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A general repeat inspection plan for dependent multicharacteristic critical components
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European Journal of Operational Research 191 (2008) 374–385 www.elsevier.com/locate/ejor
Stochastics and Statistics
A general repeat inspection plan for dependent multicharacteristic critical components S.O. Duffuaa *, Mehmood Khan Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 1 December 2005; accepted 27 February 2007 Available online 25 March 2007
Abstract A component is critical if it causes disaster or a very high cost upon failure. Multicharacteristic critical components exist in many systems. Such components could be a part of an aircraft, space shuttle, a special weapon system or a gas ignition system. In many situations, characteristics’ failures of these components are statistically dependent. In this paper, a new inspection plan for such components is proposed. A mathematical model that depicts the plan is developed and an example demonstrating the results of the model is given. The advantage of this model over the other model where independence of the characteristics’ failure is assumed for the case of dependency is illustrated. The model resulted in an average of 32.4% reduction in cost compared to the situation where the dependency case is solved assuming statistical independence. Ó 2007 Published by Elsevier B.V. Keywords: Quality control; Inspection error; Repeat inspection plan; Dependent characteristics
1. Introduction The focus of this paper is on modeling and determining optimal inspection plan for critical components with dependent characteristics’ failures. A component is critical if it causes disaster or a very high cost upon failure. Such components can be a part of an aircraft, a space shuttle, a space shuttle system or a complex gas ignition system. For critical components, a common practice in industry is to institute multiple inspections [5]. The reason for multiple inspections is that inspections are never perfect. There is always the possibility of misclassifications. All misclassifications have costs such as the cost of misclassifying a good component as rework. In the case of critical components the cost of false acceptance is much higher than the cost of false rejection, because falsely accepted components may result in system failure, which may involve human losses. Therefore, it is perceived and shown that repeat inspections are likely to reduce the costs of the errors and increase the cost of inspection. However, the expected total cost is likely to reduce. Hence, a need exists to determine the optimal inspection plan and the optimal number of repeat inspections that minimizes the expected total cost.
*
Corresponding author. Tel.: +966 3 860 2692; fax: +966 3 860 4426. E-mail address: duff[email protected] (S.O. Duffuaa).
0377-2217/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.ejor.2007.02.033
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Ayoub et al. [1] defined mean inspection error to be the average number of defectives classified as good items by the inspector. They gave a formula for average outgoing quality (AOQ) and average total inspection (ATI) from a single sampling plan under inspection error. Collins et al. [7] considered the effects of inspection error on probability of acceptance, average outgoing quality and average total inspection. They examined these measures under both replacement and non-replacement assumptions. Bennet et al. [2] investigated the effect of error on a single sampling plan with known incoming quality. Sylla and Drury [19] proposed a model, which uses a form of SDT to predict inspector performance in order to improve system performance. They gave the concept of liability to characterize the inspector’s ability to respond to costs, penalties and probabilities involved in the inspection decision. Raouf et al. [18] were the first to develop a model for determining the optimal number of repeat inspections for multicharacteristic components to minimize the total expected cost per accepted component due to Type I error, Type II error and cost of inspection. Tang and Schneider [20] investigated the economic and statistical effects of inspection error on the complete inspection plan. They developed two models with considerations of inspection error under different rework schemes and then compared to the model without inspection error consideration. Lee [17] provided a simplified version of the model given by Raouf et al. [18] to evaluate the costs in the multiple-cycle inspection schemes for multicharacteristic components. He extended the results for the case where the probabilities of defectives are random. Chandra and Schall [5] studied the effect of replicate measurements on average outgoing quality and the average total inspection. They obtained the optimum number of replications based on total cost of inspection. Duffuaa and Raouf [9] developed three mathematical optimization models for multicharacteristic repeat inspection. Duffuaa and Raouf [10] established an optimal rule for sequencing characteristics for inspection in the plan proposed by Raouf et al. [18]. Duffuaa and Nadeem [11] extended the model proposed by Raouf et al. [18] to cases where defective rates are statistically dependent. They proposed an algorithm to determine the optimal number of repeat inspections and sequenced the characteristics for inspection in order to minimize the total expected cost. Duffuaa and Al-Najjar [12] proposed a new inspection plan for critical multicharacteristic components with variable number of inspections for different characteristics. They proposed an algorithm to determine the optimal number of repeat inspections and sequenced the characteristics for inspection in order to minimize the total expected cost. Duffuaa and Khan [13] extended the work of Duffuaa and Nadeem [11] for the case where inspectors make a number of classification errors. Duffuaa [8] investigated the statistical and economic impact of the inspector errors on the performance measures, i.e. ATI, AOQ and ETC of a complete inspection plan. He concluded that Type I and Type II errors have significant effect on the performance measures of repeat inspection plans. Chen and Labbrecht [6] proposed a model to optimize the sequence and frequency of inspections of multicharacteristic components. They used marginal analysis and gave an efficient algorithm to find the optimal plan. Hong et al. [16] developed economic screening procedures when the rejected items are reworked. Screening procedures based on the performance variable of interest and a correlated variable are considered. They considered the cost incurred by imperfect quality, reprocessing cost and inspection cost. Most of the models in the literature are based on the plan given by Raouf et al. [18] and Duffuaa and Al-Najjar [12]. Hald and Keiding [15] determined the optimal asymptotic K-shape decision sampling and decision rule. Asymptotic expressions were derived for sample size, acceptance and rejection criteria and the minimum regret by minimizing average regret. Garrison [14] developed a basic program for Walds sequential sampling for attribute inspection. This program eliminates the need for lengthy manual calculation. Brint [3] proposed a sequential sampling model for scheduling the maintenance of switchgear within a distribution network. Canty [4] developed a sequential attribute sampling inspection game for facilities that pose environmental hazard. Wei [21] in his thesis evaluated the performance of sequential sampling schemes for independent and dependent models. There are no publications for critical components having statistically dependent characteristics, pertaining to inspectors making several types of classifications. In these situations an inspector may commit a number of errors. The objective of the study in this paper is to develop a model for such components having several characteristics with known incoming quality. This model is an extension of the model given by Duffuaa and Nadeem [11], Duffuaa and Khan [13]. The classification of the characterisitics in that work was into two categories, i.e. good or defective. While the inspection plan for this paper assumes three classifications for the components
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under inspection, which are: characteristics meet specifications (good), need rework or are scrap. In this situation an inspector can make six types of errors. These are: 1. A good characteristic is classified as rework or scrap. 2. A rework characteristic is classified as good or scrap. 3. A scrap characteristic (defective) is classified as good or rework. The probabilities of misclassification are assumed to be known. The model determines the optimal number of inspections necessary to minimize the total cost, per accepted component. 2. Model description The inspection model is developed for components with several characteristics whose failure rates are statistically dependent. A characteristic can be classified in one of three classes. Theses classes are: meets specifications (good), rework or scrap. A component is rejected if one its characteristic does not meet specifications and is accepted if all of its characteristics meet the quality specifications. We denote a random variable Xi for the characteristic i, which takes the value 0 if the characteristic is scrap, 1 if the characteristic is rework and 2 if it meets specifications. The joint probability mass function of the multivariate random variable X = (x1, x2, . . . , xN) is assumed to be known. This assumption is essential in the model, however the joint probability mass function can be estimated from previous records of inspection or an experiment can be designed to estimate the joint probability mass function. The probabilities of the misclassification errors by the inspector are assumed to be known. Three types of costs are considered: (i) cost of false rejection of the components that meet specification (sent to rework or to scrap), (ii) cost of false acceptance of the components which are to be reworked or to be scrapped, and finally, (iii) cost of inspection. The cost of inspection is taken to be of two types, i.e., for the inspection at the inspection station and for the inspection at the rework station. The inspection process at the rework station is assumed to be error free and therefore it costs more to inspect there. The estimates for these costs are assumed to be available. The inspection plan is given in Fig. 1 and is described as follows: an inspector inspects one particular characteristic for each component entering the inspection process, and classifies it as meeting specifications, scrap or rework. All the accepted components and the ones that are found to be meeting specifications at rework station, go to the second inspector, who inspects the second characteristic. This chain of inspection continues until all the characteristics are inspected once. This completes one cycle of inspection. All accepted components, if necessary, go to the next cycle of inspection, and this process is repeated a total of n times before the components are finally accepted. Here n is the optimal number of inspections necessary to minimize the total cost per accepted component. The inspection plan for the jth cycle is shown in Fig. 1 (j = 1, 2, . . . , n). Finally, the accepted components will be those that are accepted in the nth cycle, and the total scrapped components will be the sum of those scrapped in the 1st, 2nd, . . . , nth cycles. Prior to stating the model the following notations are defined. Notations I an index representing a characteristic J an index representing a cycle of inspection K an index representing a stage (an inspection station) Mj number of components entering the jth cycle of inspection N number of characteristics in each component to be inspected Xi a discrete random variable which takes value 0 if characteristic i is scrap, 1 if it is rework and 2 if it is meeting specifications P(x1 ; x2 ; . . . ; xN ) joint probability mass function of the random variables Xi, i = 1, . . . , N, at the start of inspection Pi(xi) marginal probability mass function of the random variable Xi j P(x1 ; x2 ; . . . ; xN ) joint probability mass function of the random variables Xi for a component entering the jth cycle of inspection
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Incoming components for jth cycle of inspection
jth inspection of all components for 1st characteristic
Components sent to rework in the 1st stage
Components accepted in the 1st stage
Fraction of good items Components scrapped in the jth cycle
j-th inspection of all components for 2nd characteristic
Components accepted in the 2nd stage
Components sent to rework in the 2nd stage
Fraction of good items Components scrapped at rework station in jth cycle jth inspection of all components for N th characteristic
Components accepted in the N th stage
Components sent to rework in the N th stage
Fraction of good items Components accepted in the jth cycle
Fig. 1. Inspection plan for jth cycle, j = 1, 2, . . . , n.
