A Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time

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1 Joural of Na Ka, Vol. 0, No., pp.5-9 (20) 5 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te Suh-Jeq Yag, Ja-Yuar Guo, Hs-Tao Lee Departet of Idustral Maageet, Na Ka Uversty of Techology 通 訊 作 者 : 楊 肅 正 聯 絡 地 址 : 南 投 縣 草 屯 鎮 中 正 路 568 號 電 子 郵 件 :syag@kut.edu.tw 投 稿 日 期 :202 年 月 接 受 日 期 :20 年 5 月 Abstract Producto schedulg wth ateace plag to prove the producto effcecy of aches has receved cosderable atteto over the years. I two recet papers, Wag, Wag, ad Lu (20) ad Wag, Huag, J, ad Feg (202) studed the schedulg proble whch the obs ust be scheduled o parallel urelated aches wth deteroratg ateace actvtes to ze the total copleto te. Wag et al. (20) showed that the proble 2 could be solved O ( + ) te ad Wag et al. (202) showed that the proble could be solved O ( + ) f 0<, where α > 0 was the odfyg rate of ob, f t was α assged to ache after the ateace actvty was perfored. I ths ote, we proposed a ore effcet algorth ad showed that both probles studed by Wag et al. (20) ad Wag et al. (202) could be solved O ( + ) te eve though for the case of α >. Keywords: schedulg, ateace actvty, urelated parallel-ache, total copleto te. Itroducto Mateace s portat producto as t helps susta ache effcecy or product qualty. Schedulg wth ateace actvty has bee wdely studed the past decade. For research results o schedulg odels wth ateace actvtes ad dfferet ache evroets, a reader ca refer to Lee ad Leo (200), Wu ad Lee (200), J, He, ad Cheg (2006), Lee ad Wu (2008), Kuo ad Yag (2008), Low, Hsu, ad Su (2008), Hsu, Low, ad Su (200), Mosheov ad Sdey (200), Zhao, Tag, ad Cheg (2009), Lodree ad Geger (200), Yag ad Yag (200a; 200b; 200c), Zhao ad Tag (200), ad Yag (20; 202). I a recet paper, Wag, Wag, ad Lu (20) studed schedulg probles wth a deteroratg ateace actvty o a detcal parallel-ache settg. They assued that at ost oe ateace actvty s allowed o each ache durg the plag horzo to chage ts producto rate. They further assued that the ateace actvty ca be perfored edately after copletg the processg of ay ob, ad the

2 6 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te durato of the ateace actvty o a ache s depedet o ts rug te (Kubz & Strusevch, 2006). If ob J s processed before the ateace actvty, ts actual processg te s p (.e., ts oral processg te), ad f ob J s processed after the ateace actvty, ts actual processg te s α p, where α > 0 s the odfyg rate of ob J. The obectve was to ze the total copleto te. They frst showed that the two detcal parallel-ache schedulg 6 proble ca be solved O ( ) te. They the exteded the proble to the case wth urelated parallel-ache ad show 2 that t ca be solved O ( + ) te. Later, Wag, Huag, J, ad Feg (202) vestgated the sae proble proposed by Wag et al. (20) o a urelated-parallel ache settg. They showed that the total copleto te zato proble ca be solved O ( + ) f 0< α, where α > 0 s the odfyg rate of ob J, f t s assged to ache M after the ateace actvty s perfored. I ths ote, we exted the proble studed by Wag et al. (20) ad Wag et al. (202) ad show that t ca be solved O ( + ) te eve though for the case of α >. 2. Methods We follow the otato ad terology used by Wag et al. (202) throughout the paper ad wll troduce addtoal otato whe eeded. The proble uder study s descrbed as follows: There are depedet obs J { J, J2,..., J} processed o parallel urelated aches M { M M M } = to be =, 2,...,. Preepto of obs s ot allowed ad each ache ca process oly oe ob at a te ad caot stad dle utl the last ob assged to t has bee fshed. We assue that at ost oe ateace actvty s allowed o each ache durg the plag horzo to chage ts producto rate. We deote that the ateace actvty of ache M s posto l (0 l, =,2,..., ) f t s scheduled betwee the copleto te of the ob that s scheduled the posto ( l + )th to the last ob ad the startg te of the ob that s scheduled the posto l th to the last ob. Let p be the oral processg te of ob J f t s assged to ache M. The the actual processg te of ob J f t s scheduled the posto kth to the last ob o ache M s gve by: p k p, k > l, α p, k l. () for =,2,...,, =,2,...,, ad 0 k, where α > 0 s the odfyg rate of ob J, f t s assged to ache M after the ateace actvty s perfored. I addto, the durato of the ateace actvty o a ache s depedet o ts rug te (Kubz & Strusevch, 2006) ad s defed by T = t + δ t o ache M for =,2,...,, where t > 0 s the basc ateace te, δ > 0 s the ateace factor, ad t s the startg te of the ateace actvty. I what follows we preset a 7 obs stace o the two urelated parallel-ache schedulg proble uder study. Fgure llustrates a feasble schedule whch obs J, J 2, J ad J 4 are processed o ache M, obs J 5, J 6 ad J 7 are processed o ache M 2, ad the rate-odfyg actvty postos are l = 2 ad l 2 =. Fgure A feasble schedule wth = 2, = 7, l = 2, ad l 2 =. Followg Wag et al. (202), we deote our schedulg proble as R T = t + δt, r,0 < α C, where r dcates the rate-odfyg actvty ad C deotes the total copleto te.. Results I ths secto, we provde a ore effcet algorth to show that both probles studed by Wag et al. (20) ad Wag et al. (202) ca be solved O ( + ) te o atter what 0< α or α >. Frst, the followg two leas are useful for fdg the optal soluto of the proble. Lea (Mott, Kadel, & Baker, 986) The uber of o-egatve teger solutos to x + x x = s ( + )! C ( +, ) = C ( +, ) =. ( )!! Lea 2 (Wag et al., 202) The uber C ( +, ) s bouded fro above by (2 )!. Next, let ( l, l2,..., l ) ad C ( l, l2,..., l ) be the rate-odfyg actvty posto vector ad the copleto te of J that s processed o oe of the urelated parallel

