Advances in Production Engineering & Management Volume 12 | Number 1 | March 2017 | pp 62-74 https://doi.Org/10.14743/apem2017.1.240 ISSN 1854-6250 Journal home: apem-journal.org Original scientific paper A genetic regulatory network-based sequencing method for mixed-model assembly lines Lv, Y.a, Zhang, J.a*, Qin, W.a aSchool of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China A B S T R A C T A R T I C L E I N F O Mixed-model sequencing to minimize work overload at stations is regarded as one of the most concerned optimization problems in assembly lines manufacturing a variety of product models simultaneously. A novel sequencing method based on the genetic regulatory network is proposed to solve this problem. First, genes, gene regulation equations and gene expression procedures are developed in the network based on its similarity with the mixed-model sequencing problem. Each two-state gene represents a binary decision variable of the mathematical model. The gene regulation equations describe decision variable interactions in the constraints and objectives. The gene expression procedure depends on the regulation equations to generate solutions, in which the value of each decision variable is indicated by the expression state of the related gene. Second, regulatory parameter optimization in the regulation equations minimizes the work overload at stations. The effectiveness of the proposed method is validated through experiments consisting of reference instances and industrial instances. The experimental results demonstrate that this method outperforms other methods in large-scale instances. © 2017 PEI, University of Maribor. All rights reserved. Keywords: Assembly line Mixed-model sequencing Work overload Genetic regulatory network Genetic algorithm *Corresponding author: zhangjie@sjtu.edu.cn (Zhang, J.) Article history: Received 11 October 2016 Revised 16 January 2017 Accepted 17 January 2017 1. Introduction Mixed-model assembly lines (MMALs) are widely applied to the industrial engineering world because they can assemble various models of products in a facultative sequence while reducing setup times [1]. Although it is possible to implement any model sequence, the model sequence causes different economic impacts in the actual environment [2]. For instance, different models require diverging processing times at stations to complete the specific assembly operations that realize customized function requirements [3]. The cycle time defines the standard time to process a product at a station, which is typically the average of the processing times of different models weighted by the model demands [4]. The processing times required in complex operations are thus greater than the cycle time, while those required in simple operations are less than the cycle time [5, 6]. If some complex operations are processed continuously by using a specific station, then assembly tasks may not be completed before the operators have reached the down-stream station border, which is regarded as a work overload situation. Although utility workers or line stoppages are adopted to deal with the work overload situation, these lead to additional costs [7]. The mixed-model sequencing (MMS) problem is thus addressed to minimize the work overload at stations, in which different models of products are arranged by performing alternately complex operations and simple ones at each station [8]. 62 A genetic regulatory network-based sequencing method for mixed-model assembly lines The MMS problem has attracted a lot of attention because of its complexity and practical value. Various types of MMS methods including exact solution procedures, heuristic procedures and meta-heuristics have been proposed by scholars. However, the existing methods can hardly achieve high-quality solution when the problem is from large-scale instances and requires acceptable computational time. This study aims to fill in this gap by proposing a novel method based on the genetic regulatory network (GRN). In a GRN, gene states are the same as decision variable values in the MMS problem. Gene regulations describe the interconnection between genes, which have an analogous function with constraints. The gene expression procedure governed by gene regulations determines gene states iteratively, which is similar to a heuristic sequencing procedure. Based on these similarities, genes are first defined in the GRN to represent decision variables. Second, gene regulation equations are developed to express not only hard constraints in the mathematical model, but also soft constraints derived from