were studied (Kong et al., "Parallel-batching Scheduling," 2020). They developed a DP algorithm for the problem with identical job sizes, and a hybrid algorithm combining heuristic with DP algorithm (H-DP) for the problem with nonidentical job sizes. In addition, there was one paper on unrelated parallel-batching machines scheduling problems with p [ir] = p ir + αt ). They considered the machine capability constraint and maintenance activity constraint, and proposed a hybrid ABC-TS algorithm combining artificial bee colony (ABC) and TS.
The single-machine serial-batching scheduling problems with linear deterioration were investigated by Pei et al. (2015Pei et al. ( , 2017 and  (see Figure 8). As is shown, Pei et al. (2015) investigated a realistic problem, namely, the coordinated production and transportation scheduling in a two-stage supply chain. Models 2.8 and 2.9 were established to describe scheduling problems with and without buffer, respectively. The buffer can store the processed batches before transportation. The difference between these two models is whether a new batch can be processed at once after the previous batch is completed or not, see t 1,b+1 = C 1b and t 1,b+1 = max{C 1b , t 1b + T}. Thereinto, t 1,b+1 denotes the starting time of batch b during the production stage, T is the round-trip time between the manufacturer and the customer. Furthermore, Pei et al. (2017) took into account sequence-dependent setup times and multiple job types, and proposed some optimal algorithms.  also analysed financial budget ωU h ≤ B h , resource constraint n j=1 u j ≤ U, resourcedependent deterioration p [j] = p j + αt − κu j with 0 ≤ u j ≤ p j κ , and setup time of the kth batch s k = θ t. The parameter ω is the unit resource cost, U h is the total resource cost of manufacture h, and B h is the financial budget. They proposed a hybrid BA-VNS algorithm combining Bat algorithm (BA) and VNS to minimise the makespan.

• Group scheduling with linear starting time-dependent functions
In group scheduling studies, it is found that almost all papers focused on single-machine cases considering linear starting time-dependent deteriorating functions. Hence, future research can be extended into more practical environments such as multi-machine and flowshop.
For group scheduling problems with p [j] = α j t and p [gj] = α gj t, the existing research is similar. Most of the problems were solved by polynomial time algorithms and B&B algorithms. In these studies, deteriorating group setup time s [g] = θ g t was usually considered, where θ g is the deteriorating indicator of group setup time (Lee and Lu 2012;Wang and Liu 2014). Some scholars investigated the other two deteriorating functions p [gj] = p gj + α gj t with α gj ≥ 0 and p [gj] = p gj (μ + νt) with constants μ, ν > 0 ( Lee and Wu 2010;Wang, Lin, and Shan 2008). Most of them considered deteriorating group-dependent setup times and proposed polynomial time algorithms. Although some scholars introduced two-agent decisions and ready times to make the studied problems more practical, the related research on group scheduling with deterioration still lacks reasonable realistic background and theoretical innovation (Liu, Tang, and Zhou 2010;Liu, Yang, and Lu 2019).

Combined learning and deteriorating effects
Even though pure learning effects or deteriorating effects have caught the eye of many scholars, the research on scheduling problems with the combined effects of learning and deteriorating did not come out until 2004. Lee (2004) first introduced combined effects in singlemachine scheduling problems with the objective to minimise the makespan and total flow time. Thereafter, a growing number of researchers took into account combined effects in the domain of machine scheduling. In addition, some related problems were explored on the basis of actual production processes, e.g. the production of porcelain craftworks (Wang and Cheng 2007) and glass crafts (Cheng, Wu, and Lee 2008). This section summarises scheduling problems with combined effects, most of which can be represented as combinations of time-dependent deterioration and position-based learning effects.