3 7 aches based o ( ) l, l2,..., l. We deote by the subscrpt r [] a ob scheduled the rth posto to the last ob o ache M, for =,2,..., ad r =,2,...,, where s the uber of obs assged to be processed o ache ad =. The, we obta that = l Cll (,,..., l) = rα p + ( r+ δlp ) + lt 2 [] r [] r [] r = r= r= l + = M. (2) Let w r [ ] be the weght of ob J f t s scheduled the posto rth to the last ob processed o ache ca be rewrtte as follows: where 2 [ r] = r= = M. The, (2) C ( l, l,..., l ) = w + lt, () w r [ ] rαr [ ] pr [ ], =,2,...,, r l, ( r + δl) p[ r], =,2,...,, r > l. I addto, we defe y s = f J s the posto sth to the last ob processed o M ad y s = 0 otherwse. The, the R T = t + δt, r,0 < α C proble ca be forulated as follows: where Mze subect to w s s s = = s= = (4) w y + lt. ( 5 ) ys =, =,2,...,, s =, 2,..., l (6) = ys, =,2,...,, s = l+, l+ 2,..., (7) = ys =, =,2,..., (8) = s= y y... y, =, 2,..., (9) 2 = = = s { 0, } y, =,2,...,, =,2,...,, s=,2,..., (0) sα p, =,2,...,, =,2,...,, s l, ( s + δ l ) p, =,2,...,, =,2,...,, s > l. () Costrat (6) akes sure that each posto ( s l, =,2,..., ) o each ache s take by oe ob. Costrat (7) akes sure that each posto ( s l +, =, 2,..., ) o each ache s take by at ost oe ob. Costrat (8) akes sure that each ob s scheduled exactly oce. Costrat (9) esures that o every ache, the uassged postos ust precede all the assged postos, so that y s = f ad oly f ob J s fact the sth to the last ob o ache M. Fro (), we see that w w2 L wl ad, respectvely. w w L w ad thus 2 ( l+ ) ( l+ 2) y... y y = = = l ad y ( ) ( 2)... l+ y l+ y = = = Moreover, fro costrats (6) ad (7), we kow that y ( ) l = y ( ) ( ) = = l+ for, 2,...,. equalty y y 2... y = = = = So, the holds, for =, 2,...,. Hece, costrat (9) ca be reoved fro the forulato wthout affectg the optal soluto of the proble. As a result, the R T = t + δt, r,0 < α C proble ca be forulated as the followg assget proble ad, therefore, ca be solved O ( ) te (Brucker, 200). Mze w y + lt s s = = s= = subect to (6), (7), (8), ad (0). I addto, sce there are obs to be assged to urelated parallel aches, we have that 0 l+ l l. Let l ( l l l) 0 eas that l + l l+ =. By Lea, the uber of oegatve teger solutos to l+ l l+ = s C ( +, ). By Lea 2, the uber C ( +, ) s bouded fro above by (2 )!. Thus, we coclude that the followg theore holds. Theore. The R T = t + δt, r,0 < α C proble ca be solved O ( ) te. Note that f the uber of aches s fxed, the the R T = t + δt, r,0 < α C proble s polyoally solvable. 4. Coclusos I ths ote, we exteded the proble studed by Wag et al. (20) ad Wag et al. (202). We provded a ore effcet algorth ad showed that the R T = t + δt, r,0 < α C proble ca be solved O ( + ) te o atter what 0< α or α >. Further research ght be to cosder the proble wth ult-ateace actvtes, other shop settgs or optzg other perforace easures.

