# Critical Review on the Objective Function of Flexible Job Shop Scheduling

Authors: Fengque Pei; Jiaxuan Zhang; Song Mei; Haojie Song
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The nature of construction and formulation of the multi-objective functions in the flexible job shop (FJS) provides great insights about the study of the technique and the mechanism in scheduling helping to prioritize efforts for optimizing the manufacturing execution performance, which draw the focus of the researchers at home and abroad. From the perspective of the model, the solution method and the objective function, this paper reviews and analyzes the related academic papers and conferences articles during past few years. As a result, the construction forms of the multi-objective function and the scheduling scenarios show strongly positive correlation with the Makespan, the delivery tardiness, the energy consumption or the cost. Consequently, facing with the characteristic in a general way, the detailed construction analysis on the function for the flexible job shop scheduling problem (FJSP) is critical reviewed, and a total 6 types with 49 kinds classification are discussed. Obviously, the main goals are to summarize, analyze, discuss, and synthesize the existing achievements, the current research status, and the ongoing studies on the FJSP functions construction, and additionally to give useful insight into future research. Combined with the development trend of the intelligent manufacturing and the operations research, arguing for the formulation of the multi-obj. function, this review is expected to contribute to the future research of FJSP.


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## Body

## 1. Introduction

Facing the emerging manufacturing plans of Germany, the United States and Japan, China put forward the “intelligent manufacturing” strategy, supported by the novel and emerging manufacturing technologies, to promote the comprehensive development of manufacturing execution performance. As the materialized center, a shop floor is not only collecting and distributing data, but also directly providing the manufacturing services. Obviously, optimizing the shop floor scheduling methods and cooperative abilities, and enhancing the effectiveness and availability of manufacturing execution is a crucial part to strengthen the competitiveness of the industry in the constantly changing and rapidly responsive market.Essentially, the shop scheduling refers to the arrangement of the resources, the processing time and the processing sequence under the constraints such as the process route, the resource situation and the delivery date as far as possible, so as to obtain the optimal product manufacturing time or cost [1]. This concept can increase significantly the use of existing production capacity in the shop floor. To date, step is taken and a number of publications are associated with the shop scheduling. Apparently, the flexible job shop (FJS) is becoming an important field to explore the new model of shop scheduling and intelligent manufacturing. The major advantage of the scheduling in the FJS is balancing the equipment utilization rate, increasing the production efficiency, reducing the production energy consumption, and improving the production continuity and diversity both locally and globally. In fact, it is widely applied in various industries such as the semiconductor wafer industry, the parts processing industry, and the complex product assembly industry. Among other types, the multi-obj. scheduling is the best candidate in the flexible job shop scheduling problem (FJSP), which is used to solve the conflicts and asynchronism among the multiple objectives. According to the accuracy of the solutions, it can be divided into the approximate technique (the iteration and construction method) and the precise method (the branch and bound, programming, etc.).Indeed, the aims of this research are to summarize, analyze, discuss, and synthesize the existing achievements, the current research status, and the ongoing studies on the FJSP functions construction, and additionally to give useful insight into future research. Combined with the development trend of the intelligent manufacturing and the operations research, arguing for the formulation of the multi-obj. function, this review is expected to contribute to the future research of FJSP. Up to now, a massive number of articles associated with FJSP have been published, and in this papers, the academic article published during 2019–2021 are analyzed. From the perspective of the multi-obj. function construction, the problem description, the classification and the solution ideas are summarized. In addition, the construction forms and application scenarios of the several main objective functions such as the Makespan, the delivery tardiness, the energy consumption, the cost, the equipment utilization rate are respectively analyzed.The rest of the paper is organized as follows: In second section, the multi-obj. FJSP are described; The third section presents the detailed construction of 5 main objective functions (the Makespan-, the delivery-tardiness-, the energy-consumption-, the cost-, the equipment-utilization-/the workload-rate-oriented functions). The fourth section simply summarized the unconventional objective function for some emerging scenarios or the new problems and the last section concludes the paper and considers the future work (on the selection, the construction, the evaluation and the optimization of the objective function).


## 2. The Multi-Obj. FJSP

According to FJSP has some important fields of application and it is widely discussed. This section describes the multi-obj. FJSP itself in a general way, and argues for its characteristic.


### 2.1. The Description of Multi-Obj. FJSP

According to the different nature of the shop floor, the scheduling problems can be identify into the flow shop scheduling, the mixed-flow shop scheduling, the job shop scheduling and the FJSP. Compared with the other three candidates, the actual FJSP shows more complex, stochastic dynamics, flexible and multiple objectives, and has certainly become an important and hot issue in the scholars [2, 3].The multi-obj. problem herein is a typical optimization combination problem, which can be generally described as a functional relationship composed ofN decision variable parameters, M objective functions, and q+p constraints. The multi-obj. FJSP can be expressed as several jobs are processed continuously on several processes, and there are mc facilities in the process c, and each c contains at least one process, namely c>1. The schematic diagram is shown in Figure 1.Figure 1
The diagram of the FJSP.The mathematical expression as follows:(1)minF=f1x,f2x,…,fMxTs.tgkx<0,hvx=0.In the formula,x=x1,x2,…xNT is an N-dimensional decision variable, F is an M-dimensional objective vector, k=1,2,…,q,gix represents q inequality constraints; v=1,2,…,p,hix denotes p equality constraints. (Notationsin Table 1).Table 1
Some notations of the symbols.
No.SymbolNotationsDetails or examples1NThe decision variables—2MObj. Function—3cThe number of the processescc>14mcThe number of the facilitiesmc5xN -Dimensional sub decision vectorx=x1,x2,…xNT,N-d6FM-Dimensional objective vectorM-d7gkxThe inequality constraints,qk=1,2,…,q8hvxThe equality constraintspv=1,2,…,p9CmaxMakespan10DDelivery tardiness11S或PSlack time or process time12EEnergy consumption13CCost14WWorkload15ERError rate16IImpact on the process17HHuman resources18TVariation in the tooling or material usage19RRobust and reliability20BBatch size21UMachine utilization22PUPunishment value