certain sequencing rules. The importance of soft constraints is weighted by regulatory parameters in the equations. Third, the gene expression procedure is designed to indicate a heuristic procedure that is specified by regulatory parameters. Finally, the regulatory parameters are optimized to obtain the optimal solutions with minimum work overload at stations. Thereupon, the key contribution is the extension of GRN applications to assembly line scheduling. A series of computational experiments are conducted to validate the effectiveness of this GRN-based method. The remainder of this paper is organized as follows. A relevant literature review of MMS methods and GRNs is offered in Section 2. A mathematical model of MMS problems is presented in Section 3. A GRN-based sequencing method is given in Section 4. Section 5 contains the experimental results and discussions. Conclusions and future research directions are discussed in Section 6. 2. Literature review 2.1 Mixed-model sequencing problem In terms of sequencing in MMALs, Boysen et al. [9] provided an integrated review to discuss three fundamental approaches, i.e., MMS, car sequencing and level scheduling. Of these approaches, the MMS methods aim at minimizing sequence-dependent work overload based on a detailed scheduling in MMALs, which have been widely investigated by researchers [10]. These MMS methods can be further classified into four major classes: branch-and-bound computation, exact solution procedure, heuristic procedure, and meta-heuristic [11-13]. The focus of the existing literature was on heuristic procedure and meta-heuristic because it is impractical to implement other methods for large-scale instances [14-16]. For instance, Cano et al. [17] used a scatter search method that selects from 20 priority rules to generate hyper heuristic procedures. Gujjula et al. [18] proposed a heuristic method based on Vogel's approximation method to address large-scale MMS problems. Cortez and Costa [19] developed a set of fast constructive heuristics, two local search procedures and a meta-heuristic to deal with a specific problem featured with worker-dependent processing times. Well-known meta-heuristics such as genetic algorithm and ant colony algorithm have also been employed to solve a variety of MMS problems [20-25]. In general, most heuristic methods used greedy priority rules to construct rapidly a model sequence by appending iteratively alternative models into it. These methods chose products with minimum objective function values at each iteration, which led to intensive increases of the work overload in the last part of the model sequence. Hence, these approaches based on heuristic procedures could obtain good solutions rather than optimal ones. Although other heuristic methods were considered to improve the solution quality (e.g., the work overload increases caused by remaining model copies were taken into account at each iteration), they resulted in the computational effort to be dramatically increased for large-scale problems. Alternatively, meta-heuristic methods could find optimal solutions or near-optimal solutions by globally searching among all the feasible ones. However, the global search caused the computational effort to be increased with the number of model copies as well as the number of stations. Advances in Production Engineering & Management 12(1) 2017 63 Lv, Zhang, Qin Consequently, it is difficult to generate a near-optimal model sequence with an acceptable computational effort for real large-scale instances in the actual manufacturing environment. In this paper, we propose a novel GRN-based method to solve MMS problems. The critical factors of this method include the description of the MMS problem by using a GRN and the integration of certain validated sequencing rules in the GRN. Based on such a GRN, the proposed method can solve the MMS problem more efficiently than meta-heuristics without compromising the solution quality, especially for large-scale instances. 