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Design of random inspection rate for a flexible assembly system: a heuristic genetic algorithm approach
PII:
Microelectron. Reliab., Vol. 38, No. 4, pp. 545±551, 1998 # 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0026-2714/98 $19.00 + 0.00 S0026-2714(97)00216-3
DESIGN OF RANDOM INSPECTION RATE FOR A FLEXIBLE ASSEMBLY SYSTEM: A HEURISTIC GENETIC ALGORITHM APPROACH CHINHO LIN, JONG-MAU YEH and JIEH-RERN DING Graduate School of Industrial Management, National Cheng Kung University, Tainan, Taiwan 70101, Republic of China (Received 15 September 1996; in revised form 27 October 1997) AbstractÐBecause of parts failure in a ¯exible assembly system (FAS), in-process quality control is necessary. To maximize pro®t, it is important to derive the inspection rate for each cell. This paper proposes an analytical model using an open queuing network theory for an FAS with feedback and reworking. The problem is solved by a modi®ed genetic algorithm which has two major characteristics. First, the probability of mutation is variable rather than ®xed in processing the search. Second, a speci®ed mutation operator with calculating decoded values is proposed according to the status of the solution. These modi®cations enhance the eciency and accuracy of the algorithm in empirical experiments. # 1998 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
A ¯exible manufacturing system (FMS) is a necessary response to intensi®ed rapid technological changes and the shortening of life cycles of products [1, 16]. In order to estimate the performance of this manufacturing system, the queuing network theory has been used extensively [2, 16]. If a ¯exible assembly (FAS) of manufacturing is unreliable, the failed operation will cause products to fail, which in turn will increase costs [13, 16, 17]. If an inspection of the system is to be taken, then an in-process quality control strategy is necessary [12, 13]. To maximize total pro®t, it is important to determine the inspection rate for each cell. T`his paper proposes an analytical approach that combines open queuing network theory, mathematical programming, and a genetic algorithm for the FAS. Because a small shift in inspection rate in any one manufacturing cell will force a serial cost change in every unit, calculating its eort is too complex to solve directly. Therefore, Lin et al. [17] proposed an enumerative search procedure. In their one-step-at-a-time algorithm, one of the inspection rates shifts small steps from zero to one in every iteration, and all feasible inspection rate combinations must be tested to ®nd an optimal solution. As a result, this algorithm takes too much time in calculation to ®nd the optimal solution; that is, in order to improve the accuracy of the solution, the time spent on the calculations must increase rapidly. Therefore, this paper proposes a modi®ed genetic algorithm that searches for the feasible region of 545
inspection rate ®rst, so that a near optimal solution can be found in a short period of time with fewer calculations.
RESEARCH METHODOLOGY
In this section, we ®rst de®ne the functions of an unreliable manufacturing cell and the movements of parts in the cell, and then we derive preliminary results to characterize the entire FAS, based on the model proposed by Lin [16] and Lin et al. [17]. We next propose the theory of a genetic algorithm and discuss its relevant applications. An unreliable manufacturing cell with inspection, reworking and scrapping The FAS that we consider consists of several manufacturing cells with processing, inspection, and reworking functions in each cell. Parts of dierent types will ¯ow through various cells in the FAS [13, 16, 17] following pre-speci®ed routes. A part will leave the cell if it passes or skips inspection. If a defect is detected, two possibilities arise. When the defect is due to processing in the current cell, the unit is routed to the reworking area and then is recirculated back to the working area for future processing. Otherwise, it is scrapped. The ¯ow is shown in Fig. 1 [16, 17]. To construct a pro®t-maximization model that seeks a near-optimal quality control policy for an unreliable FAS, the operational parameters of the manufacturing cell are de®ned as follows:
546
Chinho Lin et al.
To evaluate the performance of the FAS in terms of random inspection rate, one needs to derive the formulae of the K(N), tN, VN, ty, Vy, (ERk)nk(l), (EDk)nk(l), (EFk)nk(l), (Dk)nk(l), (ESk)nk(l), (AOQk)nk(l), and Wk,nk(l). All of these formulae can be found in the studies of Lin (1992) and Lin et al. (1994). Relevant revenue and costs de®ne as follows:
REVk: CIkl:
the revenue for a type-k part, the cost of inspecting a type-k part in its lth visitation cell, the waiting cost of a type-k part per unit time, the post-sales failure cost of a type-k part, the cost of scrapping a type-k part in its lth visitation cell, the cost of reworking a type-k part in its lth visitation cell, and the cost of processing a type-k part in its lth visitation cell.
CWk:
Fig. 1. Flow of parts in a cell of FAS.
