A GA Mechanism for Optimizing the Design of attribute-double-sampling-plan

Transcription

1 A GA Mechanism for Optimizing the Design of attribute-double-sampling-plan Tao-ming Cheng *, Yen-liang Chen Department of Construction Engineering, Chaoyang University of Technology, Taiwan, R.O.C. Abstract An attribute double sampling plan can be performed when the acceptance parameters are known. These include first sample size, second sample size, first acceptance number, first rejectable number, and second acceptance number. The acceptance parameters must match the predefined probability 1-α of accepting a lot if the lot proportion defective is at the acceptable quality level (AQL) and β of accepting a lot if the lot proportion defective is at the rejectable quality level (RQL). In addition, the parameters must be all nonnegative integers and thus the system can not be solved as a closed-form solution. As a result, the trial-and-error method is usually used to seek the solutions. This paper presents a genetic-algorithms-based mechanism for facilitating the ADSP design process. Objectives of minimizing both the deviations of fitting AQL-α and RQL-β and the total sample sizes are traded off in the optimization process. Case studies show that the new mechanism can effectively locate the acceptance parameters and therefore facilitate the task of ADSP design. Keywords: attribute double sampling plan, genetic algorithms 1. Introduction An acceptance-sampling (AS) plan is a statement of the sample size to be used and the associated acceptance for sentencing an individual lot. There are different ways for classifying AS plans. One major classification is by variables and attributes. Variables are quality characteristics that are measured on a numerical scale and attributes are quality characteristics that are expressed on go and no-go basis. Performing variables sampling plan (VSP) requires basic statistical knowledge such as the calculation of standard deviation and the decision parameter as well as checking the quality index table. Contrast to VSP, attributes sampling plan (ASP) is easy to be used and does not involve statistical calculation for processing data. Construction inspectors usually do not have statistical background necessary for data processing in VSP [1]. * Corresponding author. Address: 168 Gifeng E. Rd., Wufeng, Taichung County 413, Taiwan, Tel.: Ext. 438; Fax: address: tmcheng@mail.cyut.edu.tw 1

2 Hence, an ASP plays an important role in designing quality assurance specifications in construction. ASP can be categorized as single or double sampling depending on the number of samples taken. In a single sampling plan, the decision to accept or reject a lot is made based on one sample. However, in a double sampling plan, a second sample may be required before a lot can be sentenced. A lot would be accepted or rejected depending on whether the first sample conforms to the specified requirements. Otherwise, the second sample has to be taken before a decision is made. Since the sampling phase is divided into tow stages, as a result, performing an attribute double sampling plan (ADSP) usually (not always) uses a smaller sample size and is commonly used in designing quality assurance specifications [, 3]. To properly design an ADSP, the users first have to focus on certain points on the operation characteristics (OC) curve which plots the probability of accepting the lot versus the lot fraction defective. These points include the AQL-α and RQL-β. AQL (acceptable quality level) represents the poorest level of quality for the producer s process that the consumer would consider to accept the product. RQL (rejectable quality level) would lead the consumer to reject the product. The probability of accepting a lot at the acceptable quality level (AQL) is 1-α and that of accepting a lot at the rejectable quality level (RQL) is β. After the AQL-α and RQL-β points being decided, a proper combination of acceptance parameters including first sample size (N1), second sample size (N), first acceptance number, first rejectable number, and second acceptance number, then can be selected to match these two points on OC curve. In addition, the binominal distribution is applied to develop the sampling equations for solving the acceptance parameters. However, the parameters must be all nonnegative integers and thus the system does not have a closed-form solution. As a result, the trial-and-error method is usually used to seek the solutions [3]. In order to reduce the number of examining the possible parameter combinations when trial-and-error method is employed, Duncan presented two tables for determining ADSP with α = 0.05, β = 0.10, and N1 = N or N [4]. Then, Chow et al. implemented Duncan s tables by developing a computer program [5]. In addition, Soundararajan and Vijayaraghavan introduced the procedures for constructing ADSP with α = 0.05 and β = 0.10 [6]. Moreover, Olorunniwo and Salas developed a computer program for facilitating the ADSP deign with any value of α and β but limited to N = kn1 where k is an integer [7]. Exhaustive search of possible acceptance parameter could be extremely timeconsuming. Moreover, infeasible solution can be obtained if the fitting of OC curve is strictly limited by both the AQL-α and RQL-β. Hence, a more efficient mechanism is needed for facilitating the design of ADSP process. In this study, the genetic algorithm is applied for optimizing the design process. Objectives of minimizing both the deviations of fitting AQL-α and RQL-β and the total sample sizes are traded off in the optimization process. Details of this new mechanism,

