Algebraic expression of system configurations and performance metrics for mixed-model assembly systems

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1 IIE Transactons , Copyrght C IIE ISSN: X prnt / onne DOI: 0080/074087X Agebrac expresson of system confguratons and performance metrcs for mxed-mode assemby systems ANDRES G ABAD, WEIHONG GUO 2 and JIONGHUA JUDY JIN 2,* Escuea Superor Potecnca de Ltora, Campus Gustavo Gando Veasco, Km 305 Va Permetra, Guayaqu , Ecuador 2 Industra and Operatons Engneerng Department, Unversty of Mchgan, Ann Arbor, MI , USA E-ma: jhjn@umchedu Receved December 200 and accepted Apr 203 One of the chaenges n the desgn and operaton of a mxed mode assemby system MMAS s the hgh compexty of the staton ayout confguraton due to the varous tasks that have to be performed to produce dfferent product varants It s therefore desrabe to have an effectve way of representng compex system confguratons and anayzng system performances By overcomng the drawbacks of two wdey used representaton methods bock dagrams and adjacency matrx, ths artce proposes to use agebrac expressons to represent the confguraton of an MMAS By further extendng the agebrac confguraton operators, agebrac performance operators are defned for the frst tme to aow systematc evauaton of system performance metrcs, such as quaty conformng rates for ndvdua product types at each staton and process capabty for handng compexty nduced by product varants Therefore, the benefts of usng the proposed agebrac representaton are not ony ther effectveness n achevng a compact storage of system confguratons but aso ts abty to systematcay mpement computatona agorthms for automatcay evauatng varous system performance metrcs Exampes are gven n the artce to ustrate how the proposed agebrac representaton can be effectvey used n assstng the desgn and performance anayss of an MMAS Keywords: Agebrac expresson, mxed mode assemby, product varety, system confguraton mode, compexty, moduar assemby systems Introducton In recent decades, market demand has changed from beng fary homogenous and reatvey stabe to hghy varabe and rapdy changng As a response, producton systems have gone from a mass producton paradgm to a mass customzaton mode Impementng a mass customzaton scheme, however, requres overcomng many technoogca chaenges Da Svera et a, 200 From a producton pont of vew, the correct mpementaton of moduar producton Sturgeon, 2002 s the key to addressng these chaenges In a moduar producton assemby system, dfferent product varants are produced n a system that s referred to as a Mxed Mode Assemby System MMAS The staton confguratons of an MMAS become hghy compex because dfferent product types use dfferent producton paths n the same assemby system For exampe, the assemby system shown n Fg s used to produce three dfferent Correspondng author product types It can be seen that the frst two statons, S and S 2, are basc operaton statons, whch means that every product of any type w be processed at statons S and S 2 In contrast, statons S 3, S 4, S 5, S 6, S 7,andS 8 are consdered as varant operaton statons, whch mpes that ony certan types of products w be processed at these reated statons Specfcay, the frst product type has a producton path of S S 2 S 3 S 4 S 5 ; the second producttypefoowsthepaths S 2 S 3 S 6 ; and the thrd product type s produced va the path S S 2 S 7 S 8 It shoud aso be noted that n an MMAS, parae staton confguratons are not necessary used to dupcate the same tasks as they are used n snge-mode assemby systems for the purpose of ne baancng As shown n Fg, the path of staton 6 s used to produce a product type that s dfferent from the type produced n the parae path of staton 4 and staton 5 Ths may aso be true for two snge parae statons snce each of the two parae statons may be desgned to perform dfferent tasks to produce dfferent product components or dfferent types of the same component Consequenty, addng product varants n an MMAS ncreases the compexty n operatng and evauatng manufacturng systems X C 204 IIE

2 Agebrac expresson of MMAS confguratons 23 Fg Basc operaton statons and varant operaton statons n a mxed-mode assemby system As the demand for MMASs ncreases, the need for effectvey modeng and anayzng the operaton performance of such a system ncreases Tradtonay, assemby systems have been represented by the use of bock dagrams, as shown n Fg ; ths type of dagram shows the advantages of ntutve vsua percepton of the staton ayout confguraton Bock dagrams themseves, however, do not have the computatona capabty to permt the mathematca manpuaton of system confguratons or evauaton of system performances The use of adjacency matrces representng bock dagrams s another common method of representng assemby systems, as noted n the terature Adjacency matrces are mathematca structures that emerge n graph theory as a way to more easy manpuate and represent graphs The capabty of ntutve vsua representaton of the system confguraton sera or parae reatonshps between statons cannot, however, be preseed n the adjacency matrces representaton Furthermore, adjacency matrces tend to resut n argey sparse matrces, especay as the number of statons ncreases As a consequence, adjacency matrces are not an effectve compact method for representng an assemby system To overcome the drawbacks of bock dagrams and adjacency matrces methods, Webbnk and Hu 2005 recenty ntroduced a nove way of representng compex assemby systems by usng a strng representaton The proposed method uses characters to represent the statons and parentheses to denote whether statons are connected n a sera or parae reatonshp Athough the strng representaton provdes a compact way to represent the system confguraton, t acks a computatona capabty to use computer programs to ncorporate the system confguraton nto mathematca operators to evauate system performances Therefore, ths artce ntends to overcome the shortcomng of the strng representaton by usng an agebrac representaton The advantage of an agebrac representaton s that t not ony can store and manpuate sera or parae confguraton nformaton so that t can be easy handed by computer programs but t aso can be transferred to the mathematca operators for evauatng system performances The use of bnary operators to dea wth graph probems has been proposed n the terature Path agebras, aso known as max, + agebras or dod agebras, are mathematca structures used to sove a arge number of pathfndng and network probems n graph theory Carre, 97; Gondran and Mnoux, 983 The key dea s to use bnary operators to rewrte, n a compact way, the agorthms used to sove these probems, thus achevng a pseudo-near system of equatons The appcaton of path agebras to manufacturng systems was ntated n Cohen et a 985 Snce then, t has been used to sove numerous knds of probems n ths doman; eg, resource optmzaton Gaubert, 990, producton pannng and contro Xu and Xu, 988; Yurdaku and Odrey, 2004, and modeng of dscrete-event systems Cohen et a, 989; Cofer and Garg, 993 To the best of our knowedge, however, path agebra has not been drecty used to represent compex assemby system confguratons Zssos and Duncan 97 proposed the use of agebrac operators to represent ogc crcuts wth the advantage that the symbos stand n a one-to-one correspondence wth the physca eements of the system Furthermore, a postfx or reverse Posh notaton of ths agebrac representaton was proposed n Duncan et a 975 wth the advantage that the postfx notaton eases mpementaton n computer anguages by nherenty determnng the order n whch operatons are to be resoved Recenty, Frehet et a 2004 proposed to use Booean operators as a way to determne the system states operatve or not operatve n arbtrary staton confguratons Ths was acheved by representng sera reatons among statons by a dsjunctve AND operator and parae reatons among statons by a conjunctve OR operator As a consequence, a of the states where producton s acheved are determned when a vaue of TRUE s acheved by the Booean expresson representng the system Ths artce further extends prevous work to acheve an ntutve representaton of compex assemby system confguratons usng agebrac expressons Ths agebrac expresson w be consequenty transferred nto computatona operators for evauatng compex systems performances For the purpose of evauatng system performance for MMASs, ncreasng research has been conducted n recent years For exampe, based on nformaton entropy, Zhu et a 2008 proposed metrcs to quanttatvey assess the process compexty nduced by product varants Subsequenty, Abad and Jn 20 further studed how to assess the manufacturng system s capabty n handng such compexty through the nkage wth a communcaton system framework In ther work, a set of new metrcs was proposed, ncudng Process Capabty for Compexty PCC, Normazed Process Capabty for Compexty

