An optimization-based approach for designing ...

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IJQRM 25,8

An optimization-based approach for designing attribute acceptance sampling plans

824 Received 9 December 2007 Revised 10 April 2008 Accepted 22 April 2008

Belmiro P.M. Duarte Departamento de Engenharia Quı´mica e Biolo´gica, Instituto Superior de Engenharia de Coimbra and GEPSI-Grupo PSE, CIEPQPF, Departamento de Engenharia Quı´mica, Universidade de Coimbra, Coimbra, Portugal, and

Pedro M. Saraiva GEPSI-Grupo PSE, CIEPQPF, Departamento de Engenharia Quı´mica, Universidade de Coimbra, Coimbra, Portugal Abstract Purpose – This purpose of this paper is to present an optimization-based approach to support the design of attribute sampling plans for lot acceptance purposes, with the fraction of non-conforming items being modeled by a Poisson probability distribution function. Design/methodology/approach – The paper approach stands upon the minimization of the error of the probability of acceptance equalities in the controlled points of the operating curve (OC) with respect to sample size and acceptance number. It was applied to simple and double sampling plans, including several combinations of quality levels required by the producer and the consumer. Formulation of the design of acceptance sampling plans as an optimization problem, having as a goal the minimization of the squared error at the controlled points of the OC curve, and its subsequent solution employing GAMS. Findings – The results are in strong agreement with acceptance sampling plans available in the open literature. The papers approach in some scenarios outperforms classical sampling plans and allows one to identify the lack of feasible solutions. Originality/value – An optimization-based approach to support the design of acceptance sampling plans for attributes was conceived and tested. It allows for a general treatment of these problems, including the identification of a lack of feasible solutions, as well as making possible the determination of feasible alternatives by relaxing some model constraints. Keywords Acceptance sampling, Poisson distribution, Non-linear control systems, Quality assurance Paper type Conceptual paper

International Journal of Quality & Reliability Management Vol. 25 No. 8, 2008 pp. 824-841 q Emerald Group Publishing Limited 0265-671X DOI 10.1108/02656710810898630

1. Introduction and motivation Quality assurance has emerged in the past decades as one of the keystones in supplier-customer relationships (Evans, 2005). Its main goal is to provide customers with evidence that products do indeed reach the quality levels required and agreed upon. Acceptance sampling is related with inspection and decision making regarding product quality, and is indeed one of oldest subjects covered by statistical quality control (Montgomery, 2001). Acceptance sampling is applied as an auditing tool, in order to check whether products have or not met the quality levels required. It is thus mostly focused in product quality auditing, rather than process improvement (Mitra,

1998), although both may be interconnected. Quality inspection for acceptance purposes is nowadays extended to many industrial units and applied in many stages of manufacturing lines, from incoming materials to intermediate and final products. It is also common to find inspection activities of final product being carried out by both the producer and the purchaser, in order to check the performance of the underlying quality systems (Grant and Leavenworth, 1996). Quality inspection is thus a limited but useful tool, in the sense of helping one to judge whether products do conform to specifications or not. Indeed, it cannot assure that all accepted lots of product are conforming, since Type I and Type II risks are associated with any acceptance sampling plan. One of the main criticisms raised by Deming, regarding acceptance sampling techniques, regards precisely this issue, which is unavoidable (Deming, 1986). Nevertheless, acceptance sampling is the only alternative available to extreme product quality control policies, such as 100 percent inspection (Milligan, 1991), or, on the other hand, the assumption of a no-inspection alternative, which however can only be employed if the underlying process is perfectly controlled from a statistical point of view, and the chances of not fulfilling the required quality levels are very low (Pearn and Wu, 2007). On the other side, 100 percent inspection can only be a solution when the components manufactured are critical for overall product quality and non-destructive testing procedures are available at an affordable cost. This ends up meaning that in practice most product quality control procedures do indeed rely on some kind of sampling plan. Acceptance sampling (AS) procedures can be applied to lots of items when testing reveals non-conformance or non-conformities regarding product functional attributes. It can also be applied to variables characterizing lots, thus revealing how far product quality levels are from specifications. Both AS applications have the basic purpose of classifying a lot as accepted or rejected, given the quality levels required for it. The design of acceptance sampling plans for attributes determines essentially the size of the samples to be inspected and the number of non-conformities tolerated in order for a lot to become accepted. The theoretical basis that underlies acceptance sampling is derived from statistical theory, and was mostly developed in the decades of 1930-1960. The military industry pioneered the development and large scale implementation of acceptance sampling plans, guided by a set of standard sampling plans presented as tables (MIL-STD-105D, 1963), later on adapted by the American Society of Quality Control (ASQC) as ANSI/ASQC Z1.4-1981 (ASQC, 1981). The initial approaches employed to design acceptance sampling plans used the concept of OC curves, where the risks of acceptance were stipulated by the producer and the consumer to find feasible combinations of parameters (Grubbs, 1949). This design results from the solution of the equations modeling the probability of acceptance in the points controlled over the OC curve. The analytical solution of this problem is quite challenging, and thus several authors assumed mathematical approximations in order to derive closed-form solution formulas (Hald, 1967; Duncan, 1974). Taking advantage of the growth of calculation power, several algorithms were since then developed to handle this problem computationally. Since the system of algebraic equations resulting from controlling the quality levels stipulated has not an exact form, most of the algorithms are based on enumeration techniques to seek feasible combinations of variables (Chow et al., 1972; Olorunniwo and Salas, 1982; Taylor, 1986; Soundararajan and Vijayaraghavan, 1990). In recent years, different conceptual bases were considered for supporting the