2.2 Genetic regulatory network The GRN is a structured network that describes the regulation of gene expression in cells [26]. It has been widely applied by biologists to investigate the dynamic changes of cell morphologies, and has become a hot topic in the past few years. A GRN has at least the following three elements in common: genes, gene regulations and gene expression procedure [27]. Each gene has two alternative states (i.e., the expressed state and the unexpressed state). If a gene is in the expressed state, it has regulatory effects on the states of other genes, which is the primary form of gene regulations. Based upon these regulations, gene expression procedure iteratively converts certain genes in the unexpressed state into ones in the expressed state if there are enough positive regulatory effects on these genes. Since a cell is mainly composed of copied components (e.g., mRNAs and proteins), gene expression procedure finally determines the cell's morphology in accordance with genes in the expressed states [28]. Various formalisms have already been employed to describe GRNs, for instance, Bayesian networks, directed graphs, partial differential equations, Boolean networks, qualitative differential equations, stochastic equations, and rule-based formalisms [29]. 3. Problem description The following assumptions are taken into consideration when constructing the mathematical model: • The assembly line is a 'moving line' in which the conveyor moves at a constant speed; • The length of a station is a fixed one (measured by the product passing time), and neighboring stations do not overlap; • Products are equi-spaced on the line by launching each other after a constant time interval, which is equivalent to the cycle time; • The operation processing time at a station is not longer than the length of this station; • The impact of unfinished works on operations at succeeding stations is not taken into consideration; • Operators return with infinite velocity to the subsequent product; • The model changeover time is included in the operation processing time; • To facilitate the presentation, the notations listed in Table 1 are used in the development of the mathematical model. The mathematical model takes the following form. T K Minimize t= i fc=i T Wtk (1) (2) t=i M vt (3) s,k = 0 Vk (4) stk >0,wtk >0 Vt,k (5) 64 Advances in Production Engineering & Management 12(1) 2017 A genetic regulatory network-based sequencing method for mixed-model assembly lines M wtk >stk+^ xtmpmk -lk V^ k [6] m=l M St+l,k >stk+^ xtmpmk -wtk-c Vt, k [7] m=l Table 1 Problem's notations Notations Definitions Sets {1. Set of products in the model sequence {!.- ..,fc,... ,K} Set of stations {1... ,,m,.. ■ ,M} Set of models Parameters dm c T lk Pmk Demand for model m in the production plan Cycle time Total demand for products, T = £m=i dm Length of station k (time unit) Operation processing time of model m at station fc Variables stk wtk xt.m. Starting time for assembling the tth product at station fc Extra operation processing time for the t-th product at station fc (work overload time) Binary variable: 1, if the t-th product belongs to model m; 0, otherwise The objective of the mathematical model is to minimize the total work overload at stations. The constraints in Eq. 2 ensure that the model sequence satisfies the demand for each model. Eq. 3 makes sure that a station cannot process more than one product at the same time within each production cycle. Eq. 4 represents the initial state of stations. Eq. 5 and Eq. 6 ensure that all the operations should be processed within boundaries of their related stations. If operations on a product cannot be finished within boundaries of their related stations, then a work overload occurs. Eq. 7 makes sure that operators at station k can start processing product (t + 1) after they have completed the operations on product t or product t has left station k. 4. Genetic regulatory network-based sequencing method 4.1 Mapping between the mathematical model and the genetic regulatory network In the mathematical model presented in Section 3, the binary decision variables, the constraints and the solution are similar to the genes, the gene regulations and the gene expression procedure in a GRN, respectively. Based on these similarities, the model can be represented by using a GRN (shown in Fig. 1): (1) each gene represents a decision variable; (2) gene regulations describe constraints; (3) gene expression procedure generates solutions. MMS Problem GRN gene gene expression procedure (gene state: unexpressed^expressed) Fig. 1 Mapping between the MMS problem and the GRN Advances in Production Engineering & Management 12(1) 2017 65 Lv, Zhang, Qin In general, the differential equation method is the most suitable method to develop gene regulation equations because it can represent gene regulations in a quantitative form. In the GRN, first, regulation equations are used to describe all the constraints, and the gene expression procedure is developed based upon the regulation equations to obtain feasible solutions. Second, some soft constraints related to sequencing rules are integrated in the regulation equations to decrease