Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm

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1 Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm Binggang Wang, Yunqing Rao, Xinyu Shao, and Mengchang Wang The State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China ryq@hust.edu.cn Abstract. This paper is concerned about how to optimize the input sequence for a mixed-model assembly line (MMAL) with limited intermediate buffers. Three optimization objectives are considered simultaneously: minimizing the total production rate variation, the total setup, and the total assembly cost. The mathematical model is presented by incorporating the three objectives. Since the problem is NP-hard, a hybrid algorithm based on genetic algorithm (GA) and simulated annealing (SA), is proposed for solving the model. The performance of the proposed algorithm is compared with a genetic algorithm for different-sized sequencing problems in MMALs that consist of different number of machines and different production plans. The computational results show that the proposed hybrid algorithm finds solutions with better quality and often needs a smaller number of generations to converge to a final stable state, especially in the case of large-sized problems. Keywords: Mixed-model assembly line, Scheduling, Hybrid algorithm, Genetic algorithm. Introduction Mixed-model assembly line is a type of production line where various and different models of the same products are inter-mixed to be assembled on the same line. The MMAL considered in this paper can be described as follows: it is a conveyor system, in each stage of the line, there is only one machine. Between every two successive machines j and j-, there exists a buffer with the size B j, and jobs obey the FIFO (first in first out) rule in each buffer. Each job is to be sequentially processed on through the first machine to the last one. If there is no buffer space available for a completed job on one machine, then this job must remain on that machine until either a buffer space or the machine in the next stage becomes available. At any time, no job can be processed on more than one machine, while no machine can process more than one job simultaneously. In order to maximize the utilization of a MMAL, many researchers have investigated the sequencing problem. To make the sequencing results more reasonable and practicable, much research work has done on multi-criteria sequencing problem in MMAL. Bard et al. [] developed a mathematical model that involved two objectives: minimizing the overall line length and keeping a constant rate of parts usage, and a tabu search method was C. Xiong et al. (Eds.): ICIRA 2008, Part II, LNAI 535, pp , Springer-Verlag Berlin Heidelberg 2008

2 Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm 379 proposed for solving the model. Hyun et al. [2] considered three practically important objectives: minimizing total utility work, keeping a constant rate of parts usage and minimizing total setup cost, which was solved using genetic algorithm by proposing a new genetic evaluation and selection mechanism, called Pareto stratum-niche cubicle. A SA method was proposed by McMullen and Fraizer [3] for solving the model used by McMullen [4] and the SA was compared with the tabu search method in [4]. The same problem was also solved by McMullen [5-7] by using another three methods: genetic algorithms, Kohonen self-organizing map, and ant colony optimization, respectively, and also the three method were compared with the SA and the tabu search methods.to minimize the weighted sum of tardiness and earliness penalties and to balance the production flow of the flexible assembly line, Guo et al. [8] developed a bi-level genetic algorithm to solve the scheduling problem. Tavakkoli-Moghaddam et al. [9] applied a memetic algorithm to solve a multi-criteria sequencing problem model for a MMAL, which is a weighted sum of the three objectives: total utility work, total production rate variation, and total setup cost according to their relative importance weights. Yu et al. [0] proposed a multi-objective genetic algorithm to schedule a flexible assembly line with the objectives of leveling the parts usage rate and minimizing the makespan. It can be seen from above, though many research work has done on sequencing problem in MMALs, to our best knowledge, few work was on the sequencing problem in the MMAL described above with the three important optimization goals simultaneously: total production rate variation, total setup, and total assembly cost. In this paper, we consider the three objectives simultaneously. The remainder of this paper is organized as follows: Section 2 presents the mathematical models. The proposed hybrid algorithm is described in Section 3. Case studies and discussions are reported in Section 4. The last section is the conclusions. 2 Mathematical Models In this paper, the concept of minimum part set (MPS) is applied. MPS is a vector representing a product mix, such that (d, d 2,, d M )=(D /h, D 2 /h,, D M /h), where d m is the number of product type m to be assembled in one period, D m is the number of products type m needs to be assembled during the entire planning horizon, h is the greatest common divisor factor of D, D 2,, D M This strategy operates in a cyclical manner. 2. Minimizing the Total Production Rate Variation Cost This problem can be formalized as below, which is improved according to the model proposed by Miltenberg[]. Minimizing: x i I M i lm vim i= m= l= d I m ()

