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1 Red de Revstas Centífcas de mérca Latna, el Carbe, España y Portugal Sstema de Informacón Centífca ROS-CORRE, RODRIGO.; GLINDO-PCHECO, GIN M. NLYSIS OF CSP- NDER INFLLILE ND FLLILE INSPECTION SYSTEMS Dyna, vol. 8, núm. 83, febrero, 04, pp nversdad Naconal de Colomba Medellín, Colomba valable n: Dyna, ISSN (Prnted Verson: dyna@unalmed.edu.co nversdad Naconal de Colomba Colomba How to cte Complete ssue More nformaton about ths artcle Journal's homepage Non-Proft cademc Project, developed under the Open cces Intatve

2 NLYSIS OF CSP- NDER INFLLILE ND FLLILE INSPECTION SYSTEMS NÁLISIS DE PLNES DE MESTREO CSP- CONSIDERNDO SISTEMS DE INSPECCIÓN CON Y SIN ERROR DE MESTREO RODRIGO. ROS-CORRE Ph.D., nversdad Del Norte, rbarbosa@unnorte.edu.co GIN M. GLINDO-PCHECO Ph.D., nversdad Del Norte, ggalndo@unnorte.edu.co Receved for revew prl 0 th, 03, accepted July 9 th, 03, fnal verson ugust, st, 03 STRCT: In ths paper, we dscuss the mplementaton of Contnuous Samplng Plan (CSP- under two scenaros: ( nfallble, and ( fallble nspecton systems. For both cases, we develop an optmzaton model for desgnng a CSP- that mnmzes the total expected cost. We use Markov theory to derve the expected results from the applcaton of the CSP-. ayesan approach s used to model the nspecton system relablty. ased on the analyses for the two models, we offer a dscusson on the adverse effects of dsregardng nspecton errors when mplementng CSP-. Key words: CSP-, qualty, nspecton samplng plan, ayes, Markov, smulaton, nspecton error, optmzaton RESMEN: En el presente artículo, presentamos un análss de las mplcacones relaconadas con gnorar errores de nspeccón cuando se mplementa un plan de muestreo contnuo del prmer tpo (CSP- por sus sglas en nglés. Nuestro análss cubre dos escenaros: ( nspeccón perfecta o nfalble, e ( nspeccón mperfecta o falble. Para cada caso, presentamos un correspondente modelo de optmzacón cuyo objetvo es el de mnmzar el valor esperado del costo total. El comportamento de los planes CSP- es modelado utlzando teoría Markovana, mentras que la confabldad de los sstemas de nspeccón es modelada medante un análss ayesano. Las solucones de ambos modelos son confrontadas para establecer comparacones entre los dos escenaros. Palabras clave: Caldad Salente Promedo (CSP, Planes de Muestreo, Teorema de ayes, Cadenas de Markov, Smulacón, Error Muestral, optmzacón. INTRODCTION ualty s one of the most mportant factors consdered by customers at the moment of selectng ther supplers. In an deal world, customers would prefer supplers whose products were absolutely perfect. However, n realty, customers usually agree to tolerate a certan proporton of defectve unts. Then, the task for the supplers s to mplement nspecton polces that guarantee that the average outgong qualty level (OL of ther product, does not exceed a certan value, based on the customer s expectatons regardng the acceptable proporton of defectve unts. n nspecton polcy can be defned as nspectng 00% of the products. However, due to fnancal lmtatons, a 00% nspecton s not always vable. n alternatve s to mplement samplng nspecton plans (SIPs, n whch only a fracton of the total products s nspected. The parameters nvolved n SIPs are often selected n such a way that the total expected cost s mnmzed [-4]. n mportant assumpton when computng the expected cost of a SIP s that related to the effcency or relablty of the nspecton system. Such nspecton systems can be human nspectors or machnes (from now on we wll use ndfferently the terms nspector and nspecton system to refer to both human and machne-based nspecton systems. In both cases, the occurrence of nspecton errors s nevtable. There are two types of nspecton errors: Type I, whch refers to classfyng a defectve unt as nondefectve; and Type II, whch refers to classfyng a nondefectve unt as defectve. Dsregardng these errors,.e. assumng perfect performance of the nspector, s unrealstc and t can lead to naccuraces n the computatons of the expected SIP cost and performance. In ths paper, we use Markovan and ayesan analyss to develop two optmzaton models for desgnng a Dyna, year 8, no. 83, pp Medelln, February, 04. ISSN