4 8 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te Refereces Brucker, P. (200). Schedulg algorths. New York: Sprger-Verlag. Gaweowcz, S. (2007). Schedulg deteroratg obs subect to ob or ache avalablty costrats. Europea Joural of Operatoal Research, 80, Hsu, C. J., Low, C., & Su, C. T. (200). A sgle-ache schedulg proble wth ateace actvtes to ze ake spa. Appled Matheatcs ad Coputato, 25, J, M., He, Y., & Cheg, T. C. E. (2006). Schedulg lear deteroratg obs wth a avalablty costrat o a sgle ache. Theoretcal Coputer Scece, 62, Kubz, M. A., & Strusevch, V. A. (2006). Plag ache ateace two-ache shop schedulg. Operatos Research, 54, Kuo, W. H., & Yag, D. L. (2008). Mzg the ake spa a sgle ache schedulg proble wth the cyclc process of a agg effect. Joural of the Operatoal Research Socety, 59, Lee, C. L., & Leo, V. J. (200). Mache schedulg wth a rate-odfyg actvty. Europea Joural of Operatoal Research, 28, Lee, W. C., & Wu, C. C. (2008). Mult-ache schedulg wth deteroratg obs ad scheduled ateace. Appled Matheatcal Modellg, 2, Lodree Jr., E. J., & Geger, C. D. (200). A ote o the optal sequece posto for a rate-odfyg actvty uder sple lear deterorato. Europea Joural of Operatoal Research, 20, Low, C., Hsu, C. J., & Su, C. T. (2008). Mzg the ake spa wth a avalablty costrat o a sgle ache uder sple lear deterorato. Coputers & Matheatcs wth Applcatos, 56, Mosheov, G., & Sdey, J. B. (200). Schedulg a deteroratg ateace actvty o a sgle ache. Joural of the Operatoal Research Socety, 6, Mott, J. L., Kadel, A., & Baker, T. P. (986). Dscrete atheatcs for coputer scetsts & atheatcas. Eglewood Clffs, NJ: Pretce-Hall. Wag, L. Y., Huag, X., J, P., & Feg, E. M. (202). Urelated parallel-ache schedulg wth deteroratg ateace actvtes to ze the total copleto te. Optzato Letters, do: 0.007/s x. Wag, J. J., Wag, J. B., & Lu, F. (20). Parallel aches schedulg wth a deteroratg ateace actvty. Joural of the Operatoal Research Socety, 62, Wu, C. C., & Lee, W. C. (200). Schedulg lear deteroratg obs to ze ake spa wth a avalablty costrat o a sgle ache. Iforato Processg Letters, 87, Yag, S. J. (20). Parallel aches schedulg wth sultaeous cosderatos of posto-depedet deterorato effects ad ateace actvtes. Joural of the Chese Isttute of Idustral Egeers, 28, Yag, S. J. (20). Urelated parallel-ache schedulg wth deterorato effects ad deteroratg ult-ateace actvtes for zg the total copleto te. Appled Matheatcal Modellg, 7, Yag, S. J., & Yag, D. L. (200a). Sgle-ache schedulg probles wth agg/ deteroratg effect uder a optoal ateace actvty cosderato. INFOR: Iforato Systes ad Operatoal Research, 48, Yag, S. J., & Yag, D. L. (200b). Mzg the ake spa o sgle-ache schedulg wth agg effect ad varable ateace actvtes. Oega, 8, Yag, S. J., & Yag, D. L. (200c). Mzg total copleto te sgle-ache schedulg wth agg/deteroratg effects ad deteroratg ateace actvtes. Coputers ad Matheatcs wth Applcatos, 60, Zhao, C. L., Tag, H. Y., & Cheg, C. D. (2009). Two-parallel aches schedulg wth rate-odfyg actvtes to ze total copleto te. Europea Joural of Operatoal Research, 98, Zhao, C. L., & Tag, H. Y. (200). Sgle ache schedulg wth geeral ob-depedet agg effect ad ateace actvtes to ze ake spa. Appled Matheatcal Modellg, 4,

5 9 楊 肅 正 郭 家 源 李 新 濤 南 開 科 技 大 學 工 業 管 理 系 過 去 十 年, 許 多 學 者 投 入 於 具 維 修 策 略 以 改 善 生 產 效 率 之 生 產 排 程 研 究 Wag, Wag, ad Lu (20) 及 Wag, Huag, J, ad Feg (202) 分 別 發 表 了 有 關 個 工 作 在 台 非 等 效 平 行 機 台 生 產 環 境 下, 考 慮 具 退 化 性 維 修 之 總 完 工 時 間 最 小 化 排 程 問 題 的 研 究 成 果 2 Wag 等 人 (20) 證 明 本 問 題 能 以 多 項 式 時 間 O ( + ) 求 解, 而 Wag 等 人 (202) 證 明 若 機 器 維 修 後 之 工 作 效 率 能 提 高, 則 能 以 多 項 式 時 間 O ( + ) 求 解 本 研 究 則 證 明 不 論 機 器 維 修 後 之 工 作 效 率 是 否 能 提 高, 本 問 題 均 能 以 多 項 式 時 間 O ( + ) 求 解 生 產 排 程 機 器 維 修 非 等 效 平 行 機 台 總 完 工 時 間