A General Inspection Plan For Critical Multicharacteristic Components S.O.Duffuaa* Department of Industrial and Operations Engineering University of Michigan Ann Arbor, MI 48109-2117, USA and H.J.Al-Najjar** Macdoland, International, Riyadh, Saudi Arabia April 15, 1994 Abstract In this paper, a general inspection plan for critical multicharacteristic components is presented. In this plan the assumptioni in tlhe literature that characteristics are insIpecteld equal number of tinmes is relicaxed and allowed for different number of inspections for different characteristics depending on characteristic's defective rates and inspection cost. A mathematical model that depicts and represents the plan has been developed. A decent type algorithm is proposed to determine the optimal number of repeat inspections and sequence characteristics for inspection that minimizes the expected total cost. The expected total cost consists of the cost of false acceptence (cost if type II error). cost of false rejection (cost of type I error), and the cost of inspection. Emperical comparisons with the nmodel in the literature on randomly generated problems 1

has been conducted. The results have shown that the proposed plan performs better in terms of the expected total cost on 87 percent of the generated problems for the assumed specific parameters. The reduction in the expected total cost is up to 23.5 percent. Key words Qualty Control, Optimal Inspection Plan, Optimal Sequence, Type I Error,Type II Error. *Author for correspondence.Curently oil sabatical from King Fhad University of Petroleum and Minerals, Dhahran, 31261. Saudi Arabia 2

1 Introduction In production systems, inspection for acceptence purposes is carried out at many stages. This includes inspection of incoming material, in process inspection during the manufacturing operations and finished product inspection. Competitive forces and sometimes catastrophic failure have resulted in tight quality control of products. Componets whose failure result in catastrophy, serious hazard or very high cost are termed critical components. Such components usually have several characteristics and failure of one of them results in compenent failure. The quality requirements for such components are tight and field failure must be kept to the minimum level. Such components can be a part of an air-craft, a space shuttle or a complex gas ignition system. For critical components a common practice in industry is to institute multiple inspections. The reason for multiple inspection is that inspection is never perfect. There is always the possibility of false acceptence (type II error) and false rejection (type I error). Both errors have costs i. e the cost of false acceptence and the cost of false rejection. In case of critical components the cost of false acceptence is much higher than the cost of false rejection, because falsely accepted components may result in system failure which may involve system loss and human lives losses. Therefore it is preceived and shown that repeat (multiple ) inspections is likely to reduce the costs of the errors and increase the cost of inspection. However the expected total cost which is the the sum of the three costs is likelv to reduce[11]. Hence a need exist to determine the optimal inspection plan and the optimal number of repeat inspection that minimizes the expected total cost. Raouf and Elfeituri[12] conducted a stu ldy to investigate the factors which affect inspector accuracy and concluded that type II error is a inore realistic criterion for measuring inspector accuracy. Ayoub et al [1] presented a formulla for average outgoing quality(AOQ) and average total inspection(ATI) under inspection error. In a subsequent study, Collins et al [4] relaxed the assumption of perfect inspection of replacement and allowed defective replacement in the formula for AOQ and ATI. Case et al [3], presented similar results for Dodge's sampling plans for continous production. In [2], Bennett et al investigated the effect of inspection error on cost based single sampling plan design. 3