3 232 Abad et a NPCC, and quaty conformng matrx In ths artce, these metrcs w be drecty used to evauate manufacturng system performances Gong beyond the exstng work, ths artce further studes how to use the agebrac expresson to mathematcay represent the system confguratons n order to acheve computatona smpcty n computng those performance metrcs Ths consequenty permts a scaabe computaton capabty by decomposng a compex assemby system nto herarchca mut-eves of subsystems The detaed merts of the agebrac representaton and exampes w be shown n ater Sectons 3 to 5 of ths artce The rest of ths artce s organzed as foows In Secton 2, an agebrac representaton of assemby system confguratons s ntroduced The transferrng agorthms that are used to obtan the agebrac representaton from a tradtonay used bock dagram or an adjacency matrx are provded Secton 3 dscusses how to extend the agebrac expressons of system confguratons by defnng the performance operators for evauatng system performance metrcs Then, Secton 4 ntroduces the concept of nverse operators and ustrates how to use the nverse operators to adjust ndvdua staton requrements to mprove system performances A case study s presented n Secton 5 to show some potenta appcatons of the proposed agebrac modeng method Fnay, concusons and future work are provded n Secton 6 Tabe Three equvaent representatons of system confguraton Adjacency Agebrac Bock dagram matrx H expresson H H H H H S S 2 S 3 S S S S 2 S 3 S S S 2 S 3 S S S S 2 S 3 S S S 2 S 3 S S S S 2 S 3 S S S 2 S 3 S S S S 2 S 3 S S S 2 S 3 S S S S 2 S 3 S Agebrac representaton of assemby system confguratons In ths secton, we descrbe how to effectvey mode the staton confguratons for a genera assemby system wth a hybrd confguraton structure Specfcay, we propose the use of an agebrac expresson wth two bnary operators, and, to represent the sera and parae reatonshp between two statons, respectvey The operands of these bnary operators are two assocated statons For exampe, S S 2 s used to represent two statons wth a sera confguraton ayout, and S S 2 s used for two statons wth a parae confguraton ayout To enabe comparson wth the exstng methods of bock dagram and adjacency matrx, Tabe shows these three equvaent ways for representng fve smpe assemby system confguratons consstng of three statons abeed as S, S 2,andS 3 It can be seen that the proposed agebrac expresson keeps the expct representaton of the sera/parae confguraton, as n bock dagrams, thus provdng a better representaton than the adjacency matrx method Furthermore, mathematca computaton agorthms can be easy added nto these agebrac operators for evauatng system performance metrcs For exampe, and w be defned ater for evauatng the quaty conformng rate of the tasks performed at the gven statons Therefore, the proposed agebrac expresson method aso shows a better computatona representaton than the bock dagram method Because the bock dagram and adjacency matrx representatons are commony used to represent system confguratons n practce, t w be practca to deveop transferrng agorthms for automatcay obtanng an equvaent agebrac expresson from ether a bock dagram or an adjacency matrx The foowng two subsectons w dscuss these transferrng agorthms 2 Agebrac representaton transferred from system bock dagram Ths subsecton shows how an agebrac expresson can be drecty obtaned from a bock dagram representaton Let us consder an exampe of an assemby system wth a system bock dagram as shown n Fg 2 Fgure 3 shows the detaed step-by-step transferrng procedures At each step, every par of statons s grouped wth a sera or parae confguraton by usng operator or accordngy; ths generates an equvaent staton to represent the sub-grouped statons For exampe, we frst combne staton S 2 wth S 3 and generate the equvaent staton S 2,3 S 2 S 3, as shown n Fg 3b Next, we combne group S 2,3 wth staton S 4 and generate the equvaent staton S 2,3,4 S 2 S 3 S 4, as shown n Fg 3c By

4 Agebrac expresson of MMAS confguratons 233 Tabe 3 ustrates an exampe of how the agorthm s used to transfer the adjacency matrx of the assemby system n Fg 2 nto the agebrac expresson representaton Fg 2 A moduar assemby system confguraton performng a of the steps shown n Fg 3a to 3e the agebrac expresson of the whoe system s S,2,3,4,5 S [S 2 S 3 S 4 ] S 5 The detaed resuts of the transferred agebrac expressons at each step are summarzed n Tabe 2 The fowchart shown n Fg 4 provdes the standard procedures for transferrng a system bock dagram nto an agebrac expresson 22 Agebrac representaton transferred from adjacency matrx In an adjacency matrx H representng the confguraton of the statons n an assemby system, every coumn and every row of matrx H have been abeed accordng to the staton for whch they stand The transferrng agorthm s proposed by teratvey combnng two statons e, smutaneousy combnng two rows and two coumns of matrx H unt a statons are combned nto a snge staton snge entry n matrx H At each step, we group two statons or equvaent statons by the approprate agebrac expresson based on ther confguraton reatonshp ether sera or parae The transferrng agorthm s shown by the pseudo-code n Fg 5 23 Agebrac representaton of mutpe tasks wthn a staton When mutpe tasks are executed wthn a snge staton, as shown n Fg 6, we assume n ths artce that those tasks are executed n a sequenta order Therefore, an equvaent sera confguraton can be used for representng those tasks For exampe, f two tasks A and B are executed wthn staton k for producng dfferent components of products, the agebrac expresson of staton k s then represented as S k Sk A SB k,wheres k represents task A, B at staton k As shown n Fg 2, staton s used to produce both components and 2; thus, two dfferent tasks need to be executed at staton The detaed dscusson w be gven through an ustratve exampe n Secton 32 3 Agebrac representaton of system performance It shoud be noted that the prevousy defned agebrac operators and are many used to represent the staton ayout confguratons, whch use statons as ther operands and mpy no mathematca operatons In ths secton, we w dscuss how to assgn specfc mathematca operatons to the operators and used to evauate assemby system performances Specfcay, we w show how to use the proposed agebrac expressons to cacuate quaty conformng rates, whch w then be used to obtan the PCC foowng the procedures presented n Abad and Jn 20 To obtan the quaty conformng rates of the dfferent product types, we ntroduce the quaty performance operators, represented by addng the subscrpt nto the staton confguraton operators and ; e, and In Fg 3 Equvaent staton groupngs