CFk: CSkl: CRkl: CPkl:
r: D:
AOQÿ1: q: t,V: N: K(N),tN,VN: y: ty,Vy: R: M: T: lk: V k: qk,nk(l): (ERk)nk(l): (ESk)nk(l): (EDk)nk(l): (EFk)nk(l):
(Dk)nk(l): (AOQk)nk(l): Wk,nk(l):
the inspection percentage at the manufacturing cell, 0E rE1, the defective rate of parts before inspection, including the defects that are made in the current cell and in the preceding cells, 0E DE1, the average outgoing quality of the immediately preceding cell (i.e. the defective rate of parts entering the current cell), probability that the work in the current cell is defective, 0E qE1, the mean and variance of working time and reworking time, the number of times a part passes through the working area, the probability density function, mean and variance of N, respectively, the actual total time (i.e. working time, reworking time and inspect time) a part spends in the cell, the mean and variance of y, respectively, the number of part types to be processed by the FAS, the number of manufacturing cells in the FAS, the inspection time (a constant), the arrival rate of type-k parts, the pre-speci®ed visitation sequence (process routing) of type-k parts, the probability of manufacturing a defective type-k part in the Vk(l) visitation cell, 0E qk,nk(l)E1, the expected number of type-k parts that will be reworked in the lth visitation cell per unit time, the expected number of type-k parts that will be scrapped in the lth visitation cell per unit time, the expected number of undetected defective type-k parts exiting from the lth visitation cell per unit time, the expected ¯ow of type-k parts exiting from the lth visitation cell (and entering the next cell in the visitation sequence) per unit time, the probability that a type-k part is defective after processing and prior to inspection in its lth visitation cell, the average outgoing quality of type-k parts from their lth visitation cell per unit time, the expected total time a type-k parts spends in its lth visitation cell,
Let rkj (the inspection rate for type-k parts in the Vk(l)) be a decision variable, the pro®t-maximizing model that incorporates the random inspection into the FAS is as follows: Maximize R X
Jk
1
k1
where Jk REVk EFk k zk ÿ CIkl lk ERk k 1 rk;k 1 ÿ
zk X l2
CIkl EFk k lÿ1 ERk k l rk;k l
ÿ CWk lk ERk k 1 Wk;k 1 ÿ CWk
zk X EFk k lÿ1 ERk k l l2
Wk;k l ÿ CFk AOQk EFk k zk ÿ
zk X CSkl ESk k l ÿ CPk1 l1
lk ERk k 1 ÿ ERk k l ÿ
zk X
zk X l1
l2
CPkl EFk k lÿ1
CRkl ERk k l
2
and 0 rkj 1; for all k and j:
3
The pro®t generated by type-k parts, which is the total revenue (the ®rst term) minus inspection costs (the 2nd and 3rd terms), waiting costs (the 4th and 5th terms), post-sales failure costs (the 6th term), scrapping costs (the 7th term), processing costs (the
Random inspection rate for a ¯exible assembly system
8th and 9th term), and reworking costs (the last term). The model is a complex nonlinear optimization problem that becomes more cumbersome when we attempt to incorporate the results of the queuing network. Lin [16] and Lin et al. [17] implemented an enumerative search procedure to obtain approximate solutions of random inspection rates (i.e. rk,VK(l)). Since searching all feasible solutions to ®nd the near optimal solution takes a great eort, and since the executing times explode rapidly when greater accuracy is needed, we employ genetic algorithms to solve it.
547
bound to the loops or until further improvement fails over a given period of time. The genetic algorithm is a stochastic searching method. It hops randomly from point to point, thus allowing the solutions to escape from the local optimum in which other algorithms might land. This ability to escape the local optimum depends on the probability of reproduction, crossover, and mutation. Their values are very sensitive to the performance of implementing the GA. There is no best approach to set these values [7, 9], but only tedious trial and error, because the parameters depend greatly on the problem characteristics.
