Balancing and scheduling human-robot collaborated assembly lines with layout and objective consideration

leading to more efficient solutions for JIT production, particularly in objective Type-2 followed by Type-rw and Type-1. Moreover, the U-shaped lines featuring a high HRC level can further enhance the achievement of desired objectives compared to the straight lines with no or limited HRC.


Introduction
From the times of Henry Ford up to today's industry 5.0 era, assembly lines, as a flow-oriented manufacturing system, have been widely used in various industries such as automotive and electronics for mass-customized production.An assembly line comprises a set of tasks performed on workpieces moving from one station to another through a material handling device until a complete product is assembled.Each assembly line's most elementary optimization problem is known as the assembly line balancing problem (ALBP) (Fathi et al., 2018(Fathi et al., , 2019)).The ALBP decides on the assignment of tasks among the stations to optimize some objectives while taking precedence relations into account.According to the objective function, ALBP is classified into four types: (1) Type-1 minimizes the number of stations (NS) for a given cycle time (CT); (2) Type-2 minimizes the CT for a given NS; (3) Type-E optimizes the CT and the NS simultaneously; and (4) Type-F aims to find a feasible solution given the CT and the NS (Boysen et al., 2022).
Considering the layout, the assembly lines can be mainly divided into straight and U-shaped lines (Mukund Nilakantan and Ponnambalam, 2016).From the ALBP perspective, in straight lines, the task assignment depends on all its predecessors, while in U-shaped lines, the assignment of a task is subjected to either all its predecessors or successors (Fathi et al., 2016).In the latter, the tasks dependent on their predecessors are assigned to the line's entrance, while the tasks dependent on their successors are allocated to the line's exit.Accordingly, U-shaped assembly lines are essential for just-in-time (JIT) production (Nourmohammadi et al., 2019).
On the other hand, since the advent of industrial robots, many manufacturers have used them as the technology enabler for more agile manufacturing (Weckenborg and Spengler, 2019).The first generation of industrial robots was insulated in cages to avoid the risks of intervening in human workplaces.Recently, as a consequence of Industry 4.0, the new generation of robots known as collaborative robots (cobots) have been acquainted with the capability of operating like humans thanks to their embedded software and hardware with advanced sensing and interacting systems (Nourmohammadi et al., 2022a).With such technologies, cobots can be deemed resilient operators sharing the same workplace with humans.Moreover, the design of human-centered workplaces motivated by Industry 5.0 has recently attracted many manufacturers' attention to support humans with cobots, where humans play a central role (De Nul et al., 2021).These automation trends have convinced manufacturers to shift their manual or robotic lines toward hybrid assembly lines with human-robot collaboration (HRC), where human skill and cobot agility can be utilized in the same station.
Several studies have underlined the advantages of assembly line balancing with HRC.Dalle Mura and Dini (2019) and Weckenborg and Spengler (2019) discussed that HRC is a good option for efficient assembly line balancing.Weckenborg et al. (2020), Koltai et al. (2021), Nourmohammadi et al. (2022a), and Nourmohammadi et al. (2022b) studied how cobots can significantly enhance the productivity of manual assembly lines.Dalle Mura and Dini (2022), Stecke and Mokhtarzadeh (2022), and Weckenborg et al. (2022) have emphasized the advantages of HRC for the design of both economic and ergonomic assembly lines.From the layout perspective, Li et al. (2023), Mao et al. (2023), and Rahman et al. (2023) have dealt with HRC in the U-shaped assembly lines.To the authors' knowledge, the assumptions were that only one human and robot were allowed at each station, or the human and robot could not perform different tasks in parallel.However, this assumption (i.e., limiting the HRC level) tends to inadvertently limit the full HRC potential for a flexible and efficient utilization of resources.Moreover, from the objective point of view, in line with energy sustainability in Industry 4.0 (Ghobakhloo and Fathi, 2021), the costs of robot energy consumption are not incorporated into the cost of stations and operators in the literature.At the same time, different objectives, specifically in the U-shaped settings, are yet to be explored.
To bridge the above gaps between research and practical implementation, this study aims to deal with the joint problem of balancing and scheduling assembly lines with HRC in straight and U-shaped layouts, considering different objectives.The considered objectives are minimizing the NS (Type-1), the CT (Type-2), the cost of stations, operators, and energy consumption of robots (Type-rw).In the latter type, the HRC can benefit from incorporating multiple humans and robots with different skills and capabilities while performing the tasks at stations collaboratively and in parallel.Moreover, other real-world aspects will be considered, namely robot energy cost, robot tool changes, zoning, and technological requirements.The optimization problems will be formulated as mixed-integer linear programming (MILP) models.
Moreover, efficient lower and upper bounds for each objective type are proposed to decrease the proposed MILPs' computational dimensionality and find promising solutions.The validity of the proposed models is shown in an application case study and different test problems taken from the literature.The computational experiments provide analyses of results in terms of layout, objectives, and HRC.Overall, the main contributions of this study can be summarized as follows: • Considering straight and U-shaped assembly lines with HRC and different types of objectives, namely Type-1, Typ-2, and Type-rw, the latter includes the cost of stations, operators, and robot energy consumption.• Incorporating real-world constraints, namely multiple humans and robots with different skills per station, robot tool changes, technological requirements, and zoning of tasks.• Proposing mixed-integer linear programming models with efficient lower and upper bounds for each objective and layout while testing them on a real case and a set of test problems.• Analyzing the effect of objective, layout, and HRC scenarios.
The remainder of this paper is organized as follows.In Section 2, the related literature is reviewed.In Section 3, the description of the problem is presented.The mathematical formulations for balancing and scheduling the U-shaped assembly lines with HRC and the efficient lower and upper bounds for each objective are presented in Section 4. Section 5 presents the computational study of the proposed models while solving a real-world case and test problems with further analysis of results in terms of objective, layout, and HRC.Finally, Section 6 outlines the conclusions and future research directions.