5 234 Abad et a Tabe 2 Obtanng agebrac expresson of system confguraton Statons ncuded Reatonshp Operator used Agebrac expresson Step S 2 and S 3 Sera S 2,3 S 2 S 3 Step 2 S 4 Parae S 2,3,4 S 2,3 S 4 S 2 S 3 S 4 Step 3 S Sera S,2,3,4 S S 2,3,4 S [S 2 S 3 S 4 ] Step 4 S 5 Sera S,2,3,4 S S 2,3,4 S [S 2 S 3 S 4 ] S 5 contrast to staton confguraton operators and, quaty performance operators and use the conformng rates of the tasks assgned to the assocated statons as ther operands and convey mathematca operatons for computng quaty conformng rates of the correspondng combned tasks It shoud be noted that n a mxed-mode assemby process, the quaty conformng rate shoud be anayzed for each product type throughout a reated statons The detas of those performance metrcs w be gven n the foowng subsectons 3 Representaton of quaty metrc for a snge staton The nput mx rato at each staton s represented by vector π IN,k [π IN,k 0,,π IN,k N,πIN,k ε ] T,whereπ IN,k s the nput mx rato of part type at staton k The eement πε IN,k corresponds to the porton of nonconformng products produced at the mmedatey prevous statons; ths s consdered as the pseudo-nput of staton k for consstency of the mode representaton of the whoe manufacturng system Snce π IN,k s an nput vector of staton k contanng the percentage of every product type, t s constraned to satsfy π IN,k + πε IN,k Smary, the output of the produced part mx rato at staton k can be represented as π OUT,k [π OUT,k 0,,π OUT,k N,πε OUT,k ] T, wth π OUT,k correspondng to the output mx rato of part type produced at staton k The eement πε OUT,k corresponds to the porton of nonconformng products produced at staton k, whch s consdered as the pseudo-output of staton k for consstency of the mode representaton of the whoe manufacturng system The foowng reatonshp between the nput mx rato π IN,k and the output mx rato π OUT,k hods: π OUT,k { k } T π IN,k 2 The eement k can be consdered as a transfer functon to represent staton k s quaty performance as shown n Fg 7 Based on Abad and Jn 20, f a mxed-mode producton process s requred to produce N types of Fg 4 Fowchart for transferrng system bock dagram nto agebrac expresson

6 Agebrac expresson of MMAS confguratons 235 Fg 5 Agorthm for transferrng adjacency matrx nto agebrac expresson dfferent parts, the quaty transformaton matrx at staton k k,,m, where M s the tota number of statons can be represented by an N + N + square matrx k as ψ00 k 0 0 ψ k 0ε 0 ψ k 0 ψ k ε k, ψn,n k ψn,ε k where ψ k Prob{Producng a conformng product type at staton k} and ψε k Prob{Producng a nonconformng product type at staton k}; thus, ψ k ψε k Here, ψnn k stands for the fact that there s no rework or correcton performed on nonconformng parts enterng at staton k Aso, for consstency wth the matrx formuaton of the mode, f staton k has no producton operaton on part type, we w set ψ k andψε k 0, whch means no quaty oss at staton k for part type Snce k s a dagona matrx f the ast coumn and row are gnored, we ca staton k a dagona staton 32 Agebrac operators for quaty metrc of mutpe statons usng equvaent staton representaton The concept of the equvaent staton s defned for teratvey cacuatng the quaty transfer functon when products are manufactured by assemby subsystems wth mutpe statons, 2,, n ; ths s denoted by E, 2,, n By usng such an equvaent staton representaton, the overa quaty transfer functon, represented by the conformng matrx E, 2,, n,susedtodescrbe the output of the conformng rate after parts pass through mutpe statons, 2,, n Smary, the nput mx rato and the output mx rato for the equvaent staton can be denoted as π IN,E, 2,, n and π OUT,E, 2,, n, respectvey By extendng the quaty transfer functon from a snge staton to an equvaent staton ncudng mutpe statons, we can obtan the foowng reatonshp: π OUT,E, 2,, n { E, 2,, n } T π IN,E, 2,, n, 4 where E, 2,, n s defned n a smar way as matrx k ; e, the matrx of E, 2,, n sadagonamatrxftheast coumn and row are gnored The cacuaton of E, 2,, n w be conducted step by step by teratvey cacuatng the quaty transfer functon between two sub-grouped equvaent statons wth ether a sera or parae confguraton In the foowng dscussons, the correspondng agebrac operators w be defned for sera and parae confguratons, respectvey, and the propostons are used for the smpe case wth two statons For the genera case wth more than two statons, the agebrac operators are defned n terms of Fg 6 Agebrac representaton of mutpe tasks n a snge staton Fg 7 uaty transfer functon

7 236 Abad et a Tabe 3 Transfer of an adjacency matrx nto an agebrac expresson Adjacency matrx H Step Agorthm ne # S S 2 S 3 S 4 S 5 S S S S S Step 0 nta S S 2 S 3 S 4 S 5 S S S S S S S 2 S 3 S 4 S 5 S 0 0 S 2 S S S S S 2 S 3 S 4 S 5 S S 2 S 3 S 4 S S S 2 S 3 S 4 S 5 S 0 0 S 2 S S S S S 2 S 3 S 4 S 5 S S 2 S 3 S 4 S Step S 2,3 S 2 S 3 Lnes: 0 7 S 2 j S 3 Type: sera Operator: New row/coumn abe: S 2 S 3 Step 2 Lnes: 2 9 S 2,3,4 S 2,3 S 4 S 2,3 j S 4 k S Type: parae Operator: New row/coumn abe: S 2 S 3 S 4 Step 3 S,2,3,4 S S 2,3,4 Lnes: 0 7 S j S 2,3,4 Type: parae S S 2 [ S 3 ] S 4 S 5 S S 2 S 3 S 4 0 S Operator: New row/coumn abe: S S 2 S 3 S 4 S S 2 S 3 S 4 S 5 S S 2 S 3 S 4 [ ] 0 S S S 2 S 3 S 4 S 5 S S 2 S 3 S 4 S 5 [0] Step 4 fna S,2,3,4,5 S,2,3,4 S 5 Lnes: 0 7 S,2,3,4,5 j S 5 Type: sera Operator: New row/coumn abe: S S 2 S 3 S 4 S 5 coroares, whch are proved va the mathematca nducton method 32 Sera Confguraton Sera statons may be used to produce the same type of product, or dfferent types of products Proposton Let and j be two quaty conformng matrces correspondng to staton and staton j, respectvey The quaty conformng matrx, denoted as E, j, can be cacuated by { } E, j j j ψrs ψ sv j s, 5 where { } s the rth row and vth coumn eement of matrx E, j Justfcaton for Proposton Suppose that two statons, denoted by and j, are n a sera confguraton and staton

