A MULTIOBJECTIVE OPTIMIZATION FOR THE EWMA AND MEWMA QUALITY CONTROL CHARTS

Transcription

1 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 A MULTIOBJECTIVE OPTIMIZATION FOR THE EWMA AND MEWMA QUALITY CONTROL CHARTS Fancsco Apas Depatamento de Estadístca e Investgacón Opeatva Aplcadas y Caldad. Unvesdad Poltécnca de Valenca Valenca (Span). fapas@eo.upv.es J. Calos Gacía-Díaz Depatamento de Estadístca e Investgacón Opeatva Aplcadas y Caldad. Unvesdad Poltécnca de Valenca Valenca (Span). juagad@eo.upv.es ABSTRACT The Multvaate EWMA contol chat, MEWMA, Lowy, Woodall, Champ and Rdgon [1] and ts unvaate veson EWMA, may be desgned to effcently detect small shfts n the mean vecto of a set of p qualty chaactestcs of a poducton pocess. Howeve, ths wok pesents a method fo the optmal desgn of MEWMA and EWMA chats paametes to contol pocesses whee t s not convenent to detect small magntude shfts and, at the same tme, poweful enough to detect shfts consdeed mpotant. Ths poblem can be consdeed as a multobjectve optmzaton. Woodall [2] studed the statstcal desgn of contol chats and ecommended choosng the magntude of the shft that t s mpotant to detect as a desgn cteon fo contol chats. Fo ths pupose, he suggested defnng thee egons: ncontol, ndffeent, and out-of-contol. These egons wll be delmted by two values (A and B). The man objectve of ths pape s to fnd the best MEWMA and EWMA qualty contol chats gven the pevous egons, whee the equements fo each egon has to be balanced to decde whch soluton s bette. Fo ths pupose, fendly Wndows softwae has been developed to optmze ths poblem, usng Genetc Algothms. A compason s made among the EWMA chat desgned employng ths softwae, the typcal desgn of a EWMA chat and the Shewhat contol chat. Results show that the desgn usng ou appoach outpefoms the othe desgns. NOMENCLATURE ARL. Aveage un length. ARL. ARL when the pocess s n-contol. ARL A. ARL fo pont d = A. ARL B. ARL fo pont d = B. ARLmn. ARL mnmum desed fo d =. ARL PA. ARL desed fo pont d = A. d. Mahalanobs dstance. L. EWMA o MEWMA contol lmt. n. Sample sze.. Smootng constant fo EWMA o MEWMA contol chats. INTRODUCTION Nowadays t begns to be common to face poblems o applcatons whee the mathematcal modellng poduces a optmzaton poblem wth seveal objectves. The multobjectve optmzaton conssts of optmzng smultaneously seveal objectve functons. In many cases, some of the objectve functons epesent moe o less conflctng ctea. Obvously, n these cases no unque soluton can be found because the ente objectve functons cannot be optmzed (maxmzed o mnmzed) wthout consdeng the effect of the expemental changes n the othe esponse functons. In geneal tems, the optmzaton poblem can be fomulated as follows, beng n the numbe of decson vaables, x j, m estctons and p objectves: Fnd x (x 1, x 2,, x n ) that Maxmze / mnmze Z = ( z 1 (x), z 2 (x),, z n (x)) Subject to x F Wth F R n, F feasble egon of solutons space R n and Z = z(f) R p, Z feasble egon of objectves space R p. Many tmes the set F can be wtten as F={ x R n : g (x), x j,, j } when g functons ae the estctons. In some

2 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 cases, vaables z k ae called objectve functons o objectves. One feasble soluton x s effcent, no domnated o Paeto optmum, f thee s no anothe feasble soluton x* that mpoves the values of one objectve wthout wosenng one of the othe objectves. The set of all the effcent solutons s called effcent set o Paeto font. Conventonally multobjectve optmzaton poblems have been tackled tyng to fnd a sngle optmum soluton, usng the so-called pefeencebased methods, whch assume, explctly o mplctly, a heachy n the objectves mpotance. In the best case, these appoaches conduct to a sngle optmum soluton located n the Paeto font, although no nfomaton about the Paeto font s poduced. One of the most used methods s mnmzng weghted sums of functons. Mathematcally, ths method s expessed as: Maxmze z(x) = wk z p k= 1 k ( x) Subject to x F whee w s the weght coespondng to k objectve z k (x) and can be ntepeted as the mpotance of objectve k n compason wth the est of objectves. Now the poblem s educed to fnd P(w) whee w = (w 1, w 2,, w p ). Hence, the multobjectve poblem s now educed to a unque optmzaton poblem. The objectve of ths pape s to apply multobjectve optmzaton to the desgn of MEWMA and EWMA qualty contol chats. EWMA