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Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem with interval task times

Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem with... Adv. Manuf. https://doi.org/10.1007/s40436-019-00256-3 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem with interval task times 1,2 1 1 • • Jia-Hua Zhang Ai-Ping Li Xue-Mei Liu Received: 11 August 2018 / Revised: 14 February 2019 / Accepted: 19 April 2019 The Author(s) 2019 Abstract The type-II mixed-model assembly line balanc- p Number of models to be assembled on the line ing problem with uncertain task times is a critical problem. C Cycle time This paper addresses this issue of practical significance to w Weight of model j in total production, j ¼ 1; 2; ; p production efficiency. Herein, a robust optimization model t Nominal task time of task i for model j ij for this problem is formulated to hedge against uncertainty. ^ Deviation task time from t ij ij Moreover, the counterpart of the robust optimization model I Set of immediate predecessors of task t, is developed by duality. A hybrid genetic algorithm (HGA) t ¼ 1; 2; ; n is proposed to solve this problem. In this algorithm, a c Budget for uncertainty, which means the number of heuristic method is utilized to seed the initial population. In uncertain tasks considered in a workstation, addition, an adaptive local search procedure and a discrete 0 B c B n Levy flight are hybridized with the genetic algorithm (GA) u Continuous variable ik to enhance the performance of the algorithm. The effec- x x = 1 if task i is assigned to workstation k; ik ik tiveness of the HGA is tested on a set of benchmark otherwise, x =0; i ¼ 1; 2; ; n; k ¼ 1; 2; ; m ik instances. Furthermore, the effect of uncertainty parame- ters on production efficiency is also investigated. 1 Introduction Keywords Mixed-model assembly line  Assembly line balancing  Robust optimization  Genetic algorithm (GA) Assembly lines are production systems containing serially Uncertainty located workstations. Assembly tasks are completed in these serial workstations. The first real example of an List of symbols assembly line was developed by Henry Ford in 1913. The i, t Index of assembly tasks, i; t ¼ 1; 2; ; n production rate saw an eightfold increase with the intro- j Index of models, j ¼ 1; 2; ; p duction of the assembly line production. Since then, k Index of workstations, k ¼ 1; 2; ; m assembly lines have been widely used around the world. n Number of tasks The assembly line balancing problem (ALBP) is one of the m Number of predefined workstations important problems in the design of an assembly line. ALBP is to assign assembly tasks among the workstations to optimize production objectives. This assignment must & Jia-Hua Zhang take the precedence relationship constraint and other 1510278@tongji.edu.cn technical constraints into consideration. ALBP has been an 1 active area of research since the first mathematical model School of Mechanical Engineering, Tongji University, of ALBP was presented by Salveson [1]. The detailed Shanghai 201804, People’s Republic of China 2 reviews of research on ALBP can be found in Refs. [2–4]. Department of Mechatronics Engineering, Wuxi Vocational Institute of Arts and Technology, Yixing 214206, Jiangsu, People’s Republic of China 123 J.-H. Zhang et al. A mixed-model assembly line is capable of assembling decomposition algorithm. Pereira and Alvarez-Miranda [17] similar models from a basic product through a single investigated RSALBP-I and developed a heuristic method assembly line. With increasing competition and diversity in and an exact algorithm. customer demands, mixed-model assembly lines have To the best of our knowledge, only Al-e-hashem et al. become popular in many industries, such as cars, TVs, and [11] have used the robust optimization method to study the computers. The mixed-model assembly line balancing mixed-model assembly line balancing problem with problem (MMALBP) is classified into two types: uncertain task times, in addition to formulating a mathe- MMALBP-I and MMALBP-II [5]. The former is aimed at matical model of RMMALBP-I. However, there is no minimizing the number of workstations for a given cycle related work on RMMALBP-II, which is directly related to time, whereas the latter is aimed at minimizing the cycle the production rate of the mixed-model assembly line. The time under a given number of workstations. MMALBP-II with interval task times is studied in this In many studies of the MMALBP, task times are paper using the robust optimization method, which is called assumed to be deterministic. However, in real life, robust MMALBP-II (RMMALBP-II). In this problem, each assembly tasks are subject to various uncertainties, such as task time is represented by an interval dataset. To solve this the skill level of the operator, task complexity, and problem, the robust optimization model is formulated and a resource availability. Both stochastic mixed-model hybrid genetic algorithm (HGA) is developed. The robust assembly line balancing [6, 7] and fuzzy mixed-model optimization model for this problem is nonlinear. To assembly line balancing [8] have been proposed to deal facilitate the use of an exact algorithm, the counterpart of with uncertain task times. In stochastic mixed-model the nonlinear robust optimization model is obtained by assembly line balancing, it is assumed that task times are duality. subject to probability distribution, generally normal dis- The remainder of this paper is organized as follows: the tribution. In fuzzy mixed-model assembly line balancing, RMMALBP-II is described and the mathematical models task times are assumed to be fuzzy numbers with given are formulated in Sect. 2. The HGA is developed in membership. However, both stochastic and fuzzy task Sect. 3. In Sect. 4, computational experiments are imple- times are often impossible in practice because of insuffi- mented, along with the illustration of the results. Finally, cient preliminary information to deduct the required conclusions are drawn in Sect. 5. probability or possibility distribution functions. Soyster [9] first developed the robust optimization approach in which the probability distribution of uncertain 2 Problem description and model formulation parameters is unknown. However, with the worst-case sce- narios that may never happen in real life, this method is very 2.1 Problem description conservative. To control the degree of conservatism, Ref. [10] developed a robust optimization, in which only a subset A mixed-model assembly line with m workstations is of uncertain coefficients (only c of them) is in the worst considered. p similar models are assembled simultaneously scenarios. This approach can be extended to discrete opti- in an intermixed sequence. The ratio of the unit number of mization problems and has been applied to a variety of each model j to the overall demand is w . The precedence problems. Many researchers have adopted the robust opti- relationship of each model is predefined and all relation- mization approach to study the assembly line balancing ships can be combined into only one precedence graph with problem with uncertain task times: Al-e-hashem et al. [11] n tasks. Each task has an uncertain task time. The uncer- considered the type-I robust mixed-model assembly line tainty information about the probability distribution or balancing problem (RMMALBP-I). Hazır and Dolgui [12] fuzzy membership function is not easy to know, and each studied the type-II robust simple assembly line balancing uncertain task time can only be represented by an interval problem (RSALBP-II) and developed a Benders decompo-  ^ t  t . t is the nominal task time for task i and model j, ij ij ij sition algorithm. Gurevsky et al. [13] investigated the type-I and t corresponds to the deviation task time from t . If the ij ij robust simple assembly line balancing problem (RSALBP-I) interval t  t equals 0, it means that model j does not ij ij and designed a branch and bound algorithm. Nazarian and need task i to be assembled. The aim is to minimize cycle Ko [14] considered the uncertain task and inter-task times in time C. RSALBP-II, especially focusing on non-productive times in The RMMALBP-II has the following assumptions: workstations. Moreira et al. [15] studied RSALBP-I with (i) The task time of each model is uncertain and is heterogeneous workers under uncertain task times and represented by a given interval dataset. developed two mathematical models and a heuristic method. (ii) Each common task for different models must be Hazır and Dolgui [16] studied the robust U-shaped assembly assigned to the same workstation for economy. line balancing problem, which was solved by a Benders 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… (iii) The precedence graphs for different models are to t þ t . Constraint (6) is the nondivisibility constraint, ij ij predefined, and a combined precedence graph can which means that a task cannot be split among two or more be obtained. workstations. Constraint (7) makes sure that C has an (iv) The line is a paced line with a fixed cycle time. integer value, which is always the case in a real-world (v) The line is serial with no feeder lines or parallel environment [5]. workstations. (vi) There are no assignment restrictions of tasks 2.3 Counterpart of the nonlinear robust except precedence constraints. optimization model (vii) There are no buffers between workstations. The robust optimization model proposed in Sect. 2.2 is a nonlinear model. It can be linearized by duality [10] and 2.2 Model formulation solved by an integer solver. The counterpart of the non- linear robust optimization model is developed as follows: The RMMALBP-II can be formulated as a nonlinear robust min C ð8Þ model: min C ð1Þ subject to n p n XX X subject to w t x þ p þ cz  C; 8k; ð9Þ j ij ik ik k i¼1 j¼1 i¼1 x ¼ 1; 8i; ð2Þ ik k¼1 z þ p  w t y ; 8i; k; ð10Þ k ik j ij ik m m X X j¼1 kx  kx ; 8i 2 I ; ð3Þ ik tk t k¼1 k¼1 p  0; 8i; k; ð11Þ ik () p p n n n XX XX X x  y ; 8i; k; ð12Þ ik ik w t x þ max w t x u : u  c j ij ik j ij ik ik ik i¼1 j¼1 i¼1 j¼1 i¼1 z  0; 8k; ð13Þ C; 8k; and constraints (2), (3), (6), and (7). y , p , and z are the ik ik k ð4Þ variables used in duality. 