Metaheuristic for Solving Multi-Objective Job Shop Scheduling Problem in a Robotic Cell

This paper deals with the multi-objective job shop scheduling problem in a robotic cell (MOJRCSP). All the jobs are processed according to their operations order on workstations. Different from classical job shop scheduling problem, the studied problem considers that jobs’ transportation is handled by a robot. Also, the jobs are expected to be finished in a time window, instead of a constant due date. A mixed Integer Programming (MIP) model is proposed to formulate this problem. Due to the special characteristics of the studied problem and its NP-hard computational complexity, a metaheuristic based on Teaching Learning Based Optimization (TLBO) algorithm has been proposed. The proposed algorithm determines simultaneously the operations’ assignments on workstations, the robot assignments for transportation operations, and the robot moving sequence. The objective is to minimize the makespan and the total earliness and tardiness. Computational results further validated the effectiveness and robustness of our proposed algorithm.


I. INTRODUCTION
Intelligent production equipment has been gradually applied in recent manufacturing industry. As the core unit of intelligent workshop, the robotic cell (RC) is composed of unified information control system, material transportation robots and a set of processing equipments. The jobs can be transported between the workstation and the loading/unloading station by the robot. And the robotic cell can be connected with various production and storage equipment, and the whole unit is controlled by information control system. Due to the high efficiency and flexibility of robotic cell, it can be widely used in various modern advanced industrial production fields [1]- [4]. However, the existence of industrial robots increases the complexity of production environment. Therefore, in the process of job shop scheduling, the decision-maker should not only consider the processing arrangement of jobs, but also the task allocation and the The associate editor coordinating the review of this manuscript and approving it for publication was Muhammad Zakarya . movement of robot in the cell. Therefore, the traditional job shop scheduling can not be directly applied to this kind of advanced intelligent manufacturing production problem, so new production mode and new scheduling optimization scheme are needed to solve the scheduling problem of robotic cell. Thus, it becomes an important research topic to develop an effective collaborative optimal scheduling method (to find the optimal jobs processing plan and robot handling plan) to solve the robot cell scheduling problem. Yan et al. [5] deals with a real-time dynamic job-shop scheduling problem in a robotic cell. The studied problem can be modelled as a job shop where the jobs have to be transported between machines by robots. Elmi and Topaloglu [6] solved the cyclic job shop robotic cell scheduling problem with multiple robots. All the jobs are processed in the order of their operations on multiple machines with standard processing times and the single gripper robots. The robots perform the transportation operations of jobs between the stations.
There are different research methods to solve the scheduling problem of robotic cell. and common solutions mostly include exact algorithms and approximate algorithms. The exact algorithm usually uses the method of mathematical modeling to solve the problem. Caumond et al. [7] provided the mathematical formulation and optimal solutions for the Flexible Manufacturing Systems Scheduling Problem (FMSSP) with one vehicle. This linear formulation is different from the previously works, it takes into account the maximum number of jobs allowed in the system, limited input/output buffer capacities, empty vehicle trips and nomove-ahead trips simultaneously. Che et al. [8] addressed the multi-robot 2-cyclic scheduling problem in a no-wait robotic cell where two parts enter and leave the cell during each cycle. In his work, multiple robots are responsible for transporting parts between machines. They developed a polynomial algorithm to find the minimum number of robots for all feasible cycle times. Nattafa et al. [9] studied the scheduling problems with jobs of different families on parallel machines, where not all machines are qualified (eligible) to process all jobs. To solve this problem, an integer linear programming model and a constraint programming model are proposed.
For scheduling problem, approximate algorithms are more employed. Zhang and Xing [10] addressed the distributed limited-buffer flowshop scheduling problem with makespan minimization criterion for the first time. They successively proposed two constructive heuristics for generating fast schedule and providing good initialization of meta-heuristics. Lei [11] presented a particle swarm optimization for multi-objective job shop scheduling problem. He aims to simultaneously minimize makespan and total tardiness of jobs. Wang et al. [12] investigated the energy-efficient distributed permutation flow shop scheduling problem (DPFSP) with the objectives of makespan and energy consumption. A multi-objective whale swarm algorithm (MOWSA) is proposed to solve this energy-efficient DPFSP. Also, to enhance energy efficiency without affecting production efficiency, a new problem-dependent local search is developed to improve the exploitation capability of MOWSA. Kurdi [13] proposes an improved island model memetic algorithm with a new naturally inspired cooperation phase (IIMMA) for multi-objective job shop scheduling problem. The new cooperation phase is mainly used to improve the exploitation capabilities of algorithm. Udomsakdigool and Khachitvichyanukul [14] present ant colony algorithm for solving the multi-objective Job Shop Scheduling Problem. The objectives considered in this study include the minimization of makespan, mean flow time, and mean tardiness.
