An Optimal Switching Model Considered the Risks of Production, Quality and Due Data for Limited-Cycle with Multiple Periods

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1 A Optmal Swtchg Model Cosdered the Rss of Producto, Qualty ad Due Data for Lmted-Cycle wth Multple Perods Jg Su 1 1 Departmet of Cvl Egeerg ad Systems Maagemet, Nagoya Isttute of Techology, Goso-cho, Showa-u, Nagoya, , Japa Emal: su.jg@tech.ac.jp Hsash Yamamoto 2 2 Departmet of System Desg, Maagemet Systems Egeerg, Toyo Metropolta Uversty, 6-6 Asahgaoa, Ho Toyo, , Japa Emal: yamamoto@tmu.ac.jp Masayu Matsu 3 3 Faculty of Egeerg, Kaagawa Uversty, Yoohama, Japa Emal: matsu@aagawa-u.ac.jp Abstract. - Ths paper ams to derve a optmal swtch model cosdered the rss of producto, due date ad qualty for lmted-cycle wth multple perods. I the gobble supply cha evromet, optmal operato maagemet for horzotal tegrato of producto etwor has bee pad to atteto recetly. Due to the customer eeds of reducg cost ad delvery date shortg, prompt chage the producto pla became more mportat. I the mult perod system (For stace, producto le.) where target processg tme exsts, producto, dle ad delay rss occur repeatedly for multple perods. I such stuatos, delay of oe process may fluece the delvery date of a etre process. I ths paper, we dscuss mmum expected cost cludg producto, due date ad qualty a producto process, where the rs depeds o the prevous stuato ad occurs repeatedly throughout multple perods. Also, the polcy of optmal swtchg for parallel producto system wll be aalyzed. Keywords: operatoal research, supply cha maagemet, producto maagemet. 1. INTRODUCTION Ths paper ams to derve a optmal swtch model cosdered producto, due date ad qualty for producto system. I the gobble maretg evromet, the horzotal tegrato of cooperato by formato systems of eterprses or factores that are related to mae a product s a mportat problem Supply Cha Maagemet. O the other had, varable producto etwors s ot oly oe of the most mportat ssues maufacture maagemet ad operato research but also a sgfcat factor affectg supply cha (SC). The supply collaborate cocept requres formato ad optmzato of producto or supply etwors, characterzed by tesve commucato betwee the dstrbuted ettes. The goal s to allocate amog the collaboratg parters the producto demad. Ths capablty provdes the etre etwor wth the requred flexblty to respod qucly to demad dsrupto for the products ad servces (Y. Nof et al., 2015). Ths paper aalyzes a optmal swtchg model for parallel producto system wth multple perods SCS. O the other had, varable producto etwors s ot oly oe of the most mportat ssues maufacture ma agemet ad operato research but also a sgfcat factor affectg supply cha (SC). The supply collaborate coce pt requres formato ad optmzato of producto or sup ply etwors, characterzed by tesve commucato be twee the dstrbuted ettes. The goal s to allocate amog the collaboratg parters the producto demad. Ths ca pablty provdes the etre etwor wth the requred flexb lty to respod qucly to demad dsrupto for the products ad servces (Y. Nof et al.,