work overload at stations. Third, regulatory parameters are optimized to integrate reasonable sequencing rules and thus minimize work overload at stations. Consequently, as shown in Fig. 2, the GRN-based method contains two parts: • A GRN is developed based on the mathematical model and certain sequencing rules; • Regulatory parameters are optimized by using a genetic algorithm in order to minimize the work overload. Step 2 Regulatory Parameter Optimization optimization algorithm regulatory parameters gene expression procedure " * objective function evaluation J Fig. 2 Outline of the GRN-based method 4.2 Genetic regulatory network establishment According to decision variables of the mathematical model, genes {9tm | t = 1,2, ...,T,m = 1,2,... ,M} are generated. Each gene 9tm indicates that a product of model m is assigned to the tth position of the model sequence. Moreover, the regulation equation in Eq. 8 is developed to express constraints. Vi = fi(Xi(t),X2(t),'~, xI(t),£1,£2,-, eE) (8) where I represents the number of genes in a GRN, xt(t) Vi E {1,2, •••,/} is a binary variable that is equal to 1 if gene is in the expressed state at time t, otherwise, it is equal to 0, is a nonlinear function, vt represents the inhibition coefficient to convert gene 9i to the expressed state, £1,£2,-',£e are regulatory parameters. In terms of Eq. 2 and Eq. 3, the regulation equation first describes following constraints of each position in the model sequence: (1) A model can be selected when other models have not been selected yet. (2) A model can be selected when the demand for this model has not been satisfied at former positions. Moreover, soft constraints related to the study of Cano et al. [17] and the study of Dormer et al. [30] are also included in the regulation equation: (3) A model can be selected if it causes the least work overload at stations. (4) A model can be selected if it leads to the least idle time at stations. (5) A model can be selected if its production ratio best matches its demand ratio in the production plan. No sequencing procedure could satisfy all the soft constraints completely, and each unsatisfied case might increase the work overload at stations. The regulation equation of gene 9tm is thereby developed as follows: K K Vtrn = ¿1 + Pmk ~lk) + Stk ~Vmk) k=l k=l 66 Advances in Production Engineering & Management 12(1) 2017 A genetic regulatory network-based sequencing method for mixed-model assembly lines where vtm represents the inhibition coefficient to gene dtm, H(x) is a step function satisfying H(x) = 0 (x< 0) and H(x) = +œ (x> 0), 0 (x) is a piecewise function satisfying 0 (x) = 0 (x < 0) and 0 (x) = x (x> 0), £x, £2, £3, £4 represent regulatory parameters that combine regulation segments derived from different constraints. The first three terms of the right side of Eq. 9 indicate the inhibition to gene dtm owing to soft constraints (3) to (5), respectively. The last two terms of the right side of Eq. 9 describe constraints (1) and (2), respectively. Constraints in Eq. 4 to Eq. 7 are embodied in the calculation of stk: M 0 if Stk + ^ xtmPmk ^C m=l M M stk + ^ xtmPmk ~c if cT, the model sequence is obtained based on the gene states {xtm |t = 1,2, ...,T, m = 1,2.....M}. Table 2 Pseudo codes of gene expression procedure //initialization for t ^ 1 to T do for m ^ 1 to M do Xtm ^ 0 //all the genes are initialized in the unexpressed state next; next; //gene expression circulation for k ^ 1 to K do slk^0 // initialization of stations next; for t ^ 1 to T do //discrete time m0 ^1, v0 ^ +ot //index of the gene with minimum vtm for m ^ 1 to M do calculate vtm in Eq. 9 //calculate inhibition coefficients if vtm Evaluation: lígnt=m, thenxtm=l, elsexim=0 for t=l,2,—,T —► Eq. 1 —► fitness,, PMuation —► g'«P=g«q g'nq=gnp p,q-rand[l,7] i>=rand[l, M] If = b then g'„t=g„t for i= 1,2,—,7 Crpsspyer: PCrossover —► c=rand[l, A/] t'=l If gnt^ftand^rt ^fithen g'„t=gc(t'++) for t=l,2,---,T Fig. 4 Basic steps to solve the MMS problem by using an ICGA Table 5 Processing times in industrial instances m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 lk k=1 104 100 97 92 100 94 103 100 101 195 k=2 103 103 105 107 101 108 106 102 110 195 k=3 165 156 164 161 148 156 154 164 155 195 k=4 166 175 172 167 168 167 168 156 173 195 k=5 111 114 114 115 117 117 115 111 111 195 k=6 126 121 122 124 127 130 120 121 134 195 k=7 97 96 96 93 96 89 94 101 92 195 k=8 100 97 95 106 94 102 103 102 100 195 k=9 179 174 173 178 178 171 177 171 174 195 k=10 178 172 172 177 178 177 175 173 175 195 k=11 161 152 168 167 167 166 172 157 177 195 k=12 96 106 105 97 