0  u  1; 8i; k; ð5Þ ik x 2f0; 1g; 8i; k; ð6Þ ik 3 Proposed hybrid genetic algorithm C [ 0; C is integer: ð7Þ for RMALBP-II Objective function (1) minimizes cycle time C. Con- Type-II simple assembly line balancing problem (SALBP- straint (2) is known as the occurrence constraint that each II) is known to be nondeterministic polynomial (NP)-hard task can be assigned to a workstation. Constraint (3) is the [2]. It can be found that SALBP-II is a special case of precedence constraint to guarantee the technological RMALBP-II. Similar to SALBP-II, RMLABP-II is NP- sequencing requirements. Constraint (4) ensures that the hard. Owing to its NP-hard nature, RMALBP-II can be weighted uncertain workstation times do not exceed the time-consuming to obtain an optimal solution using exact cycle time C. The left-hand side of constraint (4) consists algorithms. In this section, an HGA is developed to solve of two parts: w t is the weighted nominal task time j ij j¼1 this problem. of each task i which can be assigned into a workstation and The genetic algorithm (GA), a popular meta-heuristic w t is the weighted deviation task time for each task j ij j¼1 algorithm, has been used to solve various assembly line i. The number of uncertain tasks considered in a worksta- balancing problems such as stochastic assembly line bal- tion is bounded by parameter c. The larger c is, the more ancing problem, fuzzy assembly line balancing problem, the deviation task times should be considered. Constraint and so on [18]. Research shows that GA is competitive (5) determines the bound of variable u , which indicates ik against the best-known constructive methods. There is no the level of an uncertain task time deviating from the published research reporting the application of the GA to nominal time. When u ¼ 0, it means that the task time for ik solve the robust assembly line balancing problem. In this task i in workstation k does not deviate from the nominal section, an HGA is proposed to solve RMMALBP-II. The time and is deterministic. When u ¼ 1, task i in work- ik flowchart of this algorithm is shown in Fig. 1. station k is under the worst case, and the task time is equal 123 J.-H. Zhang et al. This algorithm starts with the generation of an initial In this case, it is proposed to seed part of the initial pop- feasible population, which is followed by the evaluation of ulation with heuristic solutions to improve the performance each chromosome through fitness evaluation. Special of the algorithm. Based on the experimental results, 20% of genetic operators for the assembly line balancing problem the initial population can be seeded by the proposed (crossover and mutation) are performed. An adaptive local heuristic and 80% can be generated randomly. This ini- search procedure and a discrete Levy flight are used to tialization method can obtain better solutions than that of a improve the quality of solutions. This loop keeps running randomly generated population, as shown in Sect. 4.1. until the maximum iteration number is reached. Finally, the Theheuristic method is basedonPereira andAlvarez- best cycle time of RMMALBP-II is obtained. Miranda [17] and Sewell and Jacobson [22], and builds solutions from the forward or backward direction of G.Task j in 3.1 Encoding scheme and initial population  ^ G, which maximizes the priority rule ðw t þ aw t þ i i i i i2U bFjjcÞ, can be selected. U is the set of tasks that have not yet A combined precedence graph, G, is formed by combining been selected; w t is the weighted nominal task time; w t is the i i i i the precedence graphs of each model. Based on the com- weighted deviation time; jj F is the number of immediate bined precedence graph, common tasks of each model can successors of task i in the forward directionsearchorthatof be assigned to the same workstation. Each chromosome, immediate predecessors of task i in the backward direction according to their precedence constraint, is designed as a search.a, b,and c are input parameters. The recommended sequence of tasks. The number of genes in the chromosome values for a, b,and c in Ref. [22] are employed. Every random is equal to that of tasks n in G, and each gene is an integer combination of a [ {0,0.005,0.010,0.015,0.020}, b [ {0, representing a task. 0.005,0.010,0.015,0.020}, and c [ {0,0.01,0.02,0.03} is used Population initialization is a crucial step in evolutionary to select a task. The random method selects a task at random algorithms (EAs). Many researchers have proposed to seed and assures the feasibility of the precedence relationship. The EAs with good initial solutions, whenever it is possible, to precedence matrix is used to describe the precedence rela- obtain important improvement in the convergence of the tionship of tasks [23]. algorithm and the quality of the solutions [19]. However, The procedure of the initial feasible population gener- the excessive use of good solutions in the initial population ation is described as follows. can decrease the exploration capacity of the GA, thereby Step 1 Build an empty vector A; read the precedence trapping the population in local optimums quickly [20, 21]. matrix P of the combined precedence graph G. Step 2 Choose the random creation method or the heuristic method. (i) If the random creation method is chosen, go to Step 3. (ii) If the heuristic method is chosen, decide the search direction randomly. If it is a backward direction search, P = P and go to Step 3; otherwise, go to Step 3. Step 3 If j B n, store the tasks where the sum of the column of P is equal to 0 into A;if j [ n, go to Step 7. Step 4 Choose a task from A for a gene in a chromosome. (i) If it is the random creation method, choose a task from A at random for the jth gene in a chromo- some. Empty A, and go to Step 5. (ii) If it is the heuristic method, generate a random combination of input parameters a, b,and c,and choose the task that maximizes the priority rule from A. The task is assigned for the jth gene in the forward direction search or for the (n - j ? 1)th in the backward direction search. Empty A, and go to Step 5. Step 5 Update the ith row of P by putting a big number D (e.g., 999) into tasks i and 0 into other tasks. Fig. 1 Flowchart of the proposed HGA 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Step 6 Update j = j ? 1, and go to Step 3. Table 1 Precedence matrix P of G Step 7 End the procedure. Task 1 2 3 45678 91011 Repeat the above procedure and get the initial popula- 1 011 11000 00 0 tion. Figure 2 is the combined precedence graph G of an 2 000 00100 00 0 illustrative example. Table 1 is the precedence matrix P of 3 000 00010 00 0 graph G. If the heuristic method from backward search is 4 000 00010 00 0 chosen, make P = P . Table 2 shows the result of Steps 2 5 000 00010 00 0 and 3 of the heuristic method from the backward search 6 000 00001 00 0 direction. After these two steps, the element of vector A is 7 000 00000 10 0 task 11 when j = 1. Because there is only one element in 8 000 00000 01 0 A to calculate the priority rule, the (n - j ? 1)th gene (it is 9 000 00000 00 1 the nth gene now) in the chromosome is task 11 in Step 4. 10 000 00000 00 1 Table 3 is the result of Step 5. Repeat this procedure 11 000 00000 00 0 until the last task is selected into a chromosome. Table 4 is the final result of the sum of columns. Figure 3 shows some chromosomes in the initial population produced by the two methods. Table 2 Searching for the tasks to formulate a feasible task sequence from the backward direction 3.2 Evaluation procedure Task Step 2: Transposed matrix of P The evaluation procedure aims to find objective values based on the task sequence. With each chromosome being 1 0000000000 0 a feasible task sequence, the procedure decides which tasks 2 1000000000 0 can be assigned to predefined workstations respecting the 3 1000000000 0 cycle time constraint. The evaluation procedure is descri- 4 1000000000 0 bed as follows. 5 1000000000 0 Step 1 Read the nominal time matrix T and the deviation 6 0100000000 0 task time matrix V. 7 0011100000 0 Step 2 Calculate the theoretical minimum cycle time for 8 0000010000 0 the initial trial cycle time C. This initial cycle time value 9 0000001000 0 is the lower boundary (LB), and the calculation equation 10 0000000100 0 is presented as 11 0000000011 0 ceil max maxðÞ t ; t m ; c ¼ 0; Step 3: Sum of columns 4111111111 0 i i i¼1 LB ¼ ^ ^ : ceil max maxðÞ t þt ; t þ maxðÞ t =m ; c  1; i i i i i¼1 ð14Þ where t ¼ w t represents the weighted nominal i j ij j¼1 ^ ^ task time for task i and t ¼ w t represents the i j ij j¼1 weighted deviation task time for task i. If c = 0, it means that no task deviation time is con- sidered. The LB is equal to the LB of the deterministic MMALBP-II. For the cycle time set as an integer in RMMALBP-II, the LB is taken as an integer. If c C 1, it means that at least one task deviation time is considered. The indivisibility of tasks requires that C ðÞ t þ t . With i i at least one task deviation time being considered, mC  t þ maxðÞ t is obtained. i i i¼1 Step 3 Assign the tasks into predetermined workstations. Tasks are assigned to the first station in the order of the Fig. 2 Combined precedence graph G of the illustrative example 123 J.-H. Zhang et al. Table 3 Updating P after a task selected Rank the deviation time of tasks in this workstation in the descending order. The worst scenario of the worksta- Task Step 5: Matrix P tion time is equal to the sum of the nominal time assigned 123 45678 91011 to this workstation and the largest c deviation time among these tasks. Once the workstation time exceeds the cycle 1 000 00000 00 0 time, the next station is opened for assignment. If the 2 100 00000 00 0 number of workstations is equal to the given m, stop the 3 100 00000 00 0 assignment and calculate the worst scenario workstation 4 100 00000 00 0 time of each station. The maximum workstation time is 5 100 00000 00 0 defined as the bottleneck workstation time C . 6 010 00000 00 0 7 001 11000 00 0 Step 4 If C  C, go to Step 5; otherwise, update C ¼ 8 000 00100 00 0 C þ 1 and go to Step 3. 9 000 00010 00 0 Step 5 The cycle time is equal to C; end this procedure. 10 000 00001 00 0 Table 5 is an illustration of the decoding/evaluating 11 000 00000 00 D procedure. The predetermined workstation number is m =4. 3.3 Tournament selection Table 4 Final content of P after forming a feasible task sequence Task Matrix P The tournament selection strategy [24] is used to select the parent chromosomes. Some chromosomes are randomly 12 345678 910 11 chosen from the current population along with the one with 1 D 0000 00000 0 the best objective value being selected for reproduction. 20 D 000 00000 0 The tournament selection strategy works as follows. 300 D 00 00000 0 Step 1 k chromosomes in the population are selected at 4 000 D 0 00000 0 random. 5 0000 D 00000 0 Step 2 The chromosome with the best objective value 6 00000 D 0000 0 (minimum cycle time) from these selected chromosomes 7 00000 0 D 000 0 is chosen as the best one and added to the mating pool. 8 00000 00 D 00 0 Step 3 This procedure is repeated until the number of 9 00000 000 D 00 individuals in the mating pool reaches the required 10 00000 0000 D 0 population size and the population is updated through 11 00000 00000 D this procedure. Sum of columns DDDDD DDDDD D 3.4 Fragment reordering crossover Because it is designed particularly for the assembly line balancing problem [25], the fragment reordering crossover can preserve the feasibility of the offspring structure with its procedure working as follows. Fig. 3 Some chromosomes in an initial population Step 1 Two parent individuals are selected from the gene sequence complying with the precedence relation- population in order. ship. The cycle time constraint as shown in inequality Step 2 Two parents selected are divided by two (4) cannot be violated. The cycle time constraint can be randomly cut points into three sections: head, middle, described as and tail. n n n X X X Step 3 The head and tail parts of the first offspring are t x þ max t x u : u  c  C: ð15Þ i ik i ik ik ik taken from the first parent and the middle part of the first i¼1 i¼1 i¼1 offspring is filled by adding missing tasks according to the order in which they are contained in the second parent. 