Simulation optimization of operator allocation problem with learning effects and server breakdown under uncertainty

ABSTRACT This paper investigates the operator allocation problem with learning effects and server breakdown in cellular manufacturing systems (CMSs) using fuzzy computer simulation and response surface methodology (RSM). The primary contribution of this study is incorporating combined server breakdowns and learning effects in CMS under uncertainty. Machine breakdowns of all the machines as well as the probability related to each entity should be delivered in good order are considered. Also, previous studies did not consider fuzzy simulation and RSM to deal with environmental and data uncertainty in operator allocation problems. The superiority of the presented model, in comparison with the traditional one, is shown according to the number of required iterations. The proposed simulation model is run in uncertain state to obtain the total processing time. RSM algorithm identifies a fitted function in terms of the value of allocated capital to each server and total processing time. This is a practical approach for decision-makers of all Cellular Manufacturing Systems.


Introduction
Flexible manufacturing systems (FMS) and cellular manufacturing systems (CMS) are thriving to production planning and productivity. In order to implement group technology, parts having similarities in manufacturing process are manufactured in one location and the most appropriate approach is cellular manufacturing system. CMS could result in reducing set-up and lead times, work in process (WIP), material handling and improved control and production planning. All machines must be placed in cells to process identical parts. Manual cells lead to increase system flexibility and the ability of the system to cope with changes such as customer demand or changes in the product design. Cross trained operators will have extensive impacts on the performance of CMS. Cross-trained operators can operate more than one machine and output rate can be adjusted by changing the number of operators in the manufacturing cells. It is known as 'intra-cell mobility,' owing to allocation of a group of operators who are multi skilled trained to several processes in the same cell. Banerjee and Al-Maliki (1988) presented a number of structured tools and assesses their role in building a functional model for FMS.
The significance of the present study lies in the fact that it simultaneously considers learning effects and server breakdowns in CMS in order to obtain the optimum allocation of operators with the minimum TSPT. In addition, it simultaneously models both factor in a unique simulation optimization approach for operator allocation. Also, it uses integrated response surface methodology (RSM) and fuzzy simulation approach to achieve the objective of this study. Some previous investigations have considered server breakdown, but none has considered combined learning effects and server breakdowns in operator allocation problem. This is the first study using RSM by applying D-optimal design to optimize operator allocation parameters with fuzzy simulation in a cellular manufacturing system (CMS). The simulation model is run to achieve the total processing time. RSM is a robust optimization used to assess simulation options and scenarios in different levels of experiments. The RSM algorithm identifies a fitted function based on the allocated capital to each server and total processing time.