Scheduling with general processing way
According to statistics, the research on general scheduling problems remains dominant in consideration of combined effects. Reviewing the studied learningdeterioration functions, it is shown that the most popular combinations are derived from pure linear starting timedependent deterioration p [j] = α j t and pure non-linear position-based learning function p [jr] = p j r a . • Functions based on p [jr] = α j tr a and p [jr] In the last twenty years, scheduling problems with learning-deterioration functions p [jr] = α j tr a and p [jr] = (p + α j t)r a have been studied in single-machine, parallel-machine, and flowshop environments (see Figure 9). These problems were usually solved by heuristics, B&B algorithms, Moore's algorithms, and general sequence rules such as smallest deterioration rate first (SDR), the weighted SDR (WSDR), and EDD. For single-machine scheduling problems with regular performance criteria, Lee (2004) first proposed two functions p [jr] = α j tr a and p [jr] = (p + α j t)r a . Subsequently, Wang, Jiang, and Wang (2009) proposed an analogous learning-deterioration function p [jr] = α j (μ + νt)r a and considered p-s-d setup time s [r] = γ r−1 l=1 p [l] . Thereinto, μ, ν, and γ are positive constants. In unrelated parallel-machine scheduling cases,  formulated two assignment problems Rm|p [ijr] = (p ij + αt)r a |δ 1 TC + δ 2 TADC + δ 3 TML and Rm|p [ijr] = (p ij + αt)r a |δ 1 TW + δ 2 TADW + δ 3 TML, see Models 3.1 and 3.2 in Figure 9. In particular, the constraint n j=1 x ijr ≤ 1 in Model 3.2 was included due to the idle time on the machine. Additionally, some scholars investigated several problems in the flowshop environment. For example, considering the extensive use of advanced intelligent machines in Industry 4.0-based manufacturing systems, Fu et al. (2018) investigated a two-objective stochastic flow-shop scheduling problem, as seen from Model 3.3 in Figure 9. The constraints of starting time were described as t ir + p [ir] ≤ t i+1,r and t ir + p [ir] ≤ t i,r+1 . They designed a firework algorithm with a explosion spark procedure and selection solution procedure.
Based on the function p [jr] = (p + α j t)r a , some scholars regarded the actual job processing time as p [r] = (p r + α × C r−1 )r a , where the starting time t is replaced with the actual completion time C r−1 of the job in position r−1 due to no idle. The single-machine and parallel-machine scheduling problems with these learning-deterioration functions are shown in Appendix E.3 of Supplemental Material.
• Functions based on p [jr] = p j r a + αt Another form of combined effects is a linear combination of learning function p [jr] = p j r a and deteriorating function p [j] = αt. Figure 10 shows single-machine and parallel-machine cases with various functions developed from p [jr] = p j r a + αt. Initially, Wang (2007) introduced the combined effects p [jr] = p j (f (t) + βr a ) in single-machine cases with regular performance criteria. The term f (t) is a deteriorating function with f (0) > 0. To obtain optimal solutions, SPT, WSPT, and EDD algorithms were proposed. In view of the truncated effect, Niu, Wang, and Yin (2015) solved the singlemachine cases by O(n 3 )-time and O(n log n)-time algorithms, where p [jr] = p j max{r a j , ρ} + αt. In the parallelmachine environment, Huang, Wang, and Ji (2014) studied the application of p [jr] = p j r a j + αt in two multiobjective problems, while Ji et al. (2016) utilised a more practical function p [jr] = p j (M + (1 − M)r a ) + αt in two single-objective problems. The parameter M ∈ [0, 1] is incompressibility factor. With regard to algorithms, the former designed polynomial time algorithms, and the latter exploited FPTAS. Particularly, in the abovementioned literature, Niu, Wang, and Yin (2015) and Huang, Wang, and Ji (2014) both formulated these problems as assignment models, see Models 3.4 and 3.5 in Figure 10.
• Functions based on p [jr] The combined effects of general non-linear sumof-processing-time-based deteriorating and pure nonlinear position-based learning were usually applied in single-machine cases. For single-machine cases with regular performance criteria, Sun (2009) proposed a general learning-deterioration function p [jr] = p j (1 + r−1 l=1 p [l] ) α r a with α ≥ 1 and a < 0. To avoid unlimited learning, Cheng, Lee, and Wu (2010) further considered a new function p [jr] = p j (1 + r−1 l=1 log p [l] ) a r α with a < 0 and α > 0. They also analysed p-s-d setup time s [jr] = γ r−1 l=1 p [l] , where γ ≥ 0 is a normalising constant. In these two papers, SPT, WSPT, and EDD algorithms were utilised. In addition, based on another viewpoint of deterioration, p [jr] = p j ( p+ r−1 l=1 p [l] p+ n l=1 p l ) α r a with α < 0 and a < 0 was designed by Yin et al. (2010). They studied p-s-d setup time s [r] = γ r−1 l=1 p [l] and proved that a series of single-machine problems are polynomially solvable.