AND MEWMA CONTROL CHARTS The statstcal desgn of a qualty contol chat lke EWMA o MEWMA conssts of selectng thee paametes. The powe of the chat (measued though ARL) depends on these paametes, sample sze, n, poston of contol lmts, L, and a smoothng constant. EWMA (Exponentally Weghted Movng- Aveage) contol chats wee ntoduced by Robets [3] as an altenatve to Shewhat contol chats fo the detecton of small shfts n the pocess. Howeve, Shewhat contol chat only takes nto account the pesent nfomaton of the pocess and does not detect quckly changes smalle than 2σ. EWMA contol chats take nto account pesent and past nfomaton and theefoe they ae moe effcent (fast) n detectng small shfts (Montgomey [4]). A wdely used measuement of the effcency of a pocess statstcal contol method s the ARL (Aveage Run Length). The ARL s the aveage numbe of samples to take (ponts n the chat) untl an out-of-contol-sgnal appeas. In the case of EWMA, the statstcal data to chat Z, to be compaed wth contol lmts at nstant, s obtaned as a weghted aveage value accodng to paamete between the obseved value X and the smoothed value Z -1, followng expesson: Z X + 1 ) Z (1) = ( 1 As t can be obseved, weghtng s done wth paamete so that the smalle the paamete s, the geate the nfluence of past obsevatons as weght deceases geometcally n functon of. Ths s the eason why EWMA contol chats ae sad to have human memoy snce they povde weghts to data exponentally: assgnng moe weght to pesent data whch deceases as data ae fa back n the past. When = 1, then the aveage value s epesented by the Shewhat contol chat, and when =, Z s a constant equal to µ. EWMA contol chat senstvty to detect changes n the pocess depends on the value of. When tends to 1 EWMA values wll depend on the most ecent obsevatons and the behavou of the contol chat s smla to that of the Shewhat contol chat. Howeve, as tends to, the hstocal behavou of the pocess gets moe weght, and then t appoaches the behavou of nomal CUSUM chats. A ecommended value fo s.2 (Hunte [5]). Fo Z the value adopted s the nomnal aveage value µ o the samplng aveage value n n-contol pocesses. Some authos (Hunte [5], Cowde [6] and Lucas and Saccucc [7]) have studed the popetes of ths chat fo the statstcal contol of ndustal pocesses. Let's analyze the desgn of the chat. If the qualty vaable to contol s dstbuted accodng to N ( µ, σ ) n n-contol pocesses and the obsevatons ae ndependent, theefoe, the contol lmts of the EWMA contol chat ae calculated wth the appoxmate expesson: UCL = µ +L σ n 2 (2)

3 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 LCL = µ - L σ n 2 whee L and ae selected to get a gven ncontol ARL and n s the sze of the subgoup. A typcal value of L s 3, followng the cteon 3σ, of the Shewhat contol chat. If we want to obtan an n-contol ARL, ARL, of 37.4 (α=.27), then we should fx the value of =.25 and L = The fst efeence on multvaate EWMA (MEWMA) contol chats coesponds to Lowy, Woodall, Champ and Rgdon [1] who defne MEWMA as an extenson of the unvaate EWMA. Hotellng's T 2 multvaate contol chat only takes nto account cuent pocess data, wheeas MEWMA chat also ncludes past data, theeby t beng moe poweful to detect small changes n the pocess. Unvaate systems only contolled one qualty vaable o chaactestc. In multvaate systems a set of p nteelated vaables wll be contolled. In ths latte case, X 1, X 2..., ae un length vectos p whch epesent the samplng aveage values of the pocess. Let andom vectos X be ndependent and equally dstbuted followng a p-vaate nomal vaable of vecto µ and covaance matx Σ, X d N ( µ p, Σ). The pocess wll be unde contol f µ = µ and out of contol n the opposte case. Vecto Z s defned as Z = X + ( 1 ) Z 1, 1 (3) the statng vecto beng µ Z = snce the pocess s unde contol E ( Z ) = µ and covaance matx of Z s expesson s gven hee below. whose Z s the vecto X of the samplng data and s a scala value between and 1. If =1 we wll obtan Hotellng's T 2 contol chat. The statstcal data 2 chated T s defned as v 2 1 T = Z ' Z (4) Z whee 1 Z s the nvese of the vaancecovaance matx of Z. The covaance matx of Z s expessed by: Z = 2 [ 1 (1 ) ] (5) 2 The measuement of vecto shft (o dstance between two vectos) used n multvaate analyss s Mahalanobs dstance. In ou case, the dstance between the ognal mean vecto and the new mean vecto s ' 1 d = ( µ µ ) ( µ µ ). The ARL pefomance of the MEWMA