Literature review
This section reviews the relevant studies focusing only on balancing the assembly line with HRC (AL-HRC).The interested readers are referred to Boysen et al. (2021) and Battaïa and Dolgui (2022) for a recent survey on ALBP literature.
In the study performed by Dalle Mura and Dini (2019), given a CT, the cost of the assembly line in terms of the number of stations, workers, robots, and workers' skills and energy load variance, were minimized.The authors developed a genetic algorithm (GA) to assign the tasks to stations with HRC capability in a scooter assembly line.Weckenborg and Spengler (2019) minimized the costs measured by the number of stations, humans, and robots while limiting the ergonomic load of stations formulated as a mathematical optimization model.Weckenborg et al. (2020) extended their previous study by minimizing the CT in which a hybrid GA and mathematical model were proposed with customized crossover, mutation, and repair operators to deal with larger test problems.Çil et al. (2020) dealt with the minimization of CT given the NS where only one robot was allowed, and no collaboration between humans and robots was permissible.Given the CT, Yaphiar et al. (2020) minimized the total cost of resources, including humans, robots, machinery, and equipment, by proposing a mathematical model for the mixed-model assembly line.Raatz et al. (2020) dealt with the balancing and scheduling assembly tasks in a real-case assembly line by proposing a GA-based framework to optimize the CT and the ergonomic load as the primary and secondary objectives, respectively.
In another study, Li et al. (2021) used a cost-oriented approach to optimize the CT and the costs of robot purchase and human salary.A mixed-integer mathematical programming model and multi-objective migration bird optimization algorithm were proposed to solve the problem.Gualtieri et al. (2021) proposed a systematic method for designing a human-robot collaborative assembly system by optimizing the CT in an industrial case study.Koltai et al. (2021) addressed different scenarios while shifting from the manual to HRC production systems by proposing mathematical models for optimizing the NS and CT in a real case.Dalle Mura and Dini (2021) considered job rotations and HRC to optimize costs in terms of the number of stations and equipment, including the cobots and ergonomic factors regarding the energy load variance among workers.The authors developed a GA to solve the assembly line balancing in an industrial case.Vieira et al. (2021) dealt with the planning and scheduling a human-robot assembly line in an industrial case study by developing a hierarchical optimizationsimulation approach.The optimization method sequentially dealt with the planning and scheduling by minimizing the total costs and makespan.The simulation module dealt with the feasibility of the generated solutions by considering the dynamic operating situation in line layouts and the dispatching rules.Stecke and Mokhtarzadeh (2022) studied the design of HRC assembly lines to optimize efficiency and ergonomic risks by considering mobile and immobile robots and zoning constraints.Three mathematical models were proposed to solve real-case and test problems, including mixed-integer programming, constraint programming, and Benders decomposition.Nourmohammadi et al. (2022) dealt with the balancing and scheduling tasks in the HRC assembly lines to optimize the CT and the number of operators while considering multiple humans and robots at the stations.A mathematical model and simulated annealing algorithm were proposed to solve case studies and test problems.Weckenborg et al. (2022) studied the economic and ergonomic aspects of AL-HRC when cobots and exoskeletons are utilized to optimize the cost and ergonomic risk measures while parallelizing tasks was not allowed in the presence of multiple operators.Kinast et al. ( 2022) considered the assembly line balancing with cobot assignment to stations in a real case.They proposed a CP formulation of the problem while assuming that only one cobot can be assigned to each station.They also presented a GA hybridized with variable neighborhood search to solve the problem in an extensive computational study.Sikora and Weckenborg (2022) studied the balancing of the AL-HRC using Benders decomposition.The authors developed some variants of Benders decomposition algorithms and tested them on a comprehensive range of test problems from the literature.The balancing U-shaped assembly line with HRC has been recently addressed by Li et al. (2023) to minimize the CT by assuming that one robot can be assigned to each station and parallel tasks by human and robot are not allowed.Three variants of mathematical models and two meta-heuristic algorithms were proposed to solve the problem.Mao et al. (2023) studied the U-shaped assembly line with HRC by assuming that each station can be equipped with one human and one robot.The problem was formulated as mixed-integer programming and a simulated annealing algorithm with enhanced search procedure was proposed to minimize the CT.Rahman et al. (2023) dealt with semi-automated ALBP with sustainability-based objectives with limited agents per station and uncertain processing times, while a memetic algorithm was proposed to solve the problem.
Based on the above literature, considering different objectives, this study is among the first attempts to balance and schedule assembly lines with HRC in straight and U-shaped layouts.This study extends the HRC level to benefit from incorporating multiple humans and robots with different skills and capabilities while performing the tasks at stations collaboratively and in parallel.Moreover, other real-world aspects, namely robot energy cost, robot tool changes, zoning, and technological requirements, are considered while addressing a case study and test problems.The effects of layout, objectives, and HRC on the optimization results are also studied.