3 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, SIP of the type CSP- [5], whch apples for products that are manufactured through a contnuous process. The mplementaton of CSP- can be summarzed as follows:. Intally, nspect 00% of the unts untl consecutve unts are found as non-defectve.. Once consecutve non-defectve unts have been nspected, dscontnue the 00% nspecton and start to systematcally nspect only a fracton f of the unts. The fractonal nspecton contnues as long as the nspected unts are non-defectve. If a defectve unt s found, reestablsh 00% nspecton,.e. return to step. In our frst model, we consder the mnmzaton of expected costs assumng perfect performance of the nspector. Ths ntal model sets the path for developng a more realstc model n whch our objectve s to mnmze the expected costs when nspecton errors are taken nto consderaton. ased on the results obtaned for such optmzaton models, we offer a dscusson on the effects of dsregardng nspecton errors n CSP-. The remanng of the paper s organzed as follows: Secton contans our lterature revew. In secton 3, we state the descrpton of the problem, as well as the correspondng notaton and assumptons. Secton 4 presents an optmzaton model for CSP- when dsregardng nspecton errors, whereas secton 5 contans the optmzaton model when takng nto consderaton such errors. In secton 6, we dscuss the economc mpact of dsregardng nspecton errors, based on the results obtaned on sectons 4 and 5. In secton 7 we extend our dscusson by ncludng the transton costs between total and partal nspecton. Fnally n secton 8 we present some remarks, conclusons and future research drectons.. LITERTRE REVIEW In the lterature, we can fnd some papers that analyze the effect of nspecton errors n SIPs, such as that offered by [6], who use a ayesan model to evaluate the effcency of nspectors. nother nterestng work that uses a ayesan approach s that gven by [7]. In ths latter paper, the authors consder a type of SIP that accounts for the number of defects n each nspected product. The authors address the problem of analyzng the best a pror dstrbuton to model the number of defects per unt under the presence of nspecton errors. nother nterestng paper that also consders nspecton errors s that offered by [8] whch establshes a set of results for matchng Dodge-Romg sngle plans wth Dodge-Romg plans under the presence of nspecton errors. s mentoned n the prevous secton, the desgn of SIPs s often subject to economc crtera such as n [9]. In that paper, the authors offer a mathematcal model to desgn both 00% and sngle samplng plans consderng potental nspecton errors, whle mnmzng a loss functon that accounts for devatons of qualty characterstcs from a certan target value. n earler related work s offered by [0] n whch the authors develop a model for locatng nspecton statons n an n-stage producton system. The optmzaton crteron used n [0] s the cost per good unt accepted by the customer. Reference [] also studes the mpact of nspecton error from an economcal perspectve. More specfcally, n [] the author develops a mathematcal model and an algorthm for desgnng a SIP under nspecton errors, whle mnmzng the expected assocated cost. Reference [] offers a model for mnmzng nspecton costs whle mposng upper bounds on the nspecton errors. One fundamental dfference between our paper and those mentoned above, s that they do not address CSP- n partcular. Two papers that specfcally nvestgate CSP- under nspecton errors from an economc pont of vew are those gven by [4], and [3]. In the frst one, the authors present an optmal mxed polcy of precse nspecton and CSP- under the presence of nspecton errors and return cost. In the second, the authors develop a model usng a renewal reward process approach for selectng an economcally optmal decson nvolvng three alternatves: do 00% nspecton, do not nspect and do a CSP- nspecton. Ther model accounts for both types of nspecton errors. key dfference between the work gven by [3] and ours s that we assume that as long as the customer s acceptable qualty level (L s satsfed, there s no penalzaton for defectve unts delvered to the customer. Wth ths reasonable assumpton, our optmzaton models are smplfed to fndng the optmal number of nspected unts as