### 2.2. The Classification and Solution of Multi-Obj. FJSP

As shown in Figure2, the FJSP is noticeable to be divided into the single-obj. scheduling and the multi-obj. scheduling. The former is the basis of the later, while the later are subdivided into the weighted objective method, the layered objective method and the Pareto method. Further, the weighted objective method adopts the classic rules, the construction rules or sublots [4, 5]. To illustrate, Zhang et al. [6] scheduled the jobs from two agents with equal processing times and non-identical job sizes on a single parallel-batching machine by using the objective of minimize the weighted sum of the two Makespan of the jobs from two agents. Specifically, Zhang et al. proved that both the absolute worst-case ratio and the asymptotic worst-case ratio was the weight of one agent [7]. Furthermore, the layered method adopts target series smoothing optimization (i.e., priority). For example, Sun [8] used the mean square error of machine workload to optimize the equipment utilization rate as a secondary goal after the Makespan-oriented function. While, the Pareto solution is generally speaking the widely used candidate. Effect of the conflicts between the two or three more objective functions on the performance (Such as Wang et al. [9] pointed out the conflicts that on account of several machines can process one job simultaneously in welding shop, increasing the number of machines can short the processing time of operation while waste more energy consumption at the same time. The Pareto method based on WSA was proposed to optimize the total energy consumption and Makespan simultaneously) can be eased or balanced when the Pareto are employed in the FJSP. The heuristic intelligence algorithm is adopt to obtain the satisfactory non-inferior or the feasible solutions in the existing or ongoing researches, and some representative models, objectives and methods are shown in Table 2.Figure 2
The classification of the FJSP.Table 2
Some representative models, objectives and methods of multi-obj. scheduling.
AuthorScenariosDescriptionModelsMethodsMuhamad et al. [10]FJSA mathematical model was developed to optimize batch size considering completion time, lateness penalties and setup times.—Meta-heuristic-GALi et al. [11]In the welding shop FJSP, an effective DABC was proposed by considering job permutation and machine allocation simultaneously2-MIPDABCMehdi et al. [12]Aim to investigate the scheduling problem in a FJS with routing flexibility under uncertainties related to the random arrival of parts orders and machines failures, by considering reliability and maintenance constraints.—NSGA-IIZhou et al. [13]Propose an efficient scheduling method for multi-objective shop floor multi-crane scheduling problems. And the objectives of minimizing total weighted tardiness and total energy consumption were considered simultaneously.—2-e EA-LSWu et al. [14]WU studied the dual resource constrained FJSP when loading and unloading time of the fixtures was considered. It formulated a multi-objective mathematical model to jointly minimize the Makespan and the total setup time by considering the influence of resource requirement similarity among different operations.—Improved-NSGA-IILi et al. [15]On the FJS optimization of maintenance, repair and operations (MRO) service resources of complex product, Li gave a mathematical model for the service scheduling problem, and three objective functions to be optimized were proposed, which were to reduce customer waiting time, reduce excessive human resources, and maximize the cost performance index of the resources.—Improved-NSGA-II.Yan [16]The scheduling scheme evaluation algorithm integrating the combination of comprehensive weight and evaluation system—TOPSIS-G1- entropy weight methodHamid et al. [17]--MILPImproved-NSGA-IILu et al. [18]FJSsSome structural properties were identified to investigate a robust parallel-batching scheduling problem with fuzzy processing time and past-sequence-dependent delivery time.—HMV-VNSLiu et al. [19]FJS, in the steel shopA digital twin intelligent agent with cyberspace physical space integration was proposed as the unified driving source. And an open multiple objective travelling salesmen problem model was established, for constraints such as a minimum rolling unit plan, process specification and a minimum power consumption per ton of steel.—SA-PSO-paretoJens et al. [20]FJS, in semiconductor wafer fabrication facilitiesA bi-criteria scheduling problem for parallel identical batch processing machines was studied. The performance measures were the total weighted tardiness and the electricity cost where a TOU tariff was assumed. Unequal ready times of the jobs and non-identical job sizes were considered. A MILP was formulated to analyze the special case where all jobs had the same size, all due dates were zero, and the jobs were available at time zero.MILPGAs-paretoBhosale et al. [21]A loading problem in FJSIn a loading problem of FJS, part type selection and operations allocation were two critical problems. The authors believed that minimum total completion time of a product for the selected process plan and a consideration of waiting time of product and machine may be the optimum.MIPReal coded GASaid et al. [22]——Wang et al. [9]A distributed welding FJSPA multi-objective MIP model for energy-efficient scheduling of distributed welding flow shop was presented based on three sub-problems with allocating jobs among factories, scheduling the jobs in each factory and determining the amount of machines upon each job. A multi-objective whale swarm algorithm was proposed to optimize the total energy consumption and Makespan simultaneously.MIPWSAMina et al. [23]FJS, 1-machineA stochastic single machine scheduling problem in which the features of learning effect on processing times, sequence-dependent setup times, and machine configuration selection were considered simultaneously. More precisely, the machine works under a set of configurations and requires stochastic sequence-dependent setup times to switch from one configuration to another. Also, the stochastic processing time of a job was a function of its position and the machine configuration. The objective was to find the sequence of jobs and chose a configuration to process each job to minimize the Makespan.MINLPEVTDAWu et al. [24]An energy-conscious bi-objective single-machine batch scheduling problem under TOU electricity tariffs was proposed, whose objective was to simultaneously minimize total electricity cost and Makespan. The core idea was to transform the bi-objective problem into a series of single-objective problems that were fast and heuristically solved to obtain an approximate pareto front.IPε-CHAMeng et al. [25]FJS, 2-machinesA bi-objective cutting parallel machine scheduling problem aiming to minimize the total Makespan and total tardiness was addressed and the machine-job-dependent processing times were considered along with the setup times, pickup times, different delivery times, and machine eligibility constraints.MIPVNSGAIIIWang et al. [26]FJS, m-machinen (m ≥ 3)Am-machine (m ≥ 3) flow shop scheduling problem with deteriorating jobs was investigatedMIPMeta-heuristicFeng et al. [27]FJS, two-agentsA two-agent with equal processing times scheduling on a bounded parallel-batching machine were proposed. Moreover, each agent had its own objective function to be minimized. Such as the objective function of agent A was the Makespan of its jobs, and the objective function of agent B was the maximum lateness of its jobs.—ParetoDang et al. [28]The FJS consists of material handling devices such as AGVs or mobile robotsA MIP model lay in consideration of three interrelated sub-problems (the computing the sequence of operations on machines, the robot assignment for transportation, and the robot assignment for processing) was formulated to minimize the Makespan.MIPGA and TSZhao et al. [29]The work-rest scheduling of order picking in the warehousingUnder the circumstance that the previous researches focused on the sequencing, ignoring human factors. Zhao present a work-rest schedule model in parts to picker picking system. Two objectives were proposed that include minimizing the picking time and minimizing picking error rate. And workers’ fatigue, workload was taken into account in the manual order picking systems.—GA - paretoLu et al. [30]Permutation FJSLU investigated permutation flow shop manufacturing cell scheduling problems with deteriorating jobs and sequence-dependent setup times under dominant machines. In the proposed models, LU made joint decisions on part families sequencing and jobs sequencing within each family.—LU’s methodAbdelhamid [31]On facing a joint scheduling problem of production and preventive maintenance, the joint scheduling ricks at any moment to deviate from the theoretical desired performances when facing disturbances due to various causes. Thus, ABDELHAMID, based on the two strategies: Sequential and integrated, proposed a approach to study robustness of joint production and maintenance scheduling in permutation FJS.IPACO, GA, TSMeriem et al. [32]FJSP with some deteriorating jobsThe model invest the problem of scheduling a set ofn deteriorating jobs on m parallel unrelated machines subject to a deterioration effect, where the processing time of each job was a linear deterioration function of the starting time. The objective was to determine the best scheduling of the jobs, minimizing the Makespan and the total energy consumption.MILPNSGA-IILiang et al. [33]FJSP with limited buffersA self-adaptive differential evolution (DE) algorithm was designed to solve multi-objective FJSP with limited buffers. The Makespan and the largest job delay were treated as two separate objectives which were optimized simultaneously.RIPMPADEAnjana et al. [34]DFJSThe sequence-dependent setup environment was characterized by the dependence of the setup time on the current job and also on the previous job processed on that machine. To further complicate the problem, in most real-life scenarios, decision-makers had to optimize more than one performance measure while scheduling jobs on machines. ANJANA considers such a multi-objective sequence-dependent setup times DFJS. The objectives were minimization of Makespan and minimization of mean tardiness.IP{ NSGA-II; Hybrid-NSGA-II; discrete-PSO; hybrid-discrete-PSO } -paretoAydin et al. [35]A practical approach for extracting efficient rules for a more general type of DFJS in which jobs arrived at the shop at different times and machine breakdowns occur stochastically. Limited-buffer conditions were also considered. Gene expression programming combined with a simulation model was used for the design of scheduling policies.—GE-SMXu et al. [36]DFJS, multiple logistics tasksIn view of that problem such as low scale merit, inefficient resource utilization and conflict of interest between the operator and customer in the process of network operation, the scheduling strategy for multiple logistics tasks oriented to multi-origin and multi-destination on the basis of multiple-allocation-hybrid hub and spoke collaborative network (including two layers). Taking the backhaul resource, resource departure time, capability limitation, task delay penalty, etc. into account, the aim of the problem was to minimize both total cost and total time for the task set.MILPImmune-GA-ParetoShi et al. [37]DFJS,1-machine-multi-agentsWhen considering the competitive multi-agent scheduling problem on a single machine, where each agent’s cost function was to minimize its total weighted late work, the optimum candidate was to find the pareto-optimal frontier, i.e., the set of all pareto-optimal points. The 1-agent/2-agent/multi-agent were discussed respectively.PPDPParetoFan et al. [38]A two-stage HFJSP with two objective functions was proposed. One of the obj. was the simultaneous rate for the arrival of different parts of products at assembly stage and another was the on-time delivery rate based on the products delivery schedule. The function can strengthen the link between the manufacturing stage and the assembly stage.—Immune -FA- paretoSven et al. [39]A HFJSP with variable discrete production speed levels was considered with the aim of minimizing both energy costs and total tardiness. Although lower speeds can reduce energy consumption, they also increased processing times, which counteracted the objective of punctual delivery. Two new model formulations additionally taking time-of-use energy prices into account were presented and compared.MIPEPS- constraint methodChe et al. [40]HFJSThis model proposed a multi-objective MIP for energy-efficient HFJS with lot streaming in order to minimize both the production Makespan and electric power consumption.MIPGA-paretoZhou et al. [41]An energy-saving scheduling scheme was carried out the part feeding tasks of mobile robots in the automobile mixed model assembly lines. The objective of minimizing the total energy consumption of mobile robots was jointly incorporated with the operational criterions when implementing part feeding tasks. Furthermore, managerial insights were illustrated to make trade-offs between the total line-side inventory level and the energy consumption, jointly making it helpful in the greening scheduling process of the practical production.MIPMDRCIUllah et al. [42]ULLAH, first time, investigated an order oriented simultaneous sequencing and balancing problem of multi-mixed model assembly lines with an aim to minimize the variation in material usage, minimize the maximum Makespan among the multi-lines and minimize the penalty cost of the late delivery models from different orders simultaneously.—a multi-objective ABCZheng [43]The continuous HFJSP—DDICOALili et al. [44]HFJS, in semiconductor wafer shopThis model addressed a novel approach for integrating preventive maintenance (PM) into production planning of a complex manufacturing system based on availability and cost. Experimental results demonstrated the effectiveness of proposed approach, reduce personnel number while guarantee the maintenance tasks.—ACOLiu et al. [45]UFJSThe author studied the UFJSP, where the sequence among operations for a job can be changed within certain constraints.MILPGADao et al. [46] and Dao et al. [47]VCIMThe collaborative manufacturing scheduling for cost and completion time in spherical distributed manufacturing systems—Improve GALee et al. [48]—A dynamic control algorithm for production scheduling that considers machine capacity and idle time controls and aims at satisfying time related production demand and reducing energy consumption in a unified manner was proposed. . A MINLP model is developed to determine job arrival sequence for a machine and machine capacity while minimizing resulting costs of just-in-time production, machine repair, and energy consumption during machine idle time and nominal processing.MINLPALNS vs. GAZhou et al. [49]a serial production lineThis paper addressed an energy-efficient multi-objective scheduling problem of a serial production line (SPSP) integrating production and maintenance, with the criteria of minimizing the Makespan and the total energy consumption simultaneously.—HMOLEAShao et al. [50]Distributed assembly blocking FJSPA distributed assembly blocking FJSP, which consisted of two stages: Production and assembly was investigated. The first stage was processing jobs in several identical factories. Each factory had a series of machines no intermediate buffers existing between adjacent ones. The second stage assembled the processed jobs into the final products through a single machine. The objective was to minimize the maximum completion time or Makespan of all products.MILPILSLiu et al. [51]Discrete manufacturing systemThe integration of process planning and scheduling in discrete manufacturing system was studiedMILP2 Layered-GAWu et al. [52]Smart manufacturing systemIn most cases the energy consumption of the manufacturing systems were considered using average energy consumption models or simple scheduling models for the needs of discrete event simulation. Wu put forward the model on energy-efficiency scheduling based on the scenario of multi-tasks and multi-resources in smart manufacturingMILPSRSubhaa et al. [53]Cellular manufacturing systemThis cross-flow facilitates routing flexibility and paves way in the cellular manufacturing system for reduced schedule length thereby optimizing resources leading to minimized operational costMINLPSA-GANote. Please refer to Table1 for the notations of some symbols and Table 3 for abbreviations.Table 3
Some abbreviations in the article.
No.Abb.Full1FJS/FJSsThe flexible job shop2FJSPThe flexible job shop scheduling problemDFJSThe dynamic flexible job shop3HFJSThe hybrid flexible job shop4UFJSThe ultra-flexible job shop5VCIMVirtual computer-integrated manufacturing6IPInteger programming7RIPRandom integer programming8ILPInteger linear programming9MIPMixed integer programming102-MIP2-Mixed integer programming11MILPMixed integer linear programming12MINLPMixed integer non-linear programming13PPDPThe pseudo-polynomial dynamic programming14GAGenetic algorithm15DABCImproved discrete artificial bee colony algorithm16NSGA-IINon-dominated sorting genetic algorithms172-e EA-LSImproved decomposition-based 2-echelon evolutionary algorithm with local search strategies18HMV-VNSHybrid multi-verse optimizer-variable neighborhood search algorithm19SASimulated annealing algorithm20PSOParticle swarm optimization21WSAWhale swarm algorithm22EVTDAExtreme value theory-based deterministic Approximation23ε-CHAε-Constraint-based constructive heuristic algorithms24VNSGAIIINSGAIII with an embedded variable neighborhood25TSTabu search26ACOAnts colony optimization27MPADEA multi-programming hybrid self-adaptive differential evolution algorithm28GE-SMGene expression programming combined with a simulation model29FAFirefly algorithm30MDRCIMulti-obj. Disturbance and repair strategy enhanced cohort algorithm31ALNSAdaptive large neighborhood search32HMOLEAa hybrid multi-obj. Opposite-based learning evolutionary algorithm33ILSAn iterated local search34SRThe score ranking35DDICOACoyote optimization algorithm with dimension by dimension improvement36TOPSIS-G1- entropy weight methodTechnique for order preference by similarity to an ideal solution-nash-g1- entropy weight method37TOUTime-of-Use38UAVUnmanned aerial vehicleIn the above Table2, it is a consensus in academic and business that the selection and construction of multi-obj. functions are mainly reflected in the characteristics of FJSP application scenarios(as well as the scheduling scale), which will be detailed discussed in Section 2.