Genetic algorithm In a genetic algorithm, an optimal solution is derived by applying the following three operators: reproduction, crossover, and mutation [7, 11, 18]. This is a structured random search methodology which is based on the laws of natural selection (survival of the ®tness) and the genetic evolutionary process. Previously, the genetic algorithm has been used to solve many complicated problems: dispatch [3, 4], unit commitment [14], scheduling [6, 10, 22], TSP [5, 9], machine layout [8], and assembly-line balancing [20]. At ®rst, a population of M solutions encoded in ®xed-length bit-strings set of initialization chromosomes is chosen randomly. Every chromosome represents either 0 or 1 randomly. Each member of the population is then decoded to a real problem solution, and a ®tness value is assigned to it by a quality function (in this study, total pro®t function) that gives a measurement of the solution quality. The reproduction operator selects speci®c strings to construct the next generation according to their ®tness function value. There are many ways to implement this, One is to construct a roulette wheel where each string has a route slot sized in proportion to its population [11]. This means that the strings with a better-®tted value have a higher probability of being selected and contribute more in the next generation. The crossover operator then tries to rearrange the population in order to produce a new ospring. First, a mating pool is selected from the population. Second, a position index (k) among 1 and the string length l is randomly selected. Third, two new strings are created by swapping all characters between (k + 1) and l. A mutation operator tries to escape the local optimum by tuning up the chromosome value randomly. A mutation is the occasional (with small probability) random alteration of the value of the position of a string. This means changing a 1 to a 0 and vice versa. A new population is generated by these three operations; then the average ®tness is calculated to check for a converge condition. All these procedures are repeated until reaching a given upper
A REVISED GENETIC ALGORITHM
To improve the eciency of the algorithm two modi®ed steps in mutation operator are proposed in this paper: (1) a mutation operator with various mutation probability, and (2) a mutation operator with calculated decoded values. Mutation operator with various mutation probability The mutation probability is a key parameter for reducing the chances to land on a local optimum or to jump a pre-mature converge. While the search operation goes on, the mutation probability should vary, depending on a cooling schedule that has been used in simulated annealing algorithm [15, 22]. Simulated annealing is an optimization technique that simulates the physical annealing process. The mutation probability is given by the following expression [15]: Pr mutation exp ÿDf=Tk
4
where Df is the increment of ®tness function of new solution compared to the current solution and Tk is the temperature at the kth iteration that is reduced according to a cooling schedule. The following geometric cooling schedule is adopted in this paper [15]: Tk r kÿ1 T0
5
where T0 is the initial temperature and its value can be determined by trial and error method. r is the temperature reduction rate. Through an experimental study conducted previously, we determined the values of T0 and r equal to 100 and 0.9, respectively. Mutation operator with calculated decoded values If that products are produced through a sequence of operation cells, there will be more defective units ¯ow to its next cell when the inspection rate in one cell is low. Therefore, the next cell's inspection rate
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Chinho Lin et al.
Fig. 2. Sensitivity analysis for costs.
must increase to avoid the increase of its failure cost. But the increasing inspection rate of the next cell will generate more inspection costs. Thus, to minimize the total cost, we have to shift the inspection rate among the cells of the FAS depending on the tradeo between these costs. A sensitivity analysis is performed to keep track of these costs. We set inspection rate in a cell raising from 0 to 1 and ®xed the other connected cells accordingly. As shown in Fig. 2, when inspection rate increases, the total inspection cost and total reworking cost increase in a linear form; the total failure cost decreases in a linear form; and the impacts on the total scrapping cost as well as the total working cost in dierent cells vary. Since the total pro®t is the linear combination of these costs, the maximum pro®t generally takes place when the inspection rate is near to 0 or 1. To con®rm this conclusion, we did a sensitivity analysis by varying these cost. Because there are 0's or 1's in the near optimal solution, this property can be incorporated in our algorithm to enhance the eciency. The steps of the mutation operator are as follows: (1) select a randomly position index (k) among 1 and the string length l, (2) decode the string to decimal value, and (3) if the decimal value is bigger than 0.95 (or less than 0.05), then set the
value of the string on position k to 1 (or to 0); Otherwise, alter the bit value of string position k, i.e. change a 1 to a 0 and vice versa. The steps of our algorithm are shown in Fig. 3. A NUMERICAL EXAMPLE
Consider an FAS with ®ve cells where each cell contains two operating areas: one for working (which may contain an inspection station) and the other for reworking. Three types of parts are processed by Vÿ1=(1, 2, 3, 5), Vÿ2=(1, 3, 5, 4), and Vÿ3=(1, 2, 4), respectively. The process is shown in Fig. 4. Other relevant data are given in Table 1. The object is to try to derive the inspection rate for each cell to maximize the total pro®t. The precision of the solution depends upon how many bits are used to represent inspection rate. In this paper, we use a 10-digit binary string to represent the random inspection rate. For example, a binary string (0101011000)2 is decoded to 0.336 in decimal: 01010110002 0 29 1 28 0 27 1 26 0 25 1 24 1 23 0 22 0 21 0 20 =1023 = 0:336
Random inspection rate for a ¯exible assembly system
549
Fig. 3. Flow diagram of the algorithm.