In the literature of multicharacteristic critical components inspection, Raouf et al proposed the following inspection plan [12]. The plan is described as follows: an inspector inspects one particular characteristic for each component entering the inspection process, and all the accepted components go to the second inspector who inspects the second characteristic. Then all accepted components go to the third inspector who inspects the third characteristic. The chain of inspection continues until all characteristics are inspected once. This completes one cycle of inspection and this process is repeated n before the component is finally accepted. Here n is the optimal number of repeat inspections needed to minimize total expected cost. Also in [13] they developed the initial model for determining the optimal number of repeat inspections which minimizes expected total cost. In [9] Duffuaa and Raouf established an optimal rule for sequencing characteristics for inspection in the plan proposed by Raouf. In [5], Duffuaa and Nadeem extended the model in [13] to situations where characteristic's defective rates are statistically dependent. Lee [11], simplified the model in [13] and obtained simple optimality conditions for the model. Using the plan in [13] Duffuaa and Raouf proposed three models for repeat multicharacteristic inspection [8]. In their model they considered another criterion which is minimizing the probability of accepting a defective conlponent. Recently Dlffuaa and Al-Najjar [6], proposed an alternative plan,where the first illspector I)erformls n inlsI)ections o01 the first characteristic, prior to passiJ;i the accepted coniponents to the secollnd inspector who inspects the second characteristic Ir times, prior passing the accepted comIlpellents to tile third inspector. This chain of inspection is continued untill all characteristics are insl)ecte(d. In this plan inspection is done in stages and in a way decenteralized. Both plans in the literature requires e(Ilal number of inspections for different characteristics. This requirement simplifies tie Iiodel for the plan, but however it is unrealistic and not cost effective. The reason for thiat different characteristics do not have the same defective rate nor the same cost of iIlsl)ection. Therefore it is ore realistic to relax the assumption of equal number of inspections for different characterictics. The plan proposed by Duffuaa and Al-Najjar can be modified to allow different number of inspections for different 4

characteristics. The general plan proposed in this paper is as follows: the first inspector inspects one particular characteristics n1 times, then passes all accepted components to the second inspector who inspects the second characteristic n2. This chain is continued untill all characteristics are inspected, nl,n2,..., nN, where N is the number of characteristics the component has. The objective of this paper is to propose a general inspection plan for critical multicharacteristic components. Then develop a model that depicts the plan and outline an algorthm to determine the optimal inspection plan that minmizes the expected total cost per accepted component. Comparasions with the original model and the plans in the literature will be presented.The rest of the paper is organized as follows: Section 2 presents the problem and the general inspection plan. Section 3 contains model development. An algorthm for determing the optimal, n1, n1,... nN, is outlined in section 4 and comparisons with the plans and models in the literature are given in section 5. Section 6 concludes the paper. 2 PROBLEM AND PLAN DEFINITION Prior to problem definition aind model formulation,the following notation is adopted. In the notation i ranges from 1 to N anId j ranges fromi 1 to n. AI Number Of componeilts to be inspected A/l Number of componelnts enterilng the i-th stage of inspection. li,3 Number of componelts Intering The j-th cycle of stage i. N Number of characteristics in each component to be inspected P, Probability of the i-tli ch'laracteristic being defective entering the inspection. Ci Cost of inspectioil of cliaralcteristi. ni Optimal number of rel)eat iIlsl)ections for characteristic. Ca Cost of false acceptance per component. Cr Cost of false rejection per component. PG Probability of a component being nondefective entering the inspection. 5

Eli Probability of classifying the i-th nondefective characteristic in the sequence of inspection as defective (type I error). E2i Probability of classifying the i-th defective characteristic in the sequence of inspection as nondefective (type I Ierror) Pij Probability of the i-th characteristic in the sequence of inspection being defective entering the j-th cycle. PGij Probability of a component being nondefective entering the j-th cycle of the ith stage. PGini+l Probability of a component being nondefective after inspecting characteristic 1 througr i, each ni times. FRij yExpected number of falsely rejected components in the j-th cycle of the i-th stage. FAij Expected number of falsely accepted components in the j-th cycle of the i-th stage. CAi, Expected number of correctly accepted components in the j-th cycle of the i-th stag RiVj Rate of rejection of components due to i-th characteristic in the sequence of inspection of the j-th cycle. Ai Expected number of accepted components in the i-th stage. CFR, Cost of false rejection in the i-th stage. CFAi Cost of false acceptance in the i-th stage. CIi Cost of inspection in tie i-th stage. TCFR Total cost of false rejection. TCFA Total cost of false accep)taIl'ce. TCI Total cost of inspection. TA Total number of accepted conmponents. E(tc) l Expected total cost per accepted component after j-th stages of inspection. The model is developed for critical coInIonent having N characteristics requiring inspection. The incoming quality of characteristic i is Pi. A component is classified as nondefective only if all the characteristics meet quality specifications. An inspector commits type I error Eli, and type II error. E22 when he inspects characteristic i. Three different costs are 6