Literature review
Labor allocation in CMS has been examined in a variety of ways (Azadeh, Asadzadeh, Mehrangohar, & Fathi, 2014;Ertay & Ruan, 2005;Kuo & Yang, 2006;Luo, Li, Tu, Xue, & Tang, 2011;Singh, Aneja, & Rana, 1992;Yang, Chen, & Hung, 2007). Russell, Huang, and Leu (1991) considered machines and human resources using a large variety of simulation experiments. Savsar (1999) examined many resource allocation policies on performance of automated transfer production lines with serial duplicate stations through simulation. Süer and Bera (1998) assumed that the influence of lot-splitting with regard to setup times. They utilized the structures of a two-stage mathematical method generating different operator levels to identify the optimal product and operator allocations for cells. Bhat (2008) presented a new method for introducing CMS in a small-scale industry to produce part families with similar manufacturing processes to reduce lead-times and costs and also improve quality and delivery performance. It is claimed that the long lasting success of manufacturing cells depends mostly on the ability of the cell members and supervisors to identify, predict, and rectify disruptions (Fan & Gassmann, 1997). The market fluctuations can reduce the performance of manufacturing systems significantly. Recently, among many configurations, the configuration with a reminder cell in a system has been studied. The remainder cell is able to perform all the operations needed in the manufacturing system. Renna (2016) explored the effect of a hybrid preventive maintenance policy in a manufacturing system with reminder cell, and the maintenance policy is a tradeoff between the preventive and corrective approach. Renna and Ambrico (2011) proposed the cell-loading approaches in a manufacturing system including dedicated cells and a remainder cell. The remainder cell consists of machines which can manufacture all part families. Kazerooni, Chan, and Abhary (1997) considered the operational problems of FMSs like scheduling. So, simulation method would be helpful for addressing these problems and simulation techniques are used for Real-time scheduling. Negahban and Smith (2014) provided a wide-ranging review of discrete event simulation articles published between 2002 and 2013 with a focus specifically on applications in manufacturing. In their paper the literature is examined in three general classes of manufacturing system operation, manufacturing system design, and simulation language. (Nakade & Ohno, 1999) examined a production line in which multitask labor was considered and also assessed identification of an optimal operator assignment. In addition, Ertay and Ruan (2005) presented a CMS framework using data envelopment analysis (DEA) to identify the optimal operator assignment. Prashanth, Prasad, Desai, Bhatnagar, and Dasgupta (2015) considered simultaneous perturbation methods for adaptive labor staffing in service systems. The application of resilience factors and concepts for designing a human resource management system, providing a robust and flexible system against sudden changes is examined (Azadeh, Heydarian, Nemati, & Yazdanparast, 2018). Azadeh, Rezaei-Malek, Evazabadian, and Sheikhalishahi (2015) proposed decisionmaking styles of operators (as an index of operator's personal characteristics) and an innovative mathematical programming model for clustering parts, machines and workers at the same time. Azadeh, Sheikhalishahi, & Koushan, (2013) presented fuzzy data envelopment analysis (FDEA) and fuzzy computer simulation approach, in which learning effects, server breakdowns, and multi products were considered. They calculated the efficiency and the number of operators in CMS while the rate of demand and the transfer batch size were changing. Performance is defined based on time-varying distributions of job arrival, job parallelism, and task service demand by fuzzy parameters. An integrated fuzzy simulation approach is presented by (Azadeh, Hosseinabadi Farahani, Moradi, & Maghool, 2014) to achieve the most-efficient policies for job scheduling. Nakade and Nishiwaki (2008) proposed a U-shaped production line that had multifunctional labors. Azadeh, Asadzadeh, Mehrangohar, & Fathi (2014) proposed an integrated simulation and genetic algorithm (GA) for optimum operator allocation in a large multiproduct assembly shop in which simulation is used as a powerful tool in order to analyze the real performance of the system and GA is used to maximize throughput of the system. Eguia, Racero, Guerrero, and Lozano (2013) presented a new approach to simultaneously solve the cell formation and the scheduling of part families for an effective working of a RCMS (reconfigurable cellular manufacturing system) consisting of multiple reconfigurable machining cells, each of which has at least one reconfigurable machine tool (RMT), a setup station, and an automatic material handling and storage system. In the literature of CMS most of the studies deal with the systems where the service places have never failed. However, in practice we frequently see the situation where service stations may fail and can be repaired (Choudhury & Deka, 2012). The industrial need of fact-based decision support, systems thinking, and skill upgrades in the maintenance area always play a pivotal role in manufacturing system. Bokrantz, Skoogh, Berlin, and Stahre (2017) presented first empirical Delphibased scenario planning study within the maintenance realm, examining a total of 34 projections about potential changes to the internal and external environment of maintenance organizations, considering both hard (technological) and soft (social) dimensions. Breakdown time is a critical factor that will affect the calculation of optimal maintenance intervals. Maintenance managers are required to determine optimal maintenance intervals with these requirements set by management (Tam, Chan, & Price, 2006). Cluster systems are clearly prone to failures and there would be reconfiguration and/or rebooting delays to resume the operation following a failure (Ever, Gemikonakli, & Chakka, 2009). Mehta (1999) proposed a predictable scheduling approach with a single machine breakdown. They measured the effects of disruptions by the difference between planned and realized completion time of the job. Sudhesh, Savitha, & Dhamaraja (2017) considered two-heterogeneous servers queue with, server failure, system disaster and repair. A single server feedback retrial queuing system having multiple working vacations and vacation interruption subject to server breakdown is examined (Rajadurai, Saravanarajan, & Chandrasekaran, 2017). Singh, Jain, & kapur (2017) investigated a single repairable server queuing system with bulk input and state dependent rates analyzed by considering the general distributions for the repair, delay to repair and service processes.
In this paper, the entities entering each queue after a certain amount of waiting time renege from the queue. In addition, each of the servers may break down based on the failure rates and will be repaired according to the repair rates immediately.
In the literature review learning effect has been stated from different viewpoints. Learning effects in scheduling have received significant attentions recently (Biskup, 2008). Janiak and Rudek (2009) examined a scheduling model-based learning model, in which processing times are defined by S-shaped functions. Wu, Lee, and Chen (2007) modeled learning effect into account in a single machine with maximum lateness. A branch-and-bound algorithm including several dominance properties was provided to achieve the optimal solution. Moreover, two heuristic algorithms were presented. Eren and Güner (2007) examined the minimizing of total tardiness problem in a learning effect situation. An integer programming model was developed to solve the problem. In addition, the Tabu search and the simulated annealing-based methods were presented. Huang, Wang, Wang, Gao, and Wang (2010) examined the single machine scheduling problems with time dependent deterioration and exponential learning effect, that is, the actual processing time of a job depends not only on the processing times of the jobs already processed but also on its scheduled position. Buckley (2005) presented a learning model considering both the machine and human learning effects at the same time. Moreover, the position-based learning and the sum-of-processing-time-based learning models are proposed in the model. Hood and Welch (1993) proposed a nonautomated optimized RSM in which the nature of the stochastic independent function was identified, such that the algorithm had the capacity to improve during the optimization process. On the contrary, in the automated RSM algorithm, human interference within the process was not possible.
There is an insufficient amount of data in the real world. Moreover, most data are not deterministic and are defined by linguistic variables such as low, medium, high, etc. In other words, the use of a fuzzy simulation model is an improvement on using the conventional model. In addition, outputs can be improved, or even be optimized. This study presents a unique fuzzy simulation model of CMS with server breakdowns and learning effects, and RSM is considered to select the optimal strategy. Finally, a comparison was made between the features of this study and those of similar studies (Table 1).