3 the theory of developing ADSP, and the computer implementation are provided in the following sections.. Attribute Double Sampling Plans This section first describes the flow of performing ADSP, then introduces the theory of ADSP in brief and depicts how to design DASP in details..1 Procedure of performing ADSP The flow of performing ADSP is shown in Fig. 1. The notations of the parameters used in Fig. 1 are described in Table 1. The procedure begins at taking a random sample of N1 from the lot and being inspected. If X1 C1, the lot is accepted on the first sample but if X1 R1, the lot is rejected. However, if X1 is greater than C1 but less than R1, then a second sample with N item has to be taken for further inspection of the lot s quality. If the sum of X1 and X is less than or equal to C, the lot is accepted on the second sample. Otherwise, the lot should be rejected. Table 1 Description of notations used in Fig. 1 Notation Description N1 sample size of the first sample N sample size of the second sample C1 acceptance number of the first sample R1 rejected number of the first sample C acceptance number for both samples X1 the number of observed defective in the first sample X the number of observed defective in the second sample Inspect a random of N1 from the lot X1 = number of observed defective X1 = C1 Yes Accept the lot No X1 = R1 Yes Reject the lot No No Yes Inspect a random sample of N from the lot X = number of observed defective X1+X = C Fig. 1 Procedure of performing an ADSP 3

4 . Theory of ADSP When a lot size is finite, the hypergeometric distribution should be used to describe the probability of the sampling. However, if the lot size is much larger than the sample size, the binomial distribution would be more appropriate to be used to calculate the probability of sampling. In construction quality inspection, many measurements can be taken and therefore, the lot size is usually considered as much larger than the sample size [1]. As a result, the binomial distribution is used to develop the ADSP in this study. Assume that the lot fraction defective is P d, and then the probability of X defectives out of a random sample of N items can be obtained by using binomial distribution as shown in Eq. (1). In other words, the acceptance probability of a lot can be calculated by employing this equation when the specific P d is known. P N! X!( N X )! X N X ( X ) = P d (1 P ) d (1) In the ADSP, the acceptance probability (denoted as P a ) of a lot is decided from the combined samples. If P a1 and P a denote the probability of acceptance on the first and second samples, respectively, then the P a is the sum of P a1 and P a as indicated in Eq. (). The way of calculating P a1 is straightforward. The lot will be accepted in the first sampling only in the condition that the defective items (X1) are less to or equal to acceptance number (C1). However, to calculate the probability of acceptance on the second sample, two conditions should be met sequentially. The first is that the lot is neither accepted nor rejected in the first sampling and the other is that the lot is accepted in the second sampling. The calculation of P a1 and P a are listed in Eqs. (3) and (4), respectively. P = P + P () a a1 a P a 1 P X1 C ( 1) = (3) R1 = 1 a = P i C1+ 1 ( X1 = i) P[ X ( C i) ] P (4) For instance, if the acceptance parameters (N1, N, C1, R1, C) equal to (50, 100,, 4, 5) and the Pd = 0.1, the calculation of Pa1 and Pa, together with Pa are demonstrated from Eqs. (5) to (7). P a 1 = P( X1 C1 = ) = P( X = 0) + P( X = 1) + P( X 50! 0 = (0.1) (0.9) 0!50! ! 1!49! (0.1) 1 (0.9) ! (0.1)!48! = ) (0.9) 48 = 11.17% (5) 4

5 P a = = 0.008% ( X1 = 3) P( X ) = P( X1 = 3) [ P( X = 0) + P( X = 1) + P( = ) ] = P X 50! 3 (0.1) (0.9) 3!47! ! (0.1) 0! 100! 0 (0.9) ! 1 + (0.1) (0.9) 1!99! ! + (0.1)!98! (0.9) 98 (6) P P + P = % % % (7) a = a1 a = The performance of an ADSP can be conveniently summarized by means of its operating characteristic (OC) curve. The OC curve is the curve that shows the acceptance probability of a lot based on the various quality levels. Fig. shows the OC curve for ADSP when N1=50, N=100, C1=, R1=4, C=5. It can be predicted that the lot will be accepted with the chance of 11.18% when the quality level of the lot is 10% defective with the specified requirement (i.e., AQL=10%) on OC curve. Probability of acceptance, Pa Lot fraction defective, Pd Fig. OC curve for an ADSP with N1=50, N=100, C1=, R1=4, C=5.3 Design of ADSP If the acceptance parameters (N1, N, C1, R1, C) are known, the probability of accepting a lot can be given in any required quality level, however, the way of designing ADSP is opposite. In designing an ADSP, first the AQL, RQL, and the risk levels of α and β have to be considered. Then a proper combination of acceptance parameters (N1, N, C1, R1, C) have to be found to fit the predefined AQL-α and RQL-β on OC curve. The relations between AQL-α and RQL-β and acceptance parameters (N1, N, C1, R1, C) are listed in Eqs. (8) and (9). R1 = i C1+ 1 ( 1 ) = P ( X1 C1) + P ( X1 = i) P [ X ( C i) ] α (8) 5