Every chromosome is constructed with a 50-digit binary string which represents the inspection rates of ®ve cells in the FAS. Evaluation of the chromosome is accomplished by decoding the chromosome string and computing the ®tness value. We set the converging condition to be the maximum times of
iteration as 100, or the improvement of the average ®tness values between two consecutive iterations as less than 0.02. Certain parameters must be selected in the initialization of every GA problem. The population size must be determined by conducting experiments
550
Chinho Lin et al. Table 1. Data of numerical example
Product station
1
tM VM tR VR Defective rate Inspection cost Scrapped cost Reworking cost Working cost Inspection time Arrival rate Failure cost Revenue Waiting cost
0.01 0.005 0.005 0.005 0.01 16 Ð 15 12 0.0005
Product 1 2 3 0.005 0.006 0.001 0.001 0.12 16 30 15 14 0.0006 20 220 2280 1.5
0.01 0.006 0.009 0.004 0.06 15 50 10 12 0.0003
1
0.01 0.005 0.004 0.001 0.05 14 70 12 16 0.0004
0.015 0.008 0.005 0.006 0.09 10 Ð 10 12 0.0005
which depend signi®cantly on the problem being solved. No single value is suitable for dierent systems [11, 18]. In the study, the size of the population is set at 50, with crossover probability to 0.6, and the mutation probability varies according to Equations (4) and (5), with T0 equal to 100 and r equation to 0.9. We solved the problem by running on a IBM-AT compatible (Pentium 120) using a C interpreter. The near optimal solutions are r* = (0.2, 1.0, 1.0, 0.0, 1.0) with total pro®t 10044.79. For comparison, we also performed the calculations by using a conventional genetic algorithm. To avoid misleading results due to the stochastic nature of the genetic algorithm, we conducted 100 runs for each algorithm, with each run starting with dierent random population. As Table 2 shows, 29 runs of the conventional genetic algorithm (CGA) cannot converge within maximum iterations. But all runs of our algorithm (MGA) are convergent. Also, no runs of CGA can attain the near optimal solution (i.e. the total pro®t of solutions bigger than 10044.79); on the other hand, all runs of MGA are able to ®nd the near optimal solutions. The average execution times for CGA and MGA are 15.083 and 6.868 seconds, respectively. This proves that MGA obtains better solutions with great accuracy and eciency.
Table 2. The results of 100 experiments of CGA and MGA CGA Number of converge Mean of ®tness Variance of ®tness Mean of iterations Variance of iterations Mean of executing time Variance of executing time
Product 2 3 5
5
MGA
71 10041.013 4.051 65.540 21.380 15.083
100 10044.790 0.000 29.333 2.559 6.868
4.865
0.587
0.006 0.009 0.008 0.001 0.1 8 35 12 15 0.0003 30 230 200 2
0.005 0.006 0.004 0.0004 0.06 10 50 18 14 0.0004
4
1
0.01 0.007 0.002 0.0002 0.09 9 70 20 15 0.0005
0.01 0.0008 0.002 0.0005 0.01 12 Ð 25 16 0.0005
Product 3 2 0.003 0.0005 0.001 0.0003 0.05 10 15 20 18 0.0006 20 450 350 2.5
4 0.007 0.0008 0.002 0.0008 0.06 11 16 18 20 0.0005
CONCLUSION
The FAS is a powerful tool for rapid environment change by rearranging its production schedule and dispatch in a short time. The system becomes more complex when the number of cells in the FAS increases, making it more dicult to achieve the best performance. In this paper, we modify the genetic algorithm to obtain the operational characteristics of an FAS with in-process quality control. This helps system managers to handle this system in order to achieve the best performance. Despite its limitations, a genetic algorithm is an ecient search algorithm for solving a complicated problem. Hence, we propose a modi®ed genetic algorithm by varying the mutation probability and calculating the decoded values to ®nd the near optimal inspection rate in the FAS by which defective parts might be produced. The experiment shows that the proposed modi®ed algorithm achieves a great improvement in accuracy and eciency.
REFERENCES 1. Buzacott, J. A. and Yao, D. D., Management Science, 1986a, 32(9), 890±905. 2. Buzacott, J. A. and Yao, D. D., Queuing Systems, 1986b, 1(4), 5±27. 3. Chen, P. and Chang, H., IEEE Trans. on Power Systems, 1995, 10(4), 1919±1926. 4. Chen, C., Vempati, V. S. and Aljaber, N., European Journal of Operational Research, 1995, 80, 389±396. 5. Cheng, R., Gen, M. and Sasaki, M., Computers and Industrial Engineering, 1995a, 29(14), 549±553. 6. Cheng, R., Gen, M. and Tozawa, T., Computers and Industrial Engineering, 1995b, 29(14), 513±517. 7. Davis, L., Handbook of Genetic Algorithms, Van Nostrand, New York, 1991. 8. Gen, M., Ida, K. and Cheng, C., Computers and Industrial Engineering, 1995a, 29(14), 519±523. 9. Gen, M., Ida, K., Li, Y. and Kubota, E., Computers and Industrial Engineering, 1995b, 29(14), 537±541. 10. Gilkinson, J. C., Rabelo, L. C. and Bush, B. O. Computers and Industrial Engineering, 1995, 29(14), 177±181.