Problems description
This study originates from an application case in the pilot assembly line of the engine module of a well-known automotive manufacturer.The assembly line produces a mass balancing system (MBS) unit used by different engine models.It contains two machined aluminum halves with two axles assembled between them, as shown in Fig. 1 (a).Moreover,Fig. 1 (b) shows the assembly line layout of the application case with the cobots.The assembly line consists of stations with embedded sensing and interacting software and hardware that perform the tasks separately, in parallel, or collaboratively by multiple humans and robots with different skills and capabilities.
From the layout perspective, the stations in the considered assembly line can be ordered based on two main types of layouts, namely, (a) the straight line and (b) the U-shaped line, as graphically shown in Fig. 2. The stations are arranged serially in a straight line so that the tasks can only be assigned in one direction from the left to the right.On the other hand, in the U-shaped lines, the tasks are assigned both on the entrance and exit sides of the line, allowing the operators to serve in both directions, increasing flexibility and productivity (Boysen et al., 2022).The configuration of AL-HRC entails solving three joint decisions, namely, (1) assignment of tasks to stations, (2) assignment of operators (humans and robots) to stations, and (3) scheduling of tasks among operators.
The current study considers three different objective types to be optimized depending on the nature of the problem, as discussed below:   • Type-1: Minimizes the NS given a CT.Traditionally, this objective arises when a new AL-HRC has to be established.The resulting optimization problem aims to find the lowest NS according to which the target CT can be satisfied, leading to higher line efficiency.• Type-2: Minimizes the CT given an NS.Traditionally, this objective arises when an existing AL-HRC attempts to respond to customer demand by minimizing the CT to react to the market demand.The resulting optimization problem aims to find the lowest CT according to which the target NS can be satisfied.• Type-rw: Minimizes the total costs of stations, operators, and robot energy consumption given a CT.Unlike the above types that consider the theoretical aspects of AL-HRC, this objective arises when a realworld AL-HRC has to be established in which the total cost (TC) of stations, operators, and robot energy consumption is minimized.With a cost-oriented objective, a trade-off is reached between the fixed cost of stations and operators and the operational cost of cobot energy consumption while performing the tasks.The energy consumption of each robot while performing each task is calculated by considering the task time, robot standard energy consumption in a unit of time, and the cost of each unit of energy consumed in the manufacturing company.
Each of these objectives results in new types of optimization problems for balancing and scheduling AL-HRC in which some or all of the following assumptions/constraints are considered: • Precedence relations: The assignment of tasks to the stations and operators and the scheduling of tasks are subjected to the precedence relationships among tasks.• Multiple humans and robots: Each station can be equipped with multiple humans and robots thanks to HRC safety sensors and interacting technologies.A set of available operators (humans and robots) at each station can perform the assembly tasks separately, in parallel, or collaboratively.The task times by each human and robot are determined based on their related skills and capabilities.The traveling times of operators are not considered.• Collaborative tasks: These sets of tasks have to be performed by HRC in which a team consisting of a human and a robot is scheduled to perform them collaboratively.• Robot tool requirement: This constraint considers the robot tool requirement while performing different tasks.Accordingly, each robot's total number of tool changes per station is limited to a fixed number (e.g., three) to avoid a higher number of tool changes.• Zoning constraint: This constraint considers that two linked tasks (e. g., using a screwdriver to tighten two screws) are performed at the same station.• Technological requirement: This attempts to ensure that a task that requires a specific technology available at a particular station cannot be performed at other stations.
Table 1 presents the specifications of the considered optimization problems in this study in terms of layout, objective, and constraint with the related abbreviations referred to hereafter.

Mathematical models for balancing and scheduling AL-HRC
This section deals with the mathematical formulation of the optimization problems discussed in the previous section, followed by the lower and upper bounds proposed for the objectives.Table 2 shows the notations used in the mathematical modeling of different optimization problems.

Straight AL-HRC
Assuming a straight assembly line with serial stations where a set of tasks (i = 1,⋯,NT) are assigned to a group of stations (j = 1,⋯,NS) to be

Table 1
The problems and their specifications considered in this study.
performed by a set of humans (l = 1, ⋯,NH) and robots (r = 1,⋯, NR).The processing time of task i when assigned to either human l or robot r is determined based on their skills and capabilities, which are presented by th il or tr ir , respectively.The following subsections describe the mathematical formulations of BS-AL-HRC-S-1, BS-AL-HRC-S-2, and BS-AL-HRC-S-rw problems in detail.

BS-AL-HRC-S-1
The mathematical formulation for BS-AL-HRC-S-1 as a mixed-integer linear programming (MILP) is presented below.
Minimize z 1 = ∑ NS j=1 ws j (1) Table 2 The notation used in the formulation of BS-AL-HRC-S and BS-AL-HRC-U models.
Notation Definition ∑ NR r=1 rk jr − NR × ws j ≤ 0; ∀j (13) ws j ≥ ws j+1 ; ∀j (14) The objective function (1) minimizes the total number of stations.Equation (2) ensures that each task is assigned to only one operator, i.e., a human or a robot.Equation ( 3) imposes that all precedence relationships among tasks are satisfied.Equation (4) determines the processing time of each task after its assignment to either a human or a robot.The scheduling constraints are modeled with Equations ( 5) to (9).For every pair of tasks i and h, if task h is an immediate predecessor of task i, Equation ( 5) ensures that task i is started after finishing task h, in order to satisfy the precedence constraints.If two tasks i and f have no precedence relation but they are assigned to the same human or robot, either Equations ( 6) or ( 7) for the human or Equations ( 8) or ( 9) for the robot become active.For a human, if i is assigned earlier than Otherwise, if f is assigned earlier than i, then Equation ( 9) becomes ST i − ST f ≥ t f .Using a sufficiently large number M, if two tasks are assigned to different stations, they are not considered in Equation ( 5), and if two tasks are assigned to different workers, they are not considered in Equations ( 6) and ( 7) and similarly if two tasks are assigned to different robots, they are not considered in Equations ( 8) and ( 9).Equations ( 10) and ( 11) ensure that if a task is assigned to the l-th worker or r-th robot, and that worker or robot is assigned to the j-th station, then wk jl and rk jr must be equal to 1, respectively.Equation ( 12) verifies the use of stations; if at least one worker is used in the j-th station, ws j will be equal to 1. Besides, such constraint ensures that the maximum number of workers that can be assigned to each station is NH.
Similarly, Equation (13) satisfies the above condition for the robot while ensuring that the maximum number of robots assigned to each station is NR.Equation ( 14) deals with the index of stations and ensures that stations are loaded in increasing order of their indexes.Equation ( 15) is the cycle time constraint, which ensures that each task has to be completed within the given cycle time.