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 5 Evaluation procedure for the illustrative example with c =1 Chromosome 1 5 2 6 4 3 7 9 8 10 11 Step 1 t 61 2 2 7 5 356 54 ^ 0.6 0.1 0.2 0.2 0.7 0.5 0.3 0.5 0.6 0.5 0.4 Step 2 LB LB = ceil(max (7.7, 4.8848)) = 8 Step 3 Workstation (WS) WS1(1 5) WS2(2 6) WS3(4) WS4(3 7 9 8 10 11) C WS4 is bottleneck; C =28 ? 0.6 = 28.6 [ C Step 4 Update C C ¼ C þ 1 Repeat Step 5 C The cycle time of this chromosome is C =14 End the procedure Step 4 Like the building process for the first offspring, An illustration of a scramble mutation is given in Fig. 5. the head and tail of the second offspring are formed from The mutation point is 3 in the chromosome of this example. the same part of the second parent, and the middle is filled by the missing tasks according to the order in 3.6 Adaptive local search which they are contained in the first parent. Step 5 Get a new population with the offspring The pure genetic algorithm is good at global search but chromosomes. slow to converge [27]. Local search is a promising approach to improve the quality of the objective value and The fragment reordering crossover is demonstrated in convergence speed [28]. Within the proposed algorithm, a Fig. 4. local search procedure is applied to every chromosome of the population. The local search tries to transfer tasks in the bottleneck workstation to other workstations to reduce the 3.5 Scramble mutation cycle time. To tackle the increased computation time of the local search procedure, an adaptive local search scheme is Scramble mutation for the assembly line balancing prob- adopted. The basic concept of applying the local search to lem was developed by Leu et al. [26]. With scramble the population is to consider whether GA has converged to mutation, the chromosome, reconstructed drastically, still the global optimal solution or not [29]. The converging remains feasible. It works as follows. criterion for the line balancing problem is the ratio of the Step 1 The mutation point is generated randomly, and average fitness of the chromosomes to the fitness of the one chromosome from the population is divided into best chromosome less than 1.01 [30]. head and tail. The fitness value ratio (FVR), R , of the average fitness fv Step 2 The head of the chromosome is kept, and the tail of the chromosomes to the fitness of the best chromosome of the chromosome is regenerated respecting the prece- at each generation is defined as dence relationship. The precedence matrix needs to be R ¼ ; ð16Þ fv proceeded to eliminate the task already in the head of the chromosome. where a is the average fitness of the chromosomes and b f f Step 3 Get a new population with the mutation is the fitness of the best chromosome. procedure. If R [ 1.01, apply the local search; otherwise, only fv GA is implemented. The local search procedure is explained as follows. Fig. 4 Fragment reordering crossover Fig. 5 Scramble mutation 123 J.-H. Zhang et al. Step 1 Identify the workstation with the largest work- k¼ ; ð18Þ 1=b station times as the bottleneck workstation. jj v Step 2 Let n be the number of tasks in the bottleneck where u and v are drawn from normal distributions. That is, workstation. If i B n , find the ith task in the bottleneck workstation. According to the precedence relationship, u  Nð0; r Þ; ð19Þ find the earliest workstation E(i) and the latest worksta- v  Nð0; r Þ; tion L(i) to which task i can be transferred. If i [ n ,go 0 1 > b > pb to Step 6. CðÞ 1 þ b sin < B C Step 3 Rank workstations between E(i) and L(i) accord- 2 B C r ¼ ; @ A ð20Þ b1 ing to workstation times in the ascending order. ðÞ 1 þ b > C b Step 4 k is the workstation between E(i) and L(i). r ¼ 1; Transfer task i to the workstations arranged in order of Step 3. Task i can be transferred to a workstation with no where distribution parameter b [ [1, 2] and C denotes the P P T þ t þ max t x u : u  c k i j jk jk jk j2M [fig j2M [fig gamma function. k k However, the Levy flight cannot be directly used in T , and go to Step 6. T is the workstation time of the b b discrete optimization problems. In this paper, the discrete bottleneck workstation considering task deviation times. Levy flight proposed by Li et al. [32] is modified by con- T is the total nominal task time of station k. M is the set k k sidering uncertain task times and implemented on the of tasks in workstation k. chromosome with the best solution to generate a new Step 5 Update i = i ? 1, and go to Step 2. chromosome stochastically: Step 6 Get the new chromosome and end the local search procedure. x ðt þ 1Þ¼ x ðtÞ k; ð21Þ k k The application of the local search procedure is descri- where x (t) is the task at location k in the chromosome of bed in Table 6 for a chromosome. generation t; k is the new task, and x (t ? 1) is the new task at location k in the chromosome of generation t ? 1. 3.7 Discrete Levy flight The fitness values of the two chromosomes before and after the discrete Levy flight are compared, and the chro- The Levy flight can improve the performance of nature- mosome with the better solution is kept in the population. tþ1 inspired algorithms [31]. A new solution x can be The discrete Levy flight procedure is described as obtained through the Levy flight: follows. tþ1 t x ¼ x þ ak; ð17Þ k k where a is the information about the step length and k is the random step length drawn from the Levy distribution. k is calculated as Fig. 6 Chromosome for discrete Levy flight Table 6 Local search procedure for the illustrative example with c =1 Chromosome 1 2 5 6 3 4 7 8 10 9 11 Step 1 Workstation (WS) WS1 WS2 WS3 WS4 Time 11.6 12.7 14.6 9.5 Step 2 E(7), L(7) E(7) = WS2, L(7) = WS4 Step 3 Ascending order WS4(9.5) ? WS2(12.7) ? WS3(14.6) Step 4 Transferring Task 7 ? WS4 WS4 time = 9 ? 3 ? 0.5 = 12.5 \ 14.6 Step 6 New chromosome 1 2 5 6 34810 7 911 End local search procedure 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 7 Discrete Levy flight procedure for the illustrative example Table 8 Benchmark instances Table 9 Parameters of the proposed algorithm Name nm Models Product mix w Parameter Value Small size Population size 50 Mertens 7 4 2 (0.4,0.6) 0.5 Maximal iteration number 100 Bowman 8 5 2 (0.5,0.5) 0.5 Tournament size 2 Jaeschke 9 4 2 (0.8,0.2) 0.3 Crossover probability/% 80 Mansoor 11 4 2 (0.7,0.3) 0.1 Mutation probability/% 15 Jackson 11 4 2 (0.4,0.6) 0.3 Medium size Table 10 Results of Sawyer using different initializations with c =1 Mitchell 21 4 2 (0.4,0.6) 0.2 Rosizeg 25 7 4 (0.2,0.3,0.1,0.4) 0.4 Initialization method RI PI Buxey 29 6 2 (0.3,0.7) 0.3 Min Mean SD Min Mean SD Sawyer 30 8 2 (0.5,0.5) 0.2 Sawyer 46 46.7 0.67 45 46.4 0.65 Gunther 35 6 3 (0.2,0.3,0.5) 0.1 Large size Kilbridge 45 5 2 (0.9,0.1) 0.1 Warnecke 58 12 2 (0.6,0.4) 0.1 Step 2 Obtain the precedence matrix of the part before Tong 70 16 2 (0.1,0.9) 0.2 the chromosome location k. Wee-Mag 75 20 3 (0.4,0.3,0.3) 0.3 Step 3 Calculate step length k according to Eq. (18), Mukherje 94 22 2 (0.7,0.3) 0.1 where b = 1.5. Step 4 Obtain the feasible task set S using the precedence Step 1 Select location k randomly as the starting point of matrix, which makes the Levy flight obtain a feasible the Levy flight. solution. Calculate value q= minfg kðÞ t þt , where i i 123 J.-H. Zhang et al. 3.8 Elite preservation Each individual with minimum cycle time is preserved for the next generation. The fitness values of individuals from the current population and the offspring after the Levy flight are compared, and the individuals with best fitness values are preserved to form a new generation. 4 Numerical experiments Theproposedalgorithm is implementedinMATLABR2013b TM Fig. 7 Convergence behaviors of different initializations on a PC with Intel Core i5-4210U CPU, 1.70 GHz and tested on 15 benchmark instances from www.assembly-line- balancing.de. Fifteen benchmark instances are classified into i [ S. Choose task s that meetsðÞ t þ t  q; ðs 2 SÞ to small size (7–11 tasks), medium size (21–35 tasks), and large s s form the task set S . size (45–94 tasks). The original benchmark instances only Step 5 If k B 1, select task j with the minimum (t þt )in take the deterministic situation into consideration. In our j j numerical experiments, the precedence relationships and the set S as k; otherwise, choose the task with maximum task nominal task times are obtained from the benchmark instan- times in set S as k. ces. Moreover, to define the robust problem, the coefficient of Step 6 k=k ? 1, go to Step 2 until k=n. variation w is used to create the deviation task time from the Take a chromosome in Fig. 6 for example. Location 3 in nominal task time, t ¼ wt . w is randomly generated using ij ij the chromosome is randomly selected as the starting point uniform distribution in interval [0.1, 0.5] for obtaining t .The ij of the Levy flight. The discrete Levy flight is described in product mix is generated at random. w and the product mix of Table 7. each instance are given in Table 8. In each instance, the number of workstations is fixed, with the objective being to Table 11 Results of benchmark instances with c =1 Name LINGO GA HGA Improvement/% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 12 1 12 12 0 30.02 12 12 0 31.52 0 Bowman 26 1 26 26 0 41.71 26 26 0 43.02 0 Jaeschke 12 1 12 12 0 30.37 12 12 0 31.2 0 Mansoor 52 1 52 52 0 32.57 52 52 0 33.79 0 Jackson 14 1 14 14 0 37.09 14 14 0 60.11 0 Medium size Mitchell 29 1 29 29 0 49.13 29 29 0 73.71 0 Rosizeg 22 1 22 22.3 0.48 75.1 22 22 0 110.36 1.35 Buxey 61 48 62 63 0.63 212.11 61 61 0 347.67 3.17 Sawyer 45 292 47 48 1.05 151.6 45 46.4 0.65 311.83 3.33 Gunther 86 323 87 88 1.15 228.69 86 86.4 0.84 309.01 1.82 Large size Kilbridge 114 548 114 114.7 0.48 328.21 114 114 0 517.89 0.61 Warnecke N/A 138 141 2.32 987.1 137 138.6 1.1 1 720.01 1.7 Tong N/A 252 253.2 1.55 2 189.08 246 247.4 1.64 2 798.53 2.29 Wee-Mag N/A 87 87.9 0.99 1 563.5 86 87.3 0.64 2 964.67 0.68 Mukherje N/A 217 218.5 1.18 1 048.02 215 216.7 1.06 1 478.24 0.82 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 12 Results of benchmark instances with c =2 Name LINGO GA HGA Improvement /% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 14 1 14 14 0 36.73 14 14 0 39.63 0 Bowman 26 1 26 26 0 43.06 26 26 0 46.36 0 