Random inspection rate for a ¯exible assembly system 11. Goldberg, D. E., Genetic Algorithms in Search, Optimization & Machine Learning, Addison-Wesley, Reading, MA, 1989. 12. Hsu, L. F. and Tapiero, C. S., Naval Research Logistics, 1988, 35(3), 327±343. 13. Hsu, L. F. and Tapiero, C. S., The International Journal of Flexible Manufacturing Systems, 1989, 1(1), 325±346. 14. Kazarlis, S. A., Bakirtzis, A. G. and Petridis, V. IEEE Trans. on Power Systems, 1996, 11(1), 83±92. 15. Kirkpatrick, S., Cwlatt, C. D. Jr. and Vecchi, M. P., Science, 1983, 220(4598), 671±680. 16. Lin, C., Dissertation for the degree of Doctor of Philosophy, The City University of New York, 1992, pp. 5±13.
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17. Lin, C., Chien, T. W. and Hsu, L. F., Journal of Information & Optimization Sciences, 1994, 15(3), 311±336. 18. Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs, Springer, Berlin, 1992. 19. Shanthikumar, J. G. and Buzacott, J. A., International Journal of Production Research, 1981, 19(3), 255±266. 20. Tsujimura, Y., Gen, M. and Kubota, E., Computers and Industrial Engineering, 1995, 29(14), 543±547. 21. Whitt, W., Bell System Technical Journal, 1983, 62(9), 2779±2815. 22. Wong, K. P. and Wong, Y. W., IEEE Trans. on Power Systems, 1996, 11(1), 128±136.
Microelectron. Reliab., Vol. 38, No. 4, pp. 545±551, 1998 # 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0026-2714/98 $19.00 + 0.00 S0026-2714(97)00216-3
DESIGN OF RANDOM INSPECTION RATE FOR A FLEXIBLE ASSEMBLY SYSTEM: A HEURISTIC GENETIC ALGORITHM APPROACH CHINHO LIN, JONG-MAU YEH and JIEH-RERN DING Graduate School of Industrial Management, National Cheng Kung University, Tainan, Taiwan 70101, Republic of China (Received 15 September 1996; in revised form 27 October 1997) AbstractÐBecause of parts failure in a ¯exible assembly system (FAS), in-process quality control is necessary. To maximize pro®t, it is important to derive the inspection rate for each cell. This paper proposes an analytical model using an open queuing network theory for an FAS with feedback and reworking. The problem is solved by a modi®ed genetic algorithm which has two major characteristics. First, the probability of mutation is variable rather than ®xed in processing the search. Second, a speci®ed mutation operator with calculating decoded values is proposed according to the status of the solution. These modi®cations enhance the eciency and accuracy of the algorithm in empirical experiments. # 1998 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
A ¯exible manufacturing system (FMS) is a necessary response to intensi®ed rapid technological changes and the shortening of life cycles of products [1, 16]. In order to estimate the performance of this manufacturing system, the queuing network theory has been used extensively [2, 16]. If a ¯exible assembly (FAS) of manufacturing is unreliable, the failed operation will cause products to fail, which in turn will increase costs [13, 16, 17]. If an inspection of the system is to be taken, then an in-process quality control strategy is necessary [12, 13]. To maximize total pro®t, it is important to determine the inspection rate for each cell. T`his paper proposes an analytical approach that combines open queuing network theory, mathematical programming, and a genetic algorithm for the FAS. Because a small shift in inspection rate in any one manufacturing cell will force a serial cost change in every unit, calculating its eort is too complex to solve directly. Therefore, Lin et al. [17] proposed an enumerative search procedure. In their one-step-at-a-time algorithm, one of the inspection rates shifts small steps from zero to one in every iteration, and all feasible inspection rate combinations must be tested to ®nd an optimal solution. As a result, this algorithm takes too much time in calculation to ®nd the optimal solution; that is, in order to improve the accuracy of the solution, the time spent on the calculations must increase rapidly. Therefore, this paper proposes a modi®ed genetic algorithm that searches for the feasible region of 545
inspection rate ®rst, so that a near optimal solution can be found in a short period of time with fewer calculations.
RESEARCH METHODOLOGY
In this section, we ®rst de®ne the functions of an unreliable manufacturing cell and the movements of parts in the cell, and then we derive preliminary results to characterize the entire FAS, based on the model proposed by Lin [16] and Lin et al. [17]. We next propose the theory of a genetic algorithm and discuss its relevant applications. An unreliable manufacturing cell with inspection, reworking and scrapping The FAS that we consider consists of several manufacturing cells with processing, inspection, and reworking functions in each cell. Parts of dierent types will ¯ow through various cells in the FAS [13, 16, 17] following pre-speci®ed routes. A part will leave the cell if it passes or skips inspection. If a defect is detected, two possibilities arise. When the defect is due to processing in the current cell, the unit is routed to the reworking area and then is recirculated back to the working area for future processing. Otherwise, it is scrapped. The ¯ow is shown in Fig. 1 [16, 17]. To construct a pro®t-maximization model that seeks a near-optimal quality control policy for an unreliable FAS, the operational parameters of the manufacturing cell are de®ned as follows:
546
Chinho Lin et al.