chat depends only on the noncentalty paamete λ = nd 2, whee n s sample sze (Lowy, Woodall, Champ and Rgdon [1] and Lowy and Montgomey [8]). Fo the desgn of the MEWMA contol chat, the asymptotc covaance matx can be used, gven by: Z = lm Z = (6) 2 smlaly to what happened n unvaate systems fo ndvdual obsevatons. Fo sample sze othe than 1, equaton (6) coected by n wll be obtaned (Rgdon [9]) = (7) Z (2 ) n The chat dsplays an out-of-contol sgnal when 2 T > h, whee h s the contol lmt selected to obtan a gven value of ARL fo n-contol pocesses (ARL ). The compason of the ARLs (obtaned though smulaton) pesented n Rgdon's wok shows that MEWMA, usng the exact covaance matx gven by equaton (5), s somewhat hghe than MCUSUM, specally when the aveage value vecto pesents geat changes. Woodall [2] studed the statstcal desgn of contol chats and ecommended choosng the magntude of the shft that t s mpotant to detect as a desgn cteon fo contol chats. Fo ths pupose, he suggested defnng thee egons: ncontol, ndffeent, and out-of-contol. These egons wll be lmted by two values (A and B), as follows: a) In-contol egon [, A]. Ths egon coesponds to a state equvalent to one n-contol and s made up of a shft change that anges fom

4 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 d = to d = A. No shft detecton s equed n ths egon. A maxmum ARL s needed. If the chat shows an out-of-contol sgn, ths s egaded as a false alam. b) Out-of-contol egon [B, 8[, coespondng to the shft value d > B. Maxmum detecton powe s equed fom ths aea. A mnmum ARL s needed. c) Indffeent egon, ]B, A[, coveng d > A and d < B. Ths egon s ndffeent f the pocess shft s detected o not. Theefoe, gvng A and B values and the numbe of vaables to contol smultaneously we dese to fnd the paametes of EWMA and MEWMA contol chats (, L and n) that satsfy the Woodall egons. In adtton, a mnmum ncontol ARL (ARL ) s spcefed and the ARL fo d = A has to be equal to a gven one, ARL A. Ths value ARL A s aestcton that wll help to make compasons aganst X chat Hotellng s T 2 contol chat, as shown n fgue 1.., ARL 1., ARLo 1, 1, 1, A B,5 1 1,5 2 2,5 3 d Fgue 1. ARL cuve. In ths wok, the addtve utlty functon method has been employed. Ths pocedue convets the multobjectve poblem nto a optmzaton poblem wth only one objectve. Ths methos s based on defnng a functon that combnes the dffeent objectves, usng weghts that show the elatve mpotance of each objectve fo the use. Once ths functon s obtaned, the un-objectve poblem s solved. In ths case we have two objectves (p = 2) to optmze: z 1( x) = ARLo ARL and mn z 2 ( x ) = ARLB whee z1( x) s a objectve to maxmze, as t s desed to have contol chats that satsfy ARLo ARL mn and z 2 ( x ) has a negatve values because ARLB has to be mnmum. Fnally, ou optmzaton poblem s: Maxmze z(x) = wk z p k= 1 k ( x) = w 1z1( x) + w 2 z2 ( x) = w 1 ( ARLo ARLmn ) w2 ARLB Subject to ARL PA ARL tol and ARLo ARLmn A whee ARL s the n-contol ARL fo d =, ARL B s the ARL fo pont B, ARL A s the eal ARL fo pont A, ARL PA s the ARL desed (use nput) n pont A, ARL mn s the desed mnmum ARL fo d = and w 1, w 2 ae the weghts. In ou case we have employed the weghts w 1 = 1 and w 2 = 5. OPTIMUM SEARCH USING GENETICS ALGORITHMS. Genetc Algothms (GA) ae optmzaton algothms based on the natual evoluton of the speces (Holland [1], Goldbeg [11]). The seach fo the global optmum value n an optmzaton poblem s caed out when an ntal populaton (geneaton) of ndvduals passes to a new populaton (next geneaton) though the applcaton of genetc opeatos. In the ognal populaton, each ndvdual epesents a possble soluton to the optmzaton poblem, that s, a populaton of ndvduals conssts of a set of possble solutons to the poblem to optmze. The pncples, mplementaton and applcatons of GA can be followed n Bäck [12], Chambes [13] and Mchalewcz [14]. The assessment functon, efeed to as ftness functon, assgns to each ndvdual of the populaton (set of possble solutons to the poblem to optmze) the ftness value, whch ndcates the ftness of that ndvdual wth espect to the othe ndvduals of the populaton. The ftness value s a qualty value of the ndvdual and the only data pocessed by GA to seach fo the best soluton to the poblem. Its coect defnton allows fo a bette opeaton of the algothm snce to fnd the global optmum value the seach s exclusvely guded by the ftness value of the possble solutons.