Jaeschke 13 1 13 13 0 35.76 13 13 0 38.88 0 Mansoor 53 1 53 53 0 38.77 53 53 0 39.34 0 Jackson 15 1 15 15 0 45.53 15 15 0 73.21 0 Medium size Mitchell 31 1 31 31 0 51.53 31 31 0 77.44 0 Rosizeg 25 2 25 25 0 79.05 25 25 0 142.30 0 Buxey 65 59 66 67.7 0.82 240.24 65 65.7 0.48 440.23 2.95 Sawyer 48 2 094 49 51.1 1.2 158.70 48 48.6 0.84 312.23 4.89 Gunther 88 2 778 88 90.2 1.4 244.35 88 88.5 0.97 370.34 1.88 Large size Kilbridge 116 3 665 117 117 0 348.58 116 116.2 0.42 535.05 0.68 Warnecke N/A 143 144.7 1.34 1 035.84 140 142.2 1.2 1803.90 1.73 Tong N/A 265 268.7 2.70 2 379.79 258 259.4 1.08 2 906.46 3.46 Wee-Mag N/A 94 95.5 0.81 1 782.12 93 93.2 0.63 3 131.39 2.41 Mukherje N/A 221 223.6 1.71 1 226.63 219 221.4 0.97 1 806.04 0.98 Table 13 Results of benchmark instances with c =3 Name LINGO GA HGA Improvement/% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 14 1 14 14 0 38.03 14 14 0 39.89 0 Bowman 26 1 26 26 0 43.95 26 26 0 53.13 0 Jaeschke 13 1 13 13 0 35.52 13 13 0 52.50 0 Mansoor 53 1 53 53 0 31.05 53 53 0 37.30 0 Jackson 16 1 16 16 0 46.41 16 16 0 72.58 0 Medium size Mitchell 32 1 32 32 0 54.07 32 32 0 72.94 0 Rosizeg 26 1 26 26 0 77.38 26 26 0 160.70 0 Buxey 68 27 71 71.5 0.55 259.58 68 68.8 0.42 474.01 3.78 Sawyer 49 1 384 51 51.7 1.17 176.04 49 50 0.47 359.62 3.29 Gunther 90 1 452 90 91.5 1.43 281.98 90 90.7 0.95 399.67 0.87 Large size Kilbridge 117 22 790 118 118.1 0.32 351.45 118 118 0 639.35 0.08 Warnecke N/A 145 147.8 1.66 1 061.91 143 144.2 0.63 1 832.84 2.44 Tong N/A 271 272.67 1.65 2 573.45 268 269.2 1.55 3 313.65 1.27 Wee-Mag N/A 101 103.1 0.74 1 798.79 99 100 0.72 3 224.32 3.01 Mukherje N/A 227 228.3 0.95 1 308.91 224 225.4 0.92 1 833.76 1.27 123 J.-H. Zhang et al. Fig. 8 Average CPU time of benchmark instances Fig. 11 Convergence behavior of HGA and GA for Warnecke with c=1 Fig. 9 Average CPU time of small size instances Fig. 12 Average standard deviation of benchmark instances Table 14 Kruskal-Wallis test for Gap Algorithm N Median Mean rank Z value GA 150 0.022 178.2 5.53 HGA 150 0 122.8 - 5.53 Overall 300 150.5 H = 30.58DF = 1P = 0 (Not adjusted for ties) Fig. 10 Convergence behavior of HGA and GA for Rosizeg with c=1 H = 35.94DF = 1P = 0 (Adjusted for ties) minimize cycle time under different c ranging from 1 to 3. The parameters adopted in the algorithm are summarized in initialization (RI). The results are shown in Table 10. Table 9. Each instance is implemented for 10 runs. ‘‘Min’’ and ‘‘Mean’’ columns are the minimum and mean values of the solutions with the test instance of ten repli- 4.1 Comparison of different initializations cations; ‘‘SD’’ column means standard deviation of the solutions. The convergence behaviors of the average cycle In the proposed algorithm, 20% initial population is seeded time are demonstrated in Fig. 7. As shown in Table 10 and by a heuristic method designed in Sect. 3.1 and 80% initial Fig. 7, better solutions and faster convergence are obtained population is generated randomly. To test its performance, by the proposed initialization. the Sawyer instance with c ¼ 1 is initialized by two approaches: the proposed initialization (PI) and random 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Fig. 13 Combined precedence graph of problem Gunther-35 Table 15 Task times of problem Gunther-35 Tasks Model 1 Model 2 Model 3 Tasks Model 1 Model 2 Model 3 1 29292919 172120 2 3 3 3 20 17 21 20 3 5 5 6 21 10 2 2 4 20242222 101010 510 2 2 23 20 12 10 6 10181924 202628 7 2 2 2 25 10 0 0 8 910 26 467 9 20242627 4 6 7 10 35 25 20 28 40 40 40 11 23 23 25 29 4 0 0 12 35 25 20 30 3 7 8 13 20 26 25 31 3 7 8 14 400 32 111 15 18 20 20 33 38 42 45 16 28 30 32 34 2 2 2 17 045 35 222 18 044 4.2 Validation studies instances, Warnecke, Tong, Wee-Mag, and Mukherje cannot be solved optimally within 3 days calculation by The results of the proposed HGA are compared with that of LINGO. an exact algorithm and the pure GA. The detailed results of Each instance is executed ten times using the GA. For 15 instances are given in Tables 11–13. In the tables, the comparison, parameters adopted in the GA are set the same ‘‘OCT’’ column is the optimal solution found by the exact values as in the HGA. For the small size instances, both the algorithm; ‘‘CPU/s’’ column refers to the average pro- HGA and GA converge to the same optimal solutions cessing time in seconds spent by the algorithms; ‘‘Im- found by the integer solver. For the medium size instances, provement/%’’ column presents the comparative results of the HGA gets better solutions for four instances. For the the performance of the HGA with GA. large-size instances, the HGA gains better solutions for all The branch and bound algorithm embedded in the five instances. From the results, the HGA outperforms the integer solver software LINGO is used to solve the coun- GA by 83% for the number of the best solutions and terpart of the nonlinear model in Sect. 2.3. For 15 test improves the average solutions by 1.13%. 123 J.-H. Zhang et al. Fig. 14 Variation tendency of the cycle time with different c for a specific w Fig. 15 Variation tendency of the cycle time with different w for a specific c 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… The average CPU times for different algorithms are demonstrated. Its combined precedence graph is shown in described in Fig. 8. Compared with the processing time of Fig. 13, and the nominal task times of 35 tasks for three the medium and large-size instances, the average CPU time models are shown in Table 15. For product mix (0.2, 0.3, for small size instances is not obvious in Fig. 8. The pro- 0.5), six workstations are available, along with the com- cessing time of small size instances can be seen clearly in parison between the solutions of three different c (1, 2, 3) Fig. 9. As shown in Fig. 8, the processing time of LINGO and three different w (0.1, 0.2, 0.3). increases rapidly with the increase in the size of instances. Each Gunther-35 with different combinations of It indicates that the solution of large-size instances cannot parameters is solved ten times. The interval plots are be obtained by LINGO in an acceptable time span. Though adopted to show the variation tendency of the average they spend more CPU time on small size instances, the GA cycle time under different parameters. The average cycle and HGA spend less CPU time than LINGO on medium time, as described in Fig. 14, becomes larger with larger c and large-size instances. Compared with the processing for a given w. As shown in Fig. 15, the larger w makes the time of the GA, 22.22%, 69.35%, and 52.66% increases in average cycle time become larger for a specific c. c is the average CPU times of the HGA are observed, respectively, number of uncertain tasks considered and w represents the for small, medium, and large-size instances. Therefore, the task time interval. Both parameters reflect the uncertainty HGA obtains better solutions than the GA without causing level considered in the problem. Therefore, it can be con- excessive time consumption. cluded that the production efficiency will be sacrificed to Figures 10 and 11 demonstrate the convergence behav- hedge against uncertainty owing to the cycle time related to iors of the HGA and GA for medium and large-size the production efficiency of the assembly line. instances. From Figs. 10 and 11, it can be found that the HGA has better initial performance and converges at an earlier generation. This is because of the use of seeded 5 Conclusions initial population and the adaptive local search procedure in the algorithm loop. Mixed-model assembly lines are widely used in industries Concerning robustness, the HGA is more robust than the at present. With its practical benefits, the consideration of GA. As shown in Fig. 12, the average standard deviation of uncertain task times in the mixed-model assembly bal- the HGA is 0.4 and that of the GA is 0.63 for 15 instances. ancing problem is important. In this paper, MMALBP-II This is because of the reduced risk of premature conver- with interval uncertainty is considered and the robust gence using the adaptive local search and the discrete Levy optimization method is used. The robust model of this flight in the HGA. problem and its counterpart are formulated. The proposed To further compare HGA and GA, a statistical test is models are NP-hard. Therefore, an efficient HGA is carried out. The computational results are compared with developed to overcome the computational difficulties in each other in terms of Gap. Gap is the percentage differ- solving large-size problems. Experimental results show ence between the optimal cycle time (‘‘OCT’’ column in that the proposed algorithm is effective and efficient to find Tables 11–13) and the best cycle time (‘‘Min’’ column in solutions for large-size problems. It is also found that the Tables 11–13) and is calculated as Gap = (Min - OCT)/ production efficiency will be sacrificed to hedge against OCT. The lower gap value means a better performance of uncertainty. For future research, the mixed-model assem- the algorithm. With the HGA and GA converging to the bly line balancing and sequencing with interval uncertainty same optimal solutions for the small size instances and could be considered simultaneously. only one instance having the OCT solution among the large Acknowledgements This work is supported by the National Science size instances, the statistical test is implemented for the and Technology Major Project of Ministry of Science and Technology medium size instances. of China (Grant No. 2013ZX04012-071) and the Shanghai Municipal Because the normality of Gap values is violated, the Science and Technology Commission (Grant No. 15111105500). The Kruskal-Wallis test is performed. Table 14 shows the authors also want to express their gratitude to the reviewers. Their suggestions have improved this work. results of the Kruskal-Wallis test. Both p values are zero, thereby indicating that there is a statistically significant Open Access This article is distributed under the terms of the difference among these two algorithms. The result suggests Creative Commons Attribution 4.0 International License (http://crea that the HGA outperforms the GA statistically. tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a 4.3 Illustrative example link to the Creative Commons license, and indicate if changes were made. A specific example of Gunther-35 is solved by the pro- posed algorithm. The effect of c and w on the solutions is 123 J.-H. Zhang et al. 17. Pereira J, Alvarez-Miranda E (2018) An exact approach for the References robust assembly line balancing problem. Omega 78:85–98 18. Tasan SO, Tunali S (2008) A review of the current applications of 1. 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Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem with interval task times

Advances in Manufacturing , Volume OnlineFirst – Jun 1, 2019