## 2.1. The Description of Multi-Obj. FJSP

According to the different nature of the shop floor, the scheduling problems can be identify into the flow shop scheduling, the mixed-flow shop scheduling, the job shop scheduling and the FJSP. Compared with the other three candidates, the actual FJSP shows more complex, stochastic dynamics, flexible and multiple objectives, and has certainly become an important and hot issue in the scholars [2, 3].The multi-obj. problem herein is a typical optimization combination problem, which can be generally described as a functional relationship composed ofN decision variable parameters, M objective functions, and q+p constraints. The multi-obj. FJSP can be expressed as several jobs are processed continuously on several processes, and there are mc facilities in the process c, and each c contains at least one process, namely c>1. The schematic diagram is shown in Figure 1.Figure 1
The diagram of the FJSP.The mathematical expression as follows:(1)minF=f1x,f2x,…,fMxTs.tgkx<0,hvx=0.In the formula,x=x1,x2,…xNT is an N-dimensional decision variable, F is an M-dimensional objective vector, k=1,2,…,q,gix represents q inequality constraints; v=1,2,…,p,hix denotes p equality constraints. (Notationsin Table 1).Table 1
Some notations of the symbols.
No.SymbolNotationsDetails or examples1NThe decision variables—2MObj. Function—3cThe number of the processescc>14mcThe number of the facilitiesmc5xN -Dimensional sub decision vectorx=x1,x2,…xNT,N-d6FM-Dimensional objective vectorM-d7gkxThe inequality constraints,qk=1,2,…,q8hvxThe equality constraintspv=1,2,…,p9CmaxMakespan10DDelivery tardiness11S或PSlack time or process time12EEnergy consumption13CCost14WWorkload15ERError rate16IImpact on the process17HHuman resources18TVariation in the tooling or material usage19RRobust and reliability20BBatch size21UMachine utilization22PUPunishment value


## 2.2. The Classification and Solution of Multi-Obj. FJSP

As shown in Figure2, the FJSP is noticeable to be divided into the single-obj. scheduling and the multi-obj. scheduling. The former is the basis of the later, while the later are subdivided into the weighted objective method, the layered objective method and the Pareto method. Further, the weighted objective method adopts the classic rules, the construction rules or sublots [4, 5]. To illustrate, Zhang et al. [6] scheduled the jobs from two agents with equal processing times and non-identical job sizes on a single parallel-batching machine by using the objective of minimize the weighted sum of the two Makespan of the jobs from two agents. Specifically, Zhang et al. proved that both the absolute worst-case ratio and the asymptotic worst-case ratio was the weight of one agent [7]. Furthermore, the layered method adopts target series smoothing optimization (i.e., priority). For example, Sun [8] used the mean square error of machine workload to optimize the equipment utilization rate as a secondary goal after the Makespan-oriented function. While, the Pareto solution is generally speaking the widely used candidate. Effect of the conflicts between the two or three more objective functions on the performance (Such as Wang et al. [9] pointed out the conflicts that on account of several machines can process one job simultaneously in welding shop, increasing the number of machines can short the processing time of operation while waste more energy consumption at the same time. The Pareto method based on WSA was proposed to optimize the total energy consumption and Makespan simultaneously) can be eased or balanced when the Pareto are employed in the FJSP. The heuristic intelligence algorithm is adopt to obtain the satisfactory non-inferior or the feasible solutions in the existing or ongoing researches, and some representative models, objectives and methods are shown in Table 2.Figure 2
The classification of the FJSP.Table 2
Some representative models, objectives and methods of multi-obj. scheduling.
AuthorScenariosDescriptionModelsMethodsMuhamad et al. [10]FJSA mathematical model was developed to optimize batch size considering completion time, lateness penalties and setup times.—Meta-heuristic-GALi et al. [11]In the welding shop FJSP, an effective DABC was proposed by considering job permutation and machine allocation simultaneously2-MIPDABCMehdi et al. [12]Aim to investigate the scheduling problem in a FJS with routing flexibility under uncertainties related to the random arrival of parts orders and machines failures, by considering reliability and maintenance constraints.—NSGA-IIZhou et al. [13]Propose an efficient scheduling method for multi-objective shop floor multi-crane scheduling problems. And the objectives of minimizing total weighted tardiness and total energy consumption were considered simultaneously.—2-e EA-LSWu et al. [14]WU studied the dual resource constrained FJSP when loading and unloading time of the fixtures was considered. It formulated a multi-objective mathematical model to jointly minimize the Makespan and the total setup time by considering the influence of resource requirement similarity among different operations.—Improved-NSGA-IILi et al. [15]On the FJS optimization of maintenance, repair and operations (MRO) service resources of complex product, Li gave a mathematical model for the service scheduling problem, and three objective functions to be optimized were proposed, which were to reduce customer waiting time, reduce excessive human resources, and maximize the cost performance index of the resources.—Improved-NSGA-II.Yan [16]The scheduling scheme evaluation algorithm integrating the combination of comprehensive weight and evaluation system—TOPSIS-G1- entropy weight methodHamid et al. [17]--MILPImproved-NSGA-IILu et al. [18]FJSsSome structural properties were identified to investigate a robust parallel-batching scheduling problem with fuzzy processing time and past-sequence-dependent delivery time.—HMV-VNSLiu et al. [19]FJS, in the steel shopA digital twin intelligent agent with cyberspace physical space integration was proposed as the unified driving source. And an open multiple objective travelling salesmen problem model was established, for constraints such as a minimum rolling unit plan, process specification and a minimum power consumption per ton of steel.—SA-PSO-paretoJens et al. [20]FJS, in semiconductor wafer fabrication facilitiesA bi-criteria scheduling problem for parallel identical batch processing machines was studied. The performance measures were the total weighted tardiness and the electricity cost where a TOU tariff was assumed. Unequal ready times of the jobs and non-identical job sizes were considered. A MILP was formulated to analyze the special case where all jobs had the same size, all due dates were zero, and the jobs were available at time zero.MILPGAs-paretoBhosale et al. [21]A loading problem in FJSIn a loading problem of FJS, part type selection and operations allocation were two critical problems. The authors believed that minimum total completion time of a product for the selected process plan and a consideration of waiting time of product and machine may be the optimum.MIPReal coded GASaid et al. [22]——Wang et al. [9]A distributed welding FJSPA multi-objective MIP model for energy-efficient scheduling of distributed welding flow shop was presented based on three sub-problems with allocating jobs among factories, scheduling the jobs in each factory and determining the amount of machines upon each job. A multi-objective whale swarm algorithm was proposed to optimize the total energy consumption and Makespan simultaneously.MIPWSAMina et al. [23]FJS, 1-machineA stochastic single machine scheduling problem in which the features of learning effect on processing times, sequence-dependent setup times, and machine configuration selection were considered simultaneously. More precisely, the machine works under a set of configurations and requires stochastic sequence-dependent setup times to switch from one configuration to another. Also, the stochastic processing time of a job was a function of its position and the machine configuration. The objective was to find the sequence of jobs and chose a configuration to process each job to minimize the Makespan.MINLPEVTDAWu et al. [24]An energy-conscious bi-objective single-machine batch scheduling problem under TOU electricity tariffs was proposed, whose objective was to simultaneously minimize total electricity cost and Makespan. The core idea was to transform the bi-objective problem into a series of single-objective problems that were fast and heuristically solved to obtain an approximate pareto front.IPε-CHAMeng et al. [25]FJS, 2-machinesA bi-objective cutting parallel machine scheduling problem aiming to minimize the total Makespan and total tardiness was addressed and the machine-job-dependent processing times were considered along with the setup times, pickup times, different delivery times, and machine eligibility constraints.MIPVNSGAIIIWang et al. [26]FJS, m-machinen (m ≥ 3)Am-machine (m ≥ 3) flow shop scheduling problem with deteriorating jobs was investigatedMIPMeta-heuristicFeng et al. [27]FJS, two-agentsA two-agent with equal processing times scheduling on a bounded parallel-batching machine were proposed. Moreover, each agent had its own objective function to be minimized. Such as the objective function of agent A was the Makespan of its jobs, and the objective function of agent B was the maximum lateness of its jobs.—ParetoDang et al. [28]The FJS consists of material handling devices such as AGVs or mobile robotsA MIP model lay in consideration of three interrelated sub-problems (the computing the sequence of operations on machines, the robot assignment for transportation, and the robot assignment for processing) was formulated to minimize the Makespan.MIPGA and TSZhao et al. [29]The work-rest scheduling of order picking in the warehousingUnder the circumstance that the previous researches focused on the sequencing, ignoring human factors. Zhao present a work-rest schedule model in parts to picker picking system. Two objectives were proposed that include minimizing the picking time and minimizing picking error rate. And workers’ fatigue, workload was taken into account in the manual order picking systems.—GA - paretoLu et al. [30]Permutation FJSLU investigated permutation flow shop manufacturing cell scheduling problems with deteriorating jobs and sequence-dependent setup times under dominant machines. In the proposed models, LU made joint decisions on part families sequencing and jobs sequencing within each family.—LU’s methodAbdelhamid [31]On facing a joint scheduling problem of production and preventive maintenance, the joint scheduling ricks at any moment to deviate from the theoretical desired performances when facing disturbances due to various causes. Thus, ABDELHAMID, based on the two strategies: Sequential and integrated, proposed a approach to study robustness of joint production and maintenance scheduling in permutation FJS.IPACO, GA, TSMeriem et al. [32]FJSP with some deteriorating jobsThe model invest the problem of scheduling a set ofn deteriorating jobs on m parallel unrelated machines subject to a deterioration effect, where the processing time of each job was a linear deterioration function of the starting time. The objective was to determine the best scheduling of the jobs, minimizing the Makespan and the total energy consumption.MILPNSGA-IILiang et al. [33]FJSP with limited buffersA self-adaptive differential evolution (DE) algorithm was designed to solve multi-objective FJSP with limited buffers. The Makespan and the largest job delay were treated as two separate objectives which were optimized simultaneously.RIPMPADEAnjana et al. [34]DFJSThe sequence-dependent setup environment was characterized by the dependence of the setup time on the current job and also on the previous job processed on that machine. To further complicate the problem, in most real-life scenarios, decision-makers had to optimize more than one performance measure while scheduling jobs on machines. ANJANA considers such a multi-objective sequence-dependent setup times DFJS. The objectives were minimization of Makespan and minimization of mean tardiness.IP{ NSGA-II; Hybrid-NSGA-II; discrete-PSO; hybrid-discrete-PSO } -paretoAydin et al. [35]A practical approach for extracting efficient rules for a more general type of DFJS in which jobs arrived at the shop at different times and machine breakdowns occur stochastically. Limited-buffer conditions were also considered. Gene expression programming combined with a simulation model was used for the design of scheduling policies.—GE-SMXu et al. [36]DFJS, multiple logistics tasksIn view of that problem such as low scale merit, inefficient resource utilization and conflict of interest between the operator and customer in the process of network operation, the scheduling strategy for multiple logistics tasks oriented to multi-origin and multi-destination on the basis of multiple-allocation-hybrid hub and spoke collaborative network (including two layers). Taking the backhaul resource, resource departure time, capability limitation, task delay penalty, etc. into account, the aim of the problem was to minimize both total cost and total time for the task set.MILPImmune-GA-ParetoShi et al. [37]DFJS,1-machine-multi-agentsWhen considering the competitive multi-agent scheduling problem on a single machine, where each agent’s cost function was to minimize its total weighted late work, the optimum candidate was to find the pareto-optimal frontier, i.e., the set of all pareto-optimal points. The 1-agent/2-agent/multi-agent were discussed respectively.PPDPParetoFan et al. [38]A two-stage HFJSP with two objective functions was proposed. One of the obj. was the simultaneous rate for the arrival of different parts of products at assembly stage and another was the on-time delivery rate based on the products delivery schedule. The function can strengthen the link between the manufacturing stage and the assembly stage.—Immune -FA- paretoSven et al. [39]A HFJSP with variable discrete production speed levels was considered with the aim of minimizing both energy costs and total tardiness. Although lower speeds can reduce energy consumption, they also increased processing times, which counteracted the objective of punctual delivery. Two new model formulations additionally taking time-of-use energy prices into account were presented and compared.MIPEPS- constraint methodChe et al. [40]HFJSThis model proposed a multi-objective MIP for energy-efficient HFJS with lot streaming in order to minimize both the production Makespan and electric power consumption.MIPGA-paretoZhou et al. [41]An energy-saving scheduling scheme was carried out the part feeding tasks of mobile robots in the automobile mixed model assembly lines. The objective of minimizing the total energy consumption of mobile robots was jointly incorporated with the operational criterions when implementing part feeding tasks. Furthermore, managerial insights were illustrated to make trade-offs between the total line-side inventory level and the energy consumption, jointly making it helpful in the greening scheduling process of the practical production.MIPMDRCIUllah et al. [42]ULLAH, first time, investigated an order oriented simultaneous sequencing and balancing problem of multi-mixed model assembly lines with an aim to minimize the variation in material usage, minimize the maximum Makespan among the multi-lines and minimize the penalty cost of the late delivery models from different orders simultaneously.—a multi-objective ABCZheng [43]The continuous HFJSP—DDICOALili et al. [44]HFJS, in semiconductor wafer shopThis model addressed a novel approach for integrating preventive maintenance (PM) into production planning of a complex manufacturing system based on availability and cost. Experimental results demonstrated the effectiveness of proposed approach, reduce personnel number while guarantee the maintenance tasks.—ACOLiu et al. [45]UFJSThe author studied the UFJSP, where the sequence among operations for a job can be changed within certain constraints.MILPGADao et al. [46] and Dao et al. [47]VCIMThe collaborative manufacturing scheduling for cost and completion time in spherical distributed manufacturing systems—Improve GALee et al. [48]—A dynamic control algorithm for production scheduling that considers machine capacity and idle time controls and aims at satisfying time related production demand and reducing energy consumption in a unified manner was proposed. . A MINLP model is developed to determine job arrival sequence for a machine and machine capacity while minimizing resulting costs of just-in-time production, machine repair, and energy consumption during machine idle time and nominal processing.MINLPALNS vs. GAZhou et al. [49]a serial production lineThis paper addressed an energy-efficient multi-objective scheduling problem of a serial production line (SPSP) integrating production and maintenance, with the criteria of minimizing the Makespan and the total energy consumption simultaneously.—HMOLEAShao et al. [50]Distributed assembly blocking FJSPA distributed assembly blocking FJSP, which consisted of two stages: Production and assembly was investigated. The first stage was processing jobs in several identical factories. Each factory had a series of machines no intermediate buffers existing between adjacent ones. The second stage assembled the processed jobs into the final products through a single machine. The objective was to minimize the maximum completion time or Makespan of all products.MILPILSLiu et al. [51]Discrete manufacturing systemThe integration of process planning and scheduling in discrete manufacturing system was studiedMILP2 Layered-GAWu et al. [52]Smart manufacturing systemIn most cases the energy consumption of the manufacturing systems were considered using average energy consumption models or simple scheduling models for the needs of discrete event simulation. Wu put forward the model on energy-efficiency scheduling based on the scenario of multi-tasks and multi-resources in smart manufacturingMILPSRSubhaa et al. [53]Cellular manufacturing systemThis cross-flow facilitates routing flexibility and paves way in the cellular manufacturing system for reduced schedule length thereby optimizing resources leading to minimized operational costMINLPSA-GANote. Please refer to Table1 for the notations of some symbols and Table 3 for abbreviations.Table 3
Some abbreviations in the article.
No.Abb.Full1FJS/FJSsThe flexible job shop2FJSPThe flexible job shop scheduling problemDFJSThe dynamic flexible job shop3HFJSThe hybrid flexible job shop4UFJSThe ultra-flexible job shop5VCIMVirtual computer-integrated manufacturing6IPInteger programming7RIPRandom integer programming8ILPInteger linear programming9MIPMixed integer programming102-MIP2-Mixed integer programming11MILPMixed integer linear programming12MINLPMixed integer non-linear programming13PPDPThe pseudo-polynomial dynamic programming14GAGenetic algorithm15DABCImproved discrete artificial bee colony algorithm16NSGA-IINon-dominated sorting genetic algorithms172-e EA-LSImproved decomposition-based 2-echelon evolutionary algorithm with local search strategies18HMV-VNSHybrid multi-verse optimizer-variable neighborhood search algorithm19SASimulated annealing algorithm20PSOParticle swarm optimization21WSAWhale swarm algorithm22EVTDAExtreme value theory-based deterministic Approximation23ε-CHAε-Constraint-based constructive heuristic algorithms24VNSGAIIINSGAIII with an embedded variable neighborhood25TSTabu search26ACOAnts colony optimization27MPADEA multi-programming hybrid self-adaptive differential evolution algorithm28GE-SMGene expression programming combined with a simulation model29FAFirefly algorithm30MDRCIMulti-obj. Disturbance and repair strategy enhanced cohort algorithm31ALNSAdaptive large neighborhood search32HMOLEAa hybrid multi-obj. Opposite-based learning evolutionary algorithm33ILSAn iterated local search34SRThe score ranking35DDICOACoyote optimization algorithm with dimension by dimension improvement36TOPSIS-G1- entropy weight methodTechnique for order preference by similarity to an ideal solution-nash-g1- entropy weight method37TOUTime-of-Use38UAVUnmanned aerial vehicleIn the above Table2, it is a consensus in academic and business that the selection and construction of multi-obj. functions are mainly reflected in the characteristics of FJSP application scenarios(as well as the scheduling scale), which will be detailed discussed in Section 2.