Methodology
This paper aims to implement a fuzzy simulation model of a cellular manufacturing system. Therefore, various experiments are examined by response surface method to select the optimal strategy for CMS with combined service breakdowns and learning effects in a standard CMS.

Fuzzy simulation
The fuzzy simulation model is uniquely designed and applied as the real CMS under study is a fully subjective oriented production system meaning that it is more suited to this type of modeling approach. Following a start-up period, the fuzzy simulation and conventional simulation models are run for one working month and are also replicated 12 times. The fuzzy simulation model is run for three different confidence intervals, αcutsðα ¼ 0; α ¼ 0:5, and α ¼ 1), meaning that it is totally run 36 times.
Because of the lack of deterministic data, a fuzzy simulation (FS) approach is presented. In Fuzzy simulation, one point estimate shouldn't be exactly equal to a parameter such as h. A (1-β) 100% confidence interval is determined for h, in which β represents the confidence level. Hereafter, it is elaborated that how fuzzy numbers for parameters in probability density functions (probability mass functions, the discrete case) are obtained from the set of confidence intervals. For instance, X as a random variable by probability density function of f x; θ ð Þ for parameter θ is considered and also θ as an unknown estimator from a random sample X1; . . . ; Xn ð Þ is assumed. Put Y ¼ u X1; :::; Xn ð Þas a statistic estimation of θ. Given the values of these random variables Xi xi; 1 i n, a point estimate θÃ ¼ y ¼ u x1; . . . ; xn ð Þis obtained for θ. It is not expected that this point estimate is exactly equal to θ. So, a (1-β) 100% confidence interval is often computed for θ. In this confidence interval, β is often set equal to 0.10, 0.05, or 0.01. The (1-β) 100% confidence intervals can be generated for all 0.01 ≤ β ≤ 1. These confidence intervals are defined as [θ 1 (β), θ 2 (β)] for 0:01 β 1. Also, the confidence interval for β = 1 could be denoted as θ1Ã; θ2Ã ½ . Thereafter, putting these confidence intervals on top of the others gives rise to producing a triangular shaped fuzzy number θ́whose α-cuts are the confidence intervals θ α ½ ¼ θ1 α ð Þ; θ2 α ð Þ ½ for all 0:01 α 1. In this study, the fuzzy parameters are considered as a triangular-shaped fuzzy number defined through probability density function. Simulation is performed using these fuzzy inputs and the fuzzy outputs are calculated which can be directly imported to the RSM method to assess and optimize different experiments.