3 380 B. Wang et al. Subject to: M x = i, (2) im m= xim I im i= x = d m, (3) m i,m = 0 or where v im is the sequence-dependent production rate variation cost for the product model m in the ith position of the sequence. The first term in the objective function is the production ratio of model m until the ith product is produced. The second term is the demand ratio of model m. x im is if model m is assigned at the ith position in a sequence; Otherwise, x im is 0. Equation (2) is a set of position constraints indicating that every position in a sequence is occupied by exactly one product. Equation (3) imposes the restriction that all the demands must be satisfied in terms of MPS. 2.2 Minimizing the Total Setup Cost The mathematical model used by Tavakkoli-Moghaddam and Rahimi-Vahed [9] is adopted in this paper. Minimize: (4) Subject to: J I M M X C (5) j= i= m= r= imr jmr M M X im r = i, (6) m= r = M I M X im r = d m m, i= r = (7) X im r = 0, or, i, m, r (8) M X = X i=,..., I r, (9) imr ( i+ ) rp m= p= M M X = X r, (0) Imr rp m= p= where C jmr is the setup cost required when model type is changed from m to r at machine j and X imr is if model type m and r are assigned, respectively, at the ith and (i+)th position in a sequence; Otherwise, X imr is 0. Equation (6) is a set of position constraints indicating that every position in a sequence is occupied by exactly one product. Equation (7) imposes the restriction that all the demands must be satisfied in terms of MPS. Equation (9) and (0) ensure that the sequence of products must be maintained while repeating the cyclic production.

4 Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm Minimizing the Total Assembly Cost Extending from the models in literature [2], the mathematical model for this objective is formulated as follows: Minimize: Subject to: l ( S () + A PI J PI () J) () c S (2) = 0 P () S = S + A j = 2,3,... J (3) P () j P () j P () j { A ( ) A ( ) } B j + S = max S +, S + i = 2,3,... + (4) Pi () j Pi () j Pi () j Pi- j Pi- j { A ( ) A ( ) ( ) } B j + S = max S +, S +, S i > + (5) Pi () j Pi () j Pi () j Pi- j Pi- j Pi -B - j + where c l is the sequence-independent assembly cost per unit time. S P(I)J is the starting time of the last product P(I) processed on the last machine J. A P(I)J is the processing time of the last product on the last machine. j+ 3 Algorithms for Solving the Model 3. Notations POPSIZE P c P m l l max P a q q max α the maximum number of individuals in one population probability of crossover operation probability of mutation operation length of Markov chain maximum length of Markov chain probability of inferior solution being accepted times of the current best solution remains unchanged maximum times of the current best solution remains unchanged cooling rate. 3.2 Hybrid Algorithm Procedures The procedures of the proposed algorithm are described as follows. Step: Generate an initial population, Pop(0). Step2: Calculate each individual s fitness value in Pop(0), let the best solution and its objective function value be the global best S * and C *, respectively, determine the initial temperature t 0, and let generation g=0. Step3: If the terminating condition is satisfied, then output S * and C *, otherwise, continue the following steps.

5 382 B. Wang et al. Step4: Selection operations. Step5: Crossover operations. Step6: Mutation operations. Step7: Metropolis sampling process, using each individual in the temporary population after Step6 as the initial solution. Then a new temporary population, p t (g), is formed. Step8: Calculate each individual s fitness value in p t (g), and update S * and C * if necessary. Step9: Keep the best individual. Step0: Let g=j, t g =α t g-. go to Step Implementation of the Hybrid Algorithm The details of implementation of the proposed algorithm are described as follows. Encoding scheme: Job permutation based encoding scheme is widely used in many literatures for sequencing problem in MMALs, so it is also adopted in this paper. Initial population: To maintain large population diversity, individuals are generated randomly to form the initial population. When generating the individuals, the number of each type of products in each individual can not break the constraints on the demands in MPS. Fitness calculation: In order to let the individual with smaller objective function value has more chance to survive, the fitness value is calculated as follows: f(i,m,j)=/f(i,m,j). (6) where, f(i,m,j) and F(i,m,j) is the fitness and objective function value, respectively, of a feasible solution. Initial temperarure: Let Boltzman constant be, initial temperature can be calculated from the following [3]: to=δf/ln(p - a ). (7) ΔF=F ma [ F min. (8) where, F max and F min is the maximum and the minimum objective function value respectively in Pop(0). Termination criterion: For comparison s convenience, a maximum generation number, G, is set as the stopping criterion. Selection operation: Proportional selection based on fitness values is used in this paper. This process is repeated until POPSIZE solutions are selected. Crossover operation: The Modified order crossover (modox) operator is adopted in this paper. The parents for performing crossover operation are selected by probability, P c. After crossover operation, the best two among the four solutions, two parents and two offsprings, are selected to replace the original two parents. Mutation operation: To improve the solutions diversity, mutation operations are made on the selected individuals, controlled by probability, P m. Here, the INV mutation operator is adopted.