## 3. The Formulation of Some Common Function in Multi-Obj. FJSP

Section3 devoted to the basic classification of some common function in multi-obj. FJSP and the function compositions and meanings are discussed.Actually, according the review, in the multi-obj. FJSP, the commonly used objective functions includes the Makespan-, the delivery-tardiness-, the idle-time- or the process-time-, the energy-consumption-, the cost-, the workload-, the error-rate-, the impact-on-the-process-, the utilization- or the replace-times-, and the method-robustness and reliability-oriented functions. Clearly, above all, the Makespan-, the delivery-tardiness-, the idle-time- or the process-time-oriented functions have obvious temporal correlation, which need to describe the relationship between the time sequence and the process state during the function construction. On the other hand, the energy-consumption- and the cost-oriented functions are structured based on the flow (the energy flow or the capital flow). Even though, in some literature, the energy flow and the capital flow transform and express each other (the electricity or power is an important part of the capital flow). And in numerous scenarios, they measure each other, and make interactive calculation (Such as in Schulz [39] research, the energy consumption is measured by the TOU, and in Norelhouda [54] article, the cost is the product of the total energy consumption volume and the unit energy consumption cost). What’s more, the indicators such as the workload-, the error-rate-, the utilization- or the replace-times-oriented function have explicit representation equations, and the calculation methods are mostly mathematical statistics. However, there are no unified standards for the impact-on-the-process-, and the method-robustness and reliability-oriented functions, which are implicit representations. Therefore, to overcome the limitations, some other measurement models, such as Taguchi method and SNR, should be employed. After all, the main objective functions of FJSP are shown in Table 4.Table 4
The common objectives in the FJS.
No.Abbr.Symbol1The MakespanCmax2The delivery tardiness [18, 25, 27, 33, 37, 39]D3The idle time or process time [18, 30, 36]I or P4The energy consumptionE5The cost [17, 41]C6The workloadW7The error rateER8The impact on the processI9Human resourcesH10The variation in the tooling or material usageT11The robust and reliability [55]R12The batch sizeB13The machine utilization [56]U14Punishment valuePUNote. Please refer to Table1 for the notations of some symbols and Table 3 for abbreviation.