8 Agebrac expresson of MMAS confguratons 237 drecty precedes staton j Based onequaton 2 wehave and π OUT, { } T π IN, 6 π OUT, j { j } T π IN, j 7 The output mx rato of staton, denoted by π OUT,,s consdered as the nput mx rato of staton j, denoted by π IN, j Combnng Equaton 6 and Equaton 7 yeds: π OUT, j { j } T { } T π IN,, π OUT, j { j } T π IN, Hence, Equaton 5 n Proposton s justfed We can further extend Proposton to appy operator to a sera confguraton wth more than two statons, as gven by Coroary Coroary For a sera confguraton among statons, 2,, n, the quaty conformng matrx, denoted as E,2,,n, can be cacuated by the matrx mutpcaton: E,2,,n 2 n 2 n 8 The detaed justfcaton for Equaton 8 s performed va mathematca nducton, whch s gven n Appendx A 322 Parae Confguraton Parae statons may be constructed to perform the same task and act as redundant statons or to perform dfferent tasks that are used to produce dfferent types of products or dfferent parts used n the same product type Dependng on whether the performed tasks are the same, the quaty conformng matrx s cacuated accordng to Proposton 2 and Proposton 3, respectvey Proposton 2 Let and j be two quaty conformng matrces correspondng to parae connected staton and staton j If both statons and j are used to perform the same tasks wth producton rate of r and r j, respectvey, the quaty conformng matrx, denoted as E, j, can be cacuated by { } E, j j ω ψ + ω jψ j, 9 where { } s the rth row and vth coumn eement of matrx E, j ω and ω j are the probabty of parts passng through staton and staton j, respectvey, whch are usuay determned to satsfy the condtons of ω /ω j r /r j and ω + ω j Proposton 3 Let and j be two quaty conformng matrces correspondng to parae connected staton and staton j, respectvey If statons and j are used to perform dfferent tasks, the quaty conformng matrx, denoted as E, j, can be cacuated by E, j j { { mn ψ,ψ j }, If ν N + { max ψ,ψ j, 0, If ν N + s } where { } s the rth row and vth coumn eement of matrx E, j Justfcaton for Proposton 3 Two parae connected statons, denoted by and j, are used to perform dfferent tasks, whch can be further categorzed by whether the outputs of the parae statons are the fna products wth ndvdua types eg, varant statons n Fg or ntermedate parts that w be used together for the foowng staton eg, parae statons before staton 5 n Fg 2 For the case of producng ntermedate parts, t can be further cassfed by whether the outputs of the parae statons are used for the same type of products Fgure 8 shows these three possbe cases n the parae confguratons, whch are defned as foows Staton and staton j are used to produce type and type of fna products, respectvey, as shown n Fg 8a 2 Staton and staton j areusedtoproducetype and type of the ntermedate parts, respectvey, whch w be used together n the next staton as shown n Fg 8b Fg 8 System confguraton ayout assocated wth S S j

9 238 Abad et a 3 Staton and staton j are used to produce the same type of ntermedate parts, whch w be used together n the next staton as shown n Fg 8c The foowng dscusson w frst show that Proposton 3 s appcabe to a three cases that have two parae statons Afterwards, a genera concuson for n-parae statons s gven n Coroary 2, whch w be proved va mathematca nducton n Appendx B Case Two types of products are produced: suppose that the outputs of statons and j correspond to fna products wth type and type, respectvey The quaty conformng matrx at staton s ψ 0 ψ ε 0 ψ ψ 0 0 ε ψ 0 ψε The quaty conformng matrx at staton j s ψ j 0 ψ j 0 0 ε j 0 ψ j ψ j 0 ψ j ψ j 2 ε ε The nput mx rato s gven as π IN,E, j π IN, π IN, j [ π IN,,π IN,,πIN, ε [ π IN, j,π IN, j,πin, ε j ] T 3 Snce the proporton of conformng products of type comng out of equvaentstaton E,j s excusvey determned by the conformng matrx and nput proporton at staton,wehave OUT,E, j π π OUT, π IN, ψ π IN, mn ψ, π IN, mn ψ T,ψj π mn 0, 0 π mn0, 0 π mn IN,E, j IN,E, j IN,E, j ε ] T ψ,ψj 4 Based on the smar prncpe, the output of product type can be obtaned by Equaton 5: π OUT,E,j π OUT, j π IN, j mn,ψ j IN, j π T mn 0, 0 mn ψ,ψj mn0, 0 ψ j π IN, j mn ψ,ψj π IN,E,j π IN,E,j π IN,E,j ε 5 To cacuate the proporton of nonconformng products comng out of equvaent staton E, j, t s necessary to consder the nonconformng parts comng from statons and j and the transferred nonconformng parts from the nput generated from the prevous statons Based on Equaton 3, we have π OUT,E,j ε π OUT, ε π IN, π IN, + πε OUT, j + π ψε + π IN, j max ψ IN,E, j ε ψ j ε ε, 0 + π IN, j max 0,ψ j IN,E, j + π ε ε max ψ T ε,ψj ε max ψ ε,ψj ε max, + π IN,E, j ε π IN,E,j π IN,E,j π IN,E,j ε Combnng Equatons 4, 5, and 6, we have OUT, E,j π π OUT,E,j π OUT,E,j πε OUT,E,j mn ψ,ψj mn 0, 0 mn 0, 0 mn 0, 0 π IN,E,j π IN,E,j π IN,E,j ε mn ψ,ψj mn 0, 0 mn 0, 0 mn 0, 0 mn ψ,ψj mn 0, 0 mn ψ,ψj mn 0, 0 6 max ψε,ψj ε max ψ ε,ψj ε max, max ψε,ψj ε max ψ ε,ψj ε max, π IN,E,j E,j π IN,E,j 7 Hence, Equaton 0 n Proposton 3 s justfed under case Case 2 Two types of products are produced: suppose staton s used to produce product type and staton j s used to produce product type The quaty conformng matrces at staton and staton j are the same as and j gven by Equaton and Equaton 2, respectvey Based on smar consderatons as n case, π OUT,E,j π OUT,E,j OUT,E,j,andπε T T, can be derved n expressons smar to Equaton 4, Equaton 5, and Equaton 6, respectvey Hence, case 2 n Proposton 3 can be justfed n the same manner as for case Case 3 Both statons and j are used to produce dfferent components A and B that are combned n the foowng staton to produce product type Letπ OUT, and π OUT, j denote the proporton of component A and component B produced at staton and staton j, respectvey Snce both components are needed to produce a product, the