kj
Pi(xi) marginal probability mass function of the random variable Xi for a component in jth cycle entering the kth stage of inspection Eigr probability of classifying the ith characteristic in the sequence of inspection as rework when it is meeting specifications Eigs probability of classifying the ith characteristic in the sequence of inspection as scrap when it is meeting specifications Eirg probability of classifying the ith characteristic in the sequence of inspection as meeting specifications when it is rework Eirs probability of classifying the ith characteristic in the sequence of inspection as scrap when it is rework Eisg probability of classifying the ith characteristic in the sequence of inspection as meeting specifications when it is scrap Eisr probability of classifying the ith characteristic in the sequence of inspection as rework when it is scrap Mi,j number of components entering the ith stage of inspection in the jth inspection cycle PGi,j probability of a component being non-defective entering the ith stage of the jth cycle PRi,j probability that a component requires rework while entering the ith stage of the jth cycle
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PSi,j FRi,j FAi,j CAi,j FGRi,j Rk,j Ca Cfgs Cfrs Ci1 Ci2 A(j) CFR(j) CFA(j) CI(j) TCFR TCFA TCI TA E(tc)jj E( )
probability that a component is scrap while entering the ith stage of the jth cycle number of falsely rejected components entering the ith stage of the jth inspection cycle number of falsely accepted components entering the ith stage of the jth inspection cycle number of correctly accepted components entering the ith stage of the jth inspection cycle number of non-defective components falsely sent to rework in the ith stage of the jth cycle tate of rejection of components due to ith characteristic at the kth stage of the jth cycle cost of accepting a component (scrap or rework) cost of misclassifying a good component to be scrap cost of misclassifying a rework component to be scrap cost of inspecting characteristic i at the inspection station cost of inspecting characteristic i at the rework station number of accepted components in the jth cycle. cost of false rejection in the jth cycle cost of false acceptance in the jth cycle cost of inspection in the jth cycle total cost false rejection total cost false acceptance total cost inspection total number of accepted components expected total cost per accepted component after j cycles of inspection expected value of the argument inside the parenthesis
3. Development of the model We know the joint probability mass functions for Xi = 1, 2, . . . , N, at the start of inspection. From the joint probability mass function, the individual marginal probability mass functions are given by: XX XX X 1 P i ðxi Þ ¼ P ðx1 ; x2 ; . . . ; xN Þ: ð1Þ x1
x2
xi1
xiþ1
xN
The marginal probabilities for scrap, rework and good (meeting specifications) components are computed from Eq. (1) at the start of the inspection to be 1Pi(0), 1Pi(1) and 1Pi(2), respectively, where 1
P i ð2Þ ¼ 1 1 P i ð0Þ 1 P i ð1Þ:
ð2Þ
After the inspection of first characteristic, the marginal probabilities are updated using Bayes’ theorem: 2
P i ð0Þ ¼
P i ð0ÞEisg ; ½1 P i ð2Þð1 Eigs Þ þ P i ð0ÞEisg þ P i ð1ÞEirg
ð3Þ
2
P i ð1Þ ¼
P i ð1ÞEirg ½1 P i ð2Þð1 Eigs Þ þ P i ð0ÞEisg þ P i ð1ÞEirg
ð4Þ
2
P i ð2Þ ¼ 1 2 P i ð0Þ 2 P i ð1Þ:
and ð5Þ
In general, we can write the probability of being defective for the ith characteristic entering the jth cycle of inspection as follows: j1
j
P i ð0Þ ¼
½j1 P
P i ð0ÞEisg ; j1 P ð0ÞE j1 P ð1ÞE ð2Þð1 E Þ þ i igs i isg þ i irg
ð6Þ
j1
j
P i ð1Þ ¼
½j1 P
P i ð1ÞEirg ; j1 P ð0ÞE j1 P ð1ÞE i ð2Þð1 E igs Þ þ i isg þ i irg
ð7Þ
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379
and j
P i ð2Þ ¼ 1 j sP i ð0Þ j P i ð1Þ:
ð8Þ
Due to the dependency between characteristics, the marginals of the other characteristics must be updated before inspecting them. For this purpose, we update the joint probabilities first. Therefore, after inspecting ith characteristic in the first cycle the joint probability mass functions are updated as 1
P 1 ðx1 ; x2 ; . . . ; xN Þ ¼ P ðx1 ; x2 ; . . . ; xN Þ;
2
P 1 ðx1 ; x2 ; . . . ; xN Þ ¼ 1 P 1 ðx1 ; x2 ; . . . ; xN Þ 1
2
P i ðxi Þ ; P i ðxi Þ
i.e. multiply the old joint probability by the ratio of the updated and old marginals for the ith characteristic (which is just inspected). It is to be noted that the new joints satisfy the condition of being a probability mass function. The generalized form of the above relation for cycle j can be written as j
j P i ðxi Þ : P ðx1 ; x2 ; . . . ; xN Þ ¼ j1 P ðx1 ; x2 ; . . . ; xN Þ j1 P i ðxi Þ
ð9Þ
From these updated joints we obtain the marginals for the remaining characteristics. Then we proceed on to inspect the next characteristic in the 2nd stage. This process is repeated until all the characteristics are inspected. The probability that the component is good, at the end of the cycle is PGN þ1;1 ¼ N P 1 ð2; 2; . . . ; 2Þ:
ð10Þ
Before any inspection is performed, the total cost of false acceptance is EðtcÞjj¼0 ¼ C a ð1 P ð2; 2; . . . ; 2ÞÞ:
ð11Þ
The above equation does not incorporate any inspection or rejection costs as we were just at the start of the inspection process. The expected total cost per accepted component after n cycles of inspection is given by EðtcÞjj¼n ¼ ½TCFR þ TCFA þ TCI=TA:
ð12Þ
Our objective is to determine the optimal n which minimizes the expected total cost per accepted component. 3.1. Analysis of the jth cycle If M1,j is the number of components entering the first stage of the jth cycle, M 1;j ¼ M j : The probabilities of a component being non-defective, to be rework or to be scrap are given by PG1;j ¼ P ð2; 2; . . . ; 2Þ; XX X XX X P ð1; x2 ; x3 ; . . . ; xN Þ þ P ðx1 ; 1; x3 ; . . . ; xN Þ þ PR1;j ¼ x2
þ
x3
XX x1
x2
XX x1
xN
x2
X xN
X
x1
P ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
x3
XX x3
x4
xN
X
ð13Þ
P ð1; 1; x3 ; . . . ; xN Þ
xN
P ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ P ð1; 1; 1; . . . ; 1Þ;
ð14Þ
xN 2
PS 1;j ¼ 1 PG1;j PR1;j :
ð15Þ
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The probability of the first characteristic to be rework or scrap would be given by XXX X j P 1 ð1Þ ¼ P ð1; x2 ; x3 ; . . . ; xN Þ; j
P 1 ð0Þ ¼
x2
x3
x3
x2
x3
x3
XXX
xN
X
P ð0; x2 ; x3 ; . . . ; xN Þ:
xN
Thus the probability of this characteristic to be non-defective would be j
P 1 ð2Þ ¼ 1 j P 1 ð1Þ j P 1 ð0Þ:
The expected number of components falsely sent to rework is FR1;j ¼ M 1;j ðPG1;j E1gr þ PS 1;j E1sr Þ:
ð16Þ
The expected number of components falsely sent to scrap is FS 1;j ¼ M 1;j ðPG1;j E1gs þ PR1;j E1rs Þ:
ð17Þ
The expected number of components falsely accepted is FA1;j ¼ M 1;j ðj P 1 ð1ÞE1rg þ j P 1 ð0ÞE1sg þ ð1 PG1;j j P 1 ð1Þ j P 1 ð0ÞÞð1 E1gr E1gs ÞÞ:
ð18Þ
The expected number of components correctly accepted is CA1;j ¼ M 1;j PG1;j ð1 E1gs E1gs Þ:
ð19Þ
All accepted components in this stage move on to the 2nd stage where the next characteristic is inspected. So, M 2;j ¼ FA1;j þ CA1;j þ FGR1;j ;
ð20Þ
where FGR1;j ¼ M 1;j PG1;j E1gr : The probabilities that the components bear are PG2;j ¼ j P 2 ð2; 2; . . . ; 2Þ; XX X XX X j j P 2 ð1; x2 ; x3 ; . . . ; xN Þ þ P 2 ðx1 ; 1; x3 ; . . . ; xN Þ þ PR2;j ¼ x2
þ
x3
XX x1
xN
x2
XX x1
X
x1 j
P 2 ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
xN
x2
X
x3
XX x3
j
xN
X
x4
j
ð21Þ
P 2 ð1; 1; x3 ; . . . ; xN Þ
xN
P 2 ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ j P 2 ð1; 1; 1; . . . ; 1Þ;
ð22Þ
xN 2
PS 2;j ¼ 1 PG2;j PR2;j :
ð23Þ
The probability of the second characteristic to be rework or scrap would be XXX X j j P 2 ð1Þ ¼ P 2 ðx1 ; 1; x3 ; . . . ; xN Þ; j
P 2 ð0Þ ¼
x1
x3
x3
x1
x3
x3
XXX
xN
X
j
P 2 ðx1 ; 0; x3 ; . . . ; xN Þ
xN
and the components are classified as FR2;j ¼ M 2;j ðPG2;j E2gr þ PS2;j E2sr Þ;
ð24Þ
FS 2;j ¼ M 2;j ðPG2;j E2gs þ PR2;j E2rs Þ; j
j
ð25Þ j
j
j
j
j
j
FA2;j ¼ M 2;j ½ P 2 ð1ÞE2rg þ P 2 ð0ÞE2sg þ f1 PG2;j P 2 ð1Þ P 2 ð0Þ þ P 1 ð1Þ P 2 ð1Þ þ P 1 ð0Þ P 2 ð0Þg ð1 E2gr E2gs Þ;
ð26Þ
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CA2;j ¼ M 2;j PG2;j ð1 E2gs E2gs Þ:
381
ð27Þ
Similarly, we can write for the ith stage of the jth cycle, M i;j ¼ FAi1;j þ CAi1;j þ FGRi1;j ;
ð28Þ
where FGRi1;j ¼ M i1;j PGi1;j Ei1;gr ; PGi;j ¼ j P i ð2; 2; . . . ; 2Þ; XX X XX X j j PRi;j ¼ P i ð1; x2 ; x3 ; . . . ; xN Þ þ P i ðx1 ; 1; x3 ; . . . ; xN Þ þ x2
þ
x3
XX x1
x2
XX x1
xN
X
x1 j
P i ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
xN
x2
X
x3
XX x3
j
x4
xN
X
j
ð29Þ
P i ð1; 1; x3 ; . . . ; xN Þ
xN
P i ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ j P i ð1; 1; 1; . . . ; 1Þ;
ð30Þ
xN 2
PS i;j ¼ 1 PGi;j PRi;j :
ð31Þ
The probability of the ith characteristic to be rework or scrap would be XXX X j j P i ð1Þ ¼ P i ð1; x2 ; x3 ; . . . ; xN Þ; x2
j
P i ð0Þ ¼
x3
x3
XXX x2
x3
xN
X
x3
j
P i ð0; x2 ; x3 ; . . . ; xN Þ;
xN
FRi;j ¼ M i;j ðPGi;j Eigr þ PS i;j Eisr Þ;
ð32Þ
FS i;j ¼ M i;j ðPGi;j ENgs þ PRi;j ENrs Þ; FAi;j ¼ M i;j ½j P i ð1ÞEirg þ j P i ð0ÞEisg þ f1 PGi;j j P i ð1Þ j P i ð0Þ þ
ð33Þ i1 X
j
P k ð1Þj P i ð1Þ
k¼1
þ
i2 X i1 X
P k;j ð1Þj P 1 ð1Þj P i ð1Þ þ j P 1 ð1Þj P 2 ð1Þ j P i ð1Þ þ
k¼1 l¼kþ1
þ
i2 X i1 X
i1 X
j
P k ð0Þj P i ð0Þ
k¼1 j
P k ð0Þj P 1 ð0Þj P i ð0Þ þ j P 1 ð0ÞP 2;j ð0Þ j P i ð0Þgð1 Eigr Eigs Þ;
ð34Þ
k¼1 l¼kþ1
CAi;j ¼ M i;j PGi;j ð1 Eigs Eigs Þ:
ð35Þ
In a similar fashion, the next cycle can be analyzed using the updated versions of the joint and marginal probability mass functions. Next, we summarize the results of the jth cycle. 3.2. Results of the jth cycle Total number of accepted components in the jth cycle: AðjÞ ¼ M jþ1 ¼ CAN ;j þ FAN ;j þ FGRN ;j ;
ð36Þ
where FGRN ;j ¼ M N ;j PGN ;j EN ;gr : Total cost of false acceptance for the jth cycle: CFAðjÞ ¼ C a FAN ;j : Total cost of false rejection for the jth cycle:
ð37Þ
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CFRðjÞ ¼
N X
M i;j ðC fgs PGi;j Eigs þ C frs PRi;j Eirs Þ:
ð38Þ
i¼1
Cost of inspection at the inspection station, for the jth cycle: N X CI1 ðjÞ ¼ C i1 M i;j :
ð39Þ
i¼1
Cost of inspection at the rework station, for the jth cycle: N X CI2 ðjÞ ¼ C i2 M i;j ðPGi;j Eigr þ PS i;j Eisr þ PRi;j ð1 Eirg Eirs ÞÞ:
ð40Þ
i¼1
So, the total cost of inspection for the jth cycle is CIðjÞ ¼ CI1 ðjÞ þ CI2 ðjÞ:
ð41Þ
3.3. Minimization of inspection cost for jth cycle The total cost of inspection in a cycle is influenced by the sequence in which the characteristics are inspected. The characteristic with lower cost of inspection and higher rate of rejection should be inspected first, as proposed by Duffuaa and Raouf [10] for independent characteristics. In this model, we use the rule given by Duffuaa and Nadeem [11] for the dependent characteristics. So, at the beginning of the jth cycle, we compute the ratio Ratio ¼ C i =Ri;j ;
ð42Þ
where Ci, the cost of inspection for characteristic i, is given by C i ¼ C i1 þ j1 P i ð1ÞC i2 and the rejection rate for the ith characteristic in the jth cycle is Ri;j ¼ j1 P i ð0Þð1 Eisg Eisr Þ þ j1 P i ð1ÞEirs þ j1 P i ð2ÞEigs : So, we select the characteristic with the lowest ratio (42). Similarly, at the kth stage of the jth cycle we compute Ri;j;k ¼ C i =Rk;j ;
ð43Þ
where C i ¼ C i1 þ j P i ð1ÞC i2 and Rk;j ¼ j P k1 ð0Þð1 Eisg Eisr Þ þ j P k1 ð1ÞEirs þ j P k1 ð2ÞEigs and select the one among the remaining characteristics with the lowest ratio (42) to be inspected at the kth stage. 3.4. Total cost after n cycles After analyzing n cycles, we compute total cost of false acceptance TCFA, total cost of false rejection TCFR, total cost of inspection TCI and the total number of accepted components TA, as TCFA ¼ CFAðnÞ ¼ C a FAN ;n ; ð44Þ n X TCFR ¼ ½CFRðjÞ; ð45Þ j¼1
TCI ¼
n X j¼1
½CIðjÞ;
ð46Þ
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TA ¼ CAN ;n þ FAN ;n :
383
ð47Þ
Substituting TCFA, TCFR, TCI and TA in (12), we obtain the total expected cost at the end of nth cycle. Next is given a computational procedure to determine optimal n.