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Engineering; Manufacturing, Machines, Tools, Processes; Control, Robotics, Mechatronics; Nanotechnology and Microengineering
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Abstract

Adv. Manuf. https://doi.org/10.1007/s40436-019-00256-3 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem with interval task times 1,2 1 1 • • Jia-Hua Zhang Ai-Ping Li Xue-Mei Liu Received: 11 August 2018 / Revised: 14 February 2019 / Accepted: 19 April 2019 The Author(s) 2019 Abstract The type-II mixed-model assembly line balanc- p Number of models to be assembled on the line ing problem with uncertain task times is a critical problem. C Cycle time This paper addresses this issue of practical significance to w Weight of model j in total production, j ¼ 1; 2; ; p production efficiency. Herein, a robust optimization model t Nominal task time of task i for model j ij for this problem is formulated to hedge against uncertainty. ^ Deviation task time from t ij ij Moreover, the counterpart of the robust optimization model I Set of immediate predecessors of task t, is developed by duality. A hybrid genetic algorithm (HGA) t ¼ 1; 2; ; n is proposed to solve this problem. In this algorithm, a c Budget for uncertainty, which means the number of heuristic method is utilized to seed the initial population. In uncertain tasks considered in a workstation, addition, an adaptive local search procedure and a discrete 0 B c B n Levy flight are hybridized with the genetic algorithm (GA) u Continuous variable ik to enhance the performance of the algorithm. The effec- x x = 1 if task i is assigned to workstation k; ik ik tiveness of the HGA is tested on a set of benchmark otherwise, x =0; i ¼ 1; 2; ; n; k ¼ 1; 2; ; m ik instances. Furthermore, the effect of uncertainty parame- ters on production efficiency is also investigated. 1 Introduction Keywords Mixed-model assembly line  Assembly line balancing  Robust optimization  Genetic algorithm (GA) Assembly lines are production systems containing serially Uncertainty located workstations. Assembly tasks are completed in these serial workstations. The first real example of an List of symbols assembly line was developed by Henry Ford in 1913. The i, t Index of assembly tasks, i; t ¼ 1; 2; ; n production rate saw an eightfold increase with the intro- j Index of models, j ¼ 1; 2; ; p duction of the assembly line production. Since then, k Index of workstations, k ¼ 1; 2; ; m assembly lines have been widely used around the world. n Number of tasks The assembly line balancing problem (ALBP) is one of the m Number of predefined workstations important problems in the design of an assembly line. ALBP is to assign assembly tasks among the workstations to optimize production objectives. This assignment must & Jia-Hua Zhang take the precedence relationship constraint and other 1510278@tongji.edu.cn technical constraints into consideration. ALBP has been an 1 active area of research since the first mathematical model School of Mechanical Engineering, Tongji University, of ALBP was presented by Salveson [1]. The detailed Shanghai 201804, People’s Republic of China 2 reviews of research on ALBP can be found in Refs. [2–4]. Department of Mechatronics Engineering, Wuxi Vocational Institute of Arts and Technology, Yixing 214206, Jiangsu, People’s Republic of China 123 J.-H. Zhang et al. A mixed-model assembly line is capable of assembling decomposition algorithm. Pereira and Alvarez-Miranda [17] similar models from a basic product through a single investigated RSALBP-I and developed a heuristic method assembly line. With increasing competition and diversity in and an exact algorithm. customer demands, mixed-model assembly lines have To the best of our knowledge, only Al-e-hashem et al. become popular in many industries, such as cars, TVs, and [11] have used the robust optimization method to study the computers. The mixed-model assembly line balancing mixed-model assembly line balancing problem with problem (MMALBP) is classified into two types: uncertain task times, in addition to formulating a mathe- MMALBP-I and MMALBP-II [5]. The former is aimed at matical model of RMMALBP-I. However, there is no minimizing the number of workstations for a given cycle related work on RMMALBP-II, which is directly related to time, whereas the latter is aimed at minimizing the cycle the production rate of the mixed-model assembly line. The time under a given number of workstations. MMALBP-II with interval task times is studied in this In many studies of the MMALBP, task times are paper using the robust optimization method, which is called assumed to be deterministic. However, in real life, robust MMALBP-II (RMMALBP-II). In this problem, each assembly tasks are subject to various uncertainties, such as task time is represented by an interval dataset. To solve this the skill level of the operator, task complexity, and problem, the robust optimization model is formulated and a resource availability. Both stochastic mixed-model hybrid genetic algorithm (HGA) is developed. The robust assembly line balancing [6, 7] and fuzzy mixed-model optimization model for this problem is nonlinear. To assembly line balancing [8] have been proposed to deal facilitate the use of an exact algorithm, the counterpart of with uncertain task times. In stochastic mixed-model the nonlinear robust optimization model is obtained by assembly line balancing, it is assumed that task times are duality. subject to probability distribution, generally normal dis- The remainder of this paper is organized as follows: the tribution. In fuzzy mixed-model assembly line balancing, RMMALBP-II is described and the mathematical models task times are assumed to be fuzzy numbers with given are formulated in Sect. 2. The HGA is developed in membership. However, both stochastic and fuzzy task Sect. 3. In Sect. 4, computational experiments are imple- times are often impossible in practice because of insuffi- mented, along with the illustration of the results. Finally, cient preliminary information to deduct the required conclusions are drawn in Sect. 5. probability or possibility distribution functions. Soyster [9] first developed the robust optimization approach in which the probability distribution of uncertain 2 Problem description and model formulation parameters is unknown. However, with the worst-case sce- narios that may never happen in real life, this method is very 2.1 Problem description conservative. To control the degree of conservatism, Ref. [10] developed a robust optimization, in which only a subset A mixed-model assembly line with m workstations is of uncertain coefficients (only c of them) is in the worst considered. p similar models are assembled simultaneously scenarios. This approach can be extended to discrete opti- in an intermixed sequence. The ratio of the unit number of mization problems and has been applied to a variety of each model j to the overall demand is w . The precedence problems. Many researchers have adopted the robust opti- relationship of each model is predefined and all relation- mization approach to study the assembly line balancing ships can be combined into only one precedence graph with problem with uncertain task times: Al-e-hashem et al. [11] n tasks. Each task has an uncertain task time. The uncer- considered the type-I robust mixed-model assembly line tainty information about the probability distribution or balancing problem (RMMALBP-I). Hazır and Dolgui [12] fuzzy membership function is not easy to know, and each studied the type-II robust simple assembly line balancing uncertain task time can only be represented by an interval problem (RSALBP-II) and developed a Benders decompo-  ^ t  t . t is the nominal task time for task i and model j, ij ij ij sition algorithm. Gurevsky et al. [13] investigated the type-I and t corresponds to the deviation task time from t . If the ij ij robust simple assembly line balancing problem (RSALBP-I) interval t  t equals 0, it means that model j does not ij ij and designed a branch and bound algorithm. Nazarian and need task i to be assembled. The aim is to minimize cycle Ko [14] considered the uncertain task and inter-task times in time C. RSALBP-II, especially focusing on non-productive times in The RMMALBP-II has the following assumptions: workstations. Moreira et al. [15] studied RSALBP-I with (i) The task time of each model is uncertain and is heterogeneous workers under uncertain task times and represented by a given interval dataset. developed two mathematical models and a heuristic method. (ii) Each common task for different models must be Hazır and Dolgui [16] studied the robust U-shaped assembly assigned to the same workstation for economy. line balancing problem, which was solved by a Benders 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… (iii) The precedence graphs for different models are to t þ t . Constraint (6) is the nondivisibility constraint, ij ij predefined, and a combined precedence graph can which means that a task cannot be split among two or more be obtained. workstations. Constraint (7) makes sure that C has an (iv) The line is a paced line with a fixed cycle time. integer value, which is always the case in a real-world (v) The line is serial with no feeder lines or parallel environment [5]. workstations. (vi) There are no assignment restrictions of tasks 2.3 Counterpart of the nonlinear robust except precedence constraints. optimization model (vii) There are no buffers between workstations. The robust optimization model proposed in Sect. 2.2 is a nonlinear model. It can be linearized by duality [10] and 2.2 Model formulation solved by an integer solver. The counterpart of the non- linear robust optimization model is developed as follows: The RMMALBP-II can be formulated as a nonlinear robust min C ð8Þ model: min C ð1Þ subject to n p n XX X subject to w t x þ p þ cz  C; 8k; ð9Þ j ij ik ik k i¼1 j¼1 i¼1 x ¼ 1; 8i; ð2Þ ik k¼1 z þ p  w t y ; 8i; k; ð10Þ k ik j ij ik m m X X j¼1 kx  kx ; 8i 2 I ; ð3Þ ik tk t k¼1 k¼1 p  0; 8i; k; ð11Þ ik () p p n n n XX XX X x  y ; 8i; k; ð12Þ ik ik w t x þ max w t x u : u  c j ij ik j ij ik ik ik i¼1 j¼1 i¼1 j¼1 i¼1 z  0; 8k; ð13Þ C; 8k; and constraints (2), (3), (6), and (7). y , p , and z are the ik ik k ð4Þ variables used in duality. 