• Functions proposed in multi-agent systems
In many real environments such as smart grids and port scheduling, multi-agent systems were regarded as a powerful enabling technology (Yan, Ren, andZhang, 2021, Yin, Ren, andZhang 2019;Renna 2015). In recent decades, the two-agent scheduling problems have been studied widely with the commonly used learning or deteriorating functions, such as p [jr] ). These functions have been put forward in Sections 3.1, 4.1, and 4.2 (Wu 2014;Liu et al. 2013;Liu, Tang, and Zhou 2010). Moreover, there are some papers studying on two-agent scheduling problems under simultaneous consideration of deteriorating and learning effects. For instance, Wu et al. (2012) utilised B&B and SA algorithms to solve the single-machine problems, where the learning function of one agent is p [jr] = p j r a with a < 0, and the deteriorating function of the other agent is p [jr] = p j r α with α > 0. Recently, Renna (2019) further investigated two more practical situations in flexible jobshop scheduling by multi-agent system. In the case where the intervention of a different product interrupts a learning period of a machine, a combined learning-forgetting function 1 α was proposed when the learning phenomenon occurs during the forgetting period. The parameters p now and p max show the current and the maximum processing time, respectively. The other two parameters x and k represent the number of different items and same items, respectively. The proposed learning-forgetting functions are similar to that of Biel and Glock (2018), which addressed multistage scheduling problems.

Batch scheduling and group scheduling
In terms of batch and group scheduling, combined effects were usually investigated based on time-dependent deterioration and position-based learning effects. According to statistics, in recent five years, the research on batch scheduling problems with combined effects has been concerned, while group scheduling problems have received little attention.

Batch scheduling
Given the combined effects of learning and deteriorating, the existing research on batch scheduling problems was conducted in the single-machine and parallel-machine environments (see Figure 11). Under a Just-in-Time production system, Yusriski et al. (2016Yusriski et al. ( , 2018) studied single-machine integer batch scheduling problems, formulated as Models 3.6 and 3.7 in Figure 11. Thereinto, constraints (3.6b)-(3.6c) are the descriptions of processing time and due date. Although the objectives proposed in these two papers are different, both papers designed heuristic algorithms based on the Lagrange relaxation. Regarding learning-deterioration functions, the former provided /α) β , while the latter investigated a more significantly complex function. Thereinto, T [b] is the bth batch processing time, s is the setup time of batch, Q [b] is the number of jobs in batch b, and t [b] is the starting time of bth batch. Motivated by the characteristics of semiconductor manufacturing process, parallel-batching machines scheduling problems were studied in consideration of the combined effects p [ijr 1 r 2 ] = p j r a 1 + α i t and preventive maintenance (Kong et al., "A BRKGA-DE Algorithm," 2020). The parameter p [ijr 1 r 2 ] is the actual processing time of job j scheduled in the r 1 th position of a batch in the r 2 th block processed on machine i. With the aim to minimise C max , they proposed a hybrid BRKGA-DE algorithm incorporating biased random-key genetic algorithm (BRKGA) and DE. Moreover, serial-batching scheduling problems with combined effects were inspired by aluminium manufacturing process (Pei et al. 2021). They designed a novel hybrid SC-VNS algorithm combining a Society and Civilisation (SC) algorithm with VNS for parallelmachine problems. The actual job processing time is formulated as p [ijr] [l] ) α r a with α ≥ 1 and a < 0. Thereinto, x i[ϕ] [l] = 1 if job ϕ is assigned to position l on machine i.

Group scheduling
As far as we know, heretofore, group scheduling problems with combined effects have only been studied in the single-machine environment. Motivated by realistic situations in the chemical industry, Huang, Wang, and Wang (2011) considered resource consumption u g of group g and p-s-d setup time s g = f (u g ), where f denotes a decreasing function. With the objectives of minimising C max | u g ≤ U and u g |C max ≤ C, they proposed O(n log n)-time and max{O(n log n, O(ng(n)))}-time algorithms to solve the problems with p [gjr] = p gj (μ 1 G r−1 g + ν 1 )(μ 2 t + ν 2 ). Thereinto, 0 ≤ G g < 1 is the learning indicator of group g, μ 1 , ν 1 , μ 2 , ν 2 are the positive constants, and μ 1 + ν 1 = 1. There was one paper proposing an integrated scheduling model involving sequence-dependent setup time, combined effects, preventive maintenance planning, and repair time (Pan et al. 2014). They analysed three complicated learningforgetting functions and developed a particular GA to solve large-sized problems. Derived from the observations in the metal or woodcutting process, He and Sun (2015) investigated total completion time minimisation problems involving combined effects p gjr = (p gj + α g t)r a and p-s-d setup time s [g] = s g + θ g t. These problems were solved by heuristic algorithms based on an SPT rule. Additionally, under simultaneous consideration of the truncated combined effects p [gjr] = p gj max{r a g , ρ} + αt, p-s-d setup times, and release times, Fan et al. (2018) studied serial-batching group scheduling problems. In order to minimise C max , they designed a hybrid VNS-ASHLO algorithm incorporating a VNS algorithm and adaptive simplified human learning optimisation (ASHLO) algorithm.