4. Computational procedure Step 1. Compute E(tc)jj = 0 using (11). Set j = 1. Step 2. Compute jPi(0), jPi(1), jPi(2) and Mj using (6), (7), (8) and (36), respectively. Select the ith characteristic for inspection based on the ratio (43). Repeat this until all the characteristics are inspected. Step 3. Compute Mi,j, FSi,j, FAi,j and CAi,j using (28), (33), (34), (35), respectively. Step 4. Compute CFA(j), CFR(j) and CI(j) using (37), (38) and (41), respectively. Step 5. Compute TCFA(j), TCFR(j), TCI(j) and TA(j) using (44), (45), (46) and (47), respectively. Step 6. Compute E(tc)jj using (12). Step 7. If EðtcÞjj < EðtcÞjj1 , set j ¼ j þ 1 and go to step 2. Otherwise n ¼ j 1. Lemma 1. The above computational procedure (algorithm) provides a local minimum. Proof. The total expected cost E(tc)jn is a function of a single variable n. Assume the algorithm terminates at n*. Then EðtcÞjn 6 EðtcÞjn þ1 otherwise the algorithm would not have terminated at n* see step 7 of the algorithm. Also EðtcÞjn 6 EðtcÞjn 1 since if this is not true the algorithm would have terminated at n* 1 The above two arguments in this proof show that n* is a local minimum. h
5. Practical applications and example Multicharacteristic components are found in many industrial systems [13]. These systems include gas ignition systems [18], aircraft avionics systems, space shuttle and nuclear reactors. The details of some are presented below in order to demonstrate the use of the model. Printed circuit boards in aircraft avionics systems have many connections, welding points and wires that need to be inspected for quality assurance. Failure of any of these connections or wires could be catastrophic. These boards are thus treated as critical components. Gas ignition systems in heavy mechanical industries and ships have many valves, switches and circuits that need to be inspected for quality assurance. Failure of any of these switches or circuits could result in a heavy loss or fatality. Vacuum pumps on the conveyors in a food handling industry have many seals and coatings that need to be inspected for quality assurance. Failure of any of these characteristics could result in a high scale casualty. The model developed in the paper can be used to minimize the occurrence of the disasters caused by the above critical components if the characteristics’ defective rates are statistically dependent. Characteristic defective rates can be obtained from historical data or the production process capability. In order to illustrate the model presented in this paper, the following example is provided. The example is a modification of the one given in [13]. The example in [13] has been modified to induce the dependency between characteristics’ defective rates. This is done by violating the independence criterion (i.e. their joint probabilities should not be the same as the product of their marginal probabilities). Also, additional costs needed for the example are established based on the authors’ experience. A program is developed implementing the above computational procedure and is used to obtain the optimal number of repeat inspections.
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Assuming that the following data are given I
III
0 1 2
0
1
2
II
II
II
0
1
2
0
1
2
0
1
2
0.009 0.00375 0.005
0.009 0.00375 0.005
0.0072 0.03 0.004
0.00075 0.00375 0.00075
0.00075 0.00375 0.00075
0.006 0.03 0.006
0.0135 0.0005 0.0675
0.0135 0.0005 0.0675
0.108 0.004 0.54
N ¼ 3;
M ¼ 100;
Egs ¼ 0:03;
C a ¼ 100; 000;
Esg ¼ 0:1;
C i1 ¼ 100;
C i2 ¼ 5000;
C fgs ¼ 10; 000;
C frs ¼ 5000;
Egr ¼ Erg ¼ Esr ¼ Ers ¼ 0:1:
Solving this example using the proposed model gives the following results: Expected total cost per accepted component without inspection = 46000.00 Cycle 1 PG(1) A(1) Etc(1)
0.54 50 9861.40
Cycle 2 PG(2) A(2) Etc(2)
0.96 44 8778.72
Cycle 3 PG(2) A(2) Etc(2)
0.99 40 11371.81
Optimal no. of inspection = 2.
In order to demonstrate the utility of the model a set of inspection problems are generated randomly where characteristics’ defective rates are statistically dependent. The results of this model are compared to the model where independence is assumed for these examples. The results indicated that model in this paper performed better in terms of minimizing ETC. The average reduction in the cost is 32.14%. 6. Conclusion A general inspection plan for determining optimal number of inspections for critical multicharacteristic components is developed. The inspection plan takes into account six types of misclassification errors. A model depicting the inspection plan for the case where the characteristics’ defective rates are statistically dependent has been developed. A computational procedure is outlined to obtain the optimal number of repeat inspections that minimizes the total cost of inspection. The model is illustrated with the help of a numerical example. Acknowledgment The authors acknowledge the support provided by the King Fahd University of Petroleum and Minerals, Department of Systems Engineering in conducting this research.
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References [1] M.M. Ayoub, B.K. Lambert, A.G. Walvaker, Effects of two types of inspection errors on single sampling plans, Project presented at Human Factors Society, San Francisco, 1970. [2] G.K. Bennett, K.E. Case, J.W. Schmidt, The economic effects of inspector error on attribute sampling plans, Naval Research Logistics Quarterly 21 (3) (1974) 431–443. [3] A.T. Brint, Sequential inspection sampling to avoid failure critical items being in a risk condition, Journal of the Operational Research Society 51 (9) (2000) 1051–1059. [4] M. Canty, A sequential attribute sampling inspection game for item facilities, Naval Research Logistics 48 (6) (2001) 496–505. [5] J. Chandra, S. Schall, The use of repeated measurements to reduce the effect of measurement errors, IIE Transactions 20 (1) (1998) 83–87. [6] S.X. Chen, M. Labbrecht, The optimal frequency and sequencing of tests in the inspection of multicharacteristic components, IIE Transactions 29 (1997) 1039–1049. [7] R.D. Collins, K.E. Case, G.K. Bennett, The effects of inspection error on single sampling plans, International Journal of Production Research 11 (3) (1973) 289–298. [8] S.O. Duffuaa, Impact of inspection errors on performance measures of a complete repeat inspection plan, International Journal of Production Research 34 (7) (1996) 2035–2049. [9] S.O. Duffuaa, A. Raouf, Mathematical optimization models for multicharacteristic repeat inspections, Applied Mathematical Modelling 13 (1989) 408–412. [10] S.O. Duffuaa, A. Raouf, An optimal sequence in multicharacteristics inspection, Journal of Optimization Theory and Applications 67 (1) (1990) 79–86. [11] S.O. Duffuaa, I.A. Nadeem, A complete inspection plan for dependent multicharacteristic critical components, International Journal of Production Research 32 (8) (1994) 1897–1907. [12] S.O. Duffuaa, H.J. Al-Najjar, An optimal complete inspection plan for critical multicharacteristic components, Journal of the Operational Research Society 46 (1995) 930–942. [13] S.O. Duffuaa, M. Khan, An optimal repeat inspection plan with several classifications, Journal of the Operational Research Society 53 (2002) 1016–1026. [14] D.R. Garrison, Wald sequential sampling for attribute inspection, Journal of Quality Technology 16 (3) (1984) 172–174. [15] A. Hald, N. Keiding, Asymptotic properties of Bayesian decision rules for two terminal decision and multiple samplings, Journal of Royal Statistical Society, Series B (Methodological) 31 (3) (1969) 455–471. [16] S.H. Hong, S.B. Kim, H.M. Kwon, M.K. Lee, Economic design of screening procedures when the rejected items are reprocessed, European Journal of Operational Research 108 (1998) 65–73. [17] H.L. Lee, On the optimality of a simplified multicharacteristic component inspection model, IIE Transactions 20 (4) (1988) 392–398. [18] A. Raouf, J.K. Jain, P.T. Sathe, A cost minimization model for multicharacteristic component inspection, IIE Transactions 15 (3) (1983) 187–194. [19] C. Sylla, C.G. Drury, Signal detection for human error correction in quality control, Computers in Industry 26 (1995) 147–159. [20] K. Tang, H. Schneider, The effects of inspection error on a complete inspection plan, IIE Transactions 19 (4) (1987) 421–427. [21] X. Wei, Performance of sequential sampling schemes for some independent and dependent models, Ph.D. Thesis, University of Georgia, 2001.