In order to accelerate the convergence of the algorithm, many works combined the traditional approximation algorithm with neighborhood search technology which can be more efficient in solving FMS scheduling problem. To prevent premature convergence in a multi-objective optimization problem, the SA (Simulated annealing) algorithm is utilized as local search in the works of Mokhtari and Hasani [15]. It is capable of obtaining more solutions by escaping from the local minima in a wider solution space. Arnold and Sorensen [16] combined three powerful local search techniques, and implemented them in an efficient way. In their work, experiments have been made to determine how local search can be effectively combined with perturbation and pruning and how to guide the search to better solutions. Moghaddam et al. [17] proposed a new multi-objective Pareto archive particle swarm optimization (PSO) algorithm combined with genetic operators as variable neighborhood search (VNS). Their methods provided a better solution for large-sized problem instances within a reasonable computational time. Moslehi and Mahnam [18] present a new approach based on a hybridization of the particle swarm and local search algorithm to solve the multi-objective flexible job-shop scheduling problem. Zhang et al. [19] has formulated the textile dyeing process scheduling problem as a bi-objective optimization model, in which one objective is related with tardiness cost while the other objective reflects the level of pollutant emission. And they proposed a multi-objective particle swarm optimization algorithm enhanced by problem-specific local search techniques (MO-PSO-L) to seek high-quality non-dominated solutions. Luo et al. proposed a Distributed Flexible Job Shop Scheduling Problem with Transfers (DFJSPT), in which operations of a job can be processed in different factories. In order to expand the search space and accelerate the convergence speed of the solution, an Efficient Memetic Algorithm (EMA) is proposed to solve the DFJSPT with the objectives of minimizing the makespan, maximum workload, and total energy consumption of factories.
As a matter of fact, previous works mainly focus on the cases of single objective scheduling problem with robots or multi-objective scheduling problem without transported robots. These proposed solutions cannot address well our studied problem (MOJRCSP). Furthermore, most of the research methods are based on exact algorithms, and few approximate algorithms are available, especially for multi-objective manufacturing workshops with robotic cells. Therefore, in this work, we are aiming at solving the MOJRCSP with improved multi-objective teacher learning-based optimization algorithm (IMOTLBO). Since TLBO algorithm needs fewer parameters, it is simple and has strong convergence ability and fast convergence speed, so it has been applied in various fields [21]- [24], [26]- [28]. For multi-objective teacher learning-based optimization algorithm (MOTLBO) problem, up to now, the application of MOTLBO has been very limited. In [29], a modified version of the TLBO algorithm is introduced and applied for the multi-objective optimization of heat exchangers by Rao et al. Plate-fin heat exchanger and shell and tube heat exchanger are considered for the optimization. Maximization of heat exchanger effectiveness and minimization of total cost of the exchanger are considered as the objective functions. Toǧan [22] presents a procedure employing a Teaching-Learning Based Optimization (TLBO) technique for discrete optimization of planar steel frames. The design algorithm aims to obtain minimum weight frames subjected to strength and displacement requirements imposed. In this work, our optimization goal is the minimization of makespan and the total weighted earliness and tardiness. In order to accelerate the convergence of the algorithm, we combined the TLBO algorithm with a powerful local search technique: variable neighborhood descent (VND). The algorithm determines the optimal sequence of jobs operation assignment and the robot movement in scheduling. Finally, with different instances the proposed IMOTLBO is compared with multi-objective algorithms, i.e., the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [23], the Pareto-based Grouping Discrete Harmony search algorithm (PGDHS) [24] and the Multi-Objective Particle Swarm Optimization (MOPSO) [25]. The results proved the proposed algorithm's effectiveness and advantage.
Contribution and innovation of this study are highlighted as follows: • The multi-objective job shop robotic cell scheduling problem is formulated as a Mix Integer Programming model with transportation by handling robot, jobs' finished time windows are considered.
• This study proposes a new hybrid multi-objective metaheuristic which combines the TLBO algorithm with a powerful local search technique, the experimental results show that our algorithm obtains better non-dominated solutions than others. The rest of this paper is organized as follows: The problem description and mathematical model is given in Section 2. In Section 3, the details of our proposed IMOTLBO is presented for solving the MOJRCSP. Experimental results and the conclusion are given in Section 4.