Optimizationbased approach

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design of AS procedures, namely the economic aspects of sampling and inspecting a process for lot acceptance, and Bayesian statistical frameworks. Several economical models were developed and handled employing analytical strategies to derive closed-form solutions or optimization algorithms to obtain the best possible solutions (Wetherill and Chiu, 1975; Chen and Chou, 2002). Bayesian approaches stand on the assumption that the fraction of non-conforming items in a lot is described by a prior distribution, with Beta or modified Beta probability distributions usually being assumed for that purpose (Moskowitz and Berry, 1976; Tagaras and Lee, 1987; Kwon, 1996). The acceptance sampling plans thus achieved minimize the costs involved in sampling and inspection all over the domain of the fraction of non-conformities that may be indeed be present in the lots. Although this approach is effective and justifiable (Case and Chen, 1985), it requires a large amount of data regarding costs involved and to model the prior appropriately. The original OC curve based approaches require an exhaustive search of non-negative variables. Moreover, unfeasible acceptance sampling plans can be obtained, since the OC curve must simultaneously meet more than one specified quality level, passing through more than one point of the graphical representation for the probability of acceptance of the lot versus fraction of non-conforming product observed. To cope with these drawbacks, the problem of designing acceptance sampling plans, particularly double sampling plans, has recently been treated employing optimization algorithms, such as fuzzy mathematical programming (Wang and Chen, 1997) and genetic algorithms (Cheng and Chen, 2007). It is noticeable, however, that the problem requires a more general approach, in order to handle simple, double and multiple procedures, as well as to address all the possible combinations of stipulated risks. Additionally, a tool able to identify the lack of feasible sampling plans, for a given set of specifications, and subsequently prescribe alternative solutions, would also be quite helpful. This paper presents such a general-purpose framework, aimed at fulfilling the gaps just identified. We will describe its application to the design of single and double sampling plans. Our approach is based on the formulation of sampling plan design problems as optimization models, together with their subsequent solution, obtained through the use of robust optimization solvers, thus enabling us to also check about solution feasibility. This information can then be useful in relaxing some of the assumed conditions, in order to achieve feasible convenient and optimal sampling plans. The strategy employed relies upon the minimization of the squared norm of the error of the algebraic equations modeling the probability of acceptance in the points controlled over the OC curve, representing the requirements to be satisfied regarding quality levels. The squared error minimization allows us to also determine the parameters (sample size and acceptance number) that solve the probability of acceptance equations defined for the controlled points. The sample size is modeled as a continuous variable, and the choice of the acceptance number is represented by a set binary variables. The remaining parts of this paper are thus organized as follows: Section 2 presents reviews the basic concepts related with sampling theory for acceptance purposes and the terminology employed in later sections; Section 3 presents our framework for designing single sampling plans; Section 4 extends this formulation in order to deal with double sampling plans; Section 5 covers some final conclusions that can be drawn.