4 90 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, 04. the bass for desgnng an optmal CSP-, nstead on focusng on fndng drectly the parameters of the nspecton plan (see sectons 4 and 5. lso, one of the man contrbutons of our work s that t hghlghts the mpact of dsregardng nspecton errors when mplementng CSP-, whch allows vsualzng the mportance of recognzng and measurng nspecton errors. Smlar analyses have been made for attrbute SIPs, [3-7], but to our knowledge, ths ssue has not been yet addressed from the perspectve of a suppler that mplements CSP-. 3. PROLEM DESCRIPTION, NOTTION ND SSMPTIONS In ths secton we present the descrpton of our problem, as well as the correspondng notaton and assumptons. 3.. Problem Descrpton We focus our analyss on a suppler company that uses a contnuous producton scheme. The producton process has an nherent defectve fracton, whch s greater than the L specfed by the customer. In order to comply wth the customer s L, the suppler mplements an nspecton plan of the type CSP- that guarantees that the OL,.e. the expected proporton of defectve unts that are delvered to the customers, s lower than or equal to the customer s L. (The mplementaton of CSP- s as descrbed n the Introducton. The only two parameters nvolved n CSP- are and f. The values for these two parameters are specfed such that the resultng OL does not exceed the customer s L. Gven the defectve fracton of the process, dfferent combnatons of and f can be used to acheve a desred L. s mentoned before, the nspector partcpatng n CSP- s often assumed to be nfallble. However, n practce, nspecton errors are lkely to occur. In fact, accordng to the Second Law of Thermodynamcs [8] and the Prncple of ncertanty of Hesenberg [9], t s not possble to have perfect nspectors, and f they were perfect, t would be mpossble to prove t. Our objectve s to perform an analyss of the mplementaton of CSP- under two scenaros: ( assumng nfallble nspecton systems; and ( consderng the presence of nspecton errors. We then study the mpact of dsregardng such nspecton errors. 3.. ssumptons Our assumptons can be summarzed as follows: The value of the defectve fracton of the producton process s determnstc and known. Throughout CSP-, every rejected unt must be replaced by an acceptable one, accordng to the nspector crtera. ddtonal unts produced to replace rejected ones, are also nspected. s long as the OL delvered by the company s at most equal to the customer s L, there s no penalzaton for defectve unts receved by customers. The purpose of the suppler company s to desgn a CSP- that guarantees an OL lower than or equal to the customer s L. For smplcty, we perform our computatons for a shpment of unts delvered to the customer Notaton θ : event of a unt beng defectve θ : event of a unt beng non-defectve P θ : probablty of occurrence of event θ j, for j =,. Note that P( θ s the fracton defectve P θ = P θ ( j of the process, and that ( ( L : acceptable qualty level specfed by the customer. We assume that L < P( θ. Otherwse, there would be no need to mplement CSP-. OL : average outgong qualty level for a CSP- under nfallble nspecton systems OL : average outgong qualty level for a CSP- under nspecton errors : sze of the producton batch to be analyzed

5 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, S : event of the nspecton system classfyng a unt as non-defectve S : event of the nspecton system classfyng a unt as defectve : probablty of occurrence of event S k for P( S k j =,. Note that P( S = P( S : expected number of nspected unts for a CSP- under nfallble nspecton systems : expected number of nspected unts for a CSP- under fallble nspecton systems c s : cost of nspectng one unt c r : cost of rejectng one defectve unt 4. ECONOMIC NLYSIS FOR CSP- NDER IDEL INSPECTION PROCEDRES In ths secton we present a dscusson on the optmal expected cost of mplementng CSP- under deal condtons,.e. nfallble nspecton systems. s mentoned before, for CSP- to be useful, the OL must be at most equal to the consumer s L. way to assgn values to the parameters nvolved n the CSP- s by usng Markovan analyss, n whch CSP- procedure s consdered as an ergodc Markov chan. Two Markovan states are dentfed: ( 00% nspecton and ( systematc nspecton [8]. When analyzng CSP- as a Markov chan, the transton matrx would be as depcted n Table [0]. ased on Table, we can defne the steady-state probabltes x 00% and x f respectvely for the states of 00% and fractonal nspecton [0]: P( θ x00% = ( P θ + P θ x f = P ( ( P( θ ( θ + P( θ ( Recall that represents the amount of products beng nspected. Then, the expected number of nspected unts n CSP- s gven by x00% + fx f [0], whch can be expressed as follows: P ( θ + fp( θ ( θ + ( θ = P P, Table. Transton Matrx for CSP- 00% Inspecton Fractonal Inspecton 00% P Inspecton ( θ P( θ Fractonal P( θ P( θ Inspecton (3 The total expected number of rejected unts would be equal to the number of detected defectve unts,.e. P( θ. The OL would be equal to proporton of the undetected defectve unts, whch can be expressed as a functon of as follows [0]: [ ] P( θ OL( = (4 