II. PROBLEM DEFINITION AND MATHEMATICAL FORMULATION A. PROBLEM DESCRIPTION
A typical robotic cell consists of M workstations, a load/unload station and a handling robot. The framework of the RC system is illustrated in Figure 1. A number of jobs J = {J 1 , . . . J N } arrive at the load station of RC system and each of them has its own processing sequences. Each job consists of a number of n j operations O j1 , . . . O jn j specifying the job's visiting route and different processing times p ji . The number of operations of each job n j may be different and n j ≤ M .
Without loss of generality, each job starts from the load station and finishes on the unload station. An automated handling robot moves jobs from one station to another. A move task Tr ji indicates that the robot transports job j from station µ j,i−1 to station µ ji , where µ ji corresponds to the processed station of operation O ji . Before move Tr ji , if the robot is not in the position µ j,i−1 , an empty move Tr ji is needed to arrive the desired position. The robot's loaded transportation and empty move should be considered in the scheduling problem. Robot's loading time ε c and unloading time ε d are set to a constant value.
Each operation O ji starts the process on station µ ji at its arriving time s ji and leave the station at time t ji , its staying  time can be divided into three parts, p ji represents the processing time of O ji , I ji and Q ji correspond to the waiting time of O ji in the input buffer and output buffer, respectively. These variables model the operation process on the station (shown in Figure 2).
The goal is to minimize the makespan and the total tardiness and earliness simultaneously, since each job's expected finish time is confined into time windows with the given minimal and maximal time bounds.
The following assumptions are considered: • all jobs enter and leave the RC system through the load and unload stations; • the capacity of load/unload station and input/output buffer of workstation is unlimited; • workstation and robot can process at most one job at a time; • each operation can only be processed on only one workstation; • failures of machines and robot are ignored; • at time zero, all jobs are arrived at the loading station, and all workstations and robot are available; • no preemption is allowed.

OBJECTIVE FUNCTION AND CONSTRAINTS
The objective function is: The following constrains are considered: This constraint ensures that the completion time of operation O ji is equal to the arriving time plus its staying time at the station, where the staying time consists the waiting time in input buffer I ji , the processing time p ji and the waiting time in output buffer Q ji .
where j ∈ [1, J ]. These constraints ensures that the operations are performed according to the job's sequence, where O j,i−1 is denoted as the immediate previous operation of O ji in the job sequence. Constraint (4)  Constraint (5) and (6) can be considered as the special case of (4). Constraint (5) computes the completion time of the first operation of each job with t j0 = 0. Constraint (6) and (7) calculates the completion time of each job, which is equal to the time when the job is entered to the unload station, where I j,n j +1 = p j,n j +1 = Q j,n j +1 = 0.
This constraint ensures that two operations of the same job cannot be performed at the same time. If O ji is processed before O jk (U jik = 1), operation O jk must start after the robot moves the job from station µ ji to the input buffer of µ jk .
These constraints ensures that two operations on the same station cannot be processed at a time. If O ji is processed before O j i on station µ ji (X jij i = 1), then O j i will start its process after O ji enter the output buffer.
These constraints ensure that two loaded moves cannot be performed at the same time by the robot. In the case of Z jij i = 1, the loaded move Tr j i must start after the previous loaded move Tr ji is completed. When the robot finish the previous move, and if its current position µ ji is different to the starting position µ j ,i −1 of loaded move Tr ji , an empty move is required. where j, j ∈ [1, J ], i ∈ [1, n j ], i ∈ [1, n j ], with µ ji = µ j i These constraints ensure that the FIFO (First In First Out) buffer management rule is applied in the studied RC system, then all the successive operations are processed on the same order on the station, including input buffer, machine and output buffer.