considered: (i) Cost of false rejection of nondefective components, (cost of type I error), (ii) cost of false acceptance of defective components, (cost of type II error), and (iii) cost of inspection. In order to control the quality of such critical components, the following general inspection plan is proposed. The plan is applied as follows: The first inspector inspects one particular characteristic n1 times for each component entering the inspection process,(this is the first stage of inspection), all the accepted components go to the second inspector,who inspects the second characteristic n2 times (this is the second stage of inspection). This chain of inspection continues untill all characteristics are inspected, (ni, 2,..., nN). Here ni is the optimal number of repeat inspections for characteristic i necessary to minimize the expected total cost per accepted component. Stage i has ni cycles of inspection. The general inspection plan is shown in Figure 1. Finally, the accepted components will be those which are accepted at the N-th stage, and the rejected components is the sum of those rejected in the 1st, 2nd,.... N-th stages. The objective is to find the optimal number of repeat inspectons, ni for characteristic i. i = 1.... A, in order to minimize the total expected cost. The expected total cost consists of the cost of false acceptance, cost of false rejection and cost of inspection. Next a mat ihellatical ll(odel is deve'loped to depict the proposed plan and aide in finding the optimal lnumbler of repeat inspections. 3 MODEL DEVELOPMENT In the nIext three subsections the details of the model are presented.In the first subsection the ba.sic relationships are derived,followed by the general expressions in the second subsection. Then tie rlle for finding the optimal sequelnce of inspection is given. 7

Incoming Components for Inspection Rejected Components Inspection STAGE 1 in the - - of the c o0 @ 0~ OE 2 8,n Second Stage Second Characteristic i rr, u t^~~~~~~~~ al, I wZ ~ ~ ~ ~ ~ ~ ~ G I S in the of the -- j Second Stage Second Characteristic i w I8 wnI o Rejected<~ Components~~~ Inspe~Ct w a t w I a - I 0 I ur? 0 C* at <I Rejected Components eral Inspection STAGE N * -in the - -of the - -8 Nth Stage Nth Characteristic 0 CLI C E ri u TOTAL ACCEPTED COMPONENTS 8

3.1 Basic Relationships of the Model This model and the one in the literature assume characteristic defective rates are statistically independent. This assumption is not highly restrictive and apply to many situations. For example when inspecting aircraft engines,characteristics crospond to different parts of the engine, which are made at different plants. It is highly likely that the defective rates of such parts are independent. Also independence can be used as a reasonable approximation for many other situations. The probability of the i-th characteristic being defective will vary from cycle to cycle. First we shall establish the relationship between Pij and Pi. Expressing Pij in terms of Pi. Obviously Pi,i = Pi (1) Using Bayes theorem, P"12 [PIE2, + (1 - Pi)(I - El) Applying Bayes theorem again. p _ PI,2E2Z (i - [P,2E2 + (1 - PI,2)(1 - E)] Substituting for Pi.2 from equation(2) into the above formula,gives after simplifiction, P, E2 pi.3 = 2-_ L _ (4) 3 [P2 + (1 - P)(1- Eli )2] Similarly, P E3 Pf" 4 21 (5) P~4 -[?iE~, + (1 - P,)(1 - E,,)3] In general from the symmetry of expressions (2), (3), and (4) the following is deduced: Pij PE.- j- + (1-P)1- E1)](6) The probability of a characteristic being defective changes in each cycle and so the probability of a component being nondefective. Bearing this in mind, we shall establish the 9