To evaluate the performance of the FAS in terms of random inspection rate, one needs to derive the formulae of the K(N), tN, VN, ty, Vy, (ERk)nk(l), (EDk)nk(l), (EFk)nk(l), (Dk)nk(l), (ESk)nk(l), (AOQk)nk(l), and Wk,nk(l). All of these formulae can be found in the studies of Lin (1992) and Lin et al. (1994). Relevant revenue and costs de®ne as follows:
REVk: CIkl:
the revenue for a type-k part, the cost of inspecting a type-k part in its lth visitation cell, the waiting cost of a type-k part per unit time, the post-sales failure cost of a type-k part, the cost of scrapping a type-k part in its lth visitation cell, the cost of reworking a type-k part in its lth visitation cell, and the cost of processing a type-k part in its lth visitation cell.
CWk:
Fig. 1. Flow of parts in a cell of FAS.
CFk: CSkl: CRkl: CPkl:
r: D:
AOQÿ1: q: t,V: N: K(N),tN,VN: y: ty,Vy: R: M: T: lk: V k: qk,nk(l): (ERk)nk(l): (ESk)nk(l): (EDk)nk(l): (EFk)nk(l):
(Dk)nk(l): (AOQk)nk(l): Wk,nk(l):
the inspection percentage at the manufacturing cell, 0E rE1, the defective rate of parts before inspection, including the defects that are made in the current cell and in the preceding cells, 0E DE1, the average outgoing quality of the immediately preceding cell (i.e. the defective rate of parts entering the current cell), probability that the work in the current cell is defective, 0E qE1, the mean and variance of working time and reworking time, the number of times a part passes through the working area, the probability density function, mean and variance of N, respectively, the actual total time (i.e. working time, reworking time and inspect time) a part spends in the cell, the mean and variance of y, respectively, the number of part types to be processed by the FAS, the number of manufacturing cells in the FAS, the inspection time (a constant), the arrival rate of type-k parts, the pre-speci®ed visitation sequence (process routing) of type-k parts, the probability of manufacturing a defective type-k part in the Vk(l) visitation cell, 0E qk,nk(l)E1, the expected number of type-k parts that will be reworked in the lth visitation cell per unit time, the expected number of type-k parts that will be scrapped in the lth visitation cell per unit time, the expected number of undetected defective type-k parts exiting from the lth visitation cell per unit time, the expected ¯ow of type-k parts exiting from the lth visitation cell (and entering the next cell in the visitation sequence) per unit time, the probability that a type-k part is defective after processing and prior to inspection in its lth visitation cell, the average outgoing quality of type-k parts from their lth visitation cell per unit time, the expected total time a type-k parts spends in its lth visitation cell,
Let rkj (the inspection rate for type-k parts in the Vk(l)) be a decision variable, the pro®t-maximizing model that incorporates the random inspection into the FAS is as follows: Maximize R X
Jk
1
k1
where Jk REVk EFk k zk ÿ CIkl lk ERk k 1 rk;k 1 ÿ
zk X l2
CIkl EFk k lÿ1 ERk k l rk;k l
ÿ CWk lk ERk k 1 Wk;k 1 ÿ CWk
zk X EFk k lÿ1 ERk k l l2
Wk;k l ÿ CFk AOQk EFk k zk ÿ
zk X CSkl ESk k l ÿ CPk1 l1
lk ERk k 1 ÿ ERk k l ÿ
zk X
zk X l1
l2
CPkl EFk k lÿ1
CRkl ERk k l
2
and 0 rkj 1; for all k and j:
3
The pro®t generated by type-k parts, which is the total revenue (the ®rst term) minus inspection costs (the 2nd and 3rd terms), waiting costs (the 4th and 5th terms), post-sales failure costs (the 6th term), scrapping costs (the 7th term), processing costs (the
Random inspection rate for a ¯exible assembly system
8th and 9th term), and reworking costs (the last term). The model is a complex nonlinear optimization problem that becomes more cumbersome when we attempt to incorporate the results of the queuing network. Lin [16] and Lin et al. [17] implemented an enumerative search procedure to obtain approximate solutions of random inspection rates (i.e. rk,VK(l)). Since searching all feasible solutions to ®nd the near optimal solution takes a great eort, and since the executing times explode rapidly when greater accuracy is needed, we employ genetic algorithms to solve it.