### 3.1. The Makespan-Oriented Objective Function

According to the statistics of the databases such as IEEExpore, Elsevier, Springer, CNKI and Wanfang, the number of the articles with the Makespan-oriented function in the multi-obj. FJSP during the past few years is up to 1602, accounting for more than 28%, which straightforward proves that the Makespan-oriented function is the widely used candidate during the past few years as shown in Table5.Table 5
The index results of Makespan-obj. in the big-5-database during the past few years.
DatabaseKeywordsResultMakespan-orientedProportion (%)1CNKIMulti objective, job shop, scheduling56518031.862Wanfang255207.843IEEExplore12112099.174Elsevier168779447.075Springer298748816.34Total5615160228.53The Makespan-oriented function [57, 58] can be further subdivided into the total-Makespan-, the off-on-Makespan-, the time-buffer-Makespan-, the worst-case-Makespan-, the fuzzy-Makespan-, the robust-Makespan-, the weight-Makespan-oriented function to minimum the maximum-processing-time problems.According to the manufacturing process, the total-Makespan-oriented function can be calculated by the cumulative timing of all processes on the longest process chain or the cumulative finishing time of the last process on a single device [59, 60]. While the total-Makespan-oriented function, with simple structure, is easy to distort in some particular process. To overcome the various distortion, steps in the modeling have been taken [61]. For instance, the off-on process is added on the base of the total-Makespan-oriented function to handle the situations that the start/cutoff/restart time cannot be ignored [24]. Or the system covering the blocking area or the shuffle zone, like the off-on-Makespan-oriented function, the calculation range takes account all the blocking and shuffle in the whole processing cycle [62].What’s more, with the deepening research on the unascertained information, more scholars believe that the uncertainty of FJS will lead larger randomness in the Makespan-oriented function [63]. Therefore, various Makespan-oriented modeling methods based on unascertained information come into being. To name a few, the worst-case-Makespan- [64], the robust version is provided to formulate the obj. function to scale the reliability. The fuzzy-Makespan-oriented function makes a more future scaling, besides blurring the last process, the total Makespan is also taken into account to obtain the weight of the best, the most possible and the worst completion time of the process or the total Makespan [65]. In addition, on dealing with the robustness, CHEN’s [66] approach was to combine the confidence interval with the game model. While FU [65] proposed the calculation paradigm of the robust bias, which optimized the production process robustness by taking the deviation between the most possible Makespan/the worst Makespan and the best Makespan as the objective function.Besides above all, means of researches associated with the weight-Makespan-oriented function have been published and attaches the academic focus [67]. Its essence is to linear couple of some production or operation indicators such as the blocking rate or the marginal cost based on different equipment or different stages in the shop floor [68, 69]. To illustrate, Yan [70] adopted the weighted-Makespan in the mixed-flow production line, whose weight is measured by the compensation cost of the process tardiness. Up-to-date, with the rise of cloud manufacturing, the weightd-Makespan of multi-enterprise collaboration has gradually become a research trend [71], which is different from the conventional Makespan, besides the processing time, the weighted delivery time is also included in the weighted coupling of the cloud Makespan [72, 73].To sum up, the Makespan-oriented objective function can be broadly subdivided into three [74]: (1) The total-Makespan-oriented function covering some production characteristics and specific application scenarios. such as the forging shop floor and other equipment that need preheating, which takes a long start and cutoff time [75, 76]. And the off-on-Makespan-oriented function, due to the limited shuffle space in the wafer shop, the waiting sequence is needed in the formulation, namely the blocking and shuffle Makespan-oriented function. (2) For the robust-Makespan-oriented function, which covers the unascertained information, the robustness factors such as the ambiguity and the confidence interval should be introduced to eliminate the influence of the disturbance [77]. (3) The weighted-Makespan-oriented function is mostly used in multi-device, multi-organization or cloud collaborative scheduling application scenarios. This paper summarizes the construction forms of the Makespan-oriented objective function as shown in Table 6.Table 6
The construction form of Makespan-objectives.
No.The makespan-oriented functionObj. FunctionThe decision variables1The total Makespan [78], CmaxCmax−∑tT/itT/i:Σ Processing time of all processes; i: No. i process; T: All processes; θi: The robustness index of the processing sequence; maxθi: The worst processing sequence.2The worst-case Makespan [79], Cmaxw−cCmaxw−c−maxθi∑tT/i3The off-on Makespan [50], Cmaxo−oCmaxo−o−∑tT/i+tT/i:off−ontT/i:off−on: Σ Off-on time of all the processes.4The time buffer Makespan [62], Cmaxw−bCmaxw−b−∑tT/i+tb/i.tib: No. i buffer time.5The fuzzy Makespan [65], CmaxfCmaxf−c1+2×c2+c3/4c1: The best Makespan; c2: The most possible Makespan; c3: The worst Makespan.6The robustness of the Makespan [65], CmaxrCmaxr−maxc2−c1,c3−c17The expected weighted sum of Makespan [6], Cmaxw−sCmaxw−s−αCmaxA+1−αCmaxBαCmaxA: Equipment A’s Makesapn; 1−αCmaxB: Equipment B’s Makesapn.8The cloud multi-stage Makespan [72], Cmaxm−lCmaxm−l−∑t2ed−p+tmatt2ed−p: Secondary manufacturer’s completion time; tmat: The secondary manufacturer matches the transport time of the primary manufacturer.Note. Please refer to Table1 for the notations of some symbols and Table 3 for abbreviations.


### 3.2. The Tardiness-Oriented Objective Function

Besides the Makespan-oriented function, the most helpful objective functions are the delivery-tardiness-, the energy-consumption- and the cost-oriented function, which respectively accounts for more than 10%. In brief the delivery-tardiness objective function can be subdivided into the average-tardiness-, the maximum-tardiness-, the weighted-tardiness-, the blocking-tardiness-, the warehousing-waiting-, the JIT-tardiness-, and the weighted-JIT-tardiness-oriented function, etc., to meet the due time [76, 80].The average-tardiness-oriented function accumulates the tardiness of all the processes, representing the mean tardiness of all the machine in the whole shop floor or all the processes in a single device, which is widely employed among the tardiness-oriented function and becomes a critical approach to achieve the purpose of no late. While the state that the effect of tardiness cannot be characterized due to the unbalanced tardiness (It is unrealistic to tell the excessive tardiness of only one process from the relatively punctual tardiness of other means of processes in the average-tardiness-oriented function) is beyond the control. On this circumstance, the maximum-tardiness- and weighted-tardiness-oriented function are proposed to eliminate this distortion. Based on the former function, the maximum-tardiness process is the smaller-is-the-better. To illustrate, Lei Deming [81] used the minimum of the maximum-deviation or the maximum-tardiness of the due date to balance the imbalance between the number of key process and the number of non-core process in the multi-class teaching optimization theory proposed by HFJS. What’s more, the weighted-tardiness-function, takes another way, pays more attention to the core process. By assigning higher weight and increasing more influence [82], undoubtedly, the key process will be charging more responsibility in the formulation, so as to generalize the average tardiness effect in the reality. For instance, JENS [20] gave different weights to different facilities on tardiness based on the workload or the logistics quantities in the semiconductor wafer manufacturing system to focus more on what they were cared for. Intriguingly this formulation has derived and brought forward the blocking-tardiness-oriented function. Another construction form, the warehousing-wait-oriented function, is put forward to solve the minimizing the maximum-waiting-time of the storage process by the idea of minimizing the maximum-tardiness, which takes the storage as a special facility.Admittedly, the above delivery-tardiness-oriented function are all unidirectional, pay the attention on the delay or the lateness, while the JIT-tardiness-, the weighted-JIT-tardiness-oriented functions and the imbalance-level-of-the-completion-oriented function are bidirectional, which means that both the lateness and earliness are both penalized. Such as on JIT delivery, Xu and Muhammed [83] adopted the standardized operating time and the Taguchi “the target is the best” technique to measure the lateness and earliness. On the base of Xu’s research, Fan [38] assigned the weights to the core and the non-core, the blocking and the non-blocking processes. While the imbalance-level-of-the-completion-oriented function refers to the calculation of the completion time deviation between the different facilities for the same process, and the global delivery tardiness is accumulated by the unbalanced level of every single process.To summarize, the delivery-tardiness-oriented function can be further subdivided into two: (1) The unidirectional-delivery-tardiness-oriented function, represented by time, such as the average-tardiness-, the maximum-tardiness-, the weighted-tardiness-, the blocking-tardiness-, the warehousing-waiting-oriented function. The difficulty of objective function construction is to coupling the global tardiness and the key process tardiness, and balancing the minimum of the overall tardiness and the lowest blocking rate of the core equipment. (2) The bidirectional-tardiness-oriented function, represented by the punishment, such as the JIT-tardiness- and the weighted-JIT-tardiness-oriented function. These objective functions emphasize the in-time, and the construction difficulty is the quantification of the punishment. With the in-depth development of the intelligent manufacturing and the lean production, the bidirectional–JIT-oriented function will play a greater role in improving the production takt and reducing the production cost. In this paper, the construction form of the delivery-tardiness-oriented function is summarized as shown in Table7.Table 7
The construction form of delivery-objectives.
No.The delivery-oriented functionObj. FunctionThe decision variables1Mean tardiness,DmeaDmea−∑dT/i/TdiT: The delivery of each process; T: All the processes; w: The weight.2Max tardiness [81], DmaxDmax−maxdT/i3Weighted tardiness,DwDw−∑w&dT/i4Block tardinessDbDb−di/vvoldj/vdvivoldvj: The block tardiness of facility v between the process i and j.5The critical waiting time in buffer,Dw−bDw−b−maxdiw−bmaxdw−b/i: The max waiting time in buffer.6Just-in-Time deliveryDjitDjit−∑tfinish−tdue2tfinish: The finish time, both the lateness and earliness(the time can be fuzzy [84]); tdue: The standard due time.7The on- time delivery rate [38], DvoldotDTvoldot−∑w&doavoldodw: The weight; doa: The delivery early time; dod: The delivery late time.8Imbalance level of the completion [85], DiDi−Tmax−Tmin/TavgTmax: The max completion time; Tmin: The min completion time; Tavg: The average completion time.


### 3.3. The Energy Consumption-Oriented Objective Function

The growing awareness of energy efficiency and sustainable development has led to persistent attention to energy efficiency in production scheduling [86]. The growing number of published articles, especially in the past 3 years, makes energy consumption-oriented scheduling a hot research topic. Through the review, the energy-consumption-oriented function mainly includes the process-total-energy-consumption-, the distribution-total-energy-consumption-, the power-consumption-, the unit-energy-consumption-, the transition-energy-consumption-, the processing-energy-consumption-, the carbon-emission- [87] and the energy-efficiency- oriented function, etc.The process-total-energy-consumption-oriented function is calculated from the perspective of the energy consumption composition, such as the idle energy consumption, the basic energy consumption, the off-on energy consumption, the processing energy consumption. [9] With the process-total-energy-consumption-oriented function, the scholars are mainly studied the some typical process (like the welding process, the cutting process). For example, in Liu’s [51] work, the average process energy consumption demand and average process time demand in the process of parts machining were introduced to measure the process energy consumption. In addition, the carbon emission [88] are unexpectedly raised during the forming of the process-total-energy-consumption-oriented function, which transformed the solution of energy consumption into the solution of carbon emission. While the carbon-emission-function are more applicable to the electricity-related manufacturing shop floors, which involve the carbon tax trading and green certification trading, etc. [89] The carbon emission provides a new paradigm and business model for the energy consumption scheduling in manufacturing shop floors.On the other way, the shop floor logistics system employs the distribution-total-energy-consumption-oriented function to scheduling the distribution facilities and the allocation resources. Such as in the green scheduling of material delivery in the assembly line, the distribution total energy consumption is measured by the workload, the distribution distance, the real-time power, speed and the time accumulation of the mobile devices (like AGV) in Zand Liao’s [13] research. And, the TOU technology is often supplemented to calculated the power-consumption-oriented function such as the paradigm with the TOU technology and a price matrix in the two-layer batch scheduling HFJS [20].Another formulation is the unit-energy-consumption-oriented function, whose solution idea is the ratio of the total energy consumption to the total output, multiplied by the unit energy cost, to represent the cost or penalty. Its main application scenarios are the large-volume joint processing processes, like the hot rolling shop floor [19] and the forging shop floor.In addition, some researchers propose to reduce the transition energy consumption during the process, that is, to save the off-on energy consumption whose essence is to reduce the useless energy consumption and improve the energy efficiency. For example, in the parts manufacturing system, Liu [45] clustered the process and minimized the transition energy consumption, which increased the continuous working capacity of the equipment and reduced the off-on times.What’s more, compared with transition energy consumption reduction studies (some ways to cut down the useless and wasted energy), some scholars have directly studied the energy-efficiency-oriented functions, including the unit-time-energy-efficiency-, unit-production-energy-efficiency- and single-equipment-energy-efficiency-oriented function, whose objective is to introduce some parameters such as the time, the total output and the total number of facilities to the total energy consumption model and acquires an efficient-oriented new model, In this paper, the construction form of the energy-consumption-oriented function is summarized as shown in Table8.Table 8
The construction form of energy consumptions -objectives.
No.The consumptions -oriented functionObj. FunctionThe decision variables1The process total energy consumption [49], Ep−tEp−t−∑eid+eba+eoff−on+epreid: The idle energy; eba: The basic energy; eoff−on: The off-on energy; epr: The processing energy.2The total processing energy consumption [45, 51], EtpeEtpe−∑pik&tikpik: The average process energy consumption demand of the process i processing in facility Ktik: The average process time demand of the process i processing in facility K3The carbon emission [90], Eca−emiEca−emi−memi+clemi+hemi+lemimemi: The emission of the machine (processing, idle, etc); clemi: The cooling liquid emission; hemi: The handling emission; lemi: The losing emission.4The distribution total energy consumption [13, 32, 41], Ef−tEf−t−∑wl&dc+u&m+v&twl: The loading weight; dc: The distance of carrying or distribution; u&m:the power; v: The speed; t: The delivery time.5The power consumption [20, 39, 55], EcEc−∑e&ce: The electricity amount; c: The TOU price.6The unit energy consumption penalty [19], EpEp−∑Ep−t/n&cpEp−t: The processing total energy consumption; n: The total production; cp: The cost per unit energy.7The transition total energy consumption [45, 51], EtteEtte−∑eoff−oneoff−on: The off-on energy8The energy-efficiency [13, 52]——