relationship between PGij, the probability of a componnt being nondefective entering the j-th cycle of the i-th stage, and the incoming quality, Pi. Expressing PG in terms of Pi. The probability of a component being nondefective is N PG = f(1-P) (7) i=l PG1,1 = PG (8) The probability of a component being nondefective after inspecting characteristic 1, nl times is: N PGl.n+l = n(i - Pi)[(i - Pl,n+l1)] (9) 1=2 The probability of a component being defective after inspecting all characteristics is: N PGN,, Ni-,l = T( 1- Pi,ni+l) (10) 1=1 The probability of a component being defective after inspecting characteristic 1 through i - 1, ni times and characteristic i, m times and the other characteristics from i + 1 through N are not inspected is given by: i-i N PGi, [ r(1 - Pk.nk+l)][(l - Pim+)][ I (1 - Pk)] (11) k=l k=i+l When there is no inspection, the expected total cost per accepted component will simnply be the cost due to false acceptance of (efectixt' l coI)olnents and is given by: E(tc)lji, = C,,(l - PG) (12) where Ca is the cost of false acceptance peir component and PG is given by eqauation (7). The expected total cost per accepteld component, after inspecting all characteristics and characteristic i is inspected rn tilles, is gixenI as: E(tc), i=.=,,,, = [TCFR + TCFA + TCI]/TA (13) In order to determine TCFR, TCFA. TCI, and TA, an analysis of different stages of inspection is necessary. 10

3.2 Analysis of Stage (1) All the components entering stage (1) go to the first inspector, who inspects the first characteristic in each component in order to classify it as defective or nondefective. The first stage has n1 cycle of inspections. Following is the first cycle of inspection. 3.2.1 Cyle(1) Number of components entering cycle 1 is M1,i = Mi (14) The probability of a component being defective is PG1i1 = PG (15) E(number of falsely rejected components ) is FRl,1 = -Il,lPGl,1E11 = M PGE11 (16) E(number of falsely accepted components) is FA,,i =.lIl.l[P EIl + (1 - PG1i, - P)(1 - E1i)] = Ml[PIE2, +((1 - PG - P1)(1 - P1)] (17) E(nunber of correctly accepted (onlllpoiicilte) is CAl.I =.111. PGCl,(1 -Ell) = MI PG(1- -El (18) All accepted components in this cycle go to the first inspector again to inspect the first characteristic for the second time. (perform cycle 2). 11

3.2.2 Cycle(2) M1,2 = FA1 + CA1,1 = Ml[P1E21 + (1 - PG - P)(1 - El)] + M1PG(1 - El,) (19) = M[P1E2 + (1- P)( - Ell) N PGi,2 = (1- P1,2)[I( - Pi)] i=2 PG(1 - Ell)/[PE21+ (1 - P)(- Ell)] (20) FR1,2 = M1,2PG1,2Ell = MI1PG(1 -E1) (21) FAI,2 = AMI,2[Pl,2E21 + (1 - PG1,2 - P1,2)(1 - El) - M,[PE21 + (1 - P)(1 - Ell)][Pl,2E21 +(1- PG1,2- P1,2)(1 - Ell)] (22) CA1,2 = M1,2PG1,2(1- Ell) = Ai1PG(1 - El) (23) (24) Similarly. 3.2.3 Cycle(3) Al1,3 = FA,1 + CA1, = Ml[P1E21 + (1 - P)(1 - El)][2PlE21 + (1 - P1,2)(1- Ell) PG1,3 = (1 - P1,3)[1 - ) 1=2 2 FAi,3 [P1 M [P1, E2+1 - Pl, )(l- Ell)] j=1 12

[P,3E21 + (1 - PG1,3 - P1,3)(1 - El)] (27) CA1,2 = MiPG(l-E1l)3 (28) (29) 3.2.4 Cycle(ni) From the symmetry of the expressions, the nl th cycle of the first stage results can be writtern as follows: Min = l fl [Pi,3 E21 + (1 - Pj)(1 -El] (30) 3=1 ni PG n, = PG(1 - Ell)n/ i[PjE21 + (1 - P,j)(1 - El)] (31) j=1 FRl,n = M1PGE11(1- P11)nl (32) j=1I [P1,n1E21 + (1 - PG1i,1 - P1,n1)(1 - E11) (33) CA1,1h, = AI1PG(1 -- Ell)Ti (34) Tiis coInlI)letes stage one of the inspection.it has't1 cycles. 3.3 Results of Stage (1) E(nuIlliber of accepted compoIlents ) is A(l) = FA1,i + CALn, = AI [I I {Pl1jE21 + (1 - P1,)(1 - Ell)}] x [Pll E21 + (1- PGlnl -,nl)x j=1 (1 - El) + PG(1- Ell) r] (35) wliere PGI,,,, is given in equation (9). Cost of false rejection after one stag of inspection is completed is given by: Tl i CFR1 = CrZFR1, j=l 13