Two types of costs must be consdered for the mplementaton of CSP-: nspecton cost, and defectve unts cost. The expected nspecton cost would be c s, where s gven by expresson (3. On the other hand, the cost due to defectve unts nvolves the replacement of defectve detected unts by non-defectve ones. Ths mples producng and nspectng addtonal unts n order to replace the defectve ones. Note that due to the defectve fracton of our process, for producng an amount of X nondefectve unts, the expected number of unts to be produced would be X, snce we should expect P θ ( XP( θ unts to be defectve. Let us denote c p as the cost of manufacturng one unt of product. Then, the expected cost due to defectve unts would be P( θ. ddtonally, we need to nclude c ( p + cs P θ the cost of rejectng each defectve unt, denoted by c r. Then the total expected cost for CSP-, can be expressed n terms of as:

6 9 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, 04. ( = cp p ( θ cs P( θ P( θ C c s cp r ( θ (5 The expected cost for CSP- under nfallble nspecton systems, gven by equaton (5, has as ts only varable. The mnmzaton of (5 s restrcted by the fact that the OL gven n (4 must be at most equal to the customer s L. Then our optmzaton model can be expressed as: Mnmze (5 subject to: P ( θ L (6 We have that the OL s a decrement functon of the proporton of nspected unts. On the other hand, by nspectng (5 we notce that the expected cost s an ncreasng functon of. Therefore, the optmal, value of,.e., s the mnmum that allows complyng the constrant gven by (6. ny value lower than would mply an OL greater than the customer s L, whereas a value greater than would not be economcally optmal. Then, can be computed as follows: = P( θ = L (7 When solvng (7 we obtan: = P ( θ L P θ (8 ( Havng determned the value of, t s possble to compute the parameters of CSP- usng equaton (3 whch relates, P( θ, f, and, where, P( θ and would be gven. We can specfy the value of one of the parameters and then solve for the other, based on the desred value of the OL. In ths case, t s smpler to select a value of and then solve for f, as follows: ( θ + P( θ ( θ P( θ OL P f = (9 P t ths pont, we would lke to hghlght two propertes regardng the optmal expected cost and the optmal expected number of nspected unts for CSP- under nfallble nspecton. Frst we have that expresson (8 can be stated as [ L]. Therefore, we = P θ ( ncreases wth P( θ can conclude that. Ths s a reasonable result snce, ntutvely, we would expect to have to nspect more unts to guarantee a certan L for greater defectve fractons. Second, by nspectng expresson (5 we can notce that the optmal expected cost ncreases wth P( θ. gan, ths fndng s also reasonable snce the nspecton, replacement and rejecton costs ncreases wth the defectve fracton of the process. 5. EXPECTED COST FOR CSP- NDER INSPECTION ERRORS In the deal case dscussed n the prevous secton, an underlyng assumpton s that the nspecton procedure s nfallble. Therefore, t s assumed that whenever an nspected unt s classfed as non-defectve, such a unt s actually non-defectve (an analogous analyss apples for unts that are classfed as defectve. However, n practce we would expect nspecton systems not to be nfallble When we drop the nfallblty assumpton, the transton matrx gven n Table does no longer apply for CSP-, snce the probablty of an nspected unt beng classfed as defectve s not equal to the actual defectve fracton of the process. lso, recall that S s the event of the nspecton system classfyng a unt as defectve, and S as non-defectve. nytme an event S occurs, t means that the nspected unt has been accepted, whereas S mples the rejecton of the nspected unt. In our model we consder two types of condtonal probabltes: P S θ for =,, j =, gves the probablty that the nspector makes a decson S gven that the actual status of the nspected unt s θ j. Note that such condtonal probabltes are nherent to the nspector. Valdty: ( j P θ S for j =,, k =, gves the probablty that an nspected unt s θ j, gven that the nspector has classfed t as S. Predcton: ( j k k