BS-AL-HRC-S-2
The MILP formulation of BS-AL-HRC-S-2 is given as follows.
Minimizez 2 = CT (16) Equations ( 2)-( 11) ∑ NR r=1 rk jr − NR ≤ 0; ∀j (18) The objective function in ( 16) minimizes the CT.Constraints ( 17) and ( 18) guarantee that the numbers of humans and robots assigned to each station do not exceed the maximum number of humans and the maximum number of robots, respectively.Constraint ( 19) determines the completion time of each task.Constraint (20) specifies the completion time of the whole set of tasks that are assigned to a station.Constraint ( 21) is the cycle time constraint.

BS-AL-HRC-S-rw
The mathematical formulation of BS-AL-HRC-S-rw is given as follows.
Equations ( 2)-( 15) Equation ( 22) is a cost-oriented objective function regarding the costs of stations, humans, robots, and energy consumption.The positive zoning restrictions among tasks i and h, can be satisfied by Equation ( 23), while the negative zoning constraints can be ensured by Equation ( 24).Equation ( 25) considers the technological requirement between task i and station j given by a TER ij matrix.Assuming that tasks i and h are collaborative tasks to be performed by both human and robot, the Equations ( 26) to (29) ensure that these tasks are started simultaneously while one human and one robot are specifically assigned to perform them collaboratively.Equation ( 30) limits the number of tool changes for each robot at each station to RTC max while calculating the total tools switches between tasks i and h according to a given RTC ih matrix.
To avoid the non-linearity term raised by Equation ( 30), a binaryauxiliary variable (YY ihjr ) can be introduced as shown in ( 31) with ( 32)-( 34) added as its lower and upper bounds.Equation ( 30

U-shaped AL-HRC
An essential advantage of AL-HRC with a U-shaped layout is that the tasks and operators (humans and robots) can be both assigned to the entrance (forward) and the exit (backward) of the line, as shown in Fig. 2. Assuming an AL-HRC where the stations are arranged in a Ushaped layout, the BS-AL-HRC attempts to assign a set of tasks (i = 1,⋯, NT) to a set of stations (j = 1, ⋯, NS) while scheduling the tasks at each station to be performed by a group of humans (l = 1,⋯,NH) and robots (r = 1, ⋯, NR) with different competency level.The processing time of task i when assigned to either human l or robot r is determined based on their skills and capabilities, which are presented by th il or tr ir , respectively.The notations specific to the formulation of BS-AL-HRC-U are also defined in Table 2.In the following subsections, the mathematical formulations of BS-AL-HRC-U-1, BS-AL-HRC-U-2, and BS-AL-HRC-U-rw problems are described in detail.

BS-AL-HRC-U-1
The proposed MILP for BS-AL-HRC-U-1 is presented below. Minimize (38) tr ir × y ijrd ; ∀i (39) Equations ( 12)-( 15) Equation ( 36) ensures that each task can be assigned to only one operator and one direction (either forward or backward).Equations ( 37) and ( 38) satisfy the precedence relations among tasks while being assigned to the forward and backward directions, respectively.Constraint (39) determines the processing time of each task after it has been assigned to a specific human or robot.Equations ( 40) and ( 41) perform the scheduling of tasks when they have direct (immediate) precedence relationships while being assigned to the forward and backward directions, respectively.Equations ( 42) and (43) guarantee that if two tasks have no precedence relationships (neither direct nor indirect) and are performed by a human, then their start times are scheduled one after the other.Equations ( 44) and ( 45) are similar to (42) and ( 43), while a robot performs the tasks.When tasks have indirect precedence relationships, Equations ( 46) and (47) ensure that their start times are scheduled one after the other when performed by a human, while equations ( 48) and ( 49) schedule them when performed by a robot.Equation (50) ensures that if a task is assigned to a station to be performed by a human, the related human has been assigned to that station.Similarly, Equation (51) satisfies the above condition for a robot operator.

BS-AL-HRC-U-2
The proposed MILP model for BS-AL-HRC-U-2 is presented as follows: Equations ( 36)-( 51) Equations ( 17) to ( 19), ( 21) The definition of Equation ( 52) is similar to (20) in Section 4.1.2,in which the completion time of each station in terms of the times of the assigned tasks is calculated.

BS-AL-HRC-U-rw
The proposed MILP model for BS-AL-HRC-U-rw is given below.