6 Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm 383 Metropolis sampling process: Using each individual, S, in the temporary population, p t (g), as the initial solution, a local search process is performed. The scheme has the following working steps: Let l=0, current best solution S * =S *, q=0, current state S (0)=S. Generate a new solution, S, from S using INV mutation operator described in Section 3.3.7, calculate ΔC =C(S C(S). If ΔC <0, accept S, let S * = S, q=0, moreover, if C(S * )> C(S ), let S * = S IfΔC >0, accept S by the probability of P a, if S is accepted, let S (l )=S, q=t, else let S (l )=S (l). /HWl l. If termination condition (q>q max or l>l max ) satisfied, continue the following steps, else go to. Replace S with S *. Keep the best: In order to avoid losting the best solution, after Step7, if C(S * )>C(S * ), let S * = S *, else, replace the worst solution (with the biggest objective value) in the population obtained after Step7 with S *. 3.4 GA Algorithm For testing the performance of the proposed hybrid algorithm, a genetic algorithm is designed. The procedures of the GA can be inherited from Section 3.2 by deleting Step7, determine the initial temperature t 0 in Step2 and t g =α t g- in Step0, and it can be implemented by deleting the corresponding parts in Section Case Studies and Discussions The proposed hybrid algorithm and the GA are coded in C++ and implemented on an IBM ThinkPad (Intel (R) Celeron (R) M, CPU.70 GHz, 768M). Various experiments have been conducted to measure the performance of the proposed hybrid algorithm and the GA. And, the performance has been compared with each other. Each experiment is repeated 30 times. The experimental data is listed in the following Table to Table 4.We make all the experiments under the following assumptions: All the sequence-dependent production rate variation costs, v im, are the same and equal to. The sequence-independent assembly cost per unit time, c l, is set to for the sake of computational convenience. All the buffer sizes are equal to. Table. MPS Products P P2 P3 P4 P5 MPS MPS MPS

7 384 B. Wang et al. Table 2. Setup cost Products P P2 P3 P4 P5 P P P P P Table 3. The assembly time for different products on different machines Product Machine code P P P P P Table 4. The algorithm parameters Parameters GASA GA POPSIZE G P c P m l max 20 none q max 3 none P a 0.5 none Firstly, the small-sized problem is considered, in which the total number of machines in the MMAL and that of the five types of products need to be assembled in one period are small. Assuming that the MMAL has 4 machines and the total demand is 0, as listed in MPS. Computational results by the hybrid algorithm and the GA are shown in Table 5, respectively. Fig. shows their convergence curves. It can be seen, for small-sized problems, the best solution obtained by the hybrid algorithm has.5 percent improvement over that obtained by the GA. From Fig., it can be also find that both the two algorithms can converge to the optimal within 8 generations. Secondly, when the number of machines in the MMAL and the total demand of all the products in MPS2 increase to 0 and 5, respectively, the computational results of the two algorithms are also listed in Table 5, and the convergence curves are demonstrated in Fig. 2. The solution quality of the hybrid algorithm is better than that of the GA by.63 percent. The number of generations needed to converge to the optimal is 22 for the hybrid algorithm and 87 for the GA.

8 Scheduling Mixed-Model Assembly Lines with Cost Objectives by a Hybrid Algorithm 385 Total objective function values Generations Hybrid algorithm for 4 machines,mps:(2,,3,2,2) GA for 4 machines,mps:(2,,3,2,2) Fig.. Convergence curves for 4 machines and MPS Total objective function values Generations Hybrid algorithm for 0 machines,mps2:(3,4,4, 2,2) GA for 0 machines,mps2:(3,4,4, 2,2) Fig. 2. Convergence curves for 0 machines and MPS2 Table 5. Computational results Total objective machines MPS function values Best solutions algorithm Hybrid GA Hybrid algorithm GA 4 (2,,3,2,2) (3,4,4,2,2) (3,5,2,6,4)

9 386 B. Wang et al. Total objective function values Generations Hybrid algorithm for 20 machines,mps3:(3,5,2,6,4) GA for 20 machines,mps3:(3,5,2,6,4) Fig. 3. Convergence curves for 20 machines and MPS3 Finally, for the large-sized case, there are 20 machines in the MMAL and the total demand in MPS3 is 20. The last line in Table 5 shows the computational results of the two algorithms. Fig.3 shows their convergence processes. The ratio of improvement increases to 2.57 percent in terms of solution quality and the number of generations needed for the hybrid algorithm to converge to the final stable state is much smaller than that for the GA, which is 20 for the former and 5 for the latter. To summarize, we can conclude that the hybrid algorithm also performs better than the GA. 5 Conclusions Considering three cost-related objectives, minimizing the total production rate variation, the total setup, and the total assembly cost, simultaneously, we describe the mathematical model for sequencing problem in MMAL with limited intermediate buffers. A hybrid algorithm is proposed to solve the model and the algorithm performance is compared with a GA. Three computational efforts are made by the two algorithms for different sequencing problems in MMALs, each has different MPS and different number of machines. The computational results show that the hybrid algorithm can always converge to the final stable state within a smaller number of generations than the GA, especially in the case of large-sized problems and the proposed hybrid algoritm performs better than the GA in terms of the solution quality for all the test problems, and we can have more improvement as the problem size increases. Acknowledgments. This research work is supported by 863 High Technology Plan Foundation of China under Grant No.$$= and National Natural Science Foundation of China under Grant No

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