7 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, The two types of nspecton errors dscussed n the Introducton secton can be now defned n terms of the condtonal probabltes stated above, as follows: I ( ( P = P ErrorType I = P S θ (0 II ( ( P = P ErrorType II = P S θ ( y applyng ayes theorem and usng the condtonal probabltes dscussed above, we can compute the probablty of a unt beng classfed as defectve, ( ( P S, as follows: P S = P S θ P θ + P S θ P θ ( ( ( ( In a smlar way, we can compute P( S (. Then, the transton matrx for a CSP-, when consderng nspecton errors, would be as shown n Table, but replacng θ j by S j. We can perform a statstcal analyss smlar to that offered n secton 3. to compute an expresson for the expected number of nspected unts (, and the correspondng OL ( OL. For we have: P S = P S ( + fp( S ( + P( S (3 To fnd the OL, we need to compute the expected proporton of defectve unts that would be delvered to the customers. Frst, we have the expected number of defectve unts that are nspected, and that are θ. lso, when performng the nspecton, some unts wll be rejected and addtonal unts would be produced and nspected to replace the rejected ones, whch would be accepted due to error type II,.e. P( S P( S equal to P( S ( P S. mong these addtonal unts, we wll also have defectve unts that are erroneously classfed as non-defectve and are sent to the customers,.e. P( S or smply, P( θ S P( S P S ( ( ( θ P S P S. ddtonally, there wll be some defectve unts delvered to the customers whch come from those unts that are never nspected durng the fractonal nspecton. Therefore, the OL n ths case can be expressed as: OL = ( ( θ ( + ( ( θ P S P S P S P S + ( P( θ whch can be smplfed as follows: OL = ( ( θ ( θ ( θ P + P S P (4 (5 Regardng the expected cost for CSP- under nspecton errors, we have four dfferent components: ( cost of nspectng unts, ( cost of replacng rejected unts, (3 cost due to rejectng defectve unts, and (4 opportunty cost due to nspecton error Type I. The frst cost s smply c s. For the expected cost of replacng rejected unts we have that, as dscussed n secton 3., we need to produce and nspect addtonal unts to replace those that have been rejected. Note that a proporton equal to P( S of such addtonal unts would be also rejected. Then, the expected number of addtonal unts that we P S rejected must produce and nspect to replace ( unts would be P( S P( S. Therefore, the second component of the expected cost for CSP- under nspecton errors would be P( S. The expected costs related to cs + cp P S ( rejectng defectve unts would be smply c P θ S P S. r ( ( Fnally, let us denote as c I the cost of erroneously classfyng a non-defectve unt as defectve (Type I error. Then, the expected cost due to a Type I error would be cp I ( θ S P( S. Recall that we have assumed that defectve unts sent to the clents do not generate an extra cost as long as the OL s lower than or equal to the customer s L. Therefore, we do not ntroduce any cost due to errors Type II. Then, the total expected cost of mplementng CSP- under nspecton errors s gven by equaton (6.

8 94 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, 04. ( + = P( S P( S + P( S P( S ( θ ( ( C cs cs cp + cip S P S + c P S r (6 (0 The expected cost for CSP- gven by equaton (6, has as ts only varable. The mnmzaton of (6 s restrcted by the fact that the OL gven n (4 must be at most equal to the customer s L. Then our optmzaton model can be expressed as: Mnmze (6 subject to: P( θ + P ( θ S P( θ L (7 >, the OL s a decrement functon of the proporton of nspected unts. Ths s a reasonable assumpton, snce the role of the nspector s to reduce the number of defectve unts delvered to the customers (otherwse, t would be better not to use the nspector. On the other hand, the Note that by assumng that P( θ P( θ S expected cost s an ncreasng functon of. Therefore, we can proceed as before to fnd the optmal value of,.e., whch would be the mnmum that allows complyng the constrant gven by (7. Then, can be computed as follows: = ( θ ( θ ( θ P + P S P = L Then, we have: (8 (9 Havng determned the value of, t s possble to compute the parameters of CSP- usng equaton (3. To do so, we can specfy the value of one of the parameters and then solve for the other, based on the desred value of the OL. s mentoned n the prevous secton, t s smpler to select a value of and then solve for f, as follows: y nspectng expresson (0 we have the followng propertes, whch wll be stated wthout proof due to ther smplcty: Property : the optmal expected number of nspected unts,.e., s drectly proportonal to the dfference between the fracton defectve of the process and the customer s L. Property : the optmal expected number of nspected unts,.e., s nversely proportonal to the dfference ( θ ( θ P P S. Notce that, gven that a unt has been delvered to the customer, f we dd not mplement an nspecton plan at all, the probablty of such a unt resultng defectve s P( θ ; whereas