5 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 Po to the applcaton of the genetc algothm we have to code the solutons, that s, t s necessay to defne how to bette epesent each possble soluton to the poblem, an aspect that s essental fo the desgn and effcency of the GA. The genetc algothm opeates on a coded epesentaton of the solutons, equvalent to the genetc mateal of an ndvdual, and not dectly on the solutons. These paametes known as genes, fom chans efeed to as chomosomes. In ths pape the followng cossove mechansms has been employed to these chomosomes: 1 pont, 2 ponts and unfom (Bäck [12], Beasley et al [15, 16]), obtanng the best esults wth the 2 ponts opeato. Dung the last yeas, many eseaches have pad attenton to the poblems nvolved wth multobjectve optmzaton (Schaffe [17], Tabucanon [18], Fonseca and Flemng [19], Ztzle et al. [2], Coello et al. [21]). The fst multobjectve GA was the named Vecto Evaluated Genetc Algothm (VEGA), Schaffe [17]. Recently, moe pefectoned GA has been poposed,. The most mpotants ae: the Multobjetve Genetc Algothm (MOGA), Fonseca and Flemng, [22], the Nche Paeto Genetc Algothm (NPGA), Hon et al. [23], the Nondomnated Sotng Genetc Algothm (NSGA), Snvas and Deb [24], the Stength Paeto Evolutonay Algothm (SPEA), Ztzle and Thele [25] and the Paeto-Achved Evolutonay Stategy (PAES), Knowles and Cone [26]. To solve ths multobjectve optmzaton fendly Wndows softwae has been developed. RESULTS.EXAMPLE OF APPLICATION. We wll now move on to the optmum desgn of the EWMA contol chat, wth a sample sze not fxed pevously, by usng the softwae developed. We call the EWMA chat found by the softwae developed n ths wok EWMA- Regons. We wsh to compae the chat obtaned wth an X chat fo the ARL n-contol (d = ) of 5, pesentng an ARL n d = A =.25 of The pogamme enty data ae: mnmum desable ARL of 15, and ARL PA of Once the pogamme has been un we obtan the EWMA-Regons optmum contol chat ( =.91, L = 3.4, n = 5). In Fgue 2, the ARL values fo the chat "EWMA-Regons" s compaed to the ARL fo the X chat. Also we nclude the EWMA contol chat optmum to detect a shft of sze d = B = 1.5. Ths EWMA chat s called EWMA-Pont, because t s optmum fo only ths pont. Runnng the softwae developed by Apas and Gacía- Díaz [27] the "EWMA-pont" contol chat s chaactesed by the paametes L = 3.9 and =. 85 fo n = 5 and an ARLo = 5. A copy of ths softwae can be downloaded at ARL d Fgue 2. ARL compason. It can be seen how the EWMA-Regon and X contol chats have the same ARL n pont A. Ths defnes the egon of shfts we ae not nteested n detectng (d <.25). As we commented n Secton 3, the best contol scheme would be one whee the egon d <.25 has the lagest ARL value (lowest powe), and pesents the smallest ARL value (maxmum powe) fo shft magntudes d > 1.5. Compaed to the EWMA-Pont, the EWMA-Regons chat offes the advantage of poducng lowe pobablty of false alams. Ths s because ts ARL s much hghe n the egon of shfts not to be detected, d <.25. On the othe hand, t can be seen that fo shfts that ae genunely mpotant to detect, d > 1.5, both EWMA chats show vey smla powes, although the EWMA-pont s slghtly moe poweful. In the ndffeent egon,.25 < d < 1.5, the optmum EWMA s moe poweful n d = 1.5, although, as t was dscussed befoe, the ARLs n ths egon ae not mpotant. The fnal concluson to be dawn s that the EWMA chat obtaned usng the softwae developed n ths pape pactcally have the same powe fo detectng genunely mpotant shfts than the EWMA chat that s moe effcent at