CrMPGE11 j(1- Ell)j-1 (36) j=l Cost of false acceptance after stage one is completed is: CFA1 = Ca(FAl,ni) n-l 1 = CaMl 1 {P1,jE21 + (1 - Pl,j)(- El)} x L j=l [P1,nE21 + (1 - PGin - Pl,n1)(I - El)] (37) Cost of inspection after stage one is compeleted is: CI1 = C iM ij j=1 ni k-l = ClM1 E I {PIE21 + (-Pij)(1 - El)} (38) k=l 2=1 E(total cost per accepted component after one stage of inspection) is: E(tc)li= = [CFR1 + CFA4 + CI1]/A1 (39) where A1. CFR1. CFA1 and CI\ are given by equations (35), (36), (37), and equation (38) respectively. 3.4 Analysis of Stage (2) of Inspection All accepted components fromi stage onle piroc eed to the second inspector who inspects the second characteristics.Therfore the exlc)(ted llllnlll)cr of components entering stage two is Al2,1 = A1 where A1 is given by equation (37). 3.4.1 Cycle(l) Ai21 = FA1,n + CA1, nl = M1 J [P1iE21 + (1 - P1,)(1- E11)] 3=1 14

PG2,1 = PG(1- Ell)nl/ [ { Pl,E21 + (1 - Pl,j)( - Ell)}] FR2,1 = M1PGE12(1 - Ell)n FA2,1 = M1 I[{PljE21 +-(1-Plj)(l- El)}1 x [P2E22 + (1- PG2,1- P2)(1- E44) CA2,1 = MiPG(1- El)n (1-E12) 3.4.2 Cycle(2) M2,2 =FA2,1 + CA2,1 M= I {P1,jE21 + (1 - P1,j)(1 - Ell)} x [P2E22 + (1 - P2)(1- E12) Lj=1 PG2,2 = PG(1 - E1)n(1 - E12)/ I PljE21 + (1 - P,j)(1 - Ell)] [P2E22 + (1 - P2)(1 -12) FR2,2 = MPGE12(1 - Ell)nl ( - E12) FA2,2 M= MI [7 {PlE2l + (1 - Pl,j)(1 - El)} [P2E22 + (1 -P2) (1- E12)] [P2,2E22 + (1- PG22 - P22)( - E12)] CA2,2 =,IPG(1 - El )n (1 - E122 3.4.3 Cycle(n2) AI2,n,2= Al f{iPljE2 + (1 - P,j)(1 - El)} 1 {P2,jE22 + (1 - P2,j)(1 - E2)} PG(1 - El,-)'n(1- E12)n2-1 [H=n1{Pi,E21 + (1 - P, )(1 - Ejl)}] [ j 1 {P2,jE22 + (1- P2,j)(1 - E12)}] FR2,.n = 2 I1PGE12(1 - Ell)nl (l - E2)12-1 ni n2-1 FA2., = hA Ijf{Pl+E21 (1- P1,)(1 - El)} x {P2,E22 + (1- P2,j)(1 - E2)} [ +(- PG2,n2- P2,n2)(1- E12)] CA(2, n2) = MiPG(1 -E11)n'(1 -E2)2 15