Model type Objective lower bound Objective upper bound
Type-1 Equation ( 53) calculates the cost-oriented objective regarding the costs of stations, humans, robots, and robots' energy consumption.Equations ( 54) and ( 55) ensure that the positive and negative zoning constraints among tasks are satisfied, respectively.The technological requirements among tasks are guaranteed by Equation ( 56).Equations ( 57) to ( 60) ensure that HRC schedules the collaborative tasks simultaneously.Equations ( 61) to (64) satisfy the robot tool changes constraints by introducing a binary-auxiliary variable (YY ihjrd ), similar to the linearization steps taken in Section 4.1.3.

Proposed lower and upper bounds for objectives
To decrease the computational dimensionality of the proposed MILPs and find promising solutions while solving such complex models and extensions, six bounds, including three lower and three upper bounds, are introduced here to limit the objective ranges at each model type, as shown in Table 3.
In this table, for Type-1, the lower bound and the upper bound of the objective, namely NS LB and NS UB , are calculated based on the minimum       and maximum theoretical number of stations NS, while considering the minimum and maximum processing times of humans and robots (th ik , tr ir ), CT, and the maximum number of humans (NH) and robots (NR) per station, respectively.For Type-2, the lower and upper bounds of the objective, namely CT LB and CT UB , can be calculated based on the minimum and maximum theoretical CT, while considering the minimum and maximum processing times of humans and robots (th ik ,tr ir ), NS, NH, and NR, respectively.Finally, the lower bound and the upper bound of TC for Type-rw, namely TC LB and TC UB , can be calculated based on the associated cost of stations (cos j ), humans (coh k ), robots (cor r ), and robot energy consumption (En ir ).The TC LB and TC UB are based on the minimum and the total cost terms while considering the NS LB , NS UB obtained by Type-1, respectively.

Computational experiments
This section tests the application of the proposed MILP models by solving different instances of an automotive case study and the standard test problems from the literature.To this end, first, the experimental settings are presented.Then, the numerical results of the proposed MILP-based models for each problem on the case study and test problems are discussed.Third, the effect of the layout and objective on the optimization solutions is investigated.Fourth, the impact of the HRC level on the results is studied.Finally, the research implications and limitations of the study are presented.

Experimental settings
There are mainly two sets of problems considered in this study: (1) case study and (2) test problem.The case study is based on the assembly line of the MBS unit from the automotive industry, with the main characteristics presented in Table 4.This table shows the problem size, NT (number of tasks), CT (cycle time), NS (number of stations), NH (maximum number of humans), NR (maximum number of robots), instance name, and the time limit allowed to solve the related instances.
Further information about the case study is presented in the A ppendices.On the other hand, the test problems are based on the original input data existing online as the standard data set provided by Scholl et al. (1995).The rest of the data for the test problems, specifically for HRC features, were generated randomly and can be accessed in the supplementary material.The experimental settings for the test problem instances include the size of problems (small, medium, and large) and other characteristics similar to the case study, as presented in Table 4.In general, 12 problems, each with eight instances, are considered to be solved by the proposed MILP-based models.The proposed models were solved by the CPLEX solver on a Core i9 PC with a 3.10 GHz processor and 64 GB of RAM.The solver was terminated when the optimal solutions were found or the time limit shown in Table 4 was reached.The number of variables, the number of constraints, and the CPU times of the MILP-based models coded in CPLEX for all problems are reported in the supplementary material.As the problem's dimensionality increases in terms of NT, NS, CT, NH, and NR, the number of variables, constraints, and CPU times of the CPLEX models significantly increase.Over all the considered problems, the complexity of MILP models can be ranked in decreasing order of BS-AL-HRC-U-rw ≻ BS-AL-HRC-U-1 ≻ BS-AL-HRC-Srw ≻ BS-AL-HRC-U-2 ≻ BS-AL-HRC-S-2 ≻ BS-AL-HRC-S-1, in terms of the number of variables, constraints, and CPU times, simultaneously, where ≻ shows a higher complexity.

Numerical results
This section deals with the resulting solutions for BS-AL-HRC problems in the considered case study from the automotive industry and the standard test problems from the literature.Two types of models were considered while solving the instances: (1) MILP without bounds for the considered objectives (MILP) and (2) MILP with the proposed lower and upper bounds for objectives (MILP LB-UB ), as presented in Section 4.3.The subsections below compare MILP and MILP LB-UB models over the case study and test instances.

Case study instances
This section focuses on the results of the proposed MILP-based models while solving the case instances presented in Table 4.The characteristics of the case instances and the results of the MILP and MILP LB-UB models for each problem type, including the layout (straight and U-shaped) and the objectives (Type-1, Type-2, Type-rw), are shown in Table 5.In this table, the first column shows the problem type, the second column indicates the case instance number and the third column presents the given CT or NS for the related problem type.The fourth and fifth columns illustrate the maximum number of humans (NH) and the maximum number of robots (NR), respectively.The rest of columns compare the results of MILP and MILP LB-UB models in terms of the obtained NS * , CT * (second) and TC * (unit of cost) and the relative GAP(%).
Considering the objectives column under MILP LB-UB in Table 5, the performance of the MILP LB-UB, compared to MILP, was better in 10 instances, similar in 28 instances, and worse in only two instances.On the other hand, considering the GAP column, the performance of the MILP LB-UB , compared to MILP, was better in 19 instances, similar in 18 instances, and worse in three instances.MILP LB-UB could find better,  equal, and worse objectives in 25 %, 70 %, and 5 % of the case instances, respectively.Moreover, considering the GAP, MILP LB-UB was better, equal, and worse than MILP in 47.5 %, 45 %, and 7.5 % of the case instances.