### 3.4. The Cost-Oriented Objective Function

With no doubt, the cost-reduction is the ultimate goal of an enterprises in the past, now and in the future, and it is the fundamental starting point of all production operations. According to literature review, nowadays the cost-oriented objective functions is consisted of colorful types, and can be subdivided into the element-cost-, the process-cost-, the transportation-cost-, the tardiness-cost-, the inventory-cost-, the profit-oriented function.Firstly, similar to the modeling of the energy-consumption-oriented function, the cost flow is constructed and analyzed from the sight of the cost composition, which generally includes the operation cost, the equipment cost, the handling cost, the backlog cost, etc. Among them, equipment deterioration in efficiency or cost at any time or in the length of service might fails the modeling. To handle this situation, some scholars have proposed the production factors which cover the cost of deterioration effect [91] to weight of the operation time and the service age (named as the deteriorate effect coefficient for the machine). The deteriorating effect weight are the core of the modeling [92]. Parallel, the process-cost-oriented function is the modeling of the production operation and it’s the fine-grained modeling of the element-cost-oriented function, which generally includes the labor cost, the intermittent storage cost and the opportunity cost, etc.And, the tardiness cost may cause by the increased of the inventory cost of the WIP/finished goods, or some reputation loss. The calculation of inventory cost increase can be refined into (1) The raw material cost increase: the purchase cost, the transportation cost, the tariff insurance cost, the labor cost, the electricity cost (For instance, Mei [93] used the real-time power, the electricity price and the time to donate the power cost and the production cost of pulper and grinder in the pulping shop scheduling) and the management cost etc. (2) the holding cost increases such as the opportunity cost and the risk cost. (3) The order cost increase such as the purchasing order and production order. This paper summarizes the construction form of cost-oriented objective function as shown in Table 9.Table 9
The construction form of cost-objectives.
No.The cost-oriented functionObj. FunctionThe decision variables1The element cost [21], CeCe−cope.+cro+cm−h+cse+cwa+cho+cblcope.: The operation cost; cro: The robot cost; cm−h: The material handling cost; cse: The setup cost; cwa: The waiting cost; cho: The holding costs; cbl: The backlog cost2The machine deterioration cost [91], Cd−eCd−e−θ×Ceθ: The deteriorate effect coefficient, related with the ratio of the operation time to the service age.3The process cost [95], CpCp−cla+cho+cep+ctp+ctranscla: The labor cost; cho: The holding cost; cep: The earliness penalty cost; ctp: The tardiness penalty cost; ctrans: The transportation cost.Cp−copp.&cns&cspcopp.: The opportunity cost [96]; cns: The manufacturing cost of a no-split unit; csp: The manufacturing cost of a split unit.4The batching transportation cost [17], Cb−transCb−trans−cin×csl×tsl+cb−p×cp−t×ct−pcin: The intercept cost; csl: The slope cost; tsl: The slope time; cb−p: The batching product cost; cp−t: The batch transportation cost; ct−p: The penalty cost.5The tardiness cost [17], CtCt−cwip+cfin.+ctpt+cr−tcwip: The price inventory WIP; cfin: The price inventory final; ctpt: The processing tardiness price per time; cr−t: The reputation cost of tardiness.6JIT cost [48], CjitCjit−∑cjp+ce−p+ce−i+cracjp: Just-in-time penalty cost; ce−p: Energy cost for nominal processing at time t; ce−i: Energy cost for machine idle; cra: Machine repair cost.7The TOU power cost [24], CtouCtou−pba×ce−ppba: The processing time of batch b; ce−p: Energy cost for nominal processing at time t.8The inventory cost,CiCi−cit+cca+corcit: Item cost; cca: Carrying cost; cor: Ordering cost.9The resources consumption cost [54], CrcCrc−∑stataion∑facilityram×rprram: The resource quantity of a station; rpr: The resource price of a station.10The profit [97], Cpro.Cpro.−Pv−TCv×VTv×viPv: The price of a unit volume; TCv: The operation cost of a unit time; VTv: The operation time of a unit volume; vi: The volumes of a production.


### 3.5. The Utilization- Or Workload-Rate-Oriented Obj. Function

The utilization- or workload-rate-oriented obj. function, generally applied to the product-oriented or the production-efficiency-oriented shop floor, such as the SMT shop floor: the clusters production shop floor with multi-varieties and small-batch, is another main scheduling direction besides the above-mentioned function construction forms (the Makespan-, the delivery-tardiness-, the energy-consumption- and the cost-oriented obj. function). The scheduling and modeling, including the average-utilization-, the total-workload-, the maximum-workload-, the average-resource utilization-oriented function, is to obtain the best state and highest utilization of all the clusters and the construction form is shown in Table10.Table 10
The construction form of utilization/workload -objectives.
No.The utilization/workload-oriented functionObj. FunctionThe decision variables1The average machine utilization [56], UmUm−∑si×ui/∑sisi: The resource quantity of process i; ui: The utilization of process i.2The total workload of the machines,WLtWLm−∑WLiWLi: The workload of process i.3The maximal machine workload [98], WLmWLt−maxWLi4The workload balance [99]——5The average-resource utilization [100], UrUr−∑tRl/maxCTj×N∑tRl: In all the processes, the cumulative processing time of the resource Rl; maxCTj: The completion time of final process j; N: All the resource number.6The weighted machine utilization [12], Zwl+UZwl+U=∑i=1Mβ1×WLmm−WLwm2+β2×Umm−Uwm2Zwl+U: Obj. Function; β1,β2: The weight of workload and cumulative working time WLmm: The average workload of the facility m; WLwm: The average workload of all the m-type facilities; Umm: The average cumulative working time of the facility m; Uwm: The average cumulative working time of all the m-type facilities;


## 3.1. The Makespan-Oriented Objective Function

According to the statistics of the databases such as IEEExpore, Elsevier, Springer, CNKI and Wanfang, the number of the articles with the Makespan-oriented function in the multi-obj. FJSP during the past few years is up to 1602, accounting for more than 28%, which straightforward proves that the Makespan-oriented function is the widely used candidate during the past few years as shown in Table5.Table 5
The index results of Makespan-obj. in the big-5-database during the past few years.
DatabaseKeywordsResultMakespan-orientedProportion (%)1CNKIMulti objective, job shop, scheduling56518031.862Wanfang255207.843IEEExplore12112099.174Elsevier168779447.075Springer298748816.34Total5615160228.53The Makespan-oriented function [57, 58] can be further subdivided into the total-Makespan-, the off-on-Makespan-, the time-buffer-Makespan-, the worst-case-Makespan-, the fuzzy-Makespan-, the robust-Makespan-, the weight-Makespan-oriented function to minimum the maximum-processing-time problems.According to the manufacturing process, the total-Makespan-oriented function can be calculated by the cumulative timing of all processes on the longest process chain or the cumulative finishing time of the last process on a single device [59, 60]. While the total-Makespan-oriented function, with simple structure, is easy to distort in some particular process. To overcome the various distortion, steps in the modeling have been taken [61]. For instance, the off-on process is added on the base of the total-Makespan-oriented function to handle the situations that the start/cutoff/restart time cannot be ignored [24]. Or the system covering the blocking area or the shuffle zone, like the off-on-Makespan-oriented function, the calculation range takes account all the blocking and shuffle in the whole processing cycle [62].What’s more, with the deepening research on the unascertained information, more scholars believe that the uncertainty of FJS will lead larger randomness in the Makespan-oriented function [63]. Therefore, various Makespan-oriented modeling methods based on unascertained information come into being. To name a few, the worst-case-Makespan- [64], the robust version is provided to formulate the obj. function to scale the reliability. The fuzzy-Makespan-oriented function makes a more future scaling, besides blurring the last process, the total Makespan is also taken into account to obtain the weight of the best, the most possible and the worst completion time of the process or the total Makespan [65]. In addition, on dealing with the robustness, CHEN’s [66] approach was to combine the confidence interval with the game model. While FU [65] proposed the calculation paradigm of the robust bias, which optimized the production process robustness by taking the deviation between the most possible Makespan/the worst Makespan and the best Makespan as the objective function.Besides above all, means of researches associated with the weight-Makespan-oriented function have been published and attaches the academic focus [67]. Its essence is to linear couple of some production or operation indicators such as the blocking rate or the marginal cost based on different equipment or different stages in the shop floor [68, 69]. To illustrate, Yan [70] adopted the weighted-Makespan in the mixed-flow production line, whose weight is measured by the compensation cost of the process tardiness. Up-to-date, with the rise of cloud manufacturing, the weightd-Makespan of multi-enterprise collaboration has gradually become a research trend [71], which is different from the conventional Makespan, besides the processing time, the weighted delivery time is also included in the weighted coupling of the cloud Makespan [72, 73].To sum up, the Makespan-oriented objective function can be broadly subdivided into three [74]: (1) The total-Makespan-oriented function covering some production characteristics and specific application scenarios. such as the forging shop floor and other equipment that need preheating, which takes a long start and cutoff time [75, 76]. And the off-on-Makespan-oriented function, due to the limited shuffle space in the wafer shop, the waiting sequence is needed in the formulation, namely the blocking and shuffle Makespan-oriented function. (2) For the robust-Makespan-oriented function, which covers the unascertained information, the robustness factors such as the ambiguity and the confidence interval should be introduced to eliminate the influence of the disturbance [77]. (3) The weighted-Makespan-oriented function is mostly used in multi-device, multi-organization or cloud collaborative scheduling application scenarios. This paper summarizes the construction forms of the Makespan-oriented objective function as shown in Table 6.Table 6
The construction form of Makespan-objectives.
No.The makespan-oriented functionObj. FunctionThe decision variables1The total Makespan [78], CmaxCmax−∑tT/itT/i:Σ Processing time of all processes; i: No. i process; T: All processes; θi: The robustness index of the processing sequence; maxθi: The worst processing sequence.2The worst-case Makespan [79], Cmaxw−cCmaxw−c−maxθi∑tT/i3The off-on Makespan [50], Cmaxo−oCmaxo−o−∑tT/i+tT/i:off−ontT/i:off−on: Σ Off-on time of all the processes.4The time buffer Makespan [62], Cmaxw−bCmaxw−b−∑tT/i+tb/i.tib: No. i buffer time.5The fuzzy Makespan [65], CmaxfCmaxf−c1+2×c2+c3/4c1: The best Makespan; c2: The most possible Makespan; c3: The worst Makespan.6The robustness of the Makespan [65], CmaxrCmaxr−maxc2−c1,c3−c17The expected weighted sum of Makespan [6], Cmaxw−sCmaxw−s−αCmaxA+1−αCmaxBαCmaxA: Equipment A’s Makesapn; 1−αCmaxB: Equipment B’s Makesapn.8The cloud multi-stage Makespan [72], Cmaxm−lCmaxm−l−∑t2ed−p+tmatt2ed−p: Secondary manufacturer’s completion time; tmat: The secondary manufacturer matches the transport time of the primary manufacturer.Note. Please refer to Table1 for the notations of some symbols and Table 3 for abbreviations.