2. Acceptance sampling basic concepts and terminology In this paper we will focus in attributes acceptance sampling based on type B operating characteristic curves, thus assuming that the underlying manufacturing process produces a stream of lots with a large size (ten times larger) when compared to the sample size taken for inspection. This assumption allows us to treat the distribution of non-conforming items found as being continuous. The probability of observing x non-conformities or nonconforming items in a random sample of size n is thus modeled as a binomial distribution (Grubbs, 1949). Assuming a proportion p of nonconforming items or non-conformities, the probability of finding x nonconforming items is given by the continuous binomial law: ! n ð1Þ p x ð1 2 pÞx PðxÞ ¼ x and the acceptance probability, designated as P a , is then: P a ðc1 ; n; pÞ ¼ pðx # c1 ; n; pÞ ¼

c1 X

PðxÞ

ð2Þ

x¼0

with c1 standing for the acceptance number (maximum number of non-conforming items observed in the sample tolerated to judge the lot as being accepted). Assuming that the conditions of deriving a type B OC curve are guaranteed and that the probability of producing non-conforming items is low (common assumptions in real industrial processes), the Poisson probability distribution can be employed as an approximation of the binomial distribution, with: PðxÞ ¼

ðnpÞx expð2npÞ x!

ð3Þ

The OC curve represents the performance of the acceptance sampling plans by plotting the probability of accepting a lot versus its production quality, which is expressed by the proportion of non-conforming items in the lot, p. The design of acceptance sampling plans considers one or both of the following quality levels: acceptable quality level, designated by AQL, and represented by the proportion of non-conforming items that is considered to be acceptable, p1 ; limiting quality level, designated by LQL, and associated to a proportion of non-conforming items, p2 . The first level is required by the producer and the second is the worst level of quality that the consumer can tolerate. The producer’s risk, a1 , is the Type I error for the sampling plan to fail in the verification of lots of quality level equal or smaller than AQL. In view of this definition, the producer desires to reject lots of quality level p1 no more than 100ð1 2 a1 Þ% of the times. The consumer’s risk is the Type II error for the sampling plan to pass in the verification lots of quality level below LQL. The consumer requires the acceptance of lots of quality level p2 no more than 100b1 % of the times. Figure 1 illustrates the OC curve of a sampling plan, together with the quality levels and corresponding acceptance probabilities considered in the design of a sampling plan. In the following sections we describe a general approach to derive attribute simple and double sampling plans based in controlling the points ð p1 ; 1 2 a1 Þ and ð p2 ; b1 Þ of


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Figure 1. Single sampling plan OC curve for stipulated producer and consumer’s risks, and corresponding quality levels

the OC curve, including an optimization framework that we will employ to solve the complex algebraic equations representing the probability of acceptance required at those points with respect to the sample sizes and acceptance numbers. The solution of the algebraic equations thus obtained is achieved through the minimization of the sum of squared errors of the equality constraints in the points of the OC curve that are controlled. 3. Design of single sampling plans The single attribute sampling plan provides a decision rule to accept or reject a lot based on the inspection results obtained from a single random sample. The procedure corresponds to taking a random sample from the lot with size n1 and inspect each item. If the number of non-conformities or nonconforming items does not exceed the specified acceptance number, c1 , the entire lot is accepted. Otherwise, it is rejected. Here, the single sampling plan is represented as S 1 ðn1 ; c1 ). Many different acceptance plans meet the requirements of both the producer and the consumer. However, the producer is also interested in keeping the average number of items inspected to a minimum, aiming to reduce the costs of sampling and inspection, and economic aspects of the sampling plans must also be considered in practical implementations. It is out of the scope of this paper to address the economic design of attribute sampling plans. Nevertheless, the range of acceptance numbers investigated is as small as possible, aiming to achieve reduced sample sizes whenever possible. Here we introduce the nomenclature employed in the optimization problems formulated here and in the forthcoming sections, as follows:

Indices x1

– non-conforming items observed in the first sample.

x2

– non-conforming items observed in the second sample.

c1

– acceptance number for the decision based on the first sample.

c2

– acceptance number for the decision based on the second sample.

j

– points of the OC curve to be controlled.

Sets C 1 2 {c1 : 0 # c1 # c1max }

– set of acceptance numbers to judge the lot based on the first sample.

C 2 2 {c2 : c1 þ 1 # c2 # c2max } – set of acceptance numbers to judge the lot based on the second sample. Continuous variables ej

– error of the probability of acceptance equation in the point j.

n

– size of the first sample in double sampling plans, and of the sample in simple sampling plans.

m

– size of the second sample in double sampling plans.

P a1

– acceptance probability of the sampling plan S 1 ðn1 ; c1 Þ in double-sampling plans.

P a2

– acceptance probability of the second sample in double-sampling plans S 2 ðn1 ; c1 ; n2 ; c2 Þ.