f we mplemented a CSP-, such probablty would decrease to P( θ S. Then, ths property states that s nversely proportonal to the mprovement obtaned for havng an nspecton system, regardng the chances of delverng a defectve unt to the customer. Regardng the optmal expected cost, we have that t ncreases wth the proporton of rejected unts and wth the probablty of rejectng a non-defectve unt. s t depends on, t also ncreases wth P( θ and decreases wth P( θ P( θ S. L 6. CSP- NDER INFLLILE INSPECTION VS. CSP- NDER INSPECTION ERRORS Let us consder the case n whch we are nterested n mplementng a CSP- whle mnmzng the total expected cost. If we took nto consderaton the presence of nspecton errors, we would compute the optmal value for as dscussed n secton 5. However, f we neglected such nspecton errors by

9 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, assumng nfallble nspecton systems, we would compute a suboptmal value based on the analyss provded n secton 4. The dfference between and can be stated as: ( θ = P L P( θ P( θ P( θ S ( Expresson ( s always lower than or equal to zero, by our assumptons that and P( θ > L. Therefore, by mplementng a CSP- wth the expected number of nspected unts equal to <, we should expect that n the long run we wll not be able to comply wth the customer s L. y nspectng expresson ( we can formulate the followng propertes: Property 3: The dfference s drectly proportonal to the dfference between the fracton defectve and the customer s L. Ths means that even though both and ncreases wth P ( θ L, the rate at whch such an ncrement occurs for s greater than for. Ths can be verfed by examnng expressons (8 and (9, from whch we obtan that the rate at whch changes wth P( θ L. Ths P θ P( θ S s ( rate s greater than that for. P θ (, whch s equal to Property 4: The magntude of the dfference decreases wth the mprovement obtaned for havng an nspecton system, defned as P( θ P( θ S. Moreover, n the deal case n whch an accepted unt never happens to be defectve, P θ S = 0, we would have =..e. ( reasonable assumpton s that f a customer receves a shpment of unts whose OL s greater than the customer s L, all unts n such a shpment would be rejected and returned to the suppler. Therefore, f we desgn our CSP- dsregardng nspecton errors, n the long run we should expect the customer to return all of our shpments! Hence, we can conclude that usng as the optmal polcy n the presence of nspecton error, s not economcally suboptmal, but smply unvable. 7. EXTENSION CONSIDERING TRNSITION COSTS ETWEEN TOTL ND PRTIL INSPECTIONS So far we have only consdered the costs related to nspecton, rejecton and classfcaton errors. Ths approach s vald only when the cost due to the transton between total and partal nspecton s nsgnfcant. However, let us assume that we we would need to stop the producton process and reprogram the nspecton system when shftng from one type of nspecton to the other. In that case, the assumpton that the transton costs between total and partal nspectons can be neglected would no longer apply. Let us frst focus n the deal case n whch there are no nspecton errors. Let us also ntroduce the followng addtonal notaton: c : cost for shftng from 00% to systematc nspecton c : cost for shftng from systematc to 00% nspecton : number of expected unts to be nspected durng 00% nspecton : number of expected unts to be nspected durng systematc nspecton The expected number of tmes that we would shft from 00% to systematc nspecton s gven by the steadystate probablty x 00% tmes the total number of unts,, tmes the probablty of shftng from 00% to P θ,.e. systematc nspecton whch s ( ( x P θ. We also have that 00% = x. Then, 00% we have that the expected number of tmes that we would shft from 00% to systematc nspecton would be gven by pθ (. Smlarly, the expected number of tmes n whch we would shft from systematc to p θ, where 00% nspecton would be gven by (