III. PROPOSED METAHEURISTIC
Teaching learning-based optimization (TLBO) is a new intelligent optimization algorithm proposed by Rao et al. [29]. The TLBO algorithm is a kind of population based metaheuristic which simulates the teaching learning process. In standard TLBO, there are two vital stages which are teaching phase and learning phase. The algorithm can be simply described as that the learners learn from the teacher (the best learner of the current population in the teaching phase). While they learn from other learners in the learning phase.
In this work, an Improved Multi-Objective Teaching Learning Based Optimization algorithm is proposed to solve the studied problem, where class preview phase, self-studying phase and teacher promotion phase are incorporated. The framework of IMOTLBO algorithm is shown in Figure 3, and the details are described as follows.

A. ENCODING SCHEME
Encoding scheme is an essential element of any metaheuristic algorithm. In this study, a solution to the problem (student or teacher) is represented by a (len × 2) matrix where len is the sum of total number of operations and number of jobs (each job has a pseudo operation for moving the job to unload station). The first line represents the job's index, the An example for the studied problem with 3 jobs is shown in Figure 4. In this example, one solution is represented as [3 1 2 1 2 3 2 1 3] in the first line where each job has 2 processing operations. In the relative position of the second line, the first number '3-1' represents the operation O 31 and a loaded move Tr 31 which transports job 3 from the load station to its first processing station. The second number '1-1' represents the operation O 11 and the loaded move Tr 11 , since the handling robot stopped on M 2 in the preview loaded move Tr 31 , an empty move Tr 11 is required before Tr 11 in order to move the robot to the starting station of Tr 11 . The number '3-3' is a pseudo operation which indicate that the job 3 returns to the unload station. The information in the second line not only represents the scheduling sequence of operations, but also indicates the transportation sequence of handling robot. A gantt chart of the presented example is shown in Figure 5.

B. CLASS PREVIEW PHASE-POPULATION INITIALIZATION
In the teaching process, if the students have a good knowledge level before the teaching activities, it is beneficial to the teaching quality of the class. This part is defined as the class preview phase, which is responsible for generating the initial population. VOLUME 8, 2020 In this study, a well-known heuristic NEH (Nawaz, Enscore and Ham) [30] and dispatching priority rule SPT (Shortest Processing Time) [31] are applied to generate two initial solutions, in order to accelerate the convergence of the algorithm. The other solutions in the initial population are generated randomly.

C. TEACHER SELECTION-PARETO RANKING
Since our studied problem is a multi-objective optimization, more than one teachers are selected from the class according to personal knowledge level and performance. In IMOTLBO algorithm, the personal knowledge level and performance refer to objective values and ranking level of the solution. Each teacher has his own advantages, no one can completely surpass another. This process is based on Pareto dominance relationship. A same Pareto sort method of NSGA-II is applied, all solutions are divided into several Pareto fronts, the first Pareto front is considered as the teachers and they are copied into an archive.