6 R1 = i C1+ 1 ( X1 C1) + P ( X1 = i) P [ X ( C i) ] β = P (9) where N1! X X ( N 1 X ) P1 ( X ) = AQL (1 AQL ) X!( N1 X )! (10) N! X X ( N X ) P ( X ) = AQL (1 AQL ) X!( N X )! (11) N 1! X X ( N 1 X ) P3 ( X ) = RQL (1 RQL ) X!( N 1 X )! (1) N! X X ( N X ) P4 ( X ) = RQL (1 RQL ) X!( N X )! (13) Since the acceptance parameters (N1, N, C1, R1, C) must be nonnegative integers, the system does not have a closed-form solution. As a result, the trial-and-error method is usually used to seek solutions. Trial-and-error method is not only time consuming in terms of computation effort but does not guarantee that the minimum sample size can be reached. Moreover, infeasible solution can be obtained if the fitting of OC curve is strictly limited by both the AQL-α and RQL-β. Hence, a more efficient mechanism is needed to facilitate the design of ADSP process. Genetic algorithms are suitable in solving the optimal problems when potential solutions are tremendously large or even are not obvious known. Thus, it is adopted in this study to build up the mechanism. 3. Genetic algorithms for optimizing ADSP 3.1 Background Genetic algorithms (GA) is the search algorithm developed by Holland [8], which is based on the mechanics of natural selection and genetics to search through decision space for optimal solutions. The metaphor underlying GA is natural selection. In evolution, the problem that each species faces is to search for beneficial adaptations to the complicated and changing environment. In other words, each species has to change its chromosome combination to survive in the living world. In GA, a string represents a set of decisions (chromosome combination), that is a potential solution to a problem. Each string is evaluated on its performance with respect to the fitness function (objective function). The ones with better performance (fitness value) are more likely to survive than the ones with worse performance. Then the genetic information is exchanged between strings by crossover and perturbed by mutation. The result is a new generation with (usually) better survival abilities. This process is repeated until the strings in the new generation are identical, or certain termination conditions are met. A generic flow of GA operation is given in Fig. 3. Readers can find more details of GAs in [9]. 6

7 Crossover Strings Encoding Initial Population Solution New Population Selection Roulette Wheel Mutation Evaluation Offspring Decoding Solution Fitness Computation Fig. 3 Generic procedure of GA operations [9] 3. Components and process This section introduces how GA is applied in the design of ADSP. Topics include chromosome structure, fitness value, mechanism of selection, crossover, and mutation, and termination condition Chromosome structure The chromosome structure used in this study is designed so that all permutations can be represented and evaluated. The length of the chromosome is defined as the total number of parameters. For example, as shown in Fig. 4, the two strings represent that (N1=5, N=10, C1=0, R1=1, C=1) and (N1=6, N=14, C1=0, R1=3, C=3), respectively. 7

8 P N1 N C1 R1 C P N1 N C1 R1 C Fig. 4 Chromosome structure 3.. Fitness value The purpose of using GA is to decide the acceptance parameters for the sake of minimizing the sample size in ADSP and, in addition, the resulted OC curve could meet the AQL-α and RQL-β requirements as much as possible. Two objectives of minimizing both total sample size and deviation of fitting α and β risk for specific lot fraction defective are considered in the optimization process. The Pareto front approach is usually used for solving multiobjective optimization problem is thus applied in this study [6]. Pareto front are a set of solutions for the multiobjective case which cannot simply be compared with each other. In other words, no improvement can be reached in any objective function without sacrificing at least one of the other objective functions for any Pareto solution which is a solution belonging to Pareto front. The functions representing the objectives of minimizing both the total sample size and deviation of fitting α and β risk for specific lot fraction defective are described in Eqs. (14) and (15). In addition, the fitness value is defined in Eq. (16). min Z 1 = N (14) ' ' min Z = 100α 100α + 100β 100β (15) where α ' and β ' are the calculated risks (which are contrast to α and β) with required AQL and RQL when particular acceptance parameters are generated. Fi = d max d i (16) where dmax: maximal distance between di in the generation (i.e., di=min(dij, for all j) dij: distance between parent i and each of the point j of the pareto front, where d ij = ( iz j ) ( 1 Z + i 1 Z j Z ) (see Fig. 5) 8