0  u  1; 8i; k; ð5Þ ik x 2f0; 1g; 8i; k; ð6Þ ik 3 Proposed hybrid genetic algorithm C [ 0; C is integer: ð7Þ for RMALBP-II Objective function (1) minimizes cycle time C. Con- Type-II simple assembly line balancing problem (SALBP- straint (2) is known as the occurrence constraint that each II) is known to be nondeterministic polynomial (NP)-hard task can be assigned to a workstation. Constraint (3) is the [2]. It can be found that SALBP-II is a special case of precedence constraint to guarantee the technological RMALBP-II. Similar to SALBP-II, RMLABP-II is NP- sequencing requirements. Constraint (4) ensures that the hard. Owing to its NP-hard nature, RMALBP-II can be weighted uncertain workstation times do not exceed the time-consuming to obtain an optimal solution using exact cycle time C. The left-hand side of constraint (4) consists algorithms. In this section, an HGA is developed to solve of two parts: w t is the weighted nominal task time j ij j¼1 this problem. of each task i which can be assigned into a workstation and The genetic algorithm (GA), a popular meta-heuristic w t is the weighted deviation task time for each task j ij j¼1 algorithm, has been used to solve various assembly line i. The number of uncertain tasks considered in a worksta- balancing problems such as stochastic assembly line bal- tion is bounded by parameter c. The larger c is, the more ancing problem, fuzzy assembly line balancing problem, the deviation task times should be considered. Constraint and so on [18]. Research shows that GA is competitive (5) determines the bound of variable u , which indicates ik against the best-known constructive methods. There is no the level of an uncertain task time deviating from the published research reporting the application of the GA to nominal time. When u ¼ 0, it means that the task time for ik solve the robust assembly line balancing problem. In this task i in workstation k does not deviate from the nominal section, an HGA is proposed to solve RMMALBP-II. The time and is deterministic. When u ¼ 1, task i in work- ik flowchart of this algorithm is shown in Fig. 1. station k is under the worst case, and the task time is equal 123 J.-H. Zhang et al. This algorithm starts with the generation of an initial In this case, it is proposed to seed part of the initial pop- feasible population, which is followed by the evaluation of ulation with heuristic solutions to improve the performance each chromosome through fitness evaluation. Special of the algorithm. Based on the experimental results, 20% of genetic operators for the assembly line balancing problem the initial population can be seeded by the proposed (crossover and mutation) are performed. An adaptive local heuristic and 80% can be generated randomly. This ini- search procedure and a discrete Levy flight are used to tialization method can obtain better solutions than that of a improve the quality of solutions. This loop keeps running randomly generated population, as shown in Sect. 4.1. until the maximum iteration number is reached. Finally, the Theheuristic method is basedonPereira andAlvarez- best cycle time of RMMALBP-II is obtained. Miranda [17] and Sewell and Jacobson [22], and builds solutions from the forward or backward direction of G.Task j in 3.1 Encoding scheme and initial population  ^ G, which maximizes the priority rule ðw t þ aw t þ i i i i i2U bFjjcÞ, can be selected. U is the set of tasks that have not yet A combined precedence graph, G, is formed by combining been selected; w t is the weighted nominal task time; w t is the i i i i the precedence graphs of each model. Based on the com- weighted deviation time; jj F is the number of immediate bined precedence graph, common tasks of each model can successors of task i in the forward directionsearchorthatof be assigned to the same workstation. Each chromosome, immediate predecessors of task i in the backward direction according to their precedence constraint, is designed as a search.a, b,and c are input parameters. The recommended sequence of tasks. The number of genes in the chromosome values for a, b,and c in Ref. [22] are employed. Every random is equal to that of tasks n in G, and each gene is an integer combination of a [ {0,0.005,0.010,0.015,0.020}, b [ {0, representing a task. 0.005,0.010,0.015,0.020}, and c [ {0,0.01,0.02,0.03} is used Population initialization is a crucial step in evolutionary to select a task. The random method selects a task at random algorithms (EAs). Many researchers have proposed to seed and assures the feasibility of the precedence relationship. The EAs with good initial solutions, whenever it is possible, to precedence matrix is used to describe the precedence rela- obtain important improvement in the convergence of the tionship of tasks [23]. algorithm and the quality of the solutions [19]. However, The procedure of the initial feasible population gener- the excessive use of good solutions in the initial population ation is described as follows. can decrease the exploration capacity of the GA, thereby Step 1 Build an empty vector A; read the precedence trapping the population in local optimums quickly [20, 21]. matrix P of the combined precedence graph G. Step 2 Choose the random creation method or the heuristic method. (i) If the random creation method is chosen, go to Step 3. (ii) If the heuristic method is chosen, decide the search direction randomly. If it is a backward direction search, P = P and go to Step 3; otherwise, go to Step 3. Step 3 If j B n, store the tasks where the sum of the column of P is equal to 0 into A;if j [ n, go to Step 7. Step 4 Choose a task from A for a gene in a chromosome. (i) If it is the random creation method, choose a task from A at random for the jth gene in a chromo- some. Empty A, and go to Step 5. (ii) If it is the heuristic method, generate a random combination of input parameters a, b,and c,and choose the task that maximizes the priority rule from A. The task is assigned for the jth gene in the forward direction search or for the (n - j ? 1)th in the backward direction search. Empty A, and go to Step 5. Step 5 Update the ith row of P by putting a big number D (e.g., 999) into tasks i and 0 into other tasks. Fig. 1 Flowchart of the proposed HGA 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Step 6 Update j = j ? 1, and go to Step 3. Table 1 Precedence matrix P of G Step 7 End the procedure. Task 1 2 3 45678 91011 Repeat the above procedure and get the initial popula- 1 011 11000 00 0 tion. Figure 2 is the combined precedence graph G of an 2 000 00100 00 0 illustrative example. Table 1 is the precedence matrix P of 3 000 00010 00 0 graph G. If the heuristic method from backward search is 4 000 00010 00 0 chosen, make P = P . Table 2 shows the result of Steps 2 5 000 00010 00 0 and 3 of the heuristic method from the backward search 6 000 00001 00 0 direction. After these two steps, the element of vector A is 7 000 00000 10 0 task 11 when j = 1. Because there is only one element in 8 000 00000 01 0 A to calculate the priority rule, the (n - j ? 1)th gene (it is 9 000 00000 00 1 the nth gene now) in the chromosome is task 11 in Step 4. 10 000 00000 00 1 Table 3 is the result of Step 5. Repeat this procedure 11 000 00000 00 0 until the last task is selected into a chromosome. Table 4 is the final result of the sum of columns. Figure 3 shows some chromosomes in the initial population produced by the two methods. Table 2 Searching for the tasks to formulate a feasible task sequence from the backward direction 3.2 Evaluation procedure Task Step 2: Transposed matrix of P The evaluation procedure aims to find objective values based on the task sequence. With each chromosome being 1 0000000000 0 a feasible task sequence, the procedure decides which tasks 2 1000000000 0 can be assigned to predefined workstations respecting the 3 1000000000 0 cycle time constraint. The evaluation procedure is descri- 4 1000000000 0 bed as follows. 5 1000000000 0 Step 1 Read the nominal time matrix T and the deviation 6 0100000000 0 task time matrix V. 7 0011100000 0 Step 2 Calculate the theoretical minimum cycle time for 8 0000010000 0 the initial trial cycle time C. This initial cycle time value 9 0000001000 0 is the lower boundary (LB), and the calculation equation 10 0000000100 0 is presented as 11 0000000011 0 ceil max maxðÞ t ; t m ; c ¼ 0; Step 3: Sum of columns 4111111111 0 i i i¼1 LB ¼ ^ ^ : ceil max maxðÞ t þt ; t þ maxðÞ t =m ; c  1; i i i i i¼1 ð14Þ where t ¼ w t represents the weighted nominal i j ij j¼1 ^ ^ task time for task i and t ¼ w t represents the i j ij j¼1 weighted deviation task time for task i. If c = 0, it means that no task deviation time is con- sidered. The LB is equal to the LB of the deterministic MMALBP-II. For the cycle time set as an integer in RMMALBP-II, the LB is taken as an integer. If c C 1, it means that at least one task deviation time is considered. The indivisibility of tasks requires that C ðÞ t þ t . With i i at least one task deviation time being considered, mC  t þ maxðÞ t is obtained. i i i¼1 Step 3 Assign the tasks into predetermined workstations. Tasks are assigned to the first station in the order of the Fig. 2 Combined precedence graph G of the illustrative example 123 J.-H. Zhang et al. Table 3 Updating P after a task selected Rank the deviation time of tasks in this workstation in the descending order. The worst scenario of the worksta- Task Step 5: Matrix P tion time is equal to the sum of the nominal time assigned 123 45678 91011 to this workstation and the largest c deviation time among these tasks. Once the workstation time exceeds the cycle 1 000 00000 00 0 time, the next station is opened for assignment. If the 2 100 00000 00 0 number of workstations is equal to the given m, stop the 3 100 00000 00 0 assignment and calculate the worst scenario workstation 4 100 00000 00 0 time of each station. The maximum workstation time is 5 100 00000 00 0 defined as the bottleneck workstation time C . 6 010 00000 00 0 7 001 11000 00 0 Step 4 If C  C, go to Step 5; otherwise, update C ¼ 8 000 00100 00 0 C þ 1 and go to Step 3. 9 000 00010 00 0 Step 5 The cycle time is equal to C; end this procedure. 10 000 00001 00 0 Table 5 is an illustration of the decoding/evaluating 11 000 00000 00 D procedure. The predetermined workstation number is m =4. 3.3 Tournament selection Table 4 Final content of P after forming a feasible task sequence Task Matrix P The tournament selection strategy [24] is used to select the parent chromosomes. Some chromosomes are randomly 12 345678 910 11 chosen from the current population along with the one with 1 D 0000 00000 0 the best objective value being selected for reproduction. 20 D 000 00000 0 The tournament selection strategy works as follows. 300 D 00 00000 0 Step 1 k chromosomes in the population are selected at 4 000 D 0 00000 0 random. 5 0000 D 00000 0 Step 2 The chromosome with the best objective value 6 00000 D 0000 0 (minimum cycle time) from these selected chromosomes 7 00000 0 D 000 0 is chosen as the best one and added to the mating pool. 8 00000 00 D 00 0 Step 3 This procedure is repeated until the number of 9 00000 000 D 00 individuals in the mating pool reaches the required 10 00000 0000 D 0 population size and the population is updated through 11 00000 00000 D this procedure. Sum of columns DDDDD DDDDD D 3.4 Fragment reordering crossover Because it is designed particularly for the assembly line balancing problem [25], the fragment reordering crossover can preserve the feasibility of the offspring structure with its procedure working as follows. Fig. 3 Some chromosomes in an initial population Step 1 Two parent individuals are selected from the gene sequence complying with the precedence relation- population in order. ship. The cycle time constraint as shown in inequality Step 2 Two parents selected are divided by two (4) cannot be violated. The cycle time constraint can be randomly cut points into three sections: head, middle, described as and tail. n n n X X X Step 3 The head and tail parts of the first offspring are t x þ max t x u : u  c  C: ð15Þ i ik i ik ik ik taken from the first parent and the middle part of the first i¼1 i¼1 i¼1 offspring is filled by adding missing tasks according to the order in which they are contained in the second parent. 