European Journal of Operational Research 191 (2008) 374–385 www.elsevier.com/locate/ejor
Stochastics and Statistics
A general repeat inspection plan for dependent multicharacteristic critical components S.O. Duffuaa *, Mehmood Khan Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 1 December 2005; accepted 27 February 2007 Available online 25 March 2007
Abstract A component is critical if it causes disaster or a very high cost upon failure. Multicharacteristic critical components exist in many systems. Such components could be a part of an aircraft, space shuttle, a special weapon system or a gas ignition system. In many situations, characteristics’ failures of these components are statistically dependent. In this paper, a new inspection plan for such components is proposed. A mathematical model that depicts the plan is developed and an example demonstrating the results of the model is given. The advantage of this model over the other model where independence of the characteristics’ failure is assumed for the case of dependency is illustrated. The model resulted in an average of 32.4% reduction in cost compared to the situation where the dependency case is solved assuming statistical independence. Ó 2007 Published by Elsevier B.V. Keywords: Quality control; Inspection error; Repeat inspection plan; Dependent characteristics
1. Introduction The focus of this paper is on modeling and determining optimal inspection plan for critical components with dependent characteristics’ failures. A component is critical if it causes disaster or a very high cost upon failure. Such components can be a part of an aircraft, a space shuttle, a space shuttle system or a complex gas ignition system. For critical components, a common practice in industry is to institute multiple inspections [5]. The reason for multiple inspections is that inspections are never perfect. There is always the possibility of misclassifications. All misclassifications have costs such as the cost of misclassifying a good component as rework. In the case of critical components the cost of false acceptance is much higher than the cost of false rejection, because falsely accepted components may result in system failure, which may involve human losses. Therefore, it is perceived and shown that repeat inspections are likely to reduce the costs of the errors and increase the cost of inspection. However, the expected total cost is likely to reduce. Hence, a need exists to determine the optimal inspection plan and the optimal number of repeat inspections that minimizes the expected total cost.
*
Corresponding author. Tel.: +966 3 860 2692; fax: +966 3 860 4426. E-mail address: duff[email protected] (S.O. Duffuaa).
0377-2217/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.ejor.2007.02.033
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Ayoub et al. [1] defined mean inspection error to be the average number of defectives classified as good items by the inspector. They gave a formula for average outgoing quality (AOQ) and average total inspection (ATI) from a single sampling plan under inspection error. Collins et al. [7] considered the effects of inspection error on probability of acceptance, average outgoing quality and average total inspection. They examined these measures under both replacement and non-replacement assumptions. Bennet et al. [2] investigated the effect of error on a single sampling plan with known incoming quality. Sylla and Drury [19] proposed a model, which uses a form of SDT to predict inspector performance in order to improve system performance. They gave the concept of liability to characterize the inspector’s ability to respond to costs, penalties and probabilities involved in the inspection decision. Raouf et al. [18] were the first to develop a model for determining the optimal number of repeat inspections for multicharacteristic components to minimize the total expected cost per accepted component due to Type I error, Type II error and cost of inspection. Tang and Schneider [20] investigated the economic and statistical effects of inspection error on the complete inspection plan. They developed two models with considerations of inspection error under different rework schemes and then compared to the model without inspection error consideration. Lee [17] provided a simplified version of the model given by Raouf et al. [18] to evaluate the costs in the multiple-cycle inspection schemes for multicharacteristic components. He extended the results for the case where the probabilities of defectives are random. Chandra and Schall [5] studied the effect of replicate measurements on average outgoing quality and the average total inspection. They obtained the optimum number of replications based on total cost of inspection. Duffuaa and Raouf [9] developed three mathematical optimization models for multicharacteristic repeat inspection. Duffuaa and Raouf [10] established an optimal rule for sequencing characteristics for inspection in the plan proposed by Raouf et al. [18]. Duffuaa and Nadeem [11] extended the model proposed by Raouf et al. [18] to cases where defective rates are statistically dependent. They proposed an algorithm to determine the optimal number of repeat inspections and sequenced the characteristics for inspection in order to minimize the total expected cost. Duffuaa and Al-Najjar [12] proposed a new inspection plan for critical multicharacteristic components with variable number of inspections for different characteristics. They proposed an algorithm to determine the optimal number of repeat inspections and sequenced the characteristics for inspection in order to minimize the total expected cost. Duffuaa and Khan [13] extended the work of Duffuaa and Nadeem [11] for the case where inspectors make a number of classification errors. Duffuaa [8] investigated the statistical and economic impact of the inspector errors on the performance measures, i.e. ATI, AOQ and ETC of a complete inspection plan. He concluded that Type I and Type II errors have significant effect on the performance measures of repeat inspection plans. Chen and Labbrecht [6] proposed a model to optimize the sequence and frequency of inspections of multicharacteristic components. They used marginal analysis and gave an efficient algorithm to find the optimal plan. Hong et al. [16] developed economic screening procedures when the rejected items are reworked. Screening procedures based on the performance variable of interest and a correlated variable are considered. They considered the cost incurred by imperfect quality, reprocessing cost and inspection cost. Most of the models in the literature are based on the plan given by Raouf et al. [18] and Duffuaa and Al-Najjar [12]. Hald and Keiding [15] determined the optimal asymptotic K-shape decision sampling and decision rule. Asymptotic expressions were derived for sample size, acceptance and rejection criteria and the minimum regret by minimizing average regret. Garrison [14] developed a basic program for Walds sequential sampling for attribute inspection. This program eliminates the need for lengthy manual calculation. Brint [3] proposed a sequential sampling model for scheduling the maintenance of switchgear within a distribution network. Canty [4] developed a sequential attribute sampling inspection game for facilities that pose environmental hazard. Wei [21] in his thesis evaluated the performance of sequential sampling schemes for independent and dependent models. There are no publications for critical components having statistically dependent characteristics, pertaining to inspectors making several types of classifications. In these situations an inspector may commit a number of errors. The objective of the study in this paper is to develop a model for such components having several characteristics with known incoming quality. This model is an extension of the model given by Duffuaa and Nadeem [11], Duffuaa and Khan [13]. The classification of the characterisitics in that work was into two categories, i.e. good or defective. While the inspection plan for this paper assumes three classifications for the components