D. TEACHING PHASE AND LEARNING PHASE-CROSSOVER
In the preview step, the best individuals are identified as the teachers. The teachers try to improve the performance of students through the communication of knowledge. The students obtain knowledge from the teachers using the crossover operator. The PTL technique [32] is applied in this section, it can produce a pair of different permutations, even from two identical individuals (see Figure 6). The new generated solutions are compared with the original one, the best one (who dominates the others) will be accepted. If there is no dominant relationship between them, then a random chosen solution is regarded as the student.
Since the processing sequence of operations in a job cannot be changed, the operations order should be updated by a repair process after the modification. Generally, as long as the solution is modified, the sequence of operations of jobs should be rechecked by repair process to ensure the feasibility of the solution.
In the learning phase, students improve their performance by the communication among themselves. A student X interacts with another student y which is chosen randomly. This crossover is performed between students in the same manner as shown in Figure 6.

E. SELF-STUDYING PHASE-MUTATION
After the knowledge communication of the class, each student needs to review their knowledge by self study. In this phase, students complete self-improvement by a mutation operator. Each time, one of the neighborhood structures mentioned below (see Figure 7) is selected randomly as the mutation operator.

F. TEACHER PROMOTION PHASE-LOCAL SEARCH
In teaching activities, teachers continue to improve their knowledge and ability, which is beneficial to the performance of the whole class. The aim is to further improve the current optimal solutions of the entire population and archive. In this phase, a local search algorithm is applied to improve the individuals in the archive at the end of each iteration.
Usually, local search method receives one solution as input and returns one solution as output. In this work, all individuals in archive are considered as the input of local search and return a set of non-dominated solutions. This is done to keep the diversity of algorithm. IMOTLBO implements a local search technique based on the variable neighborhood descent (VND) algorithm [33], it moves continuously one solution to another position in the search space. In our implementation, we use l max = 4 different neighborhood structures (see Figure 7), these neighborhood structures are briefly explained as follows: • Exchange: This neighborhood structure exchanges the position of two operations.
• Reverse: This structure reverse the order of operations between two randomly selected position.
• Single point insert: This structure removes an operation from its current position and relocates it at a random position. • One Stage Insert: In this structure, a part of operations between two random positions are removed to another position. Inspired by the local search algorithm for E/T criterion of Gao et al. [24], an earliness and tardiness swap method is proposed to improve the individual. According to the completion time and expected due date interval of jobs, the jobs are divided into three groups in this method. These three group include the early finished jobs, the late finished jobs and on-time finished jobs, respectively. In this method, two adjacent operations on same are considered as pair from the first two operations. For all operations on all stations, if the left operation in the pair belongs to the job in group one, and the right operation in the pair belongs to the job in group two, then the position of the pair of operations on station will be interchanged. For example, as shown in Figure 5, suppose that the completion time of job 2 is before its earliest due date a 2 , the completion time of job 3 is after its latest due date b 3 , and the completion time of job 1 is between a 1 and b 1 . The operation O 22 and O 32 will swap their positions.
The pseudo code of proposed local search technique is shown in Algorithm 1. The local search receives the individuals in archive as the input. The first step is to initialize the auxiliary populations pop and localpop. Then, the local search process is applied for each individual in Arch. Different from the standard local search technique, our improved local search is used to generate several non-dominated solutions. At the beginning of each iteration, the i th solution in Arch is selected and considered as s, the temporary population localpop is initialized with the solution s. We set l to 1 and

Algorithm 1 The Pseudo-Code of Local Search Method
Input: The individuals in archive Arch Output: The offspring Archive Arch start the following search process. When a new solution s is generated by implementing the neighborhood structures. By comparing s and original solution s, there are three possibilities: 1) s dominates s, then s is replaced by s and localpop is updated, l is reset to 1 for restarting the searching process with the first neighborhood structure. 2) there are no dominance relationship between s and s , then, localpop is updated and l is reset to 1. 3) s is dominated by s, then we set l = l+1 to implement the next neighborhood structure. When the above mentioned search is finished, Earliness-Tardiness Swap is applied and update localpop with the generated solution s . At the end of each iteration, the auxiliary population pop is updated by localpop, and localpop is cleared. When all solutions in arch are selected to perform the search process, the final pop will be considered as the offspring archive.