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 5 Evaluation procedure for the illustrative example with c =1 Chromosome 1 5 2 6 4 3 7 9 8 10 11 Step 1 t 61 2 2 7 5 356 54 ^ 0.6 0.1 0.2 0.2 0.7 0.5 0.3 0.5 0.6 0.5 0.4 Step 2 LB LB = ceil(max (7.7, 4.8848)) = 8 Step 3 Workstation (WS) WS1(1 5) WS2(2 6) WS3(4) WS4(3 7 9 8 10 11) C WS4 is bottleneck; C =28 ? 0.6 = 28.6 [ C Step 4 Update C C ¼ C þ 1 Repeat Step 5 C The cycle time of this chromosome is C =14 End the procedure Step 4 Like the building process for the first offspring, An illustration of a scramble mutation is given in Fig. 5. the head and tail of the second offspring are formed from The mutation point is 3 in the chromosome of this example. the same part of the second parent, and the middle is filled by the missing tasks according to the order in 3.6 Adaptive local search which they are contained in the first parent. Step 5 Get a new population with the offspring The pure genetic algorithm is good at global search but chromosomes. slow to converge [27]. Local search is a promising approach to improve the quality of the objective value and The fragment reordering crossover is demonstrated in convergence speed [28]. Within the proposed algorithm, a Fig. 4. local search procedure is applied to every chromosome of the population. The local search tries to transfer tasks in the bottleneck workstation to other workstations to reduce the 3.5 Scramble mutation cycle time. To tackle the increased computation time of the local search procedure, an adaptive local search scheme is Scramble mutation for the assembly line balancing prob- adopted. The basic concept of applying the local search to lem was developed by Leu et al. [26]. With scramble the population is to consider whether GA has converged to mutation, the chromosome, reconstructed drastically, still the global optimal solution or not [29]. The converging remains feasible. It works as follows. criterion for the line balancing problem is the ratio of the Step 1 The mutation point is generated randomly, and average fitness of the chromosomes to the fitness of the one chromosome from the population is divided into best chromosome less than 1.01 [30]. head and tail. The fitness value ratio (FVR), R , of the average fitness fv Step 2 The head of the chromosome is kept, and the tail of the chromosomes to the fitness of the best chromosome of the chromosome is regenerated respecting the prece- at each generation is defined as dence relationship. The precedence matrix needs to be R ¼ ; ð16Þ fv proceeded to eliminate the task already in the head of the chromosome. where a is the average fitness of the chromosomes and b f f Step 3 Get a new population with the mutation is the fitness of the best chromosome. procedure. If R [ 1.01, apply the local search; otherwise, only fv GA is implemented. The local search procedure is explained as follows. Fig. 4 Fragment reordering crossover Fig. 5 Scramble mutation 123 J.-H. Zhang et al. Step 1 Identify the workstation with the largest work- k¼ ; ð18Þ 1=b station times as the bottleneck workstation. jj v Step 2 Let n be the number of tasks in the bottleneck where u and v are drawn from normal distributions. That is, workstation. If i B n , find the ith task in the bottleneck workstation. According to the precedence relationship, u  Nð0; r Þ; ð19Þ find the earliest workstation E(i) and the latest worksta- v  Nð0; r Þ; tion L(i) to which task i can be transferred. If i [ n ,go 0 1 > b > pb to Step 6. CðÞ 1 þ b sin < B C Step 3 Rank workstations between E(i) and L(i) accord- 2 B C r ¼ ; @ A ð20Þ b1 ing to workstation times in the ascending order. ðÞ 1 þ b > C b Step 4 k is the workstation between E(i) and L(i). r ¼ 1; Transfer task i to the workstations arranged in order of Step 3. Task i can be transferred to a workstation with no where distribution parameter b [ [1, 2] and C denotes the P P T þ t þ max t x u : u  c k i j jk jk jk j2M [fig j2M [fig gamma function. k k However, the Levy flight cannot be directly used in T , and go to Step 6. T is the workstation time of the b b discrete optimization problems. In this paper, the discrete bottleneck workstation considering task deviation times. Levy flight proposed by Li et al. [32] is modified by con- T is the total nominal task time of station k. M is the set k k sidering uncertain task times and implemented on the of tasks in workstation k. chromosome with the best solution to generate a new Step 5 Update i = i ? 1, and go to Step 2. chromosome stochastically: Step 6 Get the new chromosome and end the local search procedure. x ðt þ 1Þ¼ x ðtÞ k; ð21Þ k k The application of the local search procedure is descri- where x (t) is the task at location k in the chromosome of bed in Table 6 for a chromosome. generation t; k is the new task, and x (t ? 1) is the new task at location k in the chromosome of generation t ? 1. 3.7 Discrete Levy flight The fitness values of the two chromosomes before and after the discrete Levy flight are compared, and the chro- The Levy flight can improve the performance of nature- mosome with the better solution is kept in the population. tþ1 inspired algorithms [31]. A new solution x can be The discrete Levy flight procedure is described as obtained through the Levy flight: follows. tþ1 t x ¼ x þ ak; ð17Þ k k where a is the information about the step length and k is the random step length drawn from the Levy distribution. k is calculated as Fig. 6 Chromosome for discrete Levy flight Table 6 Local search procedure for the illustrative example with c =1 Chromosome 1 2 5 6 3 4 7 8 10 9 11 Step 1 Workstation (WS) WS1 WS2 WS3 WS4 Time 11.6 12.7 14.6 9.5 Step 2 E(7), L(7) E(7) = WS2, L(7) = WS4 Step 3 Ascending order WS4(9.5) ? WS2(12.7) ? WS3(14.6) Step 4 Transferring Task 7 ? WS4 WS4 time = 9 ? 3 ? 0.5 = 12.5 \ 14.6 Step 6 New chromosome 1 2 5 6 34810 7 911 End local search procedure 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 7 Discrete Levy flight procedure for the illustrative example Table 8 Benchmark instances Table 9 Parameters of the proposed algorithm Name nm Models Product mix w Parameter Value Small size Population size 50 Mertens 7 4 2 (0.4,0.6) 0.5 Maximal iteration number 100 Bowman 8 5 2 (0.5,0.5) 0.5 Tournament size 2 Jaeschke 9 4 2 (0.8,0.2) 0.3 Crossover probability/% 80 Mansoor 11 4 2 (0.7,0.3) 0.1 Mutation probability/% 15 Jackson 11 4 2 (0.4,0.6) 0.3 Medium size Table 10 Results of Sawyer using different initializations with c =1 Mitchell 21 4 2 (0.4,0.6) 0.2 Rosizeg 25 7 4 (0.2,0.3,0.1,0.4) 0.4 Initialization method RI PI Buxey 29 6 2 (0.3,0.7) 0.3 Min Mean SD Min Mean SD Sawyer 30 8 2 (0.5,0.5) 0.2 Sawyer 46 46.7 0.67 45 46.4 0.65 Gunther 35 6 3 (0.2,0.3,0.5) 0.1 Large size Kilbridge 45 5 2 (0.9,0.1) 0.1 Warnecke 58 12 2 (0.6,0.4) 0.1 Step 2 Obtain the precedence matrix of the part before Tong 70 16 2 (0.1,0.9) 0.2 the chromosome location k. Wee-Mag 75 20 3 (0.4,0.3,0.3) 0.3 Step 3 Calculate step length k according to Eq. (18), Mukherje 94 22 2 (0.7,0.3) 0.1 where b = 1.5. Step 4 Obtain the feasible task set S using the precedence Step 1 Select location k randomly as the starting point of matrix, which makes the Levy flight obtain a feasible the Levy flight. solution. Calculate value q= minfg kðÞ t þt , where i i 123 J.-H. Zhang et al. 3.8 Elite preservation Each individual with minimum cycle time is preserved for the next generation. The fitness values of individuals from the current population and the offspring after the Levy flight are compared, and the individuals with best fitness values are preserved to form a new generation. 4 Numerical experiments Theproposedalgorithm is implementedinMATLABR2013b TM Fig. 7 Convergence behaviors of different initializations on a PC with Intel Core i5-4210U CPU, 1.70 GHz and tested on 15 benchmark instances from www.assembly-line- balancing.de. Fifteen benchmark instances are classified into i [ S. Choose task s that meetsðÞ t þ t  q; ðs 2 SÞ to small size (7–11 tasks), medium size (21–35 tasks), and large s s form the task set S . size (45–94 tasks). The original benchmark instances only Step 5 If k B 1, select task j with the minimum (t þt )in take the deterministic situation into consideration. In our j j numerical experiments, the precedence relationships and the set S as k; otherwise, choose the task with maximum task nominal task times are obtained from the benchmark instan- times in set S as k. ces. Moreover, to define the robust problem, the coefficient of Step 6 k=k ? 1, go to Step 2 until k=n. variation w is used to create the deviation task time from the Take a chromosome in Fig. 6 for example. Location 3 in nominal task time, t ¼ wt . w is randomly generated using ij ij the chromosome is randomly selected as the starting point uniform distribution in interval [0.1, 0.5] for obtaining t .The ij of the Levy flight. The discrete Levy flight is described in product mix is generated at random. w and the product mix of Table 7. each instance are given in Table 8. In each instance, the number of workstations is fixed, with the objective being to Table 11 Results of benchmark instances with c =1 Name LINGO GA HGA Improvement/% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 12 1 12 12 0 30.02 12 12 0 31.52 0 Bowman 26 1 26 26 0 41.71 26 26 0 43.02 0 Jaeschke 12 1 12 12 0 30.37 12 12 0 31.2 0 Mansoor 52 1 52 52 0 32.57 52 52 0 33.79 0 Jackson 14 1 14 14 0 37.09 14 14 0 60.11 0 Medium size Mitchell 29 1 29 29 0 49.13 29 29 0 73.71 0 Rosizeg 22 1 22 22.3 0.48 75.1 22 22 0 110.36 1.35 Buxey 61 48 62 63 0.63 212.11 61 61 0 347.67 3.17 Sawyer 45 292 47 48 1.05 151.6 45 46.4 0.65 311.83 3.33 Gunther 86 323 87 88 1.15 228.69 86 86.4 0.84 309.01 1.82 Large size Kilbridge 114 548 114 114.7 0.48 328.21 114 114 0 517.89 0.61 Warnecke N/A 138 141 2.32 987.1 137 138.6 1.1 1 720.01 1.7 Tong N/A 252 253.2 1.55 2 189.08 246 247.4 1.64 2 798.53 2.29 Wee-Mag N/A 87 87.9 0.99 1 563.5 86 87.3 0.64 2 964.67 0.68 Mukherje N/A 217 218.5 1.18 1 048.02 215 216.7 1.06 1 478.24 0.82 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Table 12 Results of benchmark instances with c =2 Name LINGO GA HGA Improvement /% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 14 1 14 14 0 36.73 14 14 0 39.63 0 Bowman 26 1 26 26 0 43.06 26 26 0 