3.5 Results of Stage (2) A2 = FA2,n2 + CA2,n2 n\ n2-1 = M1 [{(Plj E21 + (1- P,,)( - Ell)} 1 {P2, jE22 + (1 - j)(I-E2)} =1. j= 1 x[P2,nE22 + (1 - PG, - P2,n)(1 - 2)] + M, PG( - E)2 (1 - -E)n n2 CFR2 = Cr FR2j j=1 n2 = CrMlPGE12(1- Ell)"' (1- E12)j=I CFA2 = Ca(FA2,n2) ni 7n2-1 =Ca'~M M -I{ P1,-E21 + (1 - Pl,j)(1 - Ell)} P2,j) (I E = CaMi [{P1, F1 + (1 1-Pl,3)(l-E1)} 1 [i {P2,jE22 + (1 - P2,)(1 - E12)} j=1 - j=l x [P2,n2E22 + (1 - PG2,n2 - P2n2)(1 - E12)] n2 CI2 = C2 M2,j j=l ni n2 k-1 = C2AIi [1{P1,jE21 + (1 -P l,)(l- Ell)} j j{P2JE22 + (1- P2,j)( -E12)) 3=1 k Lk=l j=1 3.6 Relationships needed to compute expected total cost Using the srnymmetry in the results obtained from the analysis of stages 1 and 2, the 1 ullowing general expressions needed to colnipute the expected total cost can be derived. Total nulmber of accepted comIponents after (onIp)leting NA stages of inspection, i.e, after inspecting the N-th characteristic is givenI aIs: AN M {P.jEE2 + (1 - Pk)(1 -Elk)} X L/=I j=l nN-l n P {P'.j, + (1 - PNj)(1 - EIN)} x L j=1 [PN,nNE2,N + (1 - PGN.LN - PN,nN)(1 - E1N)] + M PG (1- Elk)j (41) 16

Cost of false acceptance at each stage i, i = 1,..., N is given as: CFRi = [Cr x M x PG x Eli] [1 (1 Elk) k [ 1i (1 - Eli)k-1 (42).k=l Cost of false acceptance after completing the N-th stage of inspection is given as: N-1 nk CFAN =CaM I N {PkE2 + (1 - Pk,)(- Elk)} k=l 9=1 nN -1 {PNjE2N + (1 - PN,j)( - E1N)} x j=l [PN,nNE2N + (1 - PGN,nN - PN,nN)(1 - E1N) (43) Cost of inspection at each stage,for stage i,i = 1,..., N, is given as: i-1 nk CI, = CM Pn {PkJP2k + (1 - Pk,.)(l - Elk)}1 x [z {FJ1[Pi.jE2i + (1 - P,j)(l - Ei)} (44)'=l.=1 j=1 In order to determine the general expression for the expected total cost per accepted component, the following expressions must be determined. The expressions are, total cost of false rejection TCFR, total cost of false acceptance TCFA, total cost of inspection TCI and total number of accepted comlI)onents. TA. TCFR = y CFR, (45).: | TCFA = CFA.- = C,(FAN,lN) (46) N TCI = Ci, (47) TA = Ax = FANnN + CANnN (48) E(t) TCFA + TCFR + TCI TA The objective is to find nli, n2...... 2N that provide the minimum expected total cost. 17

3.7 Determing the Optimal Sequence of Inspection The cost of inspection is influenced by the sequence in which characteristics are ordered for inspection, i.e, the order of the stages. The following rule provedes the optimal sequence of characteristic inspection. Let C fl (R ij i3) 1,2,...,N r(i) - 1 () 1, 2...,n (50) 1 - hf(Rj) where: Rij = Pij (l - E2i) + (1 - Pij) Eli ni fl (Ri,) = EH ( - Rik-) i=l k=l f2(Rij) = f(1 -Rik) k=l (51) The characteristic with the lowest ratio is inspected first, next higher ratio second, and so on. The characteristic with the highest ratio is the N-th characteristic to be inspected. The optimality of this rule follows from the proof given by Duffuaa and Raouf[9]. Next a computational procedure is presented for finding the optimal nl, n2,.,. n N and hence the optimal inspection plan. 4 Computational Procedure The computational procedure developedl for fil(lilig the optimal parameters of the inspection plan depends on the concept of steepest (decent. At iteration i characteristic i, has been inspected r, times. The expected total cost at iteration i is: ETC(r1, r2,, rN) The decent direction i,from (r1, r2,..., r,..., rN) to (rl, rr2,,... Ti + 1,...,rN) is given by: DIN(i) = ETC(r1, r2.... r, + 1...rN) )- ETC(rl, r2,..., ri,..., rN) (52) 18