Pa

– acceptance probability of the overall sampling plan.

p1

– acceptable quality level.

p2

– limiting quality level.

p3

– acceptable quality level for the first sample in double sampling plans.

a1

– probability of rejection of AQL.

b1

– probability of acceptance of LQL.

a2

– probability of rejection of AQL based on the first sample of double sampling plans.

Binary variables wc1

– assigns the acceptance number c1 to the first sample.

wc2

– assigns the acceptance number c2 to the second sample.

Integer variables c1max – maximum acceptance number allowed for the inspection of the first sample.


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c2max – maximum acceptance number allowed for the inspection of the second sample. c1

– acceptance number for the inspection of the first sample.

c2

– acceptance number for the inspection of the first sample.

n1

– size of the first sample.

n2

– size of the second sample.

Case 1 – Stipulated producer’s risk Here we consider that the producer’s risk, a1 , and the corresponding accepted quality level, p1 , are specified. The acceptance sampling plan follows an OC curve that passes through the point ð p1 ; 1 2 a1 Þ, according to Figure 2. Since the sampling plan S 1 ðn1 ; c1 Þ depends on two variables and one single equality can be derived from the OC curve, the approach employed is to fix c1 and solve the optimization problem P 1 , with respect to n, as follows: min e21 n

s:t:

c1 X ðnp1 Þx1 expðnp1 Þ ¼ 1 2 a1 þ e 1 x1 ! x ¼0

ð4Þ

ð5Þ

1

n $ c1 þ 1

Figure 2. Single sampling plan OC curve for stipulated producer’s risk and corresponding quality level

ð6Þ

n1 ¼ dne

ð7Þ


c1 fixed Problem P 1 falls under the class of NLP problems, and was solved using the GAMS/CONOPT optimization solver, with an absolute tolerance equal to 0.0 (Brooke et al., 1998). This solver is based on the generalized reduced gradient (GRG) algorithm, denoting a remarkable efficiency for this class of problems. The value of tolerance was imposed taking into account that n is assumed to be continuous, and it is consequently straightforward to prove the existence of an exact solution for equation (5) at each value of c1 . Equation (6) establishes the lower value of the sample size, and different values of the acceptance number lead to different satisfying sampling plans. Lower values of c1 are more common in industrial usage, due to the lower costs that they involve. Table I lists the values of the optimal sample sizes obtained for different values of acceptance number, which are in strong agreement with Grubbs table (Grubbs, 1949).

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Case 2 – Stipulated producer and consumer’s risk Here we consider that the producer’s risk, a1 , the value of AQL, p1 , the consumer’s risk, b1 , and the value of the limiting quality level, p2 , are all stipulated. The OC curve thus passes through the points ð p1 ; 1 2 a1 Þ and ð p2 ; b1 Þ. Figure 1 presents the graphical representation of this problem. The optimization problem formulated to determine the sampling plan S 1 ðn1 ; c1 Þ, designated as P 2 , falls now under the class of MINLP problems. Similarly to all MINLP problems, found in the next section, it was solved using the GAMS/SBB optimization solver with an absolute tolerance of 102 6 (Brooke et al., 1998). GAMS/SBB employs a combination of the standard Branch and Bound algorithm, to handle the integer part of the problem, with a NLP solver, to deal with the non-linear structure (GAMS/CONOPT was employed to solve the NLP problems found). a1 C1

0.05

0.90

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.060 0.360 0.820 1.380 1.980 2.620 3.300 4.000 4.700 5.440 6.180 6.940 7.700 8.480 9.260 10.040

2.320 3.900 5.340 6.700 8.000 9.280 10.540 11.780 13.000 14.220 15.420 16.600 17.800 18.960 20.140 21.300

Table I. Values of n1 p1 for stipulated producer’s risk single sampling plans ð p1 ¼ 0:02Þ

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The optimization problem formulation for this case is as follows: min n;w

2 X

c1

832

s:t:

X

wc1

c1[C 1

e2j

ð8Þ

j¼1

c1 X ðnp1 Þx1 expð2np1 Þ ¼ 1 2 a1 þ e 1 x1 ! x1

c1 X X ðnp2 Þx1 expð2np2 Þ ¼ b1 þ e 2 wc1 x1 ! c1[C 1 x1

X wc1 ¼ 1

ð9Þ

ð10Þ

ð11Þ

c1[C 1

n$

X

wc1 c1 þ 1

ð12Þ

c1[C 1

n1 ¼ dne

ð13Þ

max

c1 ¼

c1 X

wc1 c1

ð14Þ

c1¼0

wc1 [ {0; 1}; ej $ 0; j [ {1; 2} This problem comprises two variables: n and c1 , with the choice of c1 being modelled by employing a binary variable, wc1 , that assigns the plan with an acceptance number c1 that minimizes the error of the equality constraints in the controlled points, with 0 # c1 # c1max . Equations (9) and (10) represent the equalities at the stipulated points of the OC curve. Equation (11) models the choice of a single acceptance number c1 [ {0; · · ·; c1max }, and equation (12) the minimum value allowed for the sample size. The error variables are restricted to positive values due to the requirements of the producer and of the consumer. The first desires the acceptance probability to be at least 100ð1 2 a1 Þ% for AQL, satisfied when P a ðn1 ; c1 ; p1 Þ 2 1 þ a1 is positive. The consumer accepts lots of quality p2 with a maximum probability of 100b1 %, so that the error of the equation must be non positive. Table II presents the sampling plans thus obtained for different ratios of p2 =p1 . One may see that problem P 2 , considering a1 ¼ 0:05 and b1 , is only feasible for values of p2 =p1 larger than 4.89 (fourth row of Table II). For those ratios of LQL/AQL there exists an OC curve that passes above or through the point ð p1 ; 1 2 a1 Þ and simultaneously below or through the point ð p2 ; b1 Þ. As the difference between p2 and p1 decreases (the ratio between them becomes smaller), there is no OC curve that is

p1

p2/p1

c1

n1p1

0.0022 0.0091 0.0154 0.0204 0.0246 0.0282 0.0312 0.0338 0.0361 0.0382 0.0400 0.0417 0.0433 0.0446 0.0459

44.84 10.96 6.51 4.89 4.06 3.55 3.21 2.96 2.77 2.62 2.50 2.40 2.31 2.24 2.18

0 1 2 3 2 2 2 2 2 2 2 2 2 2 2

0.051 0.356 0.829 1.370 0.837 * 0.845 * 0.841 * 0.845 * 0.830 * 0.840 * 0.849 * 0.833 * 0.823 * 0.848 * 0.826 *

Notes: *Solution obtained relaxing e2 $ 0 to e2 [ R

steep enough to provide the fulfillment of the requirements for both the producer and the consumer. Under such combinations of parameters, there is no feasible OC curve providing the control of acceptance probability at both points with positive errors. Therefore, all the sampling plans obtained for ratios of p2 =p1 above 4.89 were obtained satisfying only the producer desire, by keeping e1 $ 0 and relaxing the consumer requirement, assuming that it may be not met, and therefore e2 is not constrained any longer under such problem reformulations. For those scenarios, the sampling plans S 1 ðn1 ; c1 Þ based on c1 ¼ 2 were found to be optimal regardless the values of p2 =p1 , since it allows one to satisfy the condition P a ðn1 ; c1 ; p1 Þ ¼ 1 2 a1 and simultaneously minimize the difference P a ðn1 ; c1 ; p2 Þ 2 b1 . 4. Design of double sampling plans The double sampling plan involves two stages of decision regarding the acceptance of a lot. First, a random sample of size n1 is extracted from the lot and the corresponding items are inspected. If the number of non-conformities or nonconforming items, x1 , falls below or is equal to the specified acceptance number for the first sample, c1 , the lot is accepted. If the number of non-conformities or nonconforming items is greater than or equal to the specified rejection number for the first sample, r 1 , then the lot is rejected. However, if on the other hand the number of non-conformities or nonconforming items is greater than c1 but falls below r 1 , then a new random sample of size n2 is taken from the lot and the corresponding items are also inspected. Based upon the cumulative results obtained from both samples, a final decision regarding lot acceptance is then made. If the cumulative number of non-conformities or nonconforming items, x1 þ x2 , is smaller than the acceptance number for the cumulative sample, c2 , the lot is accepted. If the cumulative number of non-conformities or nonconforming items is larger than the rejection number of the two samples, r 2 , then the entire lot is rejected. In this work we will consider r 1 ¼ c2 2 1, similarly to Stewart et al. (1978), and the sampling plan can thus be compactly represented as S 1 ðn1 ; c1 ; n2 ; c2 Þ.