10 96 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, 04. = fx f. Then the total expected cost under deal nspecton would be gven by: ( = cp p ( θ cs P( θ P( θ C c s cp + c p + ( θ c p( θ r ( θ ( s mentoned before, the mnmzaton of the cost gven by ( s subject to (6. Note that from expresson ( we have the followng propertes:. + = Our fndngs presented n secton 4 regardng the optmal number of unts to be nspected stll apply. Ths means that the optmal number of unts to be nspected s. = P( θ = L To see why, let us frst consder the case n whch we nspected less than. In that case we would volate the constrant gven by (6 whch s not allowed. For the other case, let us consder that we ' ' nspect unts where >. Then, clearly the ' frst term of ( would be greater for than for. lso, C( s an ncreasng functon of both and. So f there s a combnaton of values + that allows complyng wth the constrant, such a combnaton would domnate any other that yelds to a greater values of +. Therefore, ' s always a better soluton than. C( can be expressed as: cp p ( θ cs + P( θ ( = cs + + cp r ( θ P( θ ( θ ( ( θ C + c p + c p (3 = P = L, y realzng that ( θ C n terms of. Therefore, our optmzaton problem becomes: we just need to focus on optmzng ( Mn Z ( = c p( θ ( ( θ + c p (4 y nspectng (4 we have that the optmal value of, namely s gven by: ( ( f = cp θ cp θ 0. Ths means that we would reman n 00% nspecton and we would avod shftng to systematc nspecton. Ths can be acheved by makng = and f = 0. = 0 cp ( θ cp ( θ 0 f. Ths would mply that we never enter 00% nspecton. Ths can be acheved by startng drectly the systematc nspecton, makng = 0. The value of f must be such that: = = p( θ ( θ + ( θ fx f f p p = P ( θ L P ( θ Then, the optmal value for f would be gven by: ( P( θ ( ( p( θ P θ L p θ + p θ f = (5 (6 Now let us dscuss the case n whch we consder nspecton errors. y a smlar analyss to that appled for the case wthout nspecton errors, we obtan that: C = ( + P( S P( S + P( S P( S I ( θ ( r ( ( c p( S cs cs cp + c P S P S + c P S + c p S + (7

11 arbosa-correa & Galndo-Pacheco / Dyna, year 8, no. 83, pp , February, It can be easly shown that, as n the prevous case, our fndngs for secton 5 stll apply and therefore. Then, our optmzaton problem becomes: Mn Z ( = c p( S ( ( + c p S (8 y nspectng (8 we arrve to smlar conclusons as those exposed for the prevous case: = c p S = and f = 0. ( c p( S f. Therefore 0 ( ( f = 0 cp S cp S 0. Ths can be acheved by startng drectly the systematc nspecton, makng = 0. The value of f must be such that: Then, the optmal value for f would be gven by: (9 (30 8. CONCLSIONS ND FTRE RESERCH DIRECTIONS In ths paper we have presented two models for desgnng optmal CSP- from an economc perspectve, under two scenaros: nfallble and fallble nspecton systems. Our work combnes Markovan and ayesan analyss to model the uncertan parameters of the system. We found that for both models, the optmal decson s to mplement a CSP- whose expected number of nspected unts s the mnmum requred to guarantee that the OL does not exceed the customer s L. From our fndngs, we can conclude that mplementng the model for nfallble nspecton systems when actual nspecton errors are present, results n a volaton of the customer s L. We have also extended our analyss for the cases n whch the costs for shftng between 00% and systematc nspectons cannot be neglected. Our results show that the optmal answer n ths case s ether to reman whether n 00% or n systematc nspecton, but never swtch between the two styles of nspecton wthn the same process. s future research drecton we recommend to consder the defectve fracton as stochastc rather than determnstc. In ths regard, the probablty functon of the fracton defectve can be modeled usng the Normal approxmaton to the nomal dstrbuton. lso, our analyss can be extended to other SIPs, such as CSP-, CSP-3 and other related ones. REFERENCES [] Lorenzen, T.J., Mnmum cost samplng plans usng bayesan methods, Naval Research Logstcs uarterly, 3(, pp , 985. [] Schmdt, J.W., Case, K.E. and ennett, G.K., The Choce of Varables Samplng Plans sng Cost Effectve Crtera, IIE Transactons, 6(3, pp , 974. [3] Ln, T.-Y. and Yu, H.-F., n Optmal Polcy for Csp- wth Inspecton Errors and Return Cost, Journal of the Chnese Insttute of Industral Engneers, 6(, pp , 009 [4] Yu, H.-F., Yu, W-C. and Wu, W.P., mxed nspecton polcy for CSP- and precse nspecton under nspecton errors and return cost, Computers & Industral Engneerng, 57, pp , 009 [5] Dodge, H.F., Samplng Inspecton Plan for Contnuous Producton, nnals of Mathematcal Statstcs, 4(3, 64-79, 943 [6] unno, R.d.C., Ho, L.L. and Trndade,.L.G., ayesan judgment of a dchotomous nspecton system when the true state of an evaluated tem s unknown, Computers & Industral Engneerng, 49, pp , 005. [7] Chun, Y.H. and Sumchrast, R.T., ayesan nspecton model wth the negatve bnomal pror n the presence of nspecton errors, European Journal of Operatons Research, 8, pp. 88-0, 007. [8] Sengupta, S. and Majumdar,., Some Consderatons of Dodge and Romg Sngle Samplng Plans under Inspecton

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