10 Agebrac expresson of MMAS confguratons 239 Fg 9 Exampe of parae statons and agebrac operators for quaty metrc effectve output of the conformng rate from the equvaent staton combnng statons and j s equa to the mnmum conformng rate of component A and component B produced at staton and staton j, respectvey; e, mnπ OUT,,π OUT, j For exampe, Fg 9 shows a smpe process for producng a tabe, whch conssts of three statons: staton produces the top of the tabe, staton 2 produces a set of four egs, and staton 3 assembes the top of the tabe and the four egs Suppose that the proporton of conformng rates produced at statons and 2 s ψ 95% and ψ 2 98%, respectvey In ths case, we can ony obtan a 95% conformng rate when enterng staton 3; e, the effectve output of combnng statons and 2 s equa to the combned conformng rate between the top and the set of four egs, thus yedng mnψ,ψ 2 95%, Based on Equaton 2, we see that Equaton 8 hods π OUT,E,j mn π OUT,,π OUT, j mn ψ,ψj IN,E,j π 8 Based on a smar argument, we can cacuate the proporton of nonconformng products of type produced at staton and staton j, denoted by πε OUT, and πε OUT, j, respectvey Snce every product of type must be processed by both staton and staton j, the resutant nonconformng rate of product type produced by these two statons s equa to the maxmum of the nonconformng rates of products of type at these two statons; e, maxπε OUT,,πε OUT, j Based on Equaton 2, t can be wrtten that: πε OUT,E,j max πε OUT,,πε OUT, j max ψε,ψj ε π IN,E,j ε 9 Based on Equatons 8 and 9, Equaton 0 n Proposton 3 s justfed It shoud be noted that Proposton 3 for cases 2 and 3 hods when the parae confguraton s foowed mmedatey by an assemby staton that adopts the check & assembe procedure; e, at such an assemby staton eg, staton 3 n Fg 9, the quaty of ntermedate parts to be assembed s checked before the assemby task s performed If a defectve component s found at staton 3, the staton wats for the next arrvng good component as ts repacement to compete the assemby operaton Such a check & assembe procedure removes the confct or defectve components before the assemby operaton starts, whch s sometmes essenta n order to reduce the hgh rsk of damagng expensve toos and fxtures and/or to avod the destructon of defect-free ntermedate parts In the contrary case, f the parae confguraton s mmedatey foowed by an assemby staton that s not equpped wth ths check & assembe procedure, a defect-free ntermedate part w be destroyed f t s about to be assembed wth a defectve component The quaty conformng matrx cacuaton of such a structure woud be the same as n Equaton 5 presented for the sera confguraton Wthout osng generaty of the agebrac representaton, n ths artce we consder the parae confguraton descrbed n cases 2 and 3 s foowed mmedatey by an assemby staton whch adopts the check & assembe procedure, and thus Proposton 3 hods We can further extend Proposton 3 to appy operator to a parae confguraton wth more than two statons, as gven by Coroary 2 Coroary 2 For a parae confguraton among statons, 2,, n, the quaty conformng matrx, denoted as E,2,,n, can be cacuated by E,2,,n 2 n { { mn ψ,ψ } 2,,ψn If ν N + { max ψ,ψ 2,,ψn If ν N +, } 20 where { } s the rth row and vth coumn eement of matrx E,2,,n The justfcaton of Equaton 20 s gven n Appendx B va a mathematca nducton proof Iustratve exampe: An assemby system, such as the one shownnfg2,susedtoproduceaproductconsstng of two components, each wth two dfferent varants, for a tota of four possbe product types four possbe combnatons of two components The ast coumn n Tabe 4 gves the nput mx rato of these four product types, π IN,E, 2, 3, 4, where E corresponds to an equvaent staton contanng every staton n the system For producton pannng, the nput mx rato of a products havng component wth varant and varant 2 can be cacuated by P π IN,E + π IN,E 2 and P 2 π IN,E 3 + π IN,E 4, respectvey Smary, the nput

11 240 Abad et a Tabe 4 Task assgnment, conformng rates, and nput mx rato 7 Conformng rates % Component Component Component 2 processed Varant Varant 2 Varant Varant 2 Staton and Staton Staton Staton Staton 5 and Product type Product type 2 Product type 3 Product type 4 Input mx rato n terms of component varant P 70% P 2 30% P 2 65% P 22 35% Input mx rato n terms of product type π IN,E 45% π IN,E 2 25% π IN,E 3 20% π IN,E 4 0% mx rato of a products havng component 2 wth varants and 2 can be cacuated by P 2 π IN,E + π IN,E 3 and P 22 π IN,E 2 + π IN,E 4, respectvey The resuts are gven n the ast row of Tabe 4 Tabe 4 aso descrbes the requred operatons at each staton and ther correspondng quaty conformng rates As shown n Fg 2, statons and 5 are basc operaton statons, whereas statons 2, 3, and 4 are varant operaton statons Snce staton and staton 5 have more than one task two tasks are need to produce two dfferent components, based on Fg 6, we can denote Sk as the ndvdua task for producng component at staton k In ths way, statons and 5 are represented n terms of ther assgned tasks as S S S2 and S 2 S5 S2 5, respectvey Furthermore, we can defne k, to represent the quaty conformng matrx of ndvdua task at staton k Based on Tabe 4, matrces,,,2, 5,,and 5,2 are represented as , , , , , , , The dervaton of, s gven n detaed steps n Appendx C The conformng matrx of statons and 5 can be obtaned by,,2 and 5 5, 5,2, respectvey For exampe, ψ ψ, ψ, Therefore, , For statons 2, 3, and 4 wth a snge task, the correspondng conformng matrces 2, 3,and 4 are drecty obtaned from Tabe 4 as ,