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under inspection, which are: characteristics meet specifications (good), need rework or are scrap. In this situation an inspector can make six types of errors. These are: 1. A good characteristic is classified as rework or scrap. 2. A rework characteristic is classified as good or scrap. 3. A scrap characteristic (defective) is classified as good or rework. The probabilities of misclassification are assumed to be known. The model determines the optimal number of inspections necessary to minimize the total cost, per accepted component. 2. Model description The inspection model is developed for components with several characteristics whose failure rates are statistically dependent. A characteristic can be classified in one of three classes. Theses classes are: meets specifications (good), rework or scrap. A component is rejected if one its characteristic does not meet specifications and is accepted if all of its characteristics meet the quality specifications. We denote a random variable Xi for the characteristic i, which takes the value 0 if the characteristic is scrap, 1 if the characteristic is rework and 2 if it meets specifications. The joint probability mass function of the multivariate random variable X = (x1, x2, . . . , xN) is assumed to be known. This assumption is essential in the model, however the joint probability mass function can be estimated from previous records of inspection or an experiment can be designed to estimate the joint probability mass function. The probabilities of the misclassification errors by the inspector are assumed to be known. Three types of costs are considered: (i) cost of false rejection of the components that meet specification (sent to rework or to scrap), (ii) cost of false acceptance of the components which are to be reworked or to be scrapped, and finally, (iii) cost of inspection. The cost of inspection is taken to be of two types, i.e., for the inspection at the inspection station and for the inspection at the rework station. The inspection process at the rework station is assumed to be error free and therefore it costs more to inspect there. The estimates for these costs are assumed to be available. The inspection plan is given in Fig. 1 and is described as follows: an inspector inspects one particular characteristic for each component entering the inspection process, and classifies it as meeting specifications, scrap or rework. All the accepted components and the ones that are found to be meeting specifications at rework station, go to the second inspector, who inspects the second characteristic. This chain of inspection continues until all the characteristics are inspected once. This completes one cycle of inspection. All accepted components, if necessary, go to the next cycle of inspection, and this process is repeated a total of n times before the components are finally accepted. Here n is the optimal number of inspections necessary to minimize the total cost per accepted component. The inspection plan for the jth cycle is shown in Fig. 1 (j = 1, 2, . . . , n). Finally, the accepted components will be those that are accepted in the nth cycle, and the total scrapped components will be the sum of those scrapped in the 1st, 2nd, . . . , nth cycles. Prior to stating the model the following notations are defined. Notations I an index representing a characteristic J an index representing a cycle of inspection K an index representing a stage (an inspection station) Mj number of components entering the jth cycle of inspection N number of characteristics in each component to be inspected Xi a discrete random variable which takes value 0 if characteristic i is scrap, 1 if it is rework and 2 if it is meeting specifications P(x1 ; x2 ; . . . ; xN ) joint probability mass function of the random variables Xi, i = 1, . . . , N, at the start of inspection Pi(xi) marginal probability mass function of the random variable Xi j P(x1 ; x2 ; . . . ; xN ) joint probability mass function of the random variables Xi for a component entering the jth cycle of inspection
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Incoming components for jth cycle of inspection
jth inspection of all components for 1st characteristic
Components sent to rework in the 1st stage
Components accepted in the 1st stage
Fraction of good items Components scrapped in the jth cycle
j-th inspection of all components for 2nd characteristic
Components accepted in the 2nd stage
Components sent to rework in the 2nd stage
Fraction of good items Components scrapped at rework station in jth cycle jth inspection of all components for N th characteristic
Components accepted in the N th stage
Components sent to rework in the N th stage
Fraction of good items Components accepted in the jth cycle
Fig. 1. Inspection plan for jth cycle, j = 1, 2, . . . , n.
kj
Pi(xi) marginal probability mass function of the random variable Xi for a component in jth cycle entering the kth stage of inspection Eigr probability of classifying the ith characteristic in the sequence of inspection as rework when it is meeting specifications Eigs probability of classifying the ith characteristic in the sequence of inspection as scrap when it is meeting specifications Eirg probability of classifying the ith characteristic in the sequence of inspection as meeting specifications when it is rework Eirs probability of classifying the ith characteristic in the sequence of inspection as scrap when it is rework Eisg probability of classifying the ith characteristic in the sequence of inspection as meeting specifications when it is scrap Eisr probability of classifying the ith characteristic in the sequence of inspection as rework when it is scrap Mi,j number of components entering the ith stage of inspection in the jth inspection cycle PGi,j probability of a component being non-defective entering the ith stage of the jth cycle PRi,j probability that a component requires rework while entering the ith stage of the jth cycle
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PSi,j FRi,j FAi,j CAi,j FGRi,j Rk,j Ca Cfgs Cfrs Ci1 Ci2 A(j) CFR(j) CFA(j) CI(j) TCFR TCFA TCI TA E(tc)jj E( )
probability that a component is scrap while entering the ith stage of the jth cycle number of falsely rejected components entering the ith stage of the jth inspection cycle number of falsely accepted components entering the ith stage of the jth inspection cycle number of correctly accepted components entering the ith stage of the jth inspection cycle number of non-defective components falsely sent to rework in the ith stage of the jth cycle tate of rejection of components due to ith characteristic at the kth stage of the jth cycle cost of accepting a component (scrap or rework) cost of misclassifying a good component to be scrap cost of misclassifying a rework component to be scrap cost of inspecting characteristic i at the inspection station cost of inspecting characteristic i at the rework station number of accepted components in the jth cycle. cost of false rejection in the jth cycle cost of false acceptance in the jth cycle cost of inspection in the jth cycle total cost false rejection total cost false acceptance total cost inspection total number of accepted components expected total cost per accepted component after j cycles of inspection expected value of the argument inside the parenthesis