9 d = min ( d ) i j ij Z Parent i d ij Pareto front Pareto solution j Fig. 5 Evaluate fitness of member of population Z 1 The principle behind Eq. (16) is that the closer to Pareto front for each individual solution in a generation, the more fit. However, Pareto solutions are identified in each generation and thus, a Pareto solution in a generation may disappear in the next generation. In order to maintain a tendency that the new Pareto front always move toward to the coordinate axes, each solution at Pareto front in a generation will be kept for the next generation Selection The proportional selection is used to select the strings that have better fitness value in this study. According to the fitness value generated by the previous step, the selection probability can be determined by using Eq. (17). i i = Pop _ size P F i= 1 F i where P i : the selection probability of the parent i F i : the fitness value of string i Pop_size: The population size (17) 3..4 Crossover and mutation The one-point crossover mechanism is applied to the chromosomes used in this study. The process of the one-point crossover, shown in Fig. 6, is described as follows: At first, two parent strings, P1 and P, are selected randomly. Then one randomly chosen point is used as the separator that cuts the each selected parent string into two sections. 9

10 Combine section 1 of P1 to form the head of the first offspring (O1) and use section of P to form the tail of O1. In addition, the section 1 of P and the section of P1 are used as the head and the tail of the second offspring (O), respectively. Point cut Point cut P P Section 1 Section Section 1 Section O O Section 1 of P1 Section of P Section 1 of P Fig. 6 Example of the one-point crossover Section of P1 Self-mutation [6] is used in this study. If a chromosome is selected for mutation, one gene is randomly selected for changing their values by chance. An example of selfmutation is shown in Fig. 7. P is the selected chromosome and O is the chromosome after self-mutation. Mutation Point Mutation Point P P O O Fig. 7 Example of the self-mutation Termination condition The number of iterations is usually set as the termination criterion for stopping the evolution process while performing GA operation. Ideally, the final Pareto front obtained should move toward the coordinate axes as close as possible. Therefore, it will be more suitable to determine whether the final Pareto front is reached by checking if the area below Pareto front is decreased continuously. In this study, the convergence of evolution process will be examined by the rate of decreasing the area below Pareto front. 10

11 4. The computer program A user-friendly computer implementation named attribute-double-sampling-planoptimizer (ADSPO) is developed to facilitate the task of optimizing the design of ADSP and its interface is shown in Fig. 8. There are two areas requiring users to input data. The first area is the section of Sampling Parameters including AQL, RQL, α Risk, and β Risk, which contains the data related to design ADSP parameters. In addition, users can limit the generated total sample size not exceeding to their expectation through typing data in Upper limit of sample size. The second area is the section of GA parameters, which includes Population Size, Generation, Crossover Rate, and Mutation Rate. After the data needed for these two areas are entered, the ADSPO will be ready to optimize the design of ADSP. In addition, there are three areas that report the results generated by ADSPO. Population Status indicates the performance of GA operation. Moreover, the Pareto front obtained for current generation can be observed schematically in this area. Pareto front for Current Generation shows the detailed information for each Pareto solution of the current generation during the GA operation. The information includes the acceptance parameters (N1, N, C1, R1, C) and related α and β risk. Furthermore, Population for Current Generation lists all solutions generated in current generation. Fig. 8 Interface of ADSPO 11

12 5. Results Many tests have been done to verify the accuracy and efficiency of ADSPO. Three examples are presented in this paper. The test platform is a Pentium GHz PC with 51 MB RAM. The first example, with the population of 100, generations of 00, and total sample size limited to 50, it takes ADSPO around 4 hours to complete the evolution process. In addition, the requested AQL-α and RQL-β are set to 5%-5% and 0%-10%, respectively. The final Pareto front containing 7 solutions are indicated in Table. Among them, decision maker can select the solution 1 with the least total sample size of 10 but β risk is very undesirable. On other hand, if the required α and β risks is strictly demanded, solution 7 with total sample size of 38 meets the requirement. However, solution 6 with less sample size of 37 whose β risk is only slightly higher than the requested may worth to be adopted if inspection cost could increase substantially when even one more sample is examined. Moreover, there are 789,168 possible acceptance parameters if total sample size is limited to not exceeding 50. However, as shown in Fig. 9, ADSPO searches through about 1% (100*75/789,168) of the solution space by taking 75 generations to converge to the optimal solutions Area below Pareto Front Number of Generation Fig. 9 Convergent curve of example 1 1