Response surface methodology
Response surface methodology (RSM) is a statistical method and procedure that is particularly appropriate for stochastic simulation models (Kleijnen, 1998;Myers & Montgomery, 2003). This methodology approximates the stochastic objective function with a low order polynomial. The polynomial coefficients are specified by a regression model focused on the number of observations of the stochastic objective function, such that the objective function is calculated by preparation and adjustment of points in an experimental design (Kleijnen, 1998). For example, Hood and Welch (1993) described an RSM that is applied in the nonautomated optimization of simulation models.

D-optimal approach
This approach is based on a design that is provided by a computer algorithm.
Computer-aided design is extremely helpful in situations in which the classical design is not at all effective. Moreover, the optimal design can be created by a selection process subject to the selected criteria and the given number of design runs. The computeraided procedure enables D-optimal designs to create the optimal set of experiments (Eriksson, Kettaneh-Wold, Trygg, Wikström, & Wold, 2006). The objective is to select P design points from a larger set of candidate points (Myers & Montgomery, 1995).
Equation (1) can be expressed in matrix notation as: Y is the vector of observations, e is the vector of errors, X is the matrix of the values of the design variables and B is the vector of tuning parameters. B is estimated by using the least-squares method Equation (2).
The D-optimality criterion states that the best set of points in the experiment maximizes the determinant | X T X |. 'D' signifies the determinant of the X T X matrix related to the model. From a statistical viewpoint, a D-optimal design leads to response surface models in which the maximum variance of the predicted responses should be minimized meaning that the points of the experiment minimizes the error in the response model. The merits of this approach are the possibility to use irregular shapes and the possibility including extra design points. Generally, D-optimality is one of the most important criteria in design of experiments. D-optimal approach of RSM design is used. Moreover, computer-generated D-optimal designs are required when the experimental region is not regular. Experiments that are performed should be included, qualitative factors have at least two levels, the number of runs must be decreased, specific regression models must be fitted, and mixture factors are considered in the same design (Eriksson et al., 2006). The most suitable regression model should be fitted for the responses in terms of variables to formulate the constraints of the problem used in D-optimal design.

Factors and responses
Two type of variables are always present when we conduct experiments in a design of experiments (DOE): responses and factors. The responses provide us with information regarding the investigated system and the factors are used to manipulate the system normal factors set into two or more values. They have a defined range and can be categorized into three groups based on different criteria.
As mentioned previously, the response is the general condition of a studied system during the change of the factors. It is possible to measure multiple responses that react differently to factor adjustments and a response can either be a continuous or a discrete value (Jeff & Wu, 2000). For example, a discrete scale can be an ordinal measurement with three values (good, OK, and bad). These values are difficult to process, so the use of a continuous scale is always recommended if possible. Screening design provides information regarding the important factors, after which an optimization is performed, whereby these factors are used to obtain the optimal response. The final step is the robustness test (Eriksson et al., 2006).

Fitting regression models
D-Optimal designs provide the opportunity to modify the underlying model in different ways. It is possible to remove selected experiments if the experimenter knows they are not important for the response. This allows for a reduction in the number of runs without making an impact on the investigation. A second possible modification is via the addition of single, higher-order terms. With classical designs, it is only possible to change the entire model, for example, from an interaction model to a quadratic model. In contrast, D-optimal designs allow for the addition of independent model terms (Eriksson et al., 2006). The classical RSM designs are very inefficient if the number of factors increases. The runs required for the D-optimal design are always lower, and also do not increase as fast as the classical design with an increasing number of factors (Eriksson et al., 2006). The steps of the methodology used to achieve the goal of this study is presented graphically (Figure 1).