3. Development of the model We know the joint probability mass functions for Xi = 1, 2, . . . , N, at the start of inspection. From the joint probability mass function, the individual marginal probability mass functions are given by: XX XX X 1 P i ðxi Þ ¼ P ðx1 ; x2 ; . . . ; xN Þ: ð1Þ x1
x2
xi1
xiþ1
xN
The marginal probabilities for scrap, rework and good (meeting specifications) components are computed from Eq. (1) at the start of the inspection to be 1Pi(0), 1Pi(1) and 1Pi(2), respectively, where 1
P i ð2Þ ¼ 1 1 P i ð0Þ 1 P i ð1Þ:
ð2Þ
After the inspection of first characteristic, the marginal probabilities are updated using Bayes’ theorem: 2
P i ð0Þ ¼
P i ð0ÞEisg ; ½1 P i ð2Þð1 Eigs Þ þ P i ð0ÞEisg þ P i ð1ÞEirg
ð3Þ
2
P i ð1Þ ¼
P i ð1ÞEirg ½1 P i ð2Þð1 Eigs Þ þ P i ð0ÞEisg þ P i ð1ÞEirg
ð4Þ
2
P i ð2Þ ¼ 1 2 P i ð0Þ 2 P i ð1Þ:
and ð5Þ
In general, we can write the probability of being defective for the ith characteristic entering the jth cycle of inspection as follows: j1
j
P i ð0Þ ¼
½j1 P
P i ð0ÞEisg ; j1 P ð0ÞE j1 P ð1ÞE ð2Þð1 E Þ þ i igs i isg þ i irg
ð6Þ
j1
j
P i ð1Þ ¼
½j1 P
P i ð1ÞEirg ; j1 P ð0ÞE j1 P ð1ÞE i ð2Þð1 E igs Þ þ i isg þ i irg
ð7Þ
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379
and j
P i ð2Þ ¼ 1 j sP i ð0Þ j P i ð1Þ:
ð8Þ
Due to the dependency between characteristics, the marginals of the other characteristics must be updated before inspecting them. For this purpose, we update the joint probabilities first. Therefore, after inspecting ith characteristic in the first cycle the joint probability mass functions are updated as 1
P 1 ðx1 ; x2 ; . . . ; xN Þ ¼ P ðx1 ; x2 ; . . . ; xN Þ;
2
P 1 ðx1 ; x2 ; . . . ; xN Þ ¼ 1 P 1 ðx1 ; x2 ; . . . ; xN Þ 1
2
P i ðxi Þ ; P i ðxi Þ
i.e. multiply the old joint probability by the ratio of the updated and old marginals for the ith characteristic (which is just inspected). It is to be noted that the new joints satisfy the condition of being a probability mass function. The generalized form of the above relation for cycle j can be written as j
j P i ðxi Þ : P ðx1 ; x2 ; . . . ; xN Þ ¼ j1 P ðx1 ; x2 ; . . . ; xN Þ j1 P i ðxi Þ
ð9Þ
From these updated joints we obtain the marginals for the remaining characteristics. Then we proceed on to inspect the next characteristic in the 2nd stage. This process is repeated until all the characteristics are inspected. The probability that the component is good, at the end of the cycle is PGN þ1;1 ¼ N P 1 ð2; 2; . . . ; 2Þ:
ð10Þ
Before any inspection is performed, the total cost of false acceptance is EðtcÞjj¼0 ¼ C a ð1 P ð2; 2; . . . ; 2ÞÞ:
ð11Þ
The above equation does not incorporate any inspection or rejection costs as we were just at the start of the inspection process. The expected total cost per accepted component after n cycles of inspection is given by EðtcÞjj¼n ¼ ½TCFR þ TCFA þ TCI=TA:
ð12Þ
Our objective is to determine the optimal n which minimizes the expected total cost per accepted component. 3.1. Analysis of the jth cycle If M1,j is the number of components entering the first stage of the jth cycle, M 1;j ¼ M j : The probabilities of a component being non-defective, to be rework or to be scrap are given by PG1;j ¼ P ð2; 2; . . . ; 2Þ; XX X XX X P ð1; x2 ; x3 ; . . . ; xN Þ þ P ðx1 ; 1; x3 ; . . . ; xN Þ þ PR1;j ¼ x2
þ
x3
XX x1
x2
XX x1
xN
x2
X xN
X
x1
P ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
x3
XX x3
x4
xN
X
ð13Þ
P ð1; 1; x3 ; . . . ; xN Þ
xN
P ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ P ð1; 1; 1; . . . ; 1Þ;
ð14Þ
xN 2
PS 1;j ¼ 1 PG1;j PR1;j :
ð15Þ
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The probability of the first characteristic to be rework or scrap would be given by XXX X j P 1 ð1Þ ¼ P ð1; x2 ; x3 ; . . . ; xN Þ; j
P 1 ð0Þ ¼
x2
x3
x3
x2
x3
x3
XXX
xN
X
P ð0; x2 ; x3 ; . . . ; xN Þ:
xN
Thus the probability of this characteristic to be non-defective would be j
P 1 ð2Þ ¼ 1 j P 1 ð1Þ j P 1 ð0Þ:
The expected number of components falsely sent to rework is FR1;j ¼ M 1;j ðPG1;j E1gr þ PS 1;j E1sr Þ:
ð16Þ
The expected number of components falsely sent to scrap is FS 1;j ¼ M 1;j ðPG1;j E1gs þ PR1;j E1rs Þ:
ð17Þ
The expected number of components falsely accepted is FA1;j ¼ M 1;j ðj P 1 ð1ÞE1rg þ j P 1 ð0ÞE1sg þ ð1 PG1;j j P 1 ð1Þ j P 1 ð0ÞÞð1 E1gr E1gs ÞÞ:
ð18Þ
The expected number of components correctly accepted is CA1;j ¼ M 1;j PG1;j ð1 E1gs E1gs Þ:
ð19Þ
All accepted components in this stage move on to the 2nd stage where the next characteristic is inspected. So, M 2;j ¼ FA1;j þ CA1;j þ FGR1;j ;
ð20Þ
where FGR1;j ¼ M 1;j PG1;j E1gr : The probabilities that the components bear are PG2;j ¼ j P 2 ð2; 2; . . . ; 2Þ; XX X XX X j j P 2 ð1; x2 ; x3 ; . . . ; xN Þ þ P 2 ðx1 ; 1; x3 ; . . . ; xN Þ þ PR2;j ¼ x2
þ
x3
XX x1
xN
x2
XX x1
X
x1 j
P 2 ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
xN
x2
X
x3
XX x3
j
xN
X
x4
j
ð21Þ
P 2 ð1; 1; x3 ; . . . ; xN Þ
xN
P 2 ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ j P 2 ð1; 1; 1; . . . ; 1Þ;
ð22Þ
xN 2
PS 2;j ¼ 1 PG2;j PR2;j :
ð23Þ
The probability of the second characteristic to be rework or scrap would be XXX X j j P 2 ð1Þ ¼ P 2 ðx1 ; 1; x3 ; . . . ; xN Þ; j
P 2 ð0Þ ¼
x1
x3
x3
x1
x3
x3
XXX
xN
X
j
P 2 ðx1 ; 0; x3 ; . . . ; xN Þ
xN
and the components are classified as FR2;j ¼ M 2;j ðPG2;j E2gr þ PS2;j E2sr Þ;
ð24Þ
FS 2;j ¼ M 2;j ðPG2;j E2gs þ PR2;j E2rs Þ; j
j
ð25Þ j
j
j
j
j
j
FA2;j ¼ M 2;j ½ P 2 ð1ÞE2rg þ P 2 ð0ÞE2sg þ f1 PG2;j P 2 ð1Þ P 2 ð0Þ þ P 1 ð1Þ P 2 ð1Þ þ P 1 ð0Þ P 2 ð0Þg ð1 E2gr E2gs Þ;
ð26Þ
S.O. Duffuaa, M. Khan / European Journal of Operational Research 191 (2008) 374–385
CA2;j ¼ M 2;j PG2;j ð1 E2gs E2gs Þ:
381
ð27Þ
Similarly, we can write for the ith stage of the jth cycle, M i;j ¼ FAi1;j þ CAi1;j þ FGRi1;j ;
ð28Þ
where FGRi1;j ¼ M i1;j PGi1;j Ei1;gr ; PGi;j ¼ j P i ð2; 2; . . . ; 2Þ; XX X XX X j j PRi;j ¼ P i ð1; x2 ; x3 ; . . . ; xN Þ þ P i ðx1 ; 1; x3 ; . . . ; xN Þ þ x2
þ
x3
XX x1
x2
XX x1
xN
X
x1 j
P i ðx1 ; x2 ; . . . ; xN 1 ; 1Þ
xN
x2
X
x3
XX x3
j
x4
xN
X
j
ð29Þ
P i ð1; 1; x3 ; . . . ; xN Þ
xN
P i ðx1 ; x2 ; . . . ; xN 2 ; 1; 1Þ þ j P i ð1; 1; 1; . . . ; 1Þ;
ð30Þ
xN 2
PS i;j ¼ 1 PGi;j PRi;j :
ð31Þ
The probability of the ith characteristic to be rework or scrap would be XXX X j j P i ð1Þ ¼ P i ð1; x2 ; x3 ; . . . ; xN Þ; x2
j
P i ð0Þ ¼
x3
x3
XXX x2
x3
xN
X
x3
j
P i ð0; x2 ; x3 ; . . . ; xN Þ;
xN
FRi;j ¼ M i;j ðPGi;j Eigr þ PS i;j Eisr Þ;
ð32Þ
FS i;j ¼ M i;j ðPGi;j ENgs þ PRi;j ENrs Þ; FAi;j ¼ M i;j ½j P i ð1ÞEirg þ j P i ð0ÞEisg þ f1 PGi;j j P i ð1Þ j P i ð0Þ þ
ð33Þ i1 X
j
P k ð1Þj P i ð1Þ
k¼1
þ
i2 X i1 X
P k;j ð1Þj P 1 ð1Þj P i ð1Þ þ j P 1 ð1Þj P 2 ð1Þ j P i ð1Þ þ
k¼1 l¼kþ1
þ
i2 X i1 X
i1 X
j
P k ð0Þj P i ð0Þ
k¼1 j
P k ð0Þj P 1 ð0Þj P i ð0Þ þ j P 1 ð0ÞP 2;j ð0Þ j P i ð0Þgð1 Eigr Eigs Þ;
ð34Þ
k¼1 l¼kþ1
CAi;j ¼ M i;j PGi;j ð1 Eigs Eigs Þ:
ð35Þ
In a similar fashion, the next cycle can be analyzed using the updated versions of the joint and marginal probability mass functions. Next, we summarize the results of the jth cycle. 3.2. Results of the jth cycle Total number of accepted components in the jth cycle: AðjÞ ¼ M jþ1 ¼ CAN ;j þ FAN ;j þ FGRN ;j ;