6 Invese Poblems, Desgn and Optmzaton Symposum Ro de Janeo, Bazl, 24 detectng shfts n d = 1.5. Howeve, the EWMA-Regons chats has a vey low pobablty of false alam n the n-contol egon. CONCLUSIONS. In vew of the esults dscussed n ths wok, we may daw the followng conclusons. The genetc algothms technque has poved to be a sutable method fo the optmsaton of the EWMA and MEWMA contol chats usng the egons defned by Woodall [2]. An easy-to-use softwae pogamme has been developed n a Wndows envonment, enablng the optmum paametes of these chats to be obtaned. These ae used fo contollng pocesses whee mnmum powe s equed fo detectng extemely small shfts, and maxmum powe fo detectng genunely mpotant ones. It s possble to desgn EWMA and MEWMA chats that may eveal a vey low false alam pobablty and that ae, at the same tme, genunely poweful n detectng shfts consdeed mpotant. The use of these chats would epesent an extemely sgnfcant contol tool n both pactcal stuatons as well as capable pocesses, pocesses had to adjust, o whose cost of adjustment s hgh. Acknowledgements The authos acknowledge the fnancal suppot of the Mnsty of Scence and Technology of Span, Reseach Poject Refeence DPI and Euopean FEDER fundng. REFERENCES 1. Lowy CA, Woodall WH, Champ CW, Rgdon SE. A Multvaate Exponentally Weghted Movng Aveage Contol Chat. Technometcs 1992; 34: Woodall, W.H. (1985). The statstcal desgn of qualty contol chats. The Statstcan, 34, Robets, S. W. (1959). Contol chat tests based on Geometcs Movng Aveages. Technometcs, 1, Montgomey, D.C. Intoducton to statstcal qualty contol. New Yok: 4end edn, John Wley, Hunte JS. The exponentally weghted movng aveage. Jounal of Qualty Technology 1986; 18: Cowde SV. Desgn of Exponentally Weghted Movng Aveage Schemes. Jounal of Qualty Technology 1989; 21: Lucas JM, Saccucc MS. Exponentally Weghted Movng Aveage Contol Schemes: Popetes and Enhancements. Technometcs 199; 32: Lowy CA, Montgomey DC. A evew of multvaate contol chats. IIE Tansactons 1995; 27: Rgdon SE. An ntegal equaton fo the ncontol aveage un length of a multvaate exponentally weghted movng aveage contol chat. Jounal of Statstcal Computaton and Smulaton 1995a; 52: Holland J. Adaptaton n natual and atfcal systems. Ann Abo: Unvesty of Mchgan Pess, Goldbeg DE. Genetc Algothms n Seach, Optmzaton, and Machne Leanng. Addson Wesley, Bäck T. Evolutonay algothms n theoy and pactce. Evolutonay stateges, evolutonay pogammng, genetcs algothms. New Yok: Oxfod Unvesty Pess, Chambes LD. Pactcal Handboock of Genetc Algothms: Applcatons, Vol.1. CRC Pess, Mchalewcz Z. Genetc algothms + data stuctues = evoluton pogams. Beln: Spnge, Beasley D, Bull D, Matn R. An ovevew of genetc algothms: Pat 1. Fundamentals. Unvesty Computng, 1993a; 15: Beasley D, Bull D, Matn R. An ovevew of genetc algothms: Pat 2. Fundamentals. Unvesty Computng, 1993b; 15: Schaffe JD. Multple objectve optmzaton wth vecto evaluated genetc algothms. Genetc Algothms and The Applcatons: Poc. Ist. Int. Conf. Genetc Algothms. Hllsdale, NJ: Lawence Elbaum, 1985: Tabucanon MT. Multple Ctea Decson Makng n Industy. Elseve Scence Publshes, Fonseca CM, Flemng PJ. An Ovevew of Evolutonay Algothms n Multobjectve Optmzaton. Evolutonay Computaton 1995; 3:1-16.

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