12 Agebrac expresson of MMAS confguratons The system s quaty transfer functon, based on the equvaent staton E,2,3,4,5 [ ] 5, can be cacuated by teratvey appyng the operators defned n Equaton 5 and Equaton 0 on two subgrouped statons as foows: E2,3 2 3 Equaton 5; E2,3,4 4 E2,3 Equaton 0; E,2,3,4 E2,3,4 Equaton 5; v E,2,3,4,5 E,2,3,4 5 Equaton 5 The fna resutant quaty conformng matrx s E,2,3,4, uaty and PCC for whoe assemby system Based on Equaton 4, the tota quaty conformng rate of a product types for the equvaent staton E, 2,, n, denoted as E, 2,, n, can be cacuated by E, 2,, n π IN j ψ E, 2,, n jj π OUT,E, 2,, n ε j<n 22 Therefore, the quaty conformng rate of the whoe manufacturng system can be cacuated by consderng a statons S, S 2,,S M,as E,2,,M The PCC, defned n Abad and Jn 20, s a performance metrc that assesses how we a producton process can hande the demand varety of products n a mxed mode manufacturng process PCC s cacuated based on the mutua nformaton ndex Cover and Thomas, 2006, whch s used to quantfy the amount of nformaton that two random varabes share Assume that the nput and output mx ratos are consdered as the margna probabty dstrbuton functons π IN,E, 2,, n and π OUT,E, 2,, n of two categorca random varabes, respectvey If the jont probabty matrx s denoted by π IN,OUT,E, 2,, n, where the eement n the th row and jth coumn corresponds to π IN,OUT,E, 2,, n j ψ E, 2,, n j π IN,E, 2,, n Thus, PCC can be cacuated by the mutua nformaton ndex as foows PCC π IN,OUT,E, 2,, n j, j, og π IN,OUT,E, 2,, n j, 23 π IN,E, 2,, n j π OUT,E, 2,, n A normazed vaue of PCC, rangng from zero to one, caed the NPCC, was aso proposed n Abad and Jn 20 based on the concept of coeffcent of constrant Coombs et a, 970; that s, NPCC PCC/H D, 24 where H D s the entropy of the nput demand random varabe D,gvenby H D π IN,E, 2,, n og π IN,E, 2,, n 25 Iustratve exampe: We now contnue the exampe n Secton 32 to show how to cacuate the system performance metrcs E, PCC, and NPCC for the assemby system shown n Fg 2 Based on Tabe 4, the vector of nput mx rato s represented as π IN,E,2,3,4,5 [ ] T 26 By substtutng Equaton 2 and Equaton 26 nto Equaton 22, E,2,3,4,5 s cacuated as 0666 Based on the equaton π IN,OUT,E, 2,, n ψ E, 2,, n j π IN,E, 2,, n π IN,OUT,E,2,3,4,5 j, π IN,OUT,E,2,3,4,5 s obtaned as By substtutng Equatons 2, 26, and 27 nto Equaton 23, PCC s obtaned as 2076 Furthermore, we obtan the resuts of H D 850 based on Equaton 25 and NPCC based on Equaton 24 Because NPCC s far beow one, t ndcates that ths assemby system has a far capabty of handng the mxed product varetes, but t needs to be further mproved The foowng secton w dscuss how to adjust the ndvdua staton requrements to mprove system performances by defnng nverse agebrac operators 4 Inverse agebrac operators for mprovng system performance Ths secton s used to show how to defne the nverse agebrac expressons to systematcay anayze the effect of ndvdua statons on the defned performance metrc

13 242 Abad et a of an equvaent system staton Such resuts can be used to further mprove the desgn of a manufacturng system to acheve a desred system performance or to dentfy the weakest staton n an assemby system under a partcuar performance crteron, such as quaty, throughput, etc Based on the prevousy defned agebrac operators and, the nverse operators and w be defned as foows 4 Agebrac operators and for nverse computaton of quaty conformng rate The foowng proposton w be used to descrbe the operatons correspondng to and Proposton 4 Let and j be two quaty conformng matrces correspondng to staton and staton j, respectvey, and et E, j be the equvaent staton formed by statons and j The reatonshps between each quaty operator and ts correspondng nverse operator are defned as foows For a sera confguraton, E, j j, t yeds: E, j j E, j { j } 28 and j E, j { } E, j, 29 where the arrow on the top of operator n Equaton 28 ndcates that the operand j on the rght-hand sde of the operator shoud be computed by the nverse operator Smar nterpretaton s gven to n Equaton 29 2 For a parae confguraton, E, j j, t yeds: E, j j { } E, j ψ { } ψ E, j ψ { } E, j ψ { } ψ E, j ψ unfeasbe E, j f ψ <ψ j AND v N + E, j f ψ ψ j AND v N + E, j If ψ >ψ j AND v N + E, j f ψ ψ j AND v N + otherwse 30 Justfcaton for Proposton 4 Snce operator corresponds to the conventona matrx mutpcaton operaton, shoud naturay correspond to the nverse matrx operaton Agebrac operators and are not commutatve, whch s shown n Equatons 28 and 29 For operator E,j, snce ψ ψ ψ j mnψ,ψj for v N + andψ E,j ψ ψ j maxψ,ψj for v N + ; thus Equaton 30 s justfed Iustratve exampe: We contnue the exampe of the assemby system shown n Fg 2, n whch the agebrac expresson representaton of the equvaent staton s S E,2,3,4,5 S [S 2 S 3 S 4 ] S 5 Suppose that the pant s not satsfed wth the current quaty conformng performance as gven n Equaton 2 and woud ke to ncrease the quaty conformng rate of product type, snce product type has the argest demand For exampe, the decson s made to ncrease the quaty rate from ψ E,2,3,4, to ψ E,2,3,4,5 07 Therefore, the new quaty conformng rate s represented by E,2,3,4,5,NEW as E,2,3,4,5, NEW It s assumed that t s feasbe to mprove ony staton 3 to meet the new system requrement of E,2,3,4,5,NEW The queston that remans s what must be the new quaty requrement at staton 3 n order to acheve E,2,3,4,5,NEW The step-by-step nverse operatons are ustrated as foows: S E,2,3,4,5 S [S 2 S 3 S 4 ] S 5, S S E,2,3,4,5 [S 2 S 3 S 4 ] S 5, S S E,2,3,4,5 S 5 S 2 S 3 S 4, v [S S E,2,3,4,5 S 5 ] S 4 S 2 S 3, v S 2 [[S S E,2,3,4,5 S 5 ] S 4 ] S 3 3 Now, repacng S for and usng the correspondng agebrac operators yeds: 3,NEW 2 [ E,2,3,4,5,NEW 5 4] 32 After substtutng operators and nto Equatons 28, 29, and 30, 3,NEW can be obtaned as 3,NEW By comparng 3,NEW wth 3, we can see that n order to acheve the new quaty conformng rate of ψ E,2,3,4,5 07, the quaty conformng rate of product type at staton 3 shoud be mproved from ψ 3 3,NEW 096 to ψ 09956