ð36Þ
where FGRN ;j ¼ M N ;j PGN ;j EN ;gr : Total cost of false acceptance for the jth cycle: CFAðjÞ ¼ C a FAN ;j : Total cost of false rejection for the jth cycle:
ð37Þ
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CFRðjÞ ¼
N X
M i;j ðC fgs PGi;j Eigs þ C frs PRi;j Eirs Þ:
ð38Þ
i¼1
Cost of inspection at the inspection station, for the jth cycle: N X CI1 ðjÞ ¼ C i1 M i;j :
ð39Þ
i¼1
Cost of inspection at the rework station, for the jth cycle: N X CI2 ðjÞ ¼ C i2 M i;j ðPGi;j Eigr þ PS i;j Eisr þ PRi;j ð1 Eirg Eirs ÞÞ:
ð40Þ
i¼1
So, the total cost of inspection for the jth cycle is CIðjÞ ¼ CI1 ðjÞ þ CI2 ðjÞ:
ð41Þ
3.3. Minimization of inspection cost for jth cycle The total cost of inspection in a cycle is influenced by the sequence in which the characteristics are inspected. The characteristic with lower cost of inspection and higher rate of rejection should be inspected first, as proposed by Duffuaa and Raouf [10] for independent characteristics. In this model, we use the rule given by Duffuaa and Nadeem [11] for the dependent characteristics. So, at the beginning of the jth cycle, we compute the ratio Ratio ¼ C i =Ri;j ;
ð42Þ
where Ci, the cost of inspection for characteristic i, is given by C i ¼ C i1 þ j1 P i ð1ÞC i2 and the rejection rate for the ith characteristic in the jth cycle is Ri;j ¼ j1 P i ð0Þð1 Eisg Eisr Þ þ j1 P i ð1ÞEirs þ j1 P i ð2ÞEigs : So, we select the characteristic with the lowest ratio (42). Similarly, at the kth stage of the jth cycle we compute Ri;j;k ¼ C i =Rk;j ;
ð43Þ
where C i ¼ C i1 þ j P i ð1ÞC i2 and Rk;j ¼ j P k1 ð0Þð1 Eisg Eisr Þ þ j P k1 ð1ÞEirs þ j P k1 ð2ÞEigs and select the one among the remaining characteristics with the lowest ratio (42) to be inspected at the kth stage. 3.4. Total cost after n cycles After analyzing n cycles, we compute total cost of false acceptance TCFA, total cost of false rejection TCFR, total cost of inspection TCI and the total number of accepted components TA, as TCFA ¼ CFAðnÞ ¼ C a FAN ;n ; ð44Þ n X TCFR ¼ ½CFRðjÞ; ð45Þ j¼1
TCI ¼
n X j¼1
½CIðjÞ;
ð46Þ
S.O. Duffuaa, M. Khan / European Journal of Operational Research 191 (2008) 374–385
TA ¼ CAN ;n þ FAN ;n :
383
ð47Þ
Substituting TCFA, TCFR, TCI and TA in (12), we obtain the total expected cost at the end of nth cycle. Next is given a computational procedure to determine optimal n.
4. Computational procedure Step 1. Compute E(tc)jj = 0 using (11). Set j = 1. Step 2. Compute jPi(0), jPi(1), jPi(2) and Mj using (6), (7), (8) and (36), respectively. Select the ith characteristic for inspection based on the ratio (43). Repeat this until all the characteristics are inspected. Step 3. Compute Mi,j, FSi,j, FAi,j and CAi,j using (28), (33), (34), (35), respectively. Step 4. Compute CFA(j), CFR(j) and CI(j) using (37), (38) and (41), respectively. Step 5. Compute TCFA(j), TCFR(j), TCI(j) and TA(j) using (44), (45), (46) and (47), respectively. Step 6. Compute E(tc)jj using (12). Step 7. If EðtcÞjj < EðtcÞjj1 , set j ¼ j þ 1 and go to step 2. Otherwise n ¼ j 1. Lemma 1. The above computational procedure (algorithm) provides a local minimum. Proof. The total expected cost E(tc)jn is a function of a single variable n. Assume the algorithm terminates at n*. Then EðtcÞjn 6 EðtcÞjn þ1 otherwise the algorithm would not have terminated at n* see step 7 of the algorithm. Also EðtcÞjn 6 EðtcÞjn 1 since if this is not true the algorithm would have terminated at n* 1 The above two arguments in this proof show that n* is a local minimum. h
5. Practical applications and example Multicharacteristic components are found in many industrial systems [13]. These systems include gas ignition systems [18], aircraft avionics systems, space shuttle and nuclear reactors. The details of some are presented below in order to demonstrate the use of the model. Printed circuit boards in aircraft avionics systems have many connections, welding points and wires that need to be inspected for quality assurance. Failure of any of these connections or wires could be catastrophic. These boards are thus treated as critical components. Gas ignition systems in heavy mechanical industries and ships have many valves, switches and circuits that need to be inspected for quality assurance. Failure of any of these switches or circuits could result in a heavy loss or fatality. Vacuum pumps on the conveyors in a food handling industry have many seals and coatings that need to be inspected for quality assurance. Failure of any of these characteristics could result in a high scale casualty. The model developed in the paper can be used to minimize the occurrence of the disasters caused by the above critical components if the characteristics’ defective rates are statistically dependent. Characteristic defective rates can be obtained from historical data or the production process capability. In order to illustrate the model presented in this paper, the following example is provided. The example is a modification of the one given in [13]. The example in [13] has been modified to induce the dependency between characteristics’ defective rates. This is done by violating the independence criterion (i.e. their joint probabilities should not be the same as the product of their marginal probabilities). Also, additional costs needed for the example are established based on the authors’ experience. A program is developed implementing the above computational procedure and is used to obtain the optimal number of repeat inspections.
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Assuming that the following data are given I
III
0 1 2
0
1
2
II
II
II
0
1
2
0
1
2
0
1
2
0.009 0.00375 0.005
0.009 0.00375 0.005
0.0072 0.03 0.004
0.00075 0.00375 0.00075
0.00075 0.00375 0.00075
0.006 0.03 0.006
0.0135 0.0005 0.0675
0.0135 0.0005 0.0675
0.108 0.004 0.54
N ¼ 3;
M ¼ 100;
Egs ¼ 0:03;
C a ¼ 100; 000;
Esg ¼ 0:1;
C i1 ¼ 100;
C i2 ¼ 5000;
C fgs ¼ 10; 000;
C frs ¼ 5000;
Egr ¼ Erg ¼ Esr ¼ Ers ¼ 0:1:
Solving this example using the proposed model gives the following results: Expected total cost per accepted component without inspection = 46000.00 Cycle 1 PG(1) A(1) Etc(1)
0.54 50 9861.40
Cycle 2 PG(2) A(2) Etc(2)
0.96 44 8778.72
Cycle 3 PG(2) A(2) Etc(2)
0.99 40 11371.81
Optimal no. of inspection = 2.
In order to demonstrate the utility of the model a set of inspection problems are generated randomly where characteristics’ defective rates are statistically dependent. The results of this model are compared to the model where independence is assumed for these examples. The results indicated that model in this paper performed better in terms of minimizing ETC. The average reduction in the cost is 32.14%. 6. Conclusion A general inspection plan for determining optimal number of inspections for critical multicharacteristic components is developed. The inspection plan takes into account six types of misclassification errors. A model depicting the inspection plan for the case where the characteristics’ defective rates are statistically dependent has been developed. A computational procedure is outlined to obtain the optimal number of repeat inspections that minimizes the total cost of inspection. The model is illustrated with the help of a numerical example. Acknowledgment The authors acknowledge the support provided by the King Fahd University of Petroleum and Minerals, Department of Systems Engineering in conducting this research.
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