Experiment: empirical illustration
The designed cells described in the present study are flexible. The multifunctional operators walking enable us to rapidly rebalance the production line, which is considered as U-shaped. The cells have eight stations, and can be handled by at least one operator, based on the required output for each cell. The operation times should not be balanced at every station, and the balance is obtained by operators walking through the stations. The summation of the operation times for each operator is considered almost equal. Furthermore, since each cell is divided into stations, it is acceptable to obtain the balance via the operators. When a batch size enters the cell, it is divided into transfer batch sizes. The rebalance ability is shown by the presented simulation model (Figure 2). RSM and the fuzzy simulation process were applied to formulate and optimize the present study.
Stage 1 -The number of operators (in an 8-hour working day), the lead-time of demands, and the waiting times of demands are specified as cost criteria, and operator and machine utilization and number of completed parts (per year) are specified as an index set of benefits (Wang & Kusiak, 2000). The cycle time for each operator and the rate of the server breakdown is considered fuzzy, and the time interval between arrival of operators, operating times, and walking times were also included. It is assumed that the time interval between arrivals is equal to the demand lead-time. The cell examined in this paper is of an operator-intensive manufacturing type. In addition, operators visited multiple machines once in every cycle. The following assumptions are considered: • The rebalancing nature of the operator assignments are considered to make a difference in operator efficiencies. • Machines have downtimes in simulation model which is not considered in previous studies (Figure 3). • The summations of times for multifunctional operation for all operators are assumed to be equal. • The multifunctional operations times should not be balanced. Flexibility is achieved by the walking operators from one machine to another. • The learning effect is examined with regard to all operators. • The outputs obtained included lead-time of demands, downtime, waiting time of demands, operator utilization (OU), machine utilization (MU), and the number of completed parts (per year) for each α. The aforementioned items are stated according to the standards applied in CMS (Nakade & Ohno, 1999).  Stage 2 -The input data are included and set-up times and the fuzzified processing times are modeled by triangular fuzzy number: (µ-δ, µ + δ) (Buckley & Eslami, 1997). Stage 3 -The simulation model is developed by Visual SLAM (Pritsker & O'Reilly, 1999). The experimental results are subsequently compared. Moreover, the rebalance ability for each cell is illustrated using the simulation model to obtain output changes from the cell. Eight operators and eight machines are described as RESOURCE BLOCKS, which are OP1 to OP8 and M1 to M8, respectively. The operator task is defined as LTRIB [1] in VISUAL SLAM. At the beginning, it is one and all eight tasks are being performed. The products (two types) and demands (three types) are defined by ARRAYS. Moreover, ATRIB [2] and ATRIB [3] are set to demand and part types, respectively. Demandtype = DPROB (4, 5) and Parttype = DPROB (7, 8) are considered. The DPROB (4, 5) is a type of demand selector that is chosen according to the probabilities. XX [1] is set to the number of jobs. The COLCT node is approximately equal to the total performed (completed) tasks and XX [2] is set to the number of parts. XX [task +2] is set to the number of loads, considering learning effect. The arrangements of demand types are obtained by a BATCH node. According to the learning effect Equation (3).
The setup time between tasks is influenced by learning effect as illustrated in ACTIVITY number 10 formulated 0.3×POW (xx [task +10], K). K is equal to 0.2 in this study. The presented model part types and demand levels are simultaneously considered and simulation is used to account for different operator and shift levels.

Server breakdowns in the network
The entity enters to PREEMPT node, pre-empting the machine to cease service activity and is waited to be repaired by a maintenance operator. With the use of FREE node after repair, the machine and operator resources will be freed ( Figure 3).

D-optimal design
In the present study, D-optimal design in RSM is developed. It is applicable for finding the most suitable allocation of operators to servers with minimum total system processing time (TSPT). The data with respect to working shift, number of operators in an 8hour working day (OP), wait time, parts, machine utilization (MU), operator utilization (OU), server breakdown time (SB) and consequently the response variable of the model (TSPT) is shown in Table 2.