14 Agebrac expresson of MMAS confguratons 243 Tabe 5 Conformng rate for ndvdua statons Conformng rates % Product type Product Product Product Staton processed type type 2 type 3 Fg 0 System bock dagram for producng three product types 5 Case study In ths secton, we consder a compex assemby system consstng of 20 statons for producng three product types Ko and Hu, 2008 The bock dagram of the system confguraton s shown n Fg 0 In arbtrary order, statons are numbered from S to S 20 M 20 In order to effcenty obtan the conformng rate of each product type and ther correspondng producton cyce tme, we further represent the assemby system by two sets: the assemby sub-system consstng of the basc operaton statons denoted by S B and the assemby sub-system consstng of the varant operaton statons denoted by S V In ths exampe, assemby sub-system S B contans statons S, S 2, S 3, S 4, S 5, S 6, S 7, S 8, S 9, S 0,andS, whe assemby sub-system S V contans statons S 2, S 3, S 4, S 5, S 6, S 7, S 8, S 9,andS 20 Fgure 0 shows the S B confguraton, whe dfferent S V confguratons for each product type are shown n Fg By means of the proposed agebrac representaton of the system confguraton, we can further represent equvaent sub-systems S B and S V as S B S S 2 S 3 [S 4 S 5 [S 7 S 6 S 8 S 9 S 0 S ]], 33 S V [S 2 S 4 S 3 S 5 ] [S 6 S 7 ] [S 8 S 9 S 20 ] 34 Staton, 2, and Staton 2, 2, and Staton 3, 2, and Staton 4, 2, and Staton 5, 2, and Staton 6, 2, and Staton 7, 2, and Staton 8, 2, and Staton 9, 2, and Staton 0, 2, and Staton Staton 2 and Staton 3 98 Staton 4 and Staton Staton 6 and Staton 7 99 Staton Staton Staton The equvaent staton of the whoe assemby system, denoted by S B,V,sgvenby S B,V S B S V The producton throughput s desgned to be fve unts per mnute Specfcay, two unts per mnute are aocated for product type and product type 3 and one unt per mnute for product type 2 Thus, the nput mx rato s π IN,E [ π IN,E 040 π IN,E π IN,E πε IN,E 0 ] T Based on Fg, Tabe 5 provdes the task assgnments to each staton for each product type and ther correspondng quaty conformng rates and producton cyce tmes It can be seen n Tabe 5 that staton s regarded as a redundant staton and no producton task s assgned to t Aso, Fg Bock dagram for ndvdua product types

15 244 Abad et a Fg 2 Layer-by-ayer system decomposton we assume that the staton s cyce tmes are ndependent of the product type beng processed 5 Anayss of quaty for whoe assemby system We frst compute the equvaent matrx EB correspondng to the basc operatons subsystem S B by repacng and n Equaton 33 wth and, respectvey, thus yedng EB Smary, based on Equaton 34, EV correspondng to the varant operatons subsystem S V s cacuated as EV Fnay, based on S B,V S B S V, we can cacuate EB,V EB EV, correspondng to the whoe assemby system, as EB,V Therefore, based on the proposed methodoogy, the fna conformng rates of three product types can be obtaned as ψ EB,V 8679%, ψ EB,V %, and ψ EB,V % 52 Identfcaton of weak statons va ayer-by-ayer tree decomposton We now ustrate the use of the expresson tree representaton of the assemby system S B,V to determne weak subassemby systems wth a ow process quaty Consder Fg 2a, where the system S B,V has been expressed at ts hghest eve of groupng as S B,V S B S V Sncewe are nterested n determnng sub-systems that have a ow performance, we further decompose, ayer by ayer, the equvaent staton S B, whch s the equvaent staton wth the owest process quaty, gven by As shown n Fg 2b, the owest process quaty wthn S B corresponds to S 4 Contnung wth a step-by-step decomposton, we further expand equvaent staton S 4 nto S 4,5, S 6,7,8,andS 9,0,, as shown n Fg 2c Based on ths decomposton, we have that sub-system S 6,7,8 has the owest performance at ths eve, gven by 09564, and thus further attenton shoud be gven to ths subsystem to mprove the overa process quaty of the system One shoud note that we have foowed a greedy search approach and hence we cannot cam that sub-system S 6,7,8 s a goba mnmum; nonetheess, the greedy heurstc search yeds a ocay optma souton that may cosey approxmate a goba optmum under a reasonabe short computaton tme If suffcent computatona source s avaabe, one may aso foow dynamc programmng or other optmzaton agorthms to fnd the guaranteed goba optmum or better approxmaton souton Cormen et a, Concusons and future work Ths artce ntroduces an effcent way to represent compex mxed-mode assemby system confguratons usng agebrac expressons It s appcabe to a genera hybrd asymmetrc confguraton system consstng of mxed sera and parae statons Moreover, the artce presents a systematc method to transfer the agebrac expresson nto computatona agebrac operators, whch are assgned wth specfc mathematca operatons to cacuate the system performance metrcs, such as quaty conformng rate and process capabty n handng the demand varety