13 Table Pareto solutions of example 1 Pareto Solutions Sample Size α (%) β (%) N1 N C1 R1 C

14 The second example examined with the population of 50, generations of 100, and total sample size limited to 50, it takes the ADSPO 44 minutes to complete the evolution process. The requested AQL-α and RQL-β are designed as 5%-5% and 0%- 3%, respectively. The results are listed in Table. Among the Pareto solutions, solutions 1 and whose obtained α and β risks meet the requirement. Solution 0 could be appropriate to be selected if β risk is allowed to be slightly higher than the required. In addition, as indicated in Fig. 10, about 0.1% (50*18/789,168) of the solution space is searched by ADSPO that took 18 generations to converge to the optimal solutions. Table 3 Pareto solutions of example Pareto Solutions Sample Size α (%) β (%) N1 N C1 R1 C

15 15000 Area below Pareto Front Number of Generation Fig. 10 Convergent curve of example The third example has been tested by Chow et al. and Olorunniwo and Salas. This example is tested with the population of 100, generations of 00, and total sample size limited to 150. In addition, the requested AQL-α and RQL-β are 1%-5% and 9%-10%. The results are shown in Table 3. There are four solutions (i.e., 5, 6, 7, and 8) whose α and β risks are accepted. Compared to the minimum sample size of 66 with accepted risks found in the plans presented in Olorunniwo s and Salas s work, three solutions searched by ADSPO have smaller total sample sizes. In addition, several solutions such as 3 and 4 are almost qualified for the requested risks. Moreover, as indicated in Fig. 11, about 5.9*10-6 (0*50/168,961,870) of the solution space is searched by ADSPO that took about 50 generations to converge to the Pareto solutions Area below Pareto Front Number of Generation Fig. 11 Convergent curve of example 3 15

16 Table 4 Pareto solutions of example 3 Pareto Solutions Sample Size α (%) β (%) N1 N C1 R1 C

17 6. Summary Attribute double sampling plans play an important role in constructing the quality assurance system. In designing an ADSP, the AQL, RQL, and α and β risks have to be predefined. Then a proper combination of acceptance parameters (N1, N, C1, R1, C) can be explored to fit the AQL-α and RQL-β on OC curve. In addition, the parameters must be all nonnegative integers and thus the system does not have a closed-form solution. As a result, the trial-and-error method is usually used to seek the solutions. The less the total sample sizes, the less cost and time for the quality inspection process. This is true both from the producer s and the consumer s point of view. However, the trial-and-error method not only does not guarantee that the minimum sample sizes can be reached but infeasible solution can be obtained if the fitting of OC curve is strictly limited by both the AQL-α and RQL-β. This study applied GA to optimize the design process such that the ADSP can be conveniently used for construction quality control. Objectives of minimizing both the deviations of fitting AQL-α and RQL-β and the total sample sizes are concerned in the optimization process. A computer program named ADSPO for facilitating the ADSP design process is developed. The verified results show the ADSPO can locate the optimal solutions more efficiently and accurately. References [1] L.-M. Chang and M. Hsie, Developing acceptance-sampling methods for quality construction, Journal of Construction Engineering and Management, 11(), 46-53, [] M. Hsie and L.-M. Chang, Attribute-double-sampling method for infrastructure quality assurance, Journal of Infrastructure Systems, 1(), , [3] D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons, Inc., New York, N. Y., [4] A. J. Duncan, Quality Control and Industrial Statistics, Irwin, Homewood, Illinois, [5] B. Chow, P. E. Dickinson, and H. Hughes, A computer program for the solution of double sampling plans, Journal of Quality Technology, 4(4), 05-09, 197. [6] V. Soundararajan and R. Vijayaraghavan, Procedures and tables for the construction and selection of conditional double sampling plans, Journal of Applies Statistics, 19(3), , 199. [7] F. O. Olorunniwo and J. R. Salas, An algorithm for determining double attribute sampling plans, Journal of Quality Technology, 14(3), , 198. [8] J. Holland, Adaptation in Natural and Artificial System, University of Michigan Press, Ann Arbor, Michigan, [9] M. Gen, R. Cheng, Genetic Algorithms and Engineering Optimization, Wiley, New York,