Test instances
This section concentrates on the results of the proposed MILP-based models while solving the test instances, as presented in Table 4.The characteristics of the test instances and the results of the MILP and MILP LB-UB models for each problem type are shown in Table 6.In this table, the first column shows the problem type, the second column shows the problem size and the third column indicates the test instance number.The fourth column presents the given CT or NS for the related problem type, while the fifth and sixth columns illustrate the NH and NR.The rest of columns compare the results of MILP and MILP LB-UB models in terms of the obtained NS * , CT * (second) and TC * (unit of cost) and the relative GAP(%).
To better explain the results in Table 6, Table 7 summarizes the performing status (better, equal, worse) of MILP LB-UB versus MILP in different sizes in terms of the objectives and GAP both in numerical (Num) and percentage (%).
According to Table 7, in the small size, considering the objectives (Num), the performance of the MILP LB-UB was better in 30 instances, similar in 127 instances, and worse in only three instances.On the other hand, considering the GAP (Num), MILP LB-UB was better in 22 instances, similar in 130 instances, and worse in 8 instances.In the objectives (%), MILP LB-UB was better in 19 % of instances, similar in 79 % of instances, and worse in only 2 %, while in GAP (%), MILP LB-UB was better in 14 % of instances, similar in 81 % of instances, and worse in only 5 %.
For the medium size, considering the objectives (Num), MILP LB-UB was better in 48 instances, similar in 96 instances, and worse in 16 instances.At the same time, in GAP (Num), MILP LB-UB was better in 80 instances, identical in 60 instances, and worse in 20 instances.Considering the objectives (%), MILP LB-UB was better in 30 % of instances, similar in 60 % of instances, and worse in 10 %, while in the GAP (%), MILP LB-UB was better in 50 % of instances, similar in 38 % of instances, and worse in 13 %.
For the large size, considering the objectives (Num), MILP LB-UB was better in 88 instances, similar in 20 instances, and worse in 12 instances.
At the same time, in the GAP (Num), MILP LB-UB was better in 98 instances, similar in 19 instances, and worse in only three instances.Regarding the objectives (%), MILP LB-UB was better in 73 % of instances, similar in 17 % of instances, and worse in 10 %, while in the GAP (%), MILP LB-UB was better in 82 % of instances, similar in 16 % of instances, and worse in only 3 %.
Overall, MILP LB-UB could find better, similar, and worse objectives than MILP in 166, 243, and 31 instances, respectively, while considering the GAP (Num), MILP LB-UB was better, similar, and worse in 200, 209, and 31 instances, respectively.Considering objective (%), MILP LB-UB was better, equal, and worse than MILP in 38 %, 55 %, and 7 % of all instances, while in the GAP (%), MILP LB-UB was better, similar, and worse than MILP in 45 %, 48 %, and 7 % of all instances, respectively.It is worth mentioning that while moving from the small to the large size problems, the domination of MILP LB-UB over MILP has been increasing for both the objective and GAP measures.Additionally, MILP LB-UB found at least a solution in 56 medium-to-large instances, whereas MILP could not find any solution, as reported by NA (not available) in Table 6.

Comparison of results by objective and layout
This section investigates the effect of the objective and layout on BS-AL-HRC in the case study and test instances.To measure the effects of the objective (Type-1, Type-2, Type-rw) and the layout (straight, Ushaped) on the results, three relative percent deviation (RPD) measures, namely RPD Type− 1 , RPD Type− 2 , and RPD Type− rw , are defined by Equations ( 65) to (67).For each problem type, the related RPD measures the relative percent deviation of the U-shaped results compared to the straight results in terms of the corresponding objective function.In calculating the RPD for each instance, the minimum values of the objectives by the MILP-based models are considered.The calculated RPD values for each problem type over all instances in the case study and test problems are presented in Table 8.In this table, the first and second columns show the problem size and the instance name, while under columns Type-1, Type-2, and Type-rw, the related minimum objective (NS min , CT min , TC min ) by MILP-based models for the straight and U-shaped layouts, and the calculated RPD values are shown, respectively.A positive RPD shows an improvement in the related objective by the U-shaped compared to the straight layout, while a negative RPD shows a decline in that objective.

RPD
In Table 8, for instance, in the case instance rows (i.e., C1 to C8), one can observe the related minimum objective values and RPD under columns Type-1, Type-2, and Type-rw.The rest of the rows in Table 8, i.e., T1 to T88, show the minimum objectives of each type and the obtained RPD for the small, medium, and large test instances.To better explain the results in Table 8, Table 9 compares the results by the U-shaped layout versus the straight layout for each problem type and in different sizes.Under columns Type-1, Type-2, and Type-rw, the performing status in terms of the number of better, equal, and worse objectives in terms of NS min , CT min , TC min as well as the related percentage (%), are reported.
According to Table 9, one can observe that in the case study for Type-1, the U-shaped layout outperformed the straight line in 13 % of instances, showed similar performance in 87 % of instances concerning NS min .For Type-2, the U-shaped layout was superior in 75 % of instances while showing equivalent performance in 25 % of instances regarding CT min .In Type-rw case instances, the U-shaped layout demonstrated better performance in 100 % of instances in terms of TC min .Overall, the results suggest that the U-shaped layout is most effective in Type-rw, followed by Type-2 and Type-1, in the case instances.
In small-size test instances, for Type-1, the U-shaped layout improved performance in 56 % of instances and had similar performance in 44 % of instances concerning NS min .For Type-2, the U-shaped layout showed superior performance in 91 % of instances and had similar performance in 9 % of instances concerning CT min .In Type-rw, the Ushaped layout outperformed the straight line in 100 % of instances concerning TC min .Similar to the case instances, in small-size test problems, the U-shaped layout is most effective in Type-rw, followed by Type-2 and Type-1.
In medium-size test instances, for Type-1, the U-shaped layout improved performance in 34 % of instances and had similar performance in 66 % of instances concerning NS min .For Type-2, the U-shaped layout showed superior performance in 72 % of instances, had similar performance in 19 % of instances, and performed worse in 9 % of instances concerning CT min .In Type-rw, the U-shaped layout outperformed the straight line in 56 % of instances, had a similar performance in 13 % of instances, and performed worse in 31 % of instances concerning TC min .Overall, in the medium-size, the U-shaped layout was most effective in Type-2, followed by Type-rw and Type-1.Notably, the worse performance in Type-rw was due to high GAPs in a few U-shaped results compared to their straight counterparts.
In large-size test problems, for Type-1, the U-shaped layout improved performance in 21 % of instances, had similar performance in 71 % of