16 Agebrac expresson of MMAS confguratons 245 Furthermore, the correspondng nverse agebrac operators are aso defned, whch provdes a systematc way to gude the adjustment of ndvdua statons for mprovng system performances The proposed method s restrcted to the sera and parae structure systems For other compex structure systems, such as a brdge structure system, one possbe approach s to frst decompose or transform the system nto sera and parae structures by addng pseudo-redundant statons n the system dagrams, whch s consdered as future work to be further studed The research presented n ths artce can be further extended by defnng agebrac operators for obtanng other system performance metrcs such as process cyce tmes, operatona states of a system, reabty, etc Furthermore, the agebrac expressons may be mathematcay formazed nto nterestng agebrac feds and thus provde a way to sove probems such as fndng botteneck statons and achevng ne baancng for a compcated producton system Acknowedgement The authors woud ke to gratefuy acknowedge the fnanca support of NSF-CMMI: and Genera Motors References Abad, AG and Jn, J 20 Compexty metrcs for mxed mode manufacturng systems based on nformaton theory Internatona Journa of Informaton and Decson Scences, 34, Carre, BA 97 An agebra for network routng probems IMA Journa of Apped Mathematcs, 7, Cofer, DD and Garg, VK 993 Generazed max-agebra mode for performance anayss of tmed and untmed dscrete event systems, n Amercan Contro Conference, IEEE, pp Cohen, G, Dubos, D, uadrat, JP and Vot, M 985 Lnear-systemtheoretc vew of dscrete-event processes and ts use for performance evauaton n manufacturng IEEE Transactons on Automatc Contro, 30, Cohen, G, Moer, P, uadrat, JP and Vot, M 989 Agebrac toos for the performance evauaton of dscrete event systems IEEE Proceedngs, pp Coombs, CH, Dawes, RM and Tversky, A 970 Mathematca Psychoogy: An Eementary Introducton, Prentce-Ha, Engewood Cffs, NJ Cormen, TH, Leserson, CE, Rvest, RL and Sten, C 200 Introducton to Agorthms, MIT Press, Cambrdge, MA Cover, TM and Thomas, JA 2006 Eements of Informaton Theory, Wey, New York, NY Da Svera, G, Borensten, D and Fogatto, FS 200 Mass customzaton: terature revew and research drectons Internatona Journa of Producton Economcs, 72, 3 Duncan, FG, Zssos, D and Was, M 975 A postfx notaton for ogc crcuts The Computer Journa, 8, Frehet, T, Shptan, M and Hu, SJ 2004 Productvty of paced parae-sera manufacturng nes wth and wthout crossover Journa of Manufacturng Scence and Engneerng, 26, Gaubert, S 990 An agebrac method for optmzng resources n tmed event graphs, n Proceedngs of the Nnth Internatona Conference on Anayss and Optmzaton of Systems, Sprnger, Bern, pp Gondran, M and Mnoux, M 983 Graphs and Agorthms, Wey, New York, NY Ko, J and Hu, J 2008 Baancng of manufacturng systems wth compex confguratons for deayed product dfferentaton Internatona Journa of Producton Research, 46, Sturgeon, TJ 2002 Moduar producton networks: a new Amercan mode of ndustra organzaton Industra and Corporate Change,, Webbnk, RF and Hu, SJ 2005 Automated generaton of assemby system-desgn soutons IEEE Transactons on Automaton Scence and Engneerng, 2, Xu, X and Xu, P 988 Workng on producton pannng and contro n fexbe manufacturng system wth path-agebra, n Proceedngs of the 988 IEEE Internatona Conference on Systems, Man, and Cybernetcs, IEEE Press, Pscataway, NJ, pp Yurdaku, M and Odrey, NG 2004 Deveopment of a new dod agebrac mode for manufacturng wth the schedung decson makng capabty Robotcs and Autonomous Systems, 49, Zhu, X, Hu, SJ, Koren, Y and Marn, SP 2008 Modeng of manufacturng compexty n mxed-mode assemby nes Journa of Manufacturng Scence and Engneerng, 305, Zssos, D and Duncan, FG 97 NOR and NAND operators n Booean agebra apped to swtchng crcut desgn The Computer Journa, 4, Appendces Appendx A: Justfcaton for Coroary Suppose statons, 2,, n are n a sera confguraton and that staton drecty precedes staton +,, 2,,n BasedonEquaton2wehave and π OUT, { } T π IN, π OUT,+ { j } T π IN,+ A A2 The output mx rato of staton, denoted by π OUT,,s consdered as the nput mx rato of staton +, denoted by π IN,+ Combnng Equaton A and Equaton A2, we have π OUT,n { n } T { n } T π IN,n { n } T { n } T { n 2 } T π IN,n 2 { n } T { n } T { } T π IN, { 2 n } T π IN, Hence, Equaton 8 n Coroary s justfed We have E,2,,n 2 n 2 n Appendx B: Justfcaton for Coroary 2 Suppose statons, 2,,n are n a parae confguraton Smar to the three cases n Proposton 3, there aso exst three parae confguraton cases for mutpe statons

17 246 Abad et a Fg A Three parae confguratons between statons, 2,,n Statons, 2,,n are used to produce dfferent types of fna products, respectvey, as shown n Fg Aa 2 Statons, 2,,n are used to produce dfferent types of ntermedate parts, respectvey, whch w be used together n the next staton as shown n Fg Ab 3 Statons, 2,,n are used to produce the same type of ntermedate parts, whch w be used together n the next staton as shown n Fg Ac In cases and 2, dfferent product types, denoted by, 2,, n, foow dfferent routes of the parae confguraton In the foowng dscusson we eaborate the justfcaton for Coroary 2 under cases and 2 usng mathematca nducton The justfcaton for case 3 n Coroary 2 s very smar to the justfcaton for case 3 n Proposton 3, and thus w not be dscussed further We have proved Equaton 20 for a two-staton scenaro through Equatons to 7 Assume that Equaton 20 hods when k statons are n a parae confguraton and dfferent product types foow dfferent routes Let, 2,, k denote the product types that have been produced through statons, 2,,k Based on Equaton 20 wth k statons, we have E,2,,k 2 k { { mnψ,ψ 2,,ψk }, If ν N + { maxψ,ψ 2,,ψk }, If ν N + ψ E,2,,k ψ E,2,,k ε 0 ψ E,2,,k 2 2 ψ E,2,,k 2 ε 0 ψ E,2,,k k k ψ E,2,,k k ε 0 A3 Fg A2 Equvaent staton E, 2,, k Statons, 2,,k can be represented by equvaent staton E, 2,,k as Fg A2 shows Now we have staton k + performng tasks for product type k+ Statons, 2,,k, k + are n a parae confguraton Based on Equaton A3 and Fg A2, ths parae confguraton can be represented as a parae confguraton between staton E, 2,,k and staton k + Snce now we have one more product type k+ to produce, the quaty conformng matrx E,2,,k becomes E,2,,k ψ E,2,,k 0 0 ψ E,2,,k ψ E,2,,k k k k ψ k+ k+ k+ ψ k+ k+ ε The nput mx rato s gven as 0 ψ E,2,,k ε ψ E,2,,k 2 ε ψ E,2,,k k ε, A4 A5 π IN,E,2,,k,k+ π IN,E,2,,k π [ IN,k+ ] T π IN,E,2,,k,,π IN,E,2,,k k+,πε IN,E,2,,k [ π IN,k+,,π IN,k+ k+,πε IN,k+ ] T A6 Snce the proporton of conformng products of type, 2,, k comng out of equvaent staton E, 2,, k, k + s excusvey determned by the conformng matrx and nput proporton at staton E, 2,,k, for product type, 2,,k, we have π OUT,E,2,,k,k+ π IN,E,2,,k π IN,E,2,,k π IN,E,2,,k π OUT,E,2,,k ψ E,2,,k mn mn ψ E,2,,k, ψ E,2,,k,ψ k+