46.36 0 Jaeschke 13 1 13 13 0 35.76 13 13 0 38.88 0 Mansoor 53 1 53 53 0 38.77 53 53 0 39.34 0 Jackson 15 1 15 15 0 45.53 15 15 0 73.21 0 Medium size Mitchell 31 1 31 31 0 51.53 31 31 0 77.44 0 Rosizeg 25 2 25 25 0 79.05 25 25 0 142.30 0 Buxey 65 59 66 67.7 0.82 240.24 65 65.7 0.48 440.23 2.95 Sawyer 48 2 094 49 51.1 1.2 158.70 48 48.6 0.84 312.23 4.89 Gunther 88 2 778 88 90.2 1.4 244.35 88 88.5 0.97 370.34 1.88 Large size Kilbridge 116 3 665 117 117 0 348.58 116 116.2 0.42 535.05 0.68 Warnecke N/A 143 144.7 1.34 1 035.84 140 142.2 1.2 1803.90 1.73 Tong N/A 265 268.7 2.70 2 379.79 258 259.4 1.08 2 906.46 3.46 Wee-Mag N/A 94 95.5 0.81 1 782.12 93 93.2 0.63 3 131.39 2.41 Mukherje N/A 221 223.6 1.71 1 226.63 219 221.4 0.97 1 806.04 0.98 Table 13 Results of benchmark instances with c =3 Name LINGO GA HGA Improvement/% OCT CPU/s Min Mean SD CPU/s Min Mean SD CPU/s Small size Mertens 14 1 14 14 0 38.03 14 14 0 39.89 0 Bowman 26 1 26 26 0 43.95 26 26 0 53.13 0 Jaeschke 13 1 13 13 0 35.52 13 13 0 52.50 0 Mansoor 53 1 53 53 0 31.05 53 53 0 37.30 0 Jackson 16 1 16 16 0 46.41 16 16 0 72.58 0 Medium size Mitchell 32 1 32 32 0 54.07 32 32 0 72.94 0 Rosizeg 26 1 26 26 0 77.38 26 26 0 160.70 0 Buxey 68 27 71 71.5 0.55 259.58 68 68.8 0.42 474.01 3.78 Sawyer 49 1 384 51 51.7 1.17 176.04 49 50 0.47 359.62 3.29 Gunther 90 1 452 90 91.5 1.43 281.98 90 90.7 0.95 399.67 0.87 Large size Kilbridge 117 22 790 118 118.1 0.32 351.45 118 118 0 639.35 0.08 Warnecke N/A 145 147.8 1.66 1 061.91 143 144.2 0.63 1 832.84 2.44 Tong N/A 271 272.67 1.65 2 573.45 268 269.2 1.55 3 313.65 1.27 Wee-Mag N/A 101 103.1 0.74 1 798.79 99 100 0.72 3 224.32 3.01 Mukherje N/A 227 228.3 0.95 1 308.91 224 225.4 0.92 1 833.76 1.27 123 J.-H. Zhang et al. Fig. 8 Average CPU time of benchmark instances Fig. 11 Convergence behavior of HGA and GA for Warnecke with c=1 Fig. 9 Average CPU time of small size instances Fig. 12 Average standard deviation of benchmark instances Table 14 Kruskal-Wallis test for Gap Algorithm N Median Mean rank Z value GA 150 0.022 178.2 5.53 HGA 150 0 122.8 - 5.53 Overall 300 150.5 H = 30.58DF = 1P = 0 (Not adjusted for ties) Fig. 10 Convergence behavior of HGA and GA for Rosizeg with c=1 H = 35.94DF = 1P = 0 (Adjusted for ties) minimize cycle time under different c ranging from 1 to 3. The parameters adopted in the algorithm are summarized in initialization (RI). The results are shown in Table 10. Table 9. Each instance is implemented for 10 runs. ‘‘Min’’ and ‘‘Mean’’ columns are the minimum and mean values of the solutions with the test instance of ten repli- 4.1 Comparison of different initializations cations; ‘‘SD’’ column means standard deviation of the solutions. The convergence behaviors of the average cycle In the proposed algorithm, 20% initial population is seeded time are demonstrated in Fig. 7. As shown in Table 10 and by a heuristic method designed in Sect. 3.1 and 80% initial Fig. 7, better solutions and faster convergence are obtained population is generated randomly. To test its performance, by the proposed initialization. the Sawyer instance with c ¼ 1 is initialized by two approaches: the proposed initialization (PI) and random 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… Fig. 13 Combined precedence graph of problem Gunther-35 Table 15 Task times of problem Gunther-35 Tasks Model 1 Model 2 Model 3 Tasks Model 1 Model 2 Model 3 1 29292919 172120 2 3 3 3 20 17 21 20 3 5 5 6 21 10 2 2 4 20242222 101010 510 2 2 23 20 12 10 6 10181924 202628 7 2 2 2 25 10 0 0 8 910 26 467 9 20242627 4 6 7 10 35 25 20 28 40 40 40 11 23 23 25 29 4 0 0 12 35 25 20 30 3 7 8 13 20 26 25 31 3 7 8 14 400 32 111 15 18 20 20 33 38 42 45 16 28 30 32 34 2 2 2 17 045 35 222 18 044 4.2 Validation studies instances, Warnecke, Tong, Wee-Mag, and Mukherje cannot be solved optimally within 3 days calculation by The results of the proposed HGA are compared with that of LINGO. an exact algorithm and the pure GA. The detailed results of Each instance is executed ten times using the GA. For 15 instances are given in Tables 11–13. In the tables, the comparison, parameters adopted in the GA are set the same ‘‘OCT’’ column is the optimal solution found by the exact values as in the HGA. For the small size instances, both the algorithm; ‘‘CPU/s’’ column refers to the average pro- HGA and GA converge to the same optimal solutions cessing time in seconds spent by the algorithms; ‘‘Im- found by the integer solver. For the medium size instances, provement/%’’ column presents the comparative results of the HGA gets better solutions for four instances. For the the performance of the HGA with GA. large-size instances, the HGA gains better solutions for all The branch and bound algorithm embedded in the five instances. From the results, the HGA outperforms the integer solver software LINGO is used to solve the coun- GA by 83% for the number of the best solutions and terpart of the nonlinear model in Sect. 2.3. For 15 test improves the average solutions by 1.13%. 123 J.-H. Zhang et al. Fig. 14 Variation tendency of the cycle time with different c for a specific w Fig. 15 Variation tendency of the cycle time with different w for a specific c 123 Hybrid genetic algorithm for a type-II robust mixed-model assembly line balancing problem… The average CPU times for different algorithms are demonstrated. Its combined precedence graph is shown in described in Fig. 8. Compared with the processing time of Fig. 13, and the nominal task times of 35 tasks for three the medium and large-size instances, the average CPU time models are shown in Table 15. For product mix (0.2, 0.3, for small size instances is not obvious in Fig. 8. The pro- 0.5), six workstations are available, along with the com- cessing time of small size instances can be seen clearly in parison between the solutions of three different c (1, 2, 3) Fig. 9. As shown in Fig. 8, the processing time of LINGO and three different w (0.1, 0.2, 0.3). increases rapidly with the increase in the size of instances. Each Gunther-35 with different combinations of It indicates that the solution of large-size instances cannot parameters is solved ten times. The interval plots are be obtained by LINGO in an acceptable time span. Though adopted to show the variation tendency of the average they spend more CPU time on small size instances, the GA cycle time under different parameters. The average cycle and HGA spend less CPU time than LINGO on medium time, as described in Fig. 14, becomes larger with larger c and large-size instances. Compared with the processing for a given w. As shown in Fig. 15, the larger w makes the time of the GA, 22.22%, 69.35%, and 52.66% increases in average cycle time become larger for a specific c. c is the average CPU times of the HGA are observed, respectively, number of uncertain tasks considered and w represents the for small, medium, and large-size instances. Therefore, the task time interval. Both parameters reflect the uncertainty HGA obtains better solutions than the GA without causing level considered in the problem. Therefore, it can be con- excessive time consumption. cluded that the production efficiency will be sacrificed to Figures 10 and 11 demonstrate the convergence behav- hedge against uncertainty owing to the cycle time related to iors of the HGA and GA for medium and large-size the production efficiency of the assembly line. instances. From Figs. 10 and 11, it can be found that the HGA has better initial performance and converges at an earlier generation. This is because of the use of seeded 5 Conclusions initial population and the adaptive local search procedure in the algorithm loop. Mixed-model assembly lines are widely used in industries Concerning robustness, the HGA is more robust than the at present. With its practical benefits, the consideration of GA. As shown in Fig. 12, the average standard deviation of uncertain task times in the mixed-model assembly bal- the HGA is 0.4 and that of the GA is 0.63 for 15 instances. ancing problem is important. In this paper, MMALBP-II This is because of the reduced risk of premature conver- with interval uncertainty is considered and the robust gence using the adaptive local search and the discrete Levy optimization method is used. The robust model of this flight in the HGA. problem and its counterpart are formulated. The proposed To further compare HGA and GA, a statistical test is models are NP-hard. Therefore, an efficient HGA is carried out. The computational results are compared with developed to overcome the computational difficulties in each other in terms of Gap. Gap is the percentage differ- solving large-size problems. Experimental results show ence between the optimal cycle time (‘‘OCT’’ column in that the proposed algorithm is effective and efficient to find Tables 11–13) and the best cycle time (‘‘Min’’ column in solutions for large-size problems. It is also found that the Tables 11–13) and is calculated as Gap = (Min - OCT)/ production efficiency will be sacrificed to hedge against OCT. The lower gap value means a better performance of uncertainty. For future research, the mixed-model assem- the algorithm. With the HGA and GA converging to the bly line balancing and sequencing with interval uncertainty same optimal solutions for the small size instances and could be considered simultaneously. only one instance having the OCT solution among the large Acknowledgements This work is supported by the National Science size instances, the statistical test is implemented for the and Technology Major Project of Ministry of Science and Technology medium size instances. of China (Grant No. 2013ZX04012-071) and the Shanghai Municipal Because the normality of Gap values is violated, the Science and Technology Commission (Grant No. 15111105500). The Kruskal-Wallis test is performed. Table 14 shows the authors also want to express their gratitude to the reviewers. Their suggestions have improved this work. results of the Kruskal-Wallis test. Both p values are zero, thereby indicating that there is a statistically significant Open Access This article is distributed under the terms of the difference among these two algorithms. The result suggests Creative Commons Attribution 4.0 International License (http://crea that the HGA outperforms the GA statistically. tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a 4.3 Illustrative example link to the Creative Commons license, and indicate if changes were made. A specific example of Gunther-35 is solved by the pro- posed algorithm. The effect of c and w on the solutions is 123 J.-H. Zhang et al. 17. Pereira J, Alvarez-Miranda E (2018) An exact approach for the References robust assembly line balancing problem. Omega 78:85–98 18. Tasan SO, Tunali S (2008) A review of the current applications of 1. 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Advances in ManufacturingSpringer Journals

Published: Jun 1, 2019

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