Table 9
The comparison of results by the objective and layout in different sizes.
instances, and performed worse in 8 % of instances concerning NS min .For Type-2, the U-shaped layout showed superior performance in 88 % of instances, had similar performance in 4 % of instances, and performed worse in 8 % of instances concerning CT min .In Type-rw, the U-shaped layout improved performance in 17 % of instances, had similar performance in no instances, and performed worse in 83 % of instances concerning TC min .Overall, in the large-size test problems, the U-shaped layout was most effective in Type-2, followed by Type-1 and Type-rw.As observed, the worse performance in Type-rw was primarily due to the high GAPs in a few U-shaped results compared to their straight counterparts.
Across all test problems, the U-shaped layout consistently provided higher improvement in Type-2, followed by Type-rw and Type-1.These results demonstrate that the U-shaped layout offers greater flexibility than the straight line, leading to more efficient solutions for JIT production.

Comparison of results by HRC, objective, and layout
This section investigates the combined effects of HRC, objective, and layout in the case study and test instances.To this aim, the following HRC scenarios are considered:  10.In this table, the first column shows the size, the second column shows the problem, and the rest represents the average of the minimum objective by MILP models for the HRC scenarios in the straight and U-shaped layouts with their RPD values in the last column.A positive RPD shows improvement by U-shaped compared to the straight layout, while a negative RPD indicates a disimprovement in the average of the HRC scenarios.
According to Table 10, in the case study for Type-1, the NS min Avg(HRC) in U-shaped line (1.75) is less than the straight line (2.00), showing an improvement of 13 %.For Type-2, the CT min Avg(HRC) of U-shaped line (25.60) is less than the straight line (27.45),showing an improvement of 7 %.For Type-rw, the TC min Avg(HRC) of U-shaped line (4185.25) is less than the straight line (4509.90),showing an improvement of 7 %.Thus, in the case instances, the U-shaped layout under HRC scenarios could improve the related measures in all types.
In the small size, for Type-1, the NS min Avg(HRC) in the U-shaped line are less than the straight line for all problems, showing improvement of between 11 % and 50 %.For Type-2, the CT min Avg(HRC) in the U-shaped line are less than the straight line for all problems, showing improvement of between 12 % and 26 %.For Type-rw, the TC min Avg(HRC) in the U-shaped line are less than the straight line for all problems, showing improvement of between 9 % and 28 %.Thus, in small instances, the U-shaped layout under HRC scenarios could improve the related measures in all types.
In the medium size, for Type-1, the NS min Avg(HRC) in the U-shaped line are less than or equal to the straight line for all problems, showing improvement of up to 20 %.For Type-2, the CT min Avg(HRC) in the U-shaped line are less than the straight line for all problems, showing improvement of between 4 % and 19 %.For Type-rw, the TC min Avg(HRC) in the Ushaped line are less than the straight line for all problems (except in Kilbridge), showing improvement of between 3 % and 25 %.In Kilbridge, a 2 % decline was caused by the large GAPs in the related HRC scenarios.Thus, in the medium size, the U-shaped layout under HRC scenarios could improve the average of objectives in all types in the majority of test problems.
In the large size, for Type-1, the NS min Avg(HRC) in the U-shaped line are less than or equal to the straight line for all problems, showing an improvement of up to 50 %.For Type-2, the CT min Avg(HRC) in the U-shaped line are less than the straight line for all problems, showing improvement of between 5 % and 7%.For Type-rw, the TC min Avg(HRC) in the Ushaped line are less than the straight line in Arcus, showing an improvement of 11 %.In Tonge, a 67 % decline was caused by the large GAPs in the related HRC scenarios, while in Arcus2, no value could be found due to NA status in the related HRC scenarios.Thus, in the large size, the U-shaped layout under HRC scenarios could improve the average of objectives for Type-1 and Type-2, while in Type-rw, no general conclusion can be made due to large GAPs in the related HRC scenario for the corresponding test problems.
Overall, the above results showed that in the considered HRC scenarios, the U-shaped layout could improve more than the straight line regarding the related objective at each problem type.In addition, the detailed results show that the AL-HRC (with both straight and U-shaped layouts) with higher HRC levels results in more efficient solutions for the related problem types compared with the assembly lines with no or limited HRC.