2 2015). Ths paper aalyzes a optmal swtchg model for parallel producto system wth multple perods SCS. I ay socal system or producto process wth mult perods ad predetermed target tme, system dleess ad process delay rss exsts throughout the multple perods. I such stuatos, delay of oe process sometmes affect the delvery date of the etre process. Ths d of problem s called a lmted-cycle problem wth multple perods, ad s see producto les, tme-bucet balacg, producto seat systems ad so o (Yamamoto H. et al., 2006; Matsu M., 2008) I ths paper, we dscuss mmum expected rs (cost) a parallel producto process, where the rs depeds o the prevous stuato ad occurs repeatedly throughout multple perods smart supply cha evromet. As lmted-cycle problems of producto le, Verzjl (1976) aalyzed the elemet ad costructo of the producto system. Es (2001) preseted a framewor for the aalyss of delays wth the producto system. Beders (2002) gave a revew for the org ad soluto of perod batch cotrol system. Xa ad Wu (2005) preseted a easly mplemeted hybrd algorthm for the multobjectve flexble job-shop problem. Recetly, Wu ad Zhou (2008) cocered wth the problem schedulg a set of jobs assocated wth radom leadtme o a sgle mache so as to mmze the expected maxmum lateess stochastc evromet. As lmted-cycle problems of operato maagemet, Safae ad Tavaol-Moghaddam (2009) proposed a tegrated mathematcal model of the mult-perod cell formato ad producto plag a dyamc cellular maufacturg system to mmze costs through a mxed teger programmg techque. Pora et al. (2003) proposed a mxed teger lear programmg based capactated lot szg models that cluded carryovers corporatg set-up tmes wth assocated costs. Moreo et al. El Hafs ad Ba (1998) employed a optmal multperod producto pla for a sgle product over a fte plag horzo to mmze the total vetory ad baclog costs by solvg a olear programmg problem. L et al. (2010) descrbed a optmal soluto structure by the dyamc programmg approach for a jot maufacturg ad remaufacturg system a multperod horzo. Uder ucerta codtos, the result ad effcecy of a certa producto cycle perod ad a certa process are flueced ot oly by the rss that exst the curret perod but also by the rss that exsted the foregog perods. Therefore, we dscuss the mmum expected rs of the case metoed above, whch the rs depeds o the foregog stuato ad occurs repeatedly for multple perods. Whether the process (perod or ste) satsfes the tme lmt (restrcto) usually depeds o the state of the foregog process, as see Verzjl (1976), Beders (2002), ad Wrght (1974). I ths paper, frst, the optmal swtchg problem s systematcally classfed. Next, the mathematc formulato of the total expectato cosdered producto, due date ad qualty for the producto system s proposed. Fally, the optmal swtchg pot s vestgated by umercal expermets. 2. OPTIMAL SWITCHING PROBLEM FOR PRODUCTION SYSTEM CONSITERED PRODUCTION, DUE DATE AND QUALITY Ths paper cosders cases whch the above two rss ot oly occur the sgle perod, but also multple perods repeatedly. The problem of mmzg the expected rs such a stuato s a lmted-cycle problem wth multple perods. The mult perod problem could be classfed accordg to whether the perods are depedet or ot. For ths problem, oe result s the geeral form of producto rate ad watg tme by a stato-cetered approach as dscussed Matsu el al. (1997) (2008). The explct form s obvous ad cossts of the product form the perod-depedet case, such as a sgle le, but t s utraceable the perod-depedet case such as a mxed or tadem le. The mxed le has a absorbg barrer, but the tadem le has a reflectve barrer at the ed. Ths paper presets a cost approach for the latter. The purpose of ths paper s dscussg the swtchg problem for parallel producto system show Fgure 2.

3 Fgure 1: Optmal Swtchg Problem for Producto ad Due date Rss of Multple Perods Fgure 2: Optmal Swtchg Problem for Qualty problem of Multple Perods 3. THE OPTIMAL SWITCHING MODEL 3.1 The assumpto ad otato The Optmal swtchg model for the parallel producto system wth multple perods s cosdered based o the followg assumptos: Oe product s made by a process wth processes. For =1, 2,, ; j=1, 2,, m, the producto tme of process of le j s deoted by T j whch s assumed to be statstcally depedet, respectvely. The usual processg rate s μ j1, ad the emergecy processg rate s μ j2. (3) For =1, 2,,, the target producto tme of process s deoted by T, ad the due tme of the etre process ( perods) s T. (4) s swtchg pot. (h) (5) The cost per ut tme ( ) occurs whe a process s executed before the target producto tme of the process. (h=1 meas before swtchg ad h=2 meas after swtchg) (6) The cost per ut tme ( ) occurs whe a process s executed after the target producto tme of the process (h=1 meas before swtchg ad h=2 meas after swtchg). (h) C p (7) Whe X T l > due tme ( l1 delay cost C occurs. p U ) of perods, the (8) Whe X T l < due tme ( U ) of perods, the l1 dle cost C occurs. Some otatos are also defed. s For =1, 2,,, C T, T,..., T ) : the total cost of the producto process. ( 1 2