G. PROCEDURE OF IMOTLBO
The details of the proposed IMOTLBO for solving the MOJRCSP are presented in Algorithm 2. The algorithm starts with a class of PS students, where PS is the number of students in the class. Each student corresponds to a solution of the studied problem, and has its own knowledge level.
Some of them will have better performance than others by execute class preview phase. Because there are two objectives to be considered, the Pareto ranking is applied to select the teachers (non-dominated solutions) which are saved into a teacher team (archive). Then, all of the students begin to study continuously until the stopping criterion is satisfied.
At each iteration, students learn from one teacher and communicate with other students, this step are realized through teaching phase and learning phase where PTL crossover is performed. After the communication in class, each student tries to improve his performance by using self-studying phase, where four neighborhood structures are provided as mutate operator. When all the students have completed their learning tasks in this iteration, the teacher team will be updated under the Pareto dominance relationship.
Subsequently, all teachers start their further study, in order to further improve their performance. In the teacher promotion phase, the proposed local search is implemented to all solutions in archive, and generates an offspring archive with a set of non-dominated solutions. This step can speed up the convergence of IMOTLBO. When the algorithm is terminated, the teacher solutions in archive are considered as the output of algorithm. While stopping criterion is not satisfied DO Crossover between students and teachers (Teaching phase) Crossover between students (Learning phase) Mutation for students (Self-studying phase) Update teachers Local search for teachers (Teacher promotion phase) End While Output the teachers End

IV. COMPUTATIONAL RESULTS AND COMPARISONS A. EXPERIMENTAL SETUP
To evaluate the performance of the proposed IMOTLBO algorithm, experimental evaluation and comparison with other methods were performed. Two sets of problem instances are considered: • The first data set (instance 1-12) is provided by Caumond et al. [7].
• The second data set (instance 13-30) is generated randomly by the protocol.
The Caumond benchmark set includes 12 small sized instances with size ranging from 3 jobs and 12 operations to 5 jobs and 18 operations, the number of machines is equal to 5. The generated data set is composed of 18 medium-large sized instances with size ranging from 3 jobs, 12 operations and 5 machines to 20 jobs, 110 operations and 10 machines. In this study, the processing time p ji is randomly generated between a uniform distribution [1,50]. Loaded trip duration τ rl and empty trip duration σ rl are generated in the same way as [7]. Loading time ε c and unloading time ε d are set to 1. Inspired by Gao et al. [24], the following formulas are used to generate the earliest due date and latest due date, respectively.
where T is a parameter that is equal to 0.3, n is the number of jobs, m is the number of machines.

B. MEASURE METRICS
The comparison of results of multiobjective optimization is different to the single objective method, because there is a set of non-dominated solutions, instead of a single solution. Several conditions should be respected for the search results of the approximated method: the distance to the absolute pareto optimal front is to be minimized, the diversity of the obtained solutions should be maximized and the spread of the obtained solutions should also be maximized. The following widely used performance measures are applied in this paper: • Number of non-dominated solutions This performance measure counts the total number of non-dominated solutions of the compared algorithms.
• µ d -distance of Dugardin et al. [34] This measure computes the size of distance between two non-dominated fronts. Let A and B be two non-dominated fronts to be compared and which are provided by different methods. n A and n B are the number of non-dominated solutions in front A and front B respectively. Then the µ d -distance is defined as: 2 is the cross of the corresponding rectangle, d i is the distance between a solution of A and its orthogonal projection on B. The value of d i is negative when the solution i is below the front B (or the extrapolated front), positive otherwise. This measure is the proportional improvement to the diagonal of the rectangle. It shows if front A is closer to the optimal solutions than front B. The more negative value of µ d is, the more A is better than B.  According to the Zitzler measure, C 1 computes the ratio of solutions in A which are dominated by at least one solution in B and vice versa. C 1 is defined by where a and b are the non-dominated solutions in the set A and B respectively.