Research implications
Based on the results, the following managerial insights can be presented.
• Considering the scope of problems in BS-AL-HRC, the Type-1 model usually deals with long-term decisions, particularly when a new AL-HRC is configured.The Type-2 model supports the mid-term decision to re-balance and re-schedule an existing AL-HRC.Finally, the Typerw model deals with the long-term decision of establishing an AL-HRC from the cost perspective.The Type-rw optimizes the cost of stations, operators (both number and type of HRC), and robot energy expenditure while considering real-world constraints.Moreover, considering the BS-AL-HRC problems, the decision-makers can utilize the proposed straight and U-shaped lines models to investigate which layout is more beneficial for JIT production.• According to the computational results, compared to the straight one, the U-shaped layout is more promising in obtaining efficient solutions.This can be due to the higher utilization of operators while performing tasks at the entrance and exit of the line.Furthermore, the percentage of improvement in the objective function obtained by different model types shows that Type-2 results in more improvement than Type-rw, followed by Type-1.In addition, the analysis of results showed that the AL with a high HRC level is more promising in obtaining efficient solutions than assembly lines with no or limited HRC.• Applying the proposed MILP LB-UB models in the real case from the automotive company has proven their applicability in providing high-quality solutions in small to medium-sized problems.This can be due to both efficient lower and upper bounds of objectives in the proposed MILP LB-UB and the advancement of computer technologies enabling decision-makers to solve such complex optimization problems.However, as a future research direction, embedding some initial solutions into the MILP LB-UB as a warm start and integrating MILP LB-UB with customized heuristics can further enhance the solver efficiency, particularly in addressing large-sized problems.Additionally, humans' well-being and environmental aspects while formulating and solving BS-AL-HRC are yet to be explored.

Conclusions
Owing to the growing emergence of cobots in the era of Industry 4.0, recently, many manufacturers have attempted to shift their current manual or robotic assembly lines towards a human-robot collaboration (HRC) environment.In an HRC workplace, human skills and robot capabilities are combined to experience more flexible and agile manufacturing while achieving a human-centric and resilient workplace as initiated by the Industry 5.0 trend.Such advanced assembly line design entails balancing and scheduling tasks and operators among the stations.Additionally, the consideration of line layouts can significantly affect the resulting configuration.This study deals with balancing and scheduling straight and U-shaped AL-HRC with different objectives.The three objectives are categorized into minimizing the number of stations (Type-1), cycle time (Type-2), and the costs of stations, operators, and robot energy consumption (Type-rw).In the Type-rw, the concept of HRC is enhanced by integrating multiple humans and cobots possessing diverse skills and capabilities, enabling a flexible and efficient utilization of resources while allowing separate, collaborative, and parallel execution of tasks at various stations.A few practical aspects are considered, such as robot energy consumption, robot tool changes, zoning, and technological requirements.Different mixed-integer linear programming (MILP) models with efficient lower and upper bounds for each objective are proposed and tested by solving different case instances and standard test problems from the literature.The computational results show that the proposed MILP LB-UB overcomes the conventional MILP model regarding objectives and gaps.Furthermore, analysis of results shows that the U-shaped layout consistently improved in Type-2, followed by Type-rw and Type-1, while offering greater flexibility than the straight line, leading to more efficient solutions for JIT production.Finally, the higher the HRC level in the straight and Ushaped assembly lines, the better the achievement of the desired objectives compared to the lines with no or limited HRC.
This study can be further extended by including the ergonomic risks and environmental aspects while addressing the considered problems.Moreover, another topic to explore is integrating the proposed MILP LB- UB models with initial solutions or developing customized heuristics for them.

Funding
This study was funded by the Knowledge Foundation (KKS) and Sweden's Innovation Agency through the VF-KDO, ACCURATE 4.0, and PREFER projects.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
I have shared the data in the Attach file step

Table A1
The input data for the considered case study.

•
NH1-NR1: single human -single robot, per station • NH1-NR2: single human -two robots per station • NH2-NR1: two humans -single robot per station • NH2-NR2: two humans -two robots per station Considering the above HRC scenarios, the average of the minimum objective values by MILP-based models (NS min Avg(HRC) , CT min Avg(HRC) , TC min Avg(HRC) ) at each problem type (Type-1, Type-2, Type-rw) and layout (straight, U-shaped) are compared with each other in Table

Fig. A1 .
Fig. A1.The explosive map of the MBS unit in the case study.
Set of all predecessors and successors of task i cosjCost of station j based on space and equipment costs per year coh lCost of human l based on the yearly salary corrCost of robot r based on the purchase cost per year Enir Cost of energy consumption of robot r to perform task i per year ZC + ih , ZC − ih { 1; If tasksiandhhave to be performed (ZC + ) or not be performed (ZC − ) at the same station 0; otherwise TERij { 1; If taskican not beperformedinstationj 0; otherwise JT ih { 1; If tasksiandhhave to be performed by human and robot collaboration 0; otherwise RTC ih { 1; If tasksiandhneed a robot tool change if performed by a robot 0; otherwise M A large number RTCmax Maximum tool changes allowed for each robot at each station ihjrd A binary-auxiliary variable for linearization in a U-shaped line Real positive (R + ): CT Cycle time ti Processing time of task i after being assigned to either a human or a robot STi Start time of task i Ci Completion time of task i CWj Completion time of all tasks assigned to station j A. Nourmohammadi et al.
) can now be replaced by (35) as a linear Equation.The resulting equations form a MILP for BS-AL-HRC-S-rw.

Table 4
The experimental settings in the case study and test problems.

Table 5
The results of MILP-based models for different problems and case instances.

Table 6
The results of MILP-based models for different problems and test instances.

Table 7
The comparison of MILP LB-UB vs. MILP results in different sizes.