at each point the decent is computed in all directions. Then a move is made in the direction which has the largest decent. Suppose we are at the stage where each characteristic is inspected ri. Then the steps of the algorithm are: STEP(1): Compute ETC(r, r2,..., rN) STEP(2): Find DIN(i), fori = 1,..., N. IFDIN > 0 for all i, go to step (6). Otherwise proceed. STEP(3): Find {maxi | DIN(i) I IDIN(i) < } = DIN(k). STEP(4): Inspect characteristic k and compute ETC(rl, r2,..., rk + 1, rk+l,..., rN) STEP(5): Go to step (2). STEP(6): Stop.The optimal number of inspections for each characteristics is (rl, r2,..., rN) and the total expected cost is ETC(r, r2,..., rN). The above algorithem is expected to provide a local minimum. 5 Results and Model Comparisons In order to compare the developed model with the model in the literature the following example is given. A software is developed implementing the algorithm stated above and used to obtain the optimal number of repeat. inspections. A batch consisting of 100 components is to l)e insplected.Each component has three characteristics. The data for the example is given in Table 1. The example is solved utilizing the model in the literature and the new mlo(del. 19

Table 1. Data for the example. M 100 N 8 P1 = 0.109 P2 = 0.186 P3 = 0.127 P4 = 0.212 P5 - 0.174 P6 - 0.192 P7 = 0.146 P8 =0.175 El= 0.126 E12 = 0.118 E13= 0.075 E14 0.093 E15 = 0.051 E16 = 0.129 E17 = 0.102 E18 = 0.046 E21 = 0.088 E22 = 0.121 E23 0.112 E24 = 0.088 E25 = 0.130 E26 = 0.072 E27 = 0.077 E28 = 0.136 C1 = 99 C2 = 12 C3 - 67 C4 50 C5 =76 C6 =21 C7 =14 C8 95 Ca = 523248 Cr= 733 Solving the above example using Raouf et al model and the algorithm in [14], the following results has been obtained: Optimal sequence 3, 7, 6,4.1.3. 5. 8 Optimal number of repeat insp)ectionls = 3. Minmun expected total cost = 166()(3.11 Probabilty of a component beilg iloll(tefctive is= 0.993716478 20

Solving the same example using the proposed plan and model, the following results has been obtained: Optimal sequence 2, 7, 6,4,1, 3, 5, 8 Optimal number of inspections in the order of the sequence above is 2, 3, 3, 3, 3, 2,2,3 Minmun expected total cost = 12280.25 Probabilty of a component being nondefective is= 0.99365294 The proposed model showed improvement over the model in the literature in terms of the expected total cost. Actually the saving in expected total cost amount to 23.5 percent for the above example. This led to the following detailed comparisons on randomely generated inspection problems. The parameters of the geneated problems. are N. Ci, Ca, Cr, Pi, Eli and E2i,for Z = 1, 2,..., N. The parameters N:, Ci, C,, Cr are assumed to be uniformely distributed. The other parameters Pi, Eli, E2i. are assumed to be normally distributed with mean A and variance a2 N is assumed to be a discrete uniform distribuition that takes the values 1,2,...,10. Ci is uniformely distributed between 10 and 100. i.e U(10, 100), Ca - U(100000, 1000000) and Cr ~ U(500, 1000). Pi is assumed normally distributed with mean A =.05 and variance a2 = 0.014 i.e P, - N(.05,.014). Eli -N (0.(.10.0009) and E2i - N(0.1, 0.0009) for all i. A program has been developed to generate 100 random inspection problems and solved to obtain the optimal inspection paranleters using both models. From the experiment conducted using the generated problems with the above parameters assuming the given distri21