4 C() : the producto cost of perod. : the producto tme of perod. : the producto tme of perods ( Pr X U : the probablty of delay. Pr( X U ) : the probablty of dle. T X X T l l1 β: Probablty of qualty problem occurrg producto process ( Pr( T I) ) 3.2 The assumpto ad otato The objectve of proposed model s show as followg: Z j C ( ; T, T,... T ) E Z j m( E C2 ( ;,... T ) ) C3 ( ;,... T ) s the total expected cost of le j, C ( ;,... T 1 2 ) : The producto cost, C ( ;,... T ) : The due date pealty cost, C ( ;,... T ) : The qualty pealty cost. 3 () Producto Cost From assumptos -(7) metoed Secto3.1, we ca easly see, C 1 ( ;,..., T )] C( )] Where, C() E s the expected cost of perod. I ths research, the producto tme T s assumed to be expoetal dstrbuted ad statstcally depedet, respectvely. The, for, 1,2,..., C( )] ( C p Cs ) S Cs T ] ad for, ( C p Cs ) S ] Cs T ] Cs ( C p Cs ) S ] 1 1, 2,, ). (3) C( )] ( C p Cs ) S T U ] Pr{ T U } ( C p Cs ) S T U ] Pr{ T U } Cs Cs Pr{ T U } Pr{ T U } 1 2 () Due Date Pealty Cost C2 ( ;,..., T )] Cp Pr Cs Pr where, Cp X U X U 1 ( Pr{ T U} l0 (4) X U C PrX U s (5) Pr s the delayed expected cost, s the dle expected cost. l 1U ) 1U e l! (6) 1 l ( U ) 1U Pr{ T U} 1 Pr{ T U} 1 1 e l0 l! l ( U U 1 ) 1 e l l! () Qualty Pealty Cost For, 1,2,..., C3 ( ;,..., T )] C01( max ( U I, 0)]) Pr{ T U} (8) ad for, 1, 2,, C3 ( ;,..., T )] C02 ( max ( T I, 0)]) Pr{ T U} where, C ad 01 C are the qualty cost per ut tme. 02 (7) (9) I ths research, the -cotrol tme of a producto process (I) s assumed to be expoetal dstrbuted whch the mea value s 1/λ. 4. EXPERIMENTAL CONSIDERATION

5 I ths secto, we cosder the optmal swtchg tme to mmze the total expected cost by umercal expermets, where,,,,,, C 01 C02 100,,,, C p T m 2 ad C p Table 1 Producto pealty cost by chage of the usual processg rate μ1=0.2 μ1=0.3 μ1=0.4 μ1=0.5 = = = = = = = = = = Table 2 Qualty pealty cost by chage of the usual processg rate μ1=0.2 μ1=0.3 μ1=0.4 μ1=0.5 C p tmes are 4T, 2T, 2T ad 1T, respectvely. From Fgure 3, t also ca be oted that whe usual processg rates of le 1 s costat, the optmal swtchg tme parameter decreases wth the crease of the usual processg speed of le 2. Ths s because the qualty pealty cost s large ths case, so the the behavor of total expected cost follows the behavor of qualty pealty cost. Table 3 Due date pealty cost by chage of the usual processg rate μ1=0.2 μ1=0.3 μ1=0.4 μ1=0.5 = = = = = = = = = = (Total expected cost) = = = = = = = = = = Fgure 3 show the behavor of the optmal swtchg tme by chage of the usual processg rate whe emergecy processg rates of system s 0.6. From Fgure 3, t ca be oted that whe usual processg rates of le 1 ad le 2 are 0.2, 0.3, 0.4 ad 0.5, the optmal swtchg Fgure 3: Behavor of the optmal swtchg tme by chage of the usual processg rate 5. CONCLUSIONS I ths paper, we cosdered the optmal swtchg pot (tme) to mmze the total expected cost cosdered

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