833

Table II. Values of n1 p1 for stipulated producer’s risk single sampling plans (p1 ¼ 0.10, a1 ¼ 0.05, b1 ¼ 0.10)

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Double sampling plans are viewed as being much more complex than simple sampling plans. Nevertheless, they are economically advantageous, since they require, on average, the inspection of a smaller number of items, because the decision of accepting or not a given lot can be based upon the first sample, which is always smaller than the equivalent sample in the corresponding single sampling plan. The probability of acceptance of the lot based on the first sample of size n is given by: P a1 ðn; c1 ; pÞ ¼

c1 X ðnpÞx1 expðnpÞ x1 ! x ¼0

ð15Þ

1

and the probability of acceptance based on the cumulative samples is: P a ðn; c1 ; m; c2 ; pÞ ¼ P a1 ðn; c1 ; pÞ þ P a2 ðn; c1 ; m; c2 ; pÞ

ð16Þ

with P a2 ðn; c1 ; m; c2 ; pÞ ¼

c2 2 2x1 X ðnpÞx1 expð2npÞ cX ðmpÞx2 expð2mpÞ x1 ! x2 ! x ¼c þ1 x ¼0 1

1

ð17Þ

2

Case 3 – Stipulated producer and consumer’s risk Here we will assume that the producer’s risk, a1 , the AQL, represented by the probability of non-conformities, p1 , the consumer’s risk, b1 , and the associated LQL value, represented as p2 , are all stipulated. The OC curve thus passes though the points ð p1 ; 1 2 a1 Þ and ð p2 ; b1 Þ (Figure 3). Since the problem involves two equality

Figure 3. Double sampling plan OC curves for stipulated producer and consumer’s risk of the overall sampling plan and corresponding quality levels

constraints and four variables to be determined (n1 , c1 , n2 and c2 ), the acceptance number of the first sample is fixed and included as an additional relation between n1 and n2 , similarly to Grubbs tables adapted by Mitra (1998)[1]. The variables to be determined are thus the acceptance number for the second sample, modeled by the binary variable, wc2 , and the sample size. Our optimization problem, designated as P 3 , is then defined as follows: min

2 X

n;m;wc2

835 e2j

ð18Þ

j¼1

c1 X ðnpj Þx1 expð2npj Þ ; j [ {1; 2} x1 ! x ¼0

ð19Þ

c2 2 2x1 X ðnpj Þx1 expð2npj Þ cX ðmpj Þx2 expð2mpj Þ ; j [ {1; 2} x1 ! x2 ! x ¼c þ1 x ¼0

ð20Þ

s:t: P a1;j ¼

1

P a2;j ¼

X

wc2

c2[C 2

1

1

2

P a1;1 þ P a2;1 ¼ 1 2 a1 þ e1

ð21Þ

P a1;2 þ P a2;2 ¼ b1 2 e2

ð22Þ

X wc2 ¼ 1

ð23Þ

c2 [C 2

n¼m

n $ max

ð24Þ !

max

c2 X

wc2 c2; c1

þ1

ð25Þ

c2¼0

n1 ¼ dne

ð26Þ

n2 ¼ dme

ð27Þ

max

c2 ¼

c2 X


wc2 c2 þ 1

ð28Þ

c2¼

wc2 [ {0; 1}; ej $ 0; j [ {1; 2}; c1 fixed Equation (19) represents the probability of acceptance based on the first sample for each of the OC curve points that are controlled. Equation (20) represents the probability of acceptance based only on the second sample, equations (21) and (22) the equalities

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holding at the points ð p1 ; 1 2 a1 Þ and ð p2 ; b1 Þ, respectively. Equation (23) models the choice of a single acceptance number for the cumulative samples, and equation (25) sets the minimum sample size to the maximum acceptance number plus one item. The results obtained after solving this problem are shown in Table III, comparing well with those presented in the Grubbs table for double sampling plans (n1 ¼ n2 ) (Mitra, 1998)[2]. The results obtained present lower values of c1 and equal values of c2 for equal ratios of p2 =p1 , when compared to the values listed in the Grubbs table. The designs obtained slightly outperform the sampling plans listed in such tables, requiring in most of the cases smaller samples for similar risk ratios. Table IV presents the results obtained considering 2n1 ¼ n2 , corresponding to the replacement of equation (24) in P 3 by: 2n ¼ m