Conclusions and research prospects
Through a survey of the previous studies on scheduling problems with various learning and deteriorating effects, an overall analysis of the commonly used actual processing time functions, scheduling models, and proposed algorithms is given as follows.
First, the majority of proposed learning effects can be divided into general non-linear position-based functions, the sum-of-processing-time-based functions, as well as a hybrid of the two functions. In terms of deteriorating effects, the functions depend mainly on the job starting time or the sum of processed jobs' processing time. In further research on combined effects, most papers proposed functions incorporating position-based learning effects and time-dependent deterioration.
Second, the relationships of scheduling models are depicted in consideration of various job processing ways and manufacturing environments. On the one hand, there are few studies extending batch and group scheduling problems to parallel-machine and flowshop environments. On the other hand, a number of realistic influence factors enrich the related scheduling models, including setup times, delivery times, truncated effect, resource allocation, maintenance activity, etc. However, the links between the existing research and reality are still weak.
Finally, various algorithms were proposed for different scheduling problems. The most commonly used algorithms are heuristic algorithms, which were developed based on regular sequence rules including SPT, SDR, EDD algorithms, etc. In terms of exact algorithms, B&B algorithms and DP algorithms are more popular. In addition, there are a variety of meta-heuristic algorithms, including SA algorithms, GAs, etc. Besides these algorithms, a variety of hybrid algorithms combining heuristic and meta-heuristic algorithms were designed. Sometimes, B&B algorithms and DP algorithms were also considered in hybrid algorithms.
According to the above analysis, the main points of research gaps and directions are derived from three perspectives: the studied problems, proposed algorithms, and realistic applications.
(1) Reviewing the scheduling problems with learning and deteriorating effects, various models were developed based on different settings. First, the research on scheduling problems in the more complex settings is far to be mature, such as flexible flowshop, jobshop, and openshop problems. Second, in response to the requirements in the real production processes, both multi-objective scheduling problems and dynamic scheduling problems are both valuable but not well explored, including stochastic scheduling and robust scheduling. Finally, it is a great challenge for these scheduling studies to design more subdivided and actual functions abstracted from actual manufacturing processes. In future research, it is challenging but significant for scholars to develop dynamic scheduling models considering actual learning and deteriorating functions in flexible manufacturing environments.
(2) From the view of the algorithms applied in scheduling problems, it is found that the proposed algorithms do not combine the characteristics of actual manufacturing processes, showing a lack of innovation. With the requirements of high effectiveness in solutions for large-size, multi-objective, flexible, and dynamic scheduling problems, researchers can develop more data-driven algorithms, machine-learning algorithms, and robust algorithms, etc. Further, the algorithms need to be adjusted based on the features of the focused problems, the proposed learning and deteriorating functions, the introduced realistic influence factors, etc. In addition, the application of hybrid algorithms is still necessary for solving highly complex scheduling problems efficiently. (3) In future studies, the applications of research results should be emphasised in order to offer enterprises the optimal scheduling schemes. According to the real production demand, researchers need to set some standards and give appropriate approaches to help the enterprises select efficient and economic management tools, such as a stand-alone manufacturing management system or an integrated thirdparty platform. All the proposed scheduling models and algorithms should be embedded in the common management tools, which can generate scheduling schemes and visualise production process. Moreover, a comprehensive system or platform can be developed to integrate a number of scheduling scenarios and effective algorithms, with the goal to provide specific and differentiated services interfaces for various industries.
Overall, most typical learning and deteriorating functions were proposed in combination with some specific processing characteristics in the various production scenarios. Furthermore, the relevant studies also provide theoretical framework and algorithm paradigm for solving practical problems. However, it is not a good choice for researchers to make simple extensions or combinations based on the previous processing time functions and scheduling models. To further narrow the gap between theoretical scheduling models and practical problems, some scholars have recently started to develop more creative methods such as data-driven functions and algorithms. The most critical point is that the development of scheduling problems with learning and deteriorating effects should pay more attention to the connection with practical production scenarios. Some job processing time functions that truly reflect the learning and deteriorating effects need to be further explored, in association with realistic industrial data and advanced manufacturing processes.

Disclosure statement
No potential conflict of interest was reported by the author(s).