butions, it was found that the proposed model gave better results in terms of the expected total cost in 87 percent of the generated problems. Raouf et al model gave better results on 3 percent of the generated problems. On 10 percent the two models gave the same results. In the problems where the two models agree all components have one characteristic and in that case the two plan become identical. The reduction in the total expected cost, when the proposed model performed better in comparison with Raouf et al model ranges from i to 23.5 percent,while the reduction ranges from 0.1 to 7 percent when Raouf et al performed better.The proposed model and plan tend to perfom better in terms of expected total cost when there is varability between the P's i.e the characteristic defective rates vary The above results demonstrate the superiority of the proposed approach and the resulting inspection plan over the one in the literature. 6 Conclusion A general inspection plan has been proposed for the inspection of critical components with several characteristics. The plan allows for different number of inspections for different characteristics depending on characteristic's defective rate and cost of inspection. A moel has been fornmilated which depicts the plan andil is emploved in determing the optimal Il umber of repeat inspections that miiinlizes the expec(ted total cost per accepted component. A decent type algorithml has been developed for ob)taininig the optimal number of inspections for each characteristic. Tie model and tle plani in this paper provide flexibility of variable number of inspections for different characteristics. A (esirabLle property the models and plans in the literature lack. The model has few reaistic aissumlltions and as such may have a wide range of applicability. Results on randomlyn genlerated problems have shown that the newr model and plan performed better in optimiziing insplection plans for critical components than the ones in the literature. 22

7 Acknowledgment The authors would like to acknowledge the support provided by King Fhad University of Petroleum and Minerals, Dhahran,31261, Saudi Arabia, for conducting this research. Especially the support for the sabbatical leave for the first author. Also the department of industrial and Operations Engineering,University of Michigan at Ann Arbor,is acknowledged for hosting the first author and extending it's facilities. 8 References [1] M.M. Ayoub,B.K.Lambert and A.G.Walkevar(1975) "Effects of two types of inspector error on single sampling inspection plans", A paper presented at Humam Factor Society Meeting, Baltimore, U.S.A. [2] G.K.Bennett,K.E.Case and J.W.Schmidt(1974) "The economic effects of inspector error on attribute sampling plans", Naval Research Logistics Quarterly 21, 431-443. [3] K.E. Case,G.K.Bennett and J.W.Schmidt(1973) "The Dodge CSP-1 continous sampling plan under inspection error". AIIE Trans. 5. 193-202. [4] R.D.Collins.K.E.Case and G.IK.Bennett((1972) -The effects of inspection accuracy on statistical quality control, Proceedings of the 23rd AnnuLal Conference and Convention, AIIE,Anaheim, California. U.S.A. [5]S.O.Duffuaa and I.Nadeem (1994)" A Colplete inspection plan for dependent multicharacteristic critical components". INT.J. PR()D. RES.. in press. [6] S.O.Duffuaa and H. J. Al-naj jar (1994) "An Optimal Complete Inspection Plan for multicharacteristic critical components". Techlnrlcal Report Department of Industrial and Operations Engineering, No.94-12. [7] S.O.Duffuaa and A.Raouf(1987)'A cost iiilimization model for dependent multicharacteristic inspection", Proceedings IXth Inte7rnlational Conference on Prod. Res. and Exhibition,Cincinati. Ohio, 738-746. [8] S.O.Duffuaa and A.Raouf(1989)' Models for multicharacteristic repeat inspection", App.Math.Modell 23

13, 408-412. [9] S.O.Duffuaa and A.Raouf (1990) "An optimal sequence in multicharacteristic inspection", J.of Opt. Theory and App.(JOTA) 67, No. 1, 79-86. [10] J.K.Jain(1977) "A model for determing the optimal number of inspections minimizing inspectioncost", Unpublished M.Sc.thesis, University of Windsor, Windsor, Ontario, Canada. [11] H.L.Lee(1988) "On the optimality of a simplified multicharacteristic component inspection model", IIE Trans. 20, No.4, 392-398. [12] A.Raouf and F.Elfeturi (1983) "Study of the effects of defect rates,task complexity and inspection rates on inspection accuracy, Procedings of the first Saudi Engineering Conference, Jeddah, Saudi Arabia. [13] A.Raouf,J.K.Jain and P.Y.Sathe(1983) "A cost minimization model for multicharacteristic component inspection", IIE Trans. 15, No. 3,187-194. 24