ð29Þ

These results are also in strong agreement with the Grubbs tables presented by Mitra (1998)[3], also leading to smaller sample sizes for the same ratios p2 =p1 , as compared to Grubbs tables with 2n1 ¼ n2 . Case 4 – stipulated producer and consumer’s risk of the overall plan and producer’s risk of the first sample In this case it is considered that the producer’s risk, a1 , the AQL value, p1 , the consumer’s risk, b1 , and corresponding LQL, p2 , are all stipulated. Additionally, the producer’s risk of the decision based upon the first sample, a2 , corresponding to a quality level p3 , is also specified. The sampling plan comprises two OC curves and three points to control - the first represents P a1 ðn1 ; c1 ; pÞ versus p, being denoted by OC1, and the second plotting P a2 ðn1 ; c1 ; n2 ; c2 ; pÞ versus p, denoted by OC2. Therefore, the design of double sampling plans for this case stands in controlling OC2 in the points ð p1 ; 1 2 a1 Þ and ð p2 ; b1 Þ, as well as OC1 in the point ð p3 ; 1 2 a2 Þ (see Figure 4). The problem gives origin to three equality constraints and has four variables. Here we

Table III. Values of n1 p1 for stipulated producer’s and consumer’s risk double sampling plans (p2 ¼ 0.10, n1 ¼ n2 , a1 ¼ 0.05, b1 ¼ 0.10)

p1

p2/p1

c1

c2

n1p1

0.0069 0.0124 0.0154 0.0186 0.0196 0.0232 0.0239 0.0278 0.0307 0.0338 0.0361 0.0382 0.0407 0.0452 0.0508 0.0575

14.50 8.07 6.48 5.39 5.09 4.31 4.19 3.60 3.26 2.96 2.77 2.62 2.46 2.21 1.97 1.74

0 0 0 0 1 1 1 1 2 2 2 2 2 3 4 6

1 2 3 4 5 5 5 6 8 9 10 11 12 16 20 30

0.166 0.310 0.417 0.557 0.825 0.951 0.979 1.167 1.687 1.892 2.094 2.328 2.561 3.575 4.772 7.471

p1

p2/p1

c1

c2

n1p1

0.0084 0.0133 0.0147 0.0186 0.0215 0.0235 0.0258 0.0275 0.0296 0.0312 0.0324 0.0351 0.0385 0.0410 0.0431 0.0450 0.0472

11.90 7.54 6.79 5.39 4.65 4.25 3.88 3.63 3.38 3.21 3.09 2.85 2.60 2.44 2.32 2.22 2.12

0 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

2 2 3 3 4 5 5 6 7 7 7 8 10 11 12 14 16

0.252 0.531 0.619 0.761 0.968 1.176 1.289 1.625 1.864 1.963 2.039 2.386 3.038 3.443 3.879 4.595 5.330


837

Table IV. Values of n1 p1 for stipulated producer’s and consumer’s risk double sampling plans (p2 ¼ 0.10, 2n1 ¼ n2 , a1 ¼ 0.05, b1 ¼ 0.10)

Figure 4. Double sampling plan OC curves for stipulated producer and consumer’s risk of the overall plan and producer’s risk of the decision based upon the first sample, and corresponding quality levels

assume that c1 is fixed, p1 ¼ p3 , and that there is no relation between the sizes of both samples, as opposed to Case 3. The optimization problem arising from such assumptions, designated as P 4 , is represented as follows:

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min n;m;w

3 X

c2

s:t: P a1;j ¼

838

e2j

ð30Þ

j¼1

c1 X ðnpj Þx1 expð2npj Þ ; j [ {1; 2; 3} x1 ! x ¼0

ð31Þ

1

P a2;j ¼

c2 2 2x1 X X ðnpj Þx1 expð2npj Þ cX ðmpj Þx2 expð2mpj Þ ; j [ {1; 2; 3} ð32Þ wc2 x1 ! x2 ! c2[C 2 x ¼c þ1 x ¼0 1

1

2

P a1;1 þ P a2;1 ¼ 1 2 a1 þ e1

ð33Þ

P a1;2 þ P a2;2 ¼ b1 2 e2

ð34Þ

P a1;3 ¼ 1 2 a2 þ e3

ð35Þ

X wc2 ¼ 1

ð36Þ

c2 [C 2

m$

X

wc2 c2 þ 1

ð37Þ

c2[C 2

n $ c1 þ 1

ð38Þ

n1 ¼ dne

ð39Þ

n2 ¼ m

ð40Þ

max

c2 ¼

c2 X

wc2 c2 þ 1

ð41Þ

c2¼

wc2 [ {0; 1}; ej $ 0; j [ {1; 2; 3}; c1 fixed All equations have the same meanings as before, with equation (35) modelling the equality in the point ð p3 ; 1 2 a2 Þ of the OC1 curve. One may see from Table V that these plans do require larger first sample sizes relative to the second sample. This behavior is due to the relatively high value of the probability of acceptance assumed for the decision based only upon the first sample 2 0.60.