547
bound to the loops or until further improvement fails over a given period of time. The genetic algorithm is a stochastic searching method. It hops randomly from point to point, thus allowing the solutions to escape from the local optimum in which other algorithms might land. This ability to escape the local optimum depends on the probability of reproduction, crossover, and mutation. Their values are very sensitive to the performance of implementing the GA. There is no best approach to set these values [7, 9], but only tedious trial and error, because the parameters depend greatly on the problem characteristics.
Genetic algorithm In a genetic algorithm, an optimal solution is derived by applying the following three operators: reproduction, crossover, and mutation [7, 11, 18]. This is a structured random search methodology which is based on the laws of natural selection (survival of the ®tness) and the genetic evolutionary process. Previously, the genetic algorithm has been used to solve many complicated problems: dispatch [3, 4], unit commitment [14], scheduling [6, 10, 22], TSP [5, 9], machine layout [8], and assembly-line balancing [20]. At ®rst, a population of M solutions encoded in ®xed-length bit-strings set of initialization chromosomes is chosen randomly. Every chromosome represents either 0 or 1 randomly. Each member of the population is then decoded to a real problem solution, and a ®tness value is assigned to it by a quality function (in this study, total pro®t function) that gives a measurement of the solution quality. The reproduction operator selects speci®c strings to construct the next generation according to their ®tness function value. There are many ways to implement this, One is to construct a roulette wheel where each string has a route slot sized in proportion to its population [11]. This means that the strings with a better-®tted value have a higher probability of being selected and contribute more in the next generation. The crossover operator then tries to rearrange the population in order to produce a new ospring. First, a mating pool is selected from the population. Second, a position index (k) among 1 and the string length l is randomly selected. Third, two new strings are created by swapping all characters between (k + 1) and l. A mutation operator tries to escape the local optimum by tuning up the chromosome value randomly. A mutation is the occasional (with small probability) random alteration of the value of the position of a string. This means changing a 1 to a 0 and vice versa. A new population is generated by these three operations; then the average ®tness is calculated to check for a converge condition. All these procedures are repeated until reaching a given upper
A REVISED GENETIC ALGORITHM
To improve the eciency of the algorithm two modi®ed steps in mutation operator are proposed in this paper: (1) a mutation operator with various mutation probability, and (2) a mutation operator with calculated decoded values. Mutation operator with various mutation probability The mutation probability is a key parameter for reducing the chances to land on a local optimum or to jump a pre-mature converge. While the search operation goes on, the mutation probability should vary, depending on a cooling schedule that has been used in simulated annealing algorithm [15, 22]. Simulated annealing is an optimization technique that simulates the physical annealing process. The mutation probability is given by the following expression [15]: Pr mutation exp ÿDf=Tk
4
where Df is the increment of ®tness function of new solution compared to the current solution and Tk is the temperature at the kth iteration that is reduced according to a cooling schedule. The following geometric cooling schedule is adopted in this paper [15]: Tk r kÿ1 T0
5
where T0 is the initial temperature and its value can be determined by trial and error method. r is the temperature reduction rate. Through an experimental study conducted previously, we determined the values of T0 and r equal to 100 and 0.9, respectively. Mutation operator with calculated decoded values If that products are produced through a sequence of operation cells, there will be more defective units ¯ow to its next cell when the inspection rate in one cell is low. Therefore, the next cell's inspection rate
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Fig. 2. Sensitivity analysis for costs.
must increase to avoid the increase of its failure cost. But the increasing inspection rate of the next cell will generate more inspection costs. Thus, to minimize the total cost, we have to shift the inspection rate among the cells of the FAS depending on the tradeo between these costs. A sensitivity analysis is performed to keep track of these costs. We set inspection rate in a cell raising from 0 to 1 and ®xed the other connected cells accordingly. As shown in Fig. 2, when inspection rate increases, the total inspection cost and total reworking cost increase in a linear form; the total failure cost decreases in a linear form; and the impacts on the total scrapping cost as well as the total working cost in dierent cells vary. Since the total pro®t is the linear combination of these costs, the maximum pro®t generally takes place when the inspection rate is near to 0 or 1. To con®rm this conclusion, we did a sensitivity analysis by varying these cost. Because there are 0's or 1's in the near optimal solution, this property can be incorporated in our algorithm to enhance the eciency. The steps of the mutation operator are as follows: (1) select a randomly position index (k) among 1 and the string length l, (2) decode the string to decimal value, and (3) if the decimal value is bigger than 0.95 (or less than 0.05), then set the
value of the string on position k to 1 (or to 0); Otherwise, alter the bit value of string position k, i.e. change a 1 to a 0 and vice versa. The steps of our algorithm are shown in Fig. 3. A NUMERICAL EXAMPLE
Consider an FAS with ®ve cells where each cell contains two operating areas: one for working (which may contain an inspection station) and the other for reworking. Three types of parts are processed by Vÿ1=(1, 2, 3, 5), Vÿ2=(1, 3, 5, 4), and Vÿ3=(1, 2, 4), respectively. The process is shown in Fig. 4. Other relevant data are given in Table 1. The object is to try to derive the inspection rate for each cell to maximize the total pro®t. The precision of the solution depends upon how many bits are used to represent inspection rate. In this paper, we use a 10-digit binary string to represent the random inspection rate. For example, a binary string (0101011000)2 is decoded to 0.336 in decimal: 01010110002 0 29 1 28 0 27 1 26 0 25 1 24 1 23 0 22 0 21 0 20 =1023 = 0:336
Random inspection rate for a ¯exible assembly system
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Fig. 3. Flow diagram of the algorithm.