101 100 96 104 96 195 k=13 99 101 102 101 99 101 96 102 99 195 k=14 147 155 142 154 146 143 154 153 155 195 k=15 163 152 156 152 153 152 154 156 156 195 k=16 163 185 183 178 169 173 172 182 171 195 k=17 173 179 178 169 173 178 174 175 175 195 k=18 176 167 181 180 172 173 173 168 184 195 k=19 162 150 152 152 160 151 155 148 167 195 k=20 164 161 157 159 162 160 162 158 157 195 k=21 177 161 154 168 172 170 167 149 169 195 Advances in Production Engineering & Management 12(1) 2017 69 Lv, Zhang, Qin Table 6 Production plans in industrial instances Block di d2 d3 d4 ds d6 dy de d9 Pi 30 30 30 30 30 30 30 30 30 P2 30 30 30 45 45 23 23 22 22 P3 10 10 10 60 60 30 30 30 30 P4 40 40 40 15 15 30 30 30 30 P5 40 40 40 60 60 8 8 7 7 P6 50 50 50 30 30 15 15 15 15 P7 20 20 20 75 75 15 15 15 15 P8 20 20 20 30 30 38 38 37 37 P9 70 70 70 15 15 8 8 7 7 Pi0 10 10 10 105 105 8 8 7 7 P11 10 10 10 15 15 53 53 52 52 6 P12 24 23 23 45 45 28 28 27 27 P13 37 37 36 35 35 23 23 22 22 P14 37 37 36 45 45 18 18 17 17 Pi5 24 23 23 55 55 23 23 22 22 P16 30 30 30 35 35 28 28 27 27 P17 30 30 30 55 55 18 18 17 17 P18 60 60 60 30 30 8 8 7 7 P19 10 10 10 90 90 15 15 15 15 P20 20 20 20 15 15 45 45 45 45 P21 60 60 60 15 15 15 15 15 15 P22 20 20 20 90 90 8 8 7 7 P23 10 10 10 30 30 45 45 45 45 P24 60 60 60 60 60 60 60 60 60 P25 60 60 60 90 90 45 45 45 45 P26 20 20 20 120 120 60 60 60 60 P27 80 80 80 30 30 60 60 60 60 P28 80 80 80 120 120 15 15 15 15 P29 100 100 100 60 60 30 30 30 30 P30 40 40 40 150 150 30 30 30 30 P31 40 40 40 60 60 75 75 75 75 P32 140 140 140 30 30 15 15 15 15 P33 20 20 20 210 210 15 15 15 15 P34 20 20 20 30 30 105 105 105 105 7 P35 47 47 40 90 90 55 55 55 55 P36 74 73 73 70 70 45 45 45 45 P37 74 73 73 90 90 35 35 35 35 P38 47 47 40 110 110 45 45 45 45 P39 60 60 60 70 70 55 55 55 55 P40 60 60 60 110 110 35 35 35 35 P41 120 120 120 60 60 15 15 15 15 P42 20 20 20 180 180 30 30 30 30 P43 40 40 40 30 30 90 90 90 90 P44 120 120 120 30 30 30 30 30 30 P45 40 40 40 180 180 15 15 15 15 P46 20 20 20 60 60 90 90 90 90 An Intel(R) Core(TM) i7-2720QM CPU @ 2.20 GHz and 8 GB RAM based notebook computer is used to conduct the computational experiments. Table 7 presents genetic algorithm parameters used in the GRN-based method and the ICGA method. Table 8 lists the average objective function values (Obj) obtained and the average CPU times (Tcpu) spent by the GRN-based method, the ICGA, the HH method and the Cplex solver in each block. In this table, the number of feasible solutions for each instance is evaluated based on Eq. 11 [33]. Nf = (YM dm)!/UM (dm!) (11) dm is the demand for model m in the production plan. The average number of feasible solutions in each block (Nfb) is calculated and also presented in this table as the indicator of problem scales. 70 Advances in Production Engineering & Management 12(1) 2017 A genetic regulatory network-based sequencing method for mixed-model assembly lines Table 7 Genetic algorithm parameters in different methods Method_Population size_Maximum generation_PMutation_PCrossover RCGA 50 30 0.1 0.8 ICGA 200 50 0.1 0.8 Table 8 Experimental results of different blocks Block GRN-based ICGA HH method CPLEX solver Nfb Obj Tcpu, s Obj Tcpu,s Obj Tcpu,s Obj Tcpu, s 1 3.4x103 247.7 0.4 245.2 0.6 248.6 0.5 245.2 4.2 2 8.2x105 136.4 0.5 136.7 0.7 137.1 0.6 135.0 39.3 3 3.8x107 64.5 0.6 68.3 0.9 64.6 0.8 64.3 281.3 4 1.2x107 100.4 0.6 96.5 1.2 100.7 1.1 96.2 190.8 5 5.8x106 114.6 0.6 115.2 0.9 114.7 1.5 113.2 112.2 6 5.8x10247 403.3 107.5 497.2 297.2 419.4 138.1 238410.5 7200 7 7.8x10505 856.0 201.1 924.6 578.4 875.6 216.5 245774.1 7200 As shown in Table 8, based on 'N/b column, two scenarios are considered. Block 1, 2, 3, 4 and 5 are composed of small-scale reference instances and Block 6 and 7 are composed of large-scale industrial instances. In the small-scale reference instances, the Cplex solver obtains the best results, and the other methods obtain results close to the best ones. Specifically, the results in Block 1 reveal that the GRN-based method and the HH method cannot generate the optimal solutions in some instances owing to the predetermined sequencing rules, while the ICGA can obtain the optimal ones through global searching procedure when there are a few feasible solutions. However, the results in Block 2, 3, 4 and 5 reveal that ICGA fails to obtain the optimal solutions for some instances when the number of feasible ones is increased, while the GRN-based method and the HH method generate better solutions than the ICGA. This is because the sequencing rules integrated in the GRN enable the RCGA to search among good solutions rather than