p1

p2/p1

c1

c2

n1p1

n2p1

0.0069 0.0124 0.0154 0.0186 0.0196 0.0232 0.0239 0.0278 0.0307 0.0338 0.0361 0.0382 0.0407 0.0452

14.50 8.07 6.48 5.39 5.09 4.31 4.19 3.60 3.26 2.96 2.77 2.62 2.46 2.21

0 0 1 1 1 2 2 3 3 4 5 5 6 7

4 2 6 4 4 6 6 8 7 9 10 10 11 13

0.517 0.520 1.389 1.391 1.385 2.274 2.291 3.222 3.221 4.155 5.010 5.115 6.057 7.000

0.041 0.173 0.262 0.130 0.216 0.232 0.263 0.444 0.460 0.709 0.614 0.840 0.772 1.304

5. Conclusions This paper presents a general framework to design simple and double acceptance sampling plans for attributes based on OC curves. Our framework relies on the minimization of the square norm of the error of the algebraic equations representing the probability of acceptance in the points of OC curves controlled with respect to the sample size and acceptance number. The points controlled correspond to quality level requirements of the producer and of the consumer. Our approach leads to optimization problems of the NLP or MINLP classes, solved under the GAMS computational platform. Under such problem formulations and solutions, we were able find combinations of stipulated quality levels and probabilities of acceptance for which there are no feasible sampling plans, with close alternatives being obtained by relaxing some of the constraints. Furthermore, this approach provided results which are in strong agreement with those published in standard tables, thus proving that it can be easily applied to a wide range of problems, with any combination of parameters specified, and leading to remarkable robustness and accuracy regarding the optimal solutions thus identified. Under several conditions the solutions found by our approach outperform the ones reported in the classical sampling tables that are available, thus pointing out the advantages of using modern optimization problem formulations and solvers to address the definition of the most appropriate product quality control inspection procedures. Notes 1. Adapted from Chemical Corps Engineering Agency (1953). 2. Adapted from Chemical Corps Engineering Agency (1953). 3. Adapted from Chemical Corps Engineering Agency (1953). References ASQC (1981), American National Standard – Sampling Procedures and Tables for Inspection by Attributes – ANSI/ASQC Z1.4-1981, ASQC, Milwaukee, WI.


839 Table V. Values of n1 p1 for stipulated producer’s and consumer’s risk double sampling plans (p2 ¼ 0.10, a1 ¼ 0.05, b1 ¼ 0.10, a2 ¼ 0.40, p3 ¼ p1 )

IJQRM 25,8

840

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Tagaras, G. and Lee, H.L. (1987), “Optimal Bayesian single-sampling attribute plans with modified Beta prior distribution”, Naval Research Logistics, Vol. 34 No. 6, pp. 789-801. Taylor, W.A. (1986), “A program for selecting efficient binomial double sampling plans”, Journal of Quality Technology, Vol. 18 No. 1, pp. 67-73. Wang, R.-C. and Chen, C.-H. (1997), “The Dodge-Roming double sampling plans based on fuzzy optimization”, International Journal of Quality Science, Vol. 2 No. 1, pp. 52-64. Wetherill, G.B. and Chiu, W.K. (1975), “A review of acceptance sampling schemes with emphasis on the economic aspect”, International Statistical Review, Vol. 43 No. 2, pp. 191-210. About the authors Belmiro P.M. Duarte is Assistant Professor at the Chemical and Biological Engineering Department of the Polytechnic School of Engineering at Coimbra, Portugal. He holds a PhD in Simulation and Process Control and a MSc in Organizations Management from the University of Coimbra. His main research interests are in process control, optimization, and process design under uncertainty. Belmiro P.M. Duarte is the corresponding author and can be contacted at: [email protected]. Pedro M. Saraiva is Associate Professor at the Chemical Engineering Department of the University of Coimbra. He holds a PhD in Chemical Engineering from MIT (USA). He received the Feigenbaum award from ASQ (American Society for Quality) in 1998. His main research interests are in quality management, innovation, process systems engineering and applied statistics. He is also Vice-Rector of the University of Coimbra.

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