Every chromosome is constructed with a 50-digit binary string which represents the inspection rates of ®ve cells in the FAS. Evaluation of the chromosome is accomplished by decoding the chromosome string and computing the ®tness value. We set the converging condition to be the maximum times of
iteration as 100, or the improvement of the average ®tness values between two consecutive iterations as less than 0.02. Certain parameters must be selected in the initialization of every GA problem. The population size must be determined by conducting experiments
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Chinho Lin et al. Table 1. Data of numerical example
Product station
1
tM VM tR VR Defective rate Inspection cost Scrapped cost Reworking cost Working cost Inspection time Arrival rate Failure cost Revenue Waiting cost
0.01 0.005 0.005 0.005 0.01 16 Ð 15 12 0.0005
Product 1 2 3 0.005 0.006 0.001 0.001 0.12 16 30 15 14 0.0006 20 220 2280 1.5
0.01 0.006 0.009 0.004 0.06 15 50 10 12 0.0003
1
0.01 0.005 0.004 0.001 0.05 14 70 12 16 0.0004
0.015 0.008 0.005 0.006 0.09 10 Ð 10 12 0.0005
which depend signi®cantly on the problem being solved. No single value is suitable for dierent systems [11, 18]. In the study, the size of the population is set at 50, with crossover probability to 0.6, and the mutation probability varies according to Equations (4) and (5), with T0 equal to 100 and r equation to 0.9. We solved the problem by running on a IBM-AT compatible (Pentium 120) using a C interpreter. The near optimal solutions are r* = (0.2, 1.0, 1.0, 0.0, 1.0) with total pro®t 10044.79. For comparison, we also performed the calculations by using a conventional genetic algorithm. To avoid misleading results due to the stochastic nature of the genetic algorithm, we conducted 100 runs for each algorithm, with each run starting with dierent random population. As Table 2 shows, 29 runs of the conventional genetic algorithm (CGA) cannot converge within maximum iterations. But all runs of our algorithm (MGA) are convergent. Also, no runs of CGA can attain the near optimal solution (i.e. the total pro®t of solutions bigger than 10044.79); on the other hand, all runs of MGA are able to ®nd the near optimal solutions. The average execution times for CGA and MGA are 15.083 and 6.868 seconds, respectively. This proves that MGA obtains better solutions with great accuracy and eciency.
Table 2. The results of 100 experiments of CGA and MGA CGA Number of converge Mean of ®tness Variance of ®tness Mean of iterations Variance of iterations Mean of executing time Variance of executing time
Product 2 3 5
5
MGA
71 10041.013 4.051 65.540 21.380 15.083
100 10044.790 0.000 29.333 2.559 6.868
4.865
0.587
0.006 0.009 0.008 0.001 0.1 8 35 12 15 0.0003 30 230 200 2
0.005 0.006 0.004 0.0004 0.06 10 50 18 14 0.0004
4
1
0.01 0.007 0.002 0.0002 0.09 9 70 20 15 0.0005
0.01 0.0008 0.002 0.0005 0.01 12 Ð 25 16 0.0005
Product 3 2 0.003 0.0005 0.001 0.0003 0.05 10 15 20 18 0.0006 20 450 350 2.5
4 0.007 0.0008 0.002 0.0008 0.06 11 16 18 20 0.0005
CONCLUSION
The FAS is a powerful tool for rapid environment change by rearranging its production schedule and dispatch in a short time. The system becomes more complex when the number of cells in the FAS increases, making it more dicult to achieve the best performance. In this paper, we modify the genetic algorithm to obtain the operational characteristics of an FAS with in-process quality control. This helps system managers to handle this system in order to achieve the best performance. Despite its limitations, a genetic algorithm is an ecient search algorithm for solving a complicated problem. Hence, we propose a modi®ed genetic algorithm by varying the mutation probability and calculating the decoded values to ®nd the near optimal inspection rate in the FAS by which defective parts might be produced. The experiment shows that the proposed modi®ed algorithm achieves a great improvement in accuracy and eciency.
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