## 3.2. The Tardiness-Oriented Objective Function

Besides the Makespan-oriented function, the most helpful objective functions are the delivery-tardiness-, the energy-consumption- and the cost-oriented function, which respectively accounts for more than 10%. In brief the delivery-tardiness objective function can be subdivided into the average-tardiness-, the maximum-tardiness-, the weighted-tardiness-, the blocking-tardiness-, the warehousing-waiting-, the JIT-tardiness-, and the weighted-JIT-tardiness-oriented function, etc., to meet the due time [76, 80].The average-tardiness-oriented function accumulates the tardiness of all the processes, representing the mean tardiness of all the machine in the whole shop floor or all the processes in a single device, which is widely employed among the tardiness-oriented function and becomes a critical approach to achieve the purpose of no late. While the state that the effect of tardiness cannot be characterized due to the unbalanced tardiness (It is unrealistic to tell the excessive tardiness of only one process from the relatively punctual tardiness of other means of processes in the average-tardiness-oriented function) is beyond the control. On this circumstance, the maximum-tardiness- and weighted-tardiness-oriented function are proposed to eliminate this distortion. Based on the former function, the maximum-tardiness process is the smaller-is-the-better. To illustrate, Lei Deming [81] used the minimum of the maximum-deviation or the maximum-tardiness of the due date to balance the imbalance between the number of key process and the number of non-core process in the multi-class teaching optimization theory proposed by HFJS. What’s more, the weighted-tardiness-function, takes another way, pays more attention to the core process. By assigning higher weight and increasing more influence [82], undoubtedly, the key process will be charging more responsibility in the formulation, so as to generalize the average tardiness effect in the reality. For instance, JENS [20] gave different weights to different facilities on tardiness based on the workload or the logistics quantities in the semiconductor wafer manufacturing system to focus more on what they were cared for. Intriguingly this formulation has derived and brought forward the blocking-tardiness-oriented function. Another construction form, the warehousing-wait-oriented function, is put forward to solve the minimizing the maximum-waiting-time of the storage process by the idea of minimizing the maximum-tardiness, which takes the storage as a special facility.Admittedly, the above delivery-tardiness-oriented function are all unidirectional, pay the attention on the delay or the lateness, while the JIT-tardiness-, the weighted-JIT-tardiness-oriented functions and the imbalance-level-of-the-completion-oriented function are bidirectional, which means that both the lateness and earliness are both penalized. Such as on JIT delivery, Xu and Muhammed [83] adopted the standardized operating time and the Taguchi “the target is the best” technique to measure the lateness and earliness. On the base of Xu’s research, Fan [38] assigned the weights to the core and the non-core, the blocking and the non-blocking processes. While the imbalance-level-of-the-completion-oriented function refers to the calculation of the completion time deviation between the different facilities for the same process, and the global delivery tardiness is accumulated by the unbalanced level of every single process.To summarize, the delivery-tardiness-oriented function can be further subdivided into two: (1) The unidirectional-delivery-tardiness-oriented function, represented by time, such as the average-tardiness-, the maximum-tardiness-, the weighted-tardiness-, the blocking-tardiness-, the warehousing-waiting-oriented function. The difficulty of objective function construction is to coupling the global tardiness and the key process tardiness, and balancing the minimum of the overall tardiness and the lowest blocking rate of the core equipment. (2) The bidirectional-tardiness-oriented function, represented by the punishment, such as the JIT-tardiness- and the weighted-JIT-tardiness-oriented function. These objective functions emphasize the in-time, and the construction difficulty is the quantification of the punishment. With the in-depth development of the intelligent manufacturing and the lean production, the bidirectional–JIT-oriented function will play a greater role in improving the production takt and reducing the production cost. In this paper, the construction form of the delivery-tardiness-oriented function is summarized as shown in Table7.Table 7
The construction form of delivery-objectives.
No.The delivery-oriented functionObj. FunctionThe decision variables1Mean tardiness,DmeaDmea−∑dT/i/TdiT: The delivery of each process; T: All the processes; w: The weight.2Max tardiness [81], DmaxDmax−maxdT/i3Weighted tardiness,DwDw−∑w&dT/i4Block tardinessDbDb−di/vvoldj/vdvivoldvj: The block tardiness of facility v between the process i and j.5The critical waiting time in buffer,Dw−bDw−b−maxdiw−bmaxdw−b/i: The max waiting time in buffer.6Just-in-Time deliveryDjitDjit−∑tfinish−tdue2tfinish: The finish time, both the lateness and earliness(the time can be fuzzy [84]); tdue: The standard due time.7The on- time delivery rate [38], DvoldotDTvoldot−∑w&doavoldodw: The weight; doa: The delivery early time; dod: The delivery late time.8Imbalance level of the completion [85], DiDi−Tmax−Tmin/TavgTmax: The max completion time; Tmin: The min completion time; Tavg: The average completion time.


## 3.3. The Energy Consumption-Oriented Objective Function

The growing awareness of energy efficiency and sustainable development has led to persistent attention to energy efficiency in production scheduling [86]. The growing number of published articles, especially in the past 3 years, makes energy consumption-oriented scheduling a hot research topic. Through the review, the energy-consumption-oriented function mainly includes the process-total-energy-consumption-, the distribution-total-energy-consumption-, the power-consumption-, the unit-energy-consumption-, the transition-energy-consumption-, the processing-energy-consumption-, the carbon-emission- [87] and the energy-efficiency- oriented function, etc.The process-total-energy-consumption-oriented function is calculated from the perspective of the energy consumption composition, such as the idle energy consumption, the basic energy consumption, the off-on energy consumption, the processing energy consumption. [9] With the process-total-energy-consumption-oriented function, the scholars are mainly studied the some typical process (like the welding process, the cutting process). For example, in Liu’s [51] work, the average process energy consumption demand and average process time demand in the process of parts machining were introduced to measure the process energy consumption. In addition, the carbon emission [88] are unexpectedly raised during the forming of the process-total-energy-consumption-oriented function, which transformed the solution of energy consumption into the solution of carbon emission. While the carbon-emission-function are more applicable to the electricity-related manufacturing shop floors, which involve the carbon tax trading and green certification trading, etc. [89] The carbon emission provides a new paradigm and business model for the energy consumption scheduling in manufacturing shop floors.On the other way, the shop floor logistics system employs the distribution-total-energy-consumption-oriented function to scheduling the distribution facilities and the allocation resources. Such as in the green scheduling of material delivery in the assembly line, the distribution total energy consumption is measured by the workload, the distribution distance, the real-time power, speed and the time accumulation of the mobile devices (like AGV) in Zand Liao’s [13] research. And, the TOU technology is often supplemented to calculated the power-consumption-oriented function such as the paradigm with the TOU technology and a price matrix in the two-layer batch scheduling HFJS [20].Another formulation is the unit-energy-consumption-oriented function, whose solution idea is the ratio of the total energy consumption to the total output, multiplied by the unit energy cost, to represent the cost or penalty. Its main application scenarios are the large-volume joint processing processes, like the hot rolling shop floor [19] and the forging shop floor.In addition, some researchers propose to reduce the transition energy consumption during the process, that is, to save the off-on energy consumption whose essence is to reduce the useless energy consumption and improve the energy efficiency. For example, in the parts manufacturing system, Liu [45] clustered the process and minimized the transition energy consumption, which increased the continuous working capacity of the equipment and reduced the off-on times.What’s more, compared with transition energy consumption reduction studies (some ways to cut down the useless and wasted energy), some scholars have directly studied the energy-efficiency-oriented functions, including the unit-time-energy-efficiency-, unit-production-energy-efficiency- and single-equipment-energy-efficiency-oriented function, whose objective is to introduce some parameters such as the time, the total output and the total number of facilities to the total energy consumption model and acquires an efficient-oriented new model, In this paper, the construction form of the energy-consumption-oriented function is summarized as shown in Table8.Table 8
The construction form of energy consumptions -objectives.
No.The consumptions -oriented functionObj. FunctionThe decision variables1The process total energy consumption [49], Ep−tEp−t−∑eid+eba+eoff−on+epreid: The idle energy; eba: The basic energy; eoff−on: The off-on energy; epr: The processing energy.2The total processing energy consumption [45, 51], EtpeEtpe−∑pik&tikpik: The average process energy consumption demand of the process i processing in facility Ktik: The average process time demand of the process i processing in facility K3The carbon emission [90], Eca−emiEca−emi−memi+clemi+hemi+lemimemi: The emission of the machine (processing, idle, etc); clemi: The cooling liquid emission; hemi: The handling emission; lemi: The losing emission.4The distribution total energy consumption [13, 32, 41], Ef−tEf−t−∑wl&dc+u&m+v&twl: The loading weight; dc: The distance of carrying or distribution; u&m:the power; v: The speed; t: The delivery time.5The power consumption [20, 39, 55], EcEc−∑e&ce: The electricity amount; c: The TOU price.6The unit energy consumption penalty [19], EpEp−∑Ep−t/n&cpEp−t: The processing total energy consumption; n: The total production; cp: The cost per unit energy.7The transition total energy consumption [45, 51], EtteEtte−∑eoff−oneoff−on: The off-on energy8The energy-efficiency [13, 52]——