Analysis of Variance (ANOVA)
Analysis of variance (ANOVA) is developed and executed for the selected response surface linear model (

Model summary
R-squared is a statistical measure of the closeness of the data to the fitted regression. In this experiment, R-square with value of 99.9% is predicted. Moreover, R-squared (predicted) and R-squared (adjusted) are relatively and this fact confirms the validity of the model (Table 4).

Regression equation
The regression equations are presented based on shift illustrated for this study which is the function of total processing time of system. They are shown by Equations (4-6). The optimum response variable of the model (TSPT) is estimated the main effect, the interaction effect (cross-linear effect and quadratic effect) with respect to number of operators in an 8-hour working day (OP), wait time, parts, machine utilization (MU), operator utilization (OU), server breakdown time (SB) for each working shift. The three-dimensional graphs show response surface from the side and they are called response surface plots (Figures 4-6). The stated 3D surface plots show how wait time, OP, MU and OU affect the total processing time. As shown in Figures 7-9 service breakdown has significant impact on total processing time. In these figures, response surface is curved because the model contains quadratic terms including sever breakdown that are statistically significant. Therefore, it is proven that service breakdown and learning effect must be considered and modeled simultaneously in operator allocation problem for CMS.

Model verification and validation
Model verification is obtained by comparing RSM solution points with the simulation results. This procedure compares total processing time for RSM solutions with the simulation results at exactly the same points. It is observed that when the model is run for more than 108 times in each experiment, the average values of parameters do not  differ statistically. A total of 50 experiments are found by D-optimal design. Simulation of the CMS for the optimum solution revealed that results are sufficiently close to RSM solutions. The Spearman correlation experiment was employed to validate the results and the correlations between them was 0.754 at a significance level of α = 0.05 meaning  that the stated model is validated. To verify the results of RSM, two well-known approaches are developed and analyzed as discussed in the next sections.

Discussion
The optimum solution is shown in (Table 5) (24, 4.1, 160, 1.9, 0.97, 1, 0.33). It has the minimum total system processing time equal to 47.5. By applying the proposed algorithm to an example, its validity was approved and the optimum solution, which is an allocation of operators with the minimum TSPT was obtained. In summary, in this paper service breakdown has been considered for the first time. It is capable of providing a simulation model of the system while learning effects and server breakdowns were taken into account. This model has the ability to optimize the performance by considering stated factors of queuing systems in accordance with the desired objectives. Finally, the superiority of the proposed model, in comparison with the traditional one (Azadeh, Sheikhalishahi, & Koushan, 2013), was shown according to the number of required iterations. In addition, it has been shown in this paper that integration of fuzzy simulation and RSM, which is presented for the first time for operator allocation in cellular manufacturing system is more efficient. Also, the optimal solution is found with 50 experiments by the proposed integrated approach. For the future research, other factors such as budget constraint can be imposed as a limitation in the proposed model of this study. Budget constraints are defined in the system, where allocating the maximum allowable number of servers for each queue is not possible. So, the optimal number of servers in each queue should be determined so that the total number of servers in the system could not exceed the upper limit defining based on the system budget.

Conclusions
This is the first study simultaneously considering learning effects, breakdowns, and repairs under uncertainty in CMS. In addition, a unique simulation optimization composed of fuzzy logic computer simulation, ANOVA and RSM is introduced and verified by actual data. Moreover, a unique fuzzy simulation approach is developed to incorporate uncertainty associated with modeling procedures. Also, the D-optimal design of RSM is used for optimizing operator allocation parameters. In the present study, a unique RSM is also developed and applied to optimize the stated parameters and the total processing time (TSPT). The presented simulation model was run in uncertain state to obtain the total processing time. Therefore, we identified the server affecting the entire system and other consequential results, whereby performance requires further improvement. Consequently, D-optimal design is applied to identify the best allocation of the system. Machine breakdowns of all the machines as well as the probability related to each entity should be delivered in good order are considered. The RSM algorithm identifies a fitted function in terms of the value of allocated capital to each server and total processing time. After identifying the optimum allocation which has the minimum total processing time, the fitted function suggested by RSM is used to calculate total system processing time (TSPT). In order to validate our model, we simulated the CMS for the stated solution. The result obtained is relatively close to that of RSM.

Disclosure statement
No potential conflict of interest was reported by the authors.