C. RESULTS AND COMPARISONS
In order to demonstrate the performance of the proposed IMOTLBO algorithm, it is compared with several well-known and recently published algorithm, such as NSGA-II presented by Deb et al. [23], the multi-objective particle swarm optimization (MOPSO) presented by Sha and Hsu [25] and the Pareto-based grouping discrete harmony search algorithm (PGDHS) of Gao et al. [24]. All algo- In these tables, J is the number of jobs, M represents the number of machines and S is the numbers of total operations of the tested instance. The values of µ * d , µ # d and µ b d represents the mean value, the variance and the best value of µ d -distance. The next six values represent the mean value, the variance and the best value of C 1 and C 2 respectively.    In Table 1, we can get same conclusion through the data of other lines, we find that our proposed IMOTLBO algorithm outperforms NSGA-II. Moreover, a special case appears in line 12 when µ * d = 0004, it is because the two fronts are too closed to each other (when one front overlaps another front, the d i is alternatively positive and negative, the measure µ * d is not signficant [34]). In This case the µ * d dose not depict the relative position of the two fornts. The measures C 1 /C 2 can be used to undertake the Comparison mission.
In Table2 and Table 3. The µ d shows that IMOTLBO dominates PGDHS and MOPSO. The couple C 1 /C 2 shows that fewer solution of the IMOTLBO are dominated by at least one solution of PGDHS and MOPSO (C 1 < C 2 ). IMOTLBO can produce more non-dominated solutions and require less computing time than ohter algorithms. By comparison, IMOTLBO is better than PGDHS and MOPSO. The comparison results fully prove the effectiveness, robustness and diversity of IMOTLBO algorithm.
Gantt chart helps us to observe the scheduling results intuitively, including the scheduling of operations on stations and the movements of handling robot. Two examples are shown in Figure 8 and Figure 9, corresponding to a small sized problem and a medium sized problem respectively. In these figures, X-axis represents times and Y-axis represents stations or machines.
Finally, we compared the computing times of all algorithm by graphics. The computing times of the four algorithms on Caumond benchmark are shown in Figure 10, Figure 11 shows the computing times on our generated instances. For all tested instances, the IMOTLBO algorithm shows its advantage in computing time.

V. CONCLUSION
In this paper, we focused on solving the multi-objective jobshop robotic cell scheduling problem (MOJRCSP), a Mixed Integer Programming model is proposed to model the studied problem. The objective is to minimize the makespan and total earliness and tardiness simultaneously. An improved multi-objective teaching learning based optimization algorithm is proposed, where 5 phased are implemented in order to improve the performance of algorithm. Through comparative studies with other multi-objective optimization algorithms, the proposed method outperforms the others.
XIAOHUI LI (Member, IEEE) was born in Xi'an, Shaanxi, China, in 1982. He received the B.Eng. degree from the Xi'an University of Technology, China, in 2004, the M.Eng. degree from Paul Sabatier University, France, in 2008, and the Ph.D. degree from the University of Technology of Troyes, France, in 2011. Since 2012, he has been a Lecturer with the School of Electronics and Control Engineering, Chang'an University. His research interests include optimization algorithm, scheduling problem, vehicle routing problem, UAV path planning, and multi-objective optimization.
XI YANG was born in Baoji, Shaanxi, China, in 1994. She received the bachelor's degree in measurement and control technology and instrumentation from Xi'an Aeronautical University, in 2016. She is currently pursuing the master's degree with Chang'an University. Her main research interests include intelligent optimization algorithm and workshops scheduling optimization problem. YUAN DONG was born in Xi'an, Shaanxi, China, in 1987. She received the B.E. degree in electronic and information engineering from Northwestern Polytechnical University, in 2009, and the master's and Ph.D. degrees in computer science from the University of Technology of Troyes, France, in 2013. Since 2014, she has been doing her research and teaching with Chang'an University. She has authored over ten technical papers in refereed journals and conference proceedings. Her main research interests include machine learning, pattern recognition, wireless heterogeneous networks, especially in study of image processing and decision systems.