## 3.4. The Cost-Oriented Objective Function

With no doubt, the cost-reduction is the ultimate goal of an enterprises in the past, now and in the future, and it is the fundamental starting point of all production operations. According to literature review, nowadays the cost-oriented objective functions is consisted of colorful types, and can be subdivided into the element-cost-, the process-cost-, the transportation-cost-, the tardiness-cost-, the inventory-cost-, the profit-oriented function.Firstly, similar to the modeling of the energy-consumption-oriented function, the cost flow is constructed and analyzed from the sight of the cost composition, which generally includes the operation cost, the equipment cost, the handling cost, the backlog cost, etc. Among them, equipment deterioration in efficiency or cost at any time or in the length of service might fails the modeling. To handle this situation, some scholars have proposed the production factors which cover the cost of deterioration effect [91] to weight of the operation time and the service age (named as the deteriorate effect coefficient for the machine). The deteriorating effect weight are the core of the modeling [92]. Parallel, the process-cost-oriented function is the modeling of the production operation and it’s the fine-grained modeling of the element-cost-oriented function, which generally includes the labor cost, the intermittent storage cost and the opportunity cost, etc.And, the tardiness cost may cause by the increased of the inventory cost of the WIP/finished goods, or some reputation loss. The calculation of inventory cost increase can be refined into (1) The raw material cost increase: the purchase cost, the transportation cost, the tariff insurance cost, the labor cost, the electricity cost (For instance, Mei [93] used the real-time power, the electricity price and the time to donate the power cost and the production cost of pulper and grinder in the pulping shop scheduling) and the management cost etc. (2) the holding cost increases such as the opportunity cost and the risk cost. (3) The order cost increase such as the purchasing order and production order. This paper summarizes the construction form of cost-oriented objective function as shown in Table 9.Table 9
The construction form of cost-objectives.
No.The cost-oriented functionObj. FunctionThe decision variables1The element cost [21], CeCe−cope.+cro+cm−h+cse+cwa+cho+cblcope.: The operation cost; cro: The robot cost; cm−h: The material handling cost; cse: The setup cost; cwa: The waiting cost; cho: The holding costs; cbl: The backlog cost2The machine deterioration cost [91], Cd−eCd−e−θ×Ceθ: The deteriorate effect coefficient, related with the ratio of the operation time to the service age.3The process cost [95], CpCp−cla+cho+cep+ctp+ctranscla: The labor cost; cho: The holding cost; cep: The earliness penalty cost; ctp: The tardiness penalty cost; ctrans: The transportation cost.Cp−copp.&cns&cspcopp.: The opportunity cost [96]; cns: The manufacturing cost of a no-split unit; csp: The manufacturing cost of a split unit.4The batching transportation cost [17], Cb−transCb−trans−cin×csl×tsl+cb−p×cp−t×ct−pcin: The intercept cost; csl: The slope cost; tsl: The slope time; cb−p: The batching product cost; cp−t: The batch transportation cost; ct−p: The penalty cost.5The tardiness cost [17], CtCt−cwip+cfin.+ctpt+cr−tcwip: The price inventory WIP; cfin: The price inventory final; ctpt: The processing tardiness price per time; cr−t: The reputation cost of tardiness.6JIT cost [48], CjitCjit−∑cjp+ce−p+ce−i+cracjp: Just-in-time penalty cost; ce−p: Energy cost for nominal processing at time t; ce−i: Energy cost for machine idle; cra: Machine repair cost.7The TOU power cost [24], CtouCtou−pba×ce−ppba: The processing time of batch b; ce−p: Energy cost for nominal processing at time t.8The inventory cost,CiCi−cit+cca+corcit: Item cost; cca: Carrying cost; cor: Ordering cost.9The resources consumption cost [54], CrcCrc−∑stataion∑facilityram×rprram: The resource quantity of a station; rpr: The resource price of a station.10The profit [97], Cpro.Cpro.−Pv−TCv×VTv×viPv: The price of a unit volume; TCv: The operation cost of a unit time; VTv: The operation time of a unit volume; vi: The volumes of a production.


## 3.5. The Utilization- Or Workload-Rate-Oriented Obj. Function

The utilization- or workload-rate-oriented obj. function, generally applied to the product-oriented or the production-efficiency-oriented shop floor, such as the SMT shop floor: the clusters production shop floor with multi-varieties and small-batch, is another main scheduling direction besides the above-mentioned function construction forms (the Makespan-, the delivery-tardiness-, the energy-consumption- and the cost-oriented obj. function). The scheduling and modeling, including the average-utilization-, the total-workload-, the maximum-workload-, the average-resource utilization-oriented function, is to obtain the best state and highest utilization of all the clusters and the construction form is shown in Table10.Table 10
The construction form of utilization/workload -objectives.
No.The utilization/workload-oriented functionObj. FunctionThe decision variables1The average machine utilization [56], UmUm−∑si×ui/∑sisi: The resource quantity of process i; ui: The utilization of process i.2The total workload of the machines,WLtWLm−∑WLiWLi: The workload of process i.3The maximal machine workload [98], WLmWLt−maxWLi4The workload balance [99]——5The average-resource utilization [100], UrUr−∑tRl/maxCTj×N∑tRl: In all the processes, the cumulative processing time of the resource Rl; maxCTj: The completion time of final process j; N: All the resource number.6The weighted machine utilization [12], Zwl+UZwl+U=∑i=1Mβ1×WLmm−WLwm2+β2×Umm−Uwm2Zwl+U: Obj. Function; β1,β2: The weight of workload and cumulative working time WLmm: The average workload of the facility m; WLwm: The average workload of all the m-type facilities; Umm: The average cumulative working time of the facility m; Uwm: The average cumulative working time of all the m-type facilities;


## 4. The Unconventional Objective Functions for the Emerging Scenarios or the New Problems

The previous section detailed describes the construction forms of five commonly used objective functions for FJSP to schedule the processing facilities, the logistics facilities, the orders, the processing resources, etc. Meanwhile, with the 5G/IOT/HCPS and some other new ITs developing, some new and unconventional objective functions proposed for the emerging scenarios or the new problems has jump out and occurred.(1)
The preventive maintenance in the intelligent shop floor. Some new objective functions come into being, to name a few, the minimum-waiting-time- [15], minimum-labor-usage- [15], minimum-parts-loss-oriented function [101] caused by equipment shutdown etc.(2)
The distribution and inspection in the unmanned shop floor. For example, the minimum number of charging times for UAV patrol inspection; The minimum number of conflicts [102] and the optimal cycle time of AGV scheduling [102];(3)
The human factor in a closed loop. The minimize-fatigue-strength- or the acceptable-maximum-working-hours-oriented function for the operators.The working scenarios of the objective functions mentioned above are relatively strict and narrow. And the applicability and the scalability are challenged by the new ITs, so just a briefly introduced as shown in Table11.Table 11
The construction from of the unconventional and new-objectives.
No.The new functionObj. FunctionThe decision variables1The return rate [103], in conventional mean–variance (MV) portfolio selection (the AGV or the assets scheduling), an efficient portfolio is sought in order to provide maximum return for a specified level of risk or minimum risk for a specified return.Z=∑bnxaRa−λ∑bn∑bnxaxbσabZ: Obj. Function; λ: Represents an index of investor’s risk aversion; n: is the number of assets to choose from; Ra: is the expected return for asset a; σab, represents the element in the a-th row and b-th column of the variance–covariance matrix; xa: is to determine the share of the budget invested on asset a. Finally, the sum of all xa is equal to one.2The fatigue [104]Z=ρβ/β∗+1−ρδ/δ∗Z: Obj. Function; ρ: The weight of fatigue in the target value of front line employees; β: The real fatigue; β∗: Optimal fatigue value for a single obj.; δ:the enterprise production; δ∗: Optimal value of enterprise output under a single obj.3The maximum acceptance work duration (work-rest scheduling) [29]Z−hw/r&hr/r,maxhrZ: Obj. Function; hrw:working heart rate; hrr:resting heart rate; maxhr maximum heart rate.4Total number of charges during the work [105]—UAV patrol inspection5The waiting time and human resource [15]—Preventive maintenance [77, 106]6Max the number of on-time jobs [107]—Single device multiple job scheduling7The conflicts [108], the cycle time [102]—AGV scheduling8The throughput/WIP [102]/inventory [109]——9The dynamic evaluation of service cooperation level [110]——


## 5. Conclusion and Future Work

Usually, the benchmark is employed to verify the effectiveness and reliability of a scheduling algorithm, but the selection and formulation of different objective functions can not be ignored to the operation results of the benchmark. Therefore, facing with the characteristic in a general way, the detailed construction analysis on the function for the FJSP is proposed, and a total 6 types with 49 kinds classification are reviewed, including the Makespan-, the delivery-tardiness-, the energy-consumption-, the cost-oriented function, and analyzes the components and application scenarios of the objective functions, in order to enhance the scientific nature of the selection of the objective function and the effectiveness of the scheduling process. The selection and construction of the multi-obj. FJSP function in the past few years, is a generally experience-dependence and evaluation-lack method. At the end of this paper some briefly directions, like the selection, the construction, the evaluation and the optimization of the oriented functions, are pointed out for the future studies.(1)
The selection and construction of objective function show strong relationship between the working scenes and the engineering experience. The fresh researchers or a new scene are often limited to the existing knowledge or the one-sided knowledge which leads to the lack of specific description to the unique scenes and obtain a function following the general way. Just like the research of Kong [111] and Pei [112] pointed out that the recently researchers were lack of comparative analysis of the construction of the objective function after the literature review. So the next step is to build up the basic elements of different types of shop floor scheduling objective function, such as from the perspective of the product batch (single, batch, large number), the production plan (ordering plan, inventory plan), the product continuity (continuous and intermittent pattern), the product variety and production output [112, 113]. And some special elements of special industries will be considered into account in the database collections and the toolsets.(2)
The evaluation. In the existing researches, almost all focus on the success rate of scheduling (Pareto solution), the solving speed and the functional generalization (problem size). As the first step of the researches on scheduling, there is a lack on the construction and evaluation of objective function [16]. Therefore, another future work is to build up an evaluation index system, form a set of evaluation technique based on the engineering characteristics preference, and provide a decision model for the active push of the objective function recommendation system.(3)
The optimization. The objective function evaluation technique is studied to optimize the objective function based on the index system and provide a reliable and applicable mathematical model for the scheduling. Therefore, the optimization of objective function in the future research will provide a technique model for the recommender system, and this research will certainly become an important direction. [94].



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