all the feasible ones in the regulatory parameter optimization procedure. Similarly, the HH method uses the scatter search to select from different combinations of sequencing rules and thus searches among good solutions too. However, the GRN-based method achieves better results because its weighted integration of commonly-used sequencing rules enables better searching capacity than the random combination of 20 priority rules in the HH method. For the large-scale instances, the Cplex solver fails to obtain good results in the limited CPU time, while the other methods achieve better ones in a reasonable time. The results in Block 6 and 7 reveal that the ICGA can hardly find even near-optimal solutions in an enlarged solution space, while the GRN-based method and the HH method are better than the ICGA. In addition, the results also demonstrate the GRN-based method saves the CPU time. In comparison with the ICGA method, the GRN-based method optimizes four regulatory parameters rather than the whole model sequence to decrease computational effort. This regulatory parameter optimization also demonstrates better efficiency than the scatter search on 20 priority rules in the HH method. Fig. 5 illustrates how the computational time of different methods changes with the increase of problem sizes. The CPLEX solver finds out the optimal solution by using a traversal procedure, for which the computational time increases significantly with a larger problem size. The GRN-based method, the HH method and the ICGA are based on random searching procedures that demonstrate lower increasing rates than the CPLEX solver. Moreover, the ICGA searches among all feasible solutions, whereas the GRN-based method and the HH method search among good solutions owing to the predetermined sequencing rules. For this reason, these two methods save the computational time than the ICGA. Consequently, it can be noted that the GRN-based method provides an effective means to solve the MMS problem, especially for large-scale instances. In addition, this method is also potentially applied for MMS problems in the dynamic environment by using the predictive-reactive strategy. In this strategy, the GRN-based method first provides a production plan with the minimum work overload before line production, and gives the reactive schedule within a rolling window (containing 10~20 products) once the predetermined plan is interrupted by dynamic Advances in Production Engineering & Management 12(1) 2017 71 Lv, Zhang, Qin events such as machine failures or processing time variations. Because the small-sized MMS problems are regularly solved within 1 s by using the GRN-based method, this reactive schedule realizes real-time responses for the dynamic events. In this way, by using a predictive schedule to ensure the overall performance and employing reactive schedules to make quick responses, the GRN-based method will realize efficient production in the dynamic environment. 1000 900 800 700 s 600 & 500 400 300 200 100 0 6. Conclusion This paper deals with the MMS problem in assembly lines to minimize work overload at stations. In terms of similarities between MMS and GRN, a novel MMS method based on the GRN is proposed. This method is applied to reference instances as well as industrial instances to validate its effectiveness. A Cplex solver, an ICGA and a HH method are used to benchmark the results. It is demonstrated that the GRN-based method realizes higher solution quality than other methods by integrating the sequencing rules reasonably, especially for large-sized problems. However, due to the regulatory parameter optimization that uses GA, rather than some well-designed mechanisms, the efficiency of this method requires further improvements. Thereupon, we will investigate a new parameter optimization mechanism in our future work. In addition, some other optimization problems also have the validated rules to determine its binary decision variables. The proposed method is thus potentially used for the problems in other areas, including the production scheduling problem in other manufacturing systems, the transportation scheduling problem in logistics industry and the medical device scheduling problem in healthcare industry. In our future work, we will develop new scheduling methods by extending the GRN-based approach to these areas. Acknowledgement This work was supported by the [National Natural Science Foundation of China] under Grant [number 51435009 & number U1537110]. References [1] Moradi, H., Zandieh, M., Mahdavi, I. (2011). 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