Scheduling on single machine and identical machines with rejection

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1 2014c12 $ Ê Æ Æ 118ò 14Ï Dec., 2014 Operations Research Transactions Vol.18 No.4 káýüåúó.åüs K p r 1 S 1, Á ïä káýüåúó.åüs K. éuüåœ/, ó v ^ éa \óž α. XJó kˆž, 8I zžl v ^ƒú, y²ù K Œ). XJ kó 3"ž ˆ, 8I zoóž v ^ƒú, y²t K Œ). éuó.åüs K, ïäó ü13 žˆœ/, 8I zžl v ^ƒú. éåìê 2 Ú, O Ñ ' 2 Ú 4 2/ 3 Ž{. ' c üs, Œáý, 3 Ž{, ' ã aò O êæ aò 68B20, 90B35 Scheduling on single achine and identical achines with rejection GAO Qiang 1 LU Xiwen 1, Abstract We consider scheduling probles with rejection for both single achine and identical achines in this paper. For the single achine probles, each job has penalty α ties of its processing tie. If jobs have release dates, the proble of iniizing the su of akespan and total penalty can be solved in polynoial tie. If all jobs arrive at tie zero, the proble of iniizing the su of total copletion tie and total penalty also can be solved in polynoial tie. For the identical achines probles, jobs arrive over tie at two different tie points. The objective is to iniize the su of akespan and total penalty. We design on-line approxiation algoriths with copetitive ratios 2 or 4 2/ for the two cases when the nuber of achines is two or, respectively. Keywords scheduling, rejection, on-line algorith, copetitive ratio Chinese Library Classification O Matheatics Subject Classification 68B20, 90B35 0 Ú ó 3²;üS K, kó Ñ Sü3Åìþ?1\ó. 3y ó ) ÂvFÏµ * Ä7 8µI[g, ÆÄ7 No ) 1. uànóœænæêæx, þ Departent of Matheatics, College of Science, East China University of Science and Technology, Shanghai , China ÏÕŠö Corresponding author, E-ail: xwlu@ecust.edu.cn

2 2 p r, S 18ò, Ï~ dþzós9" Ü,,Ü dè gc),, Ü Ù è?1 ï., 1n Œ±Jø ídñöž, è Œò) 1n. è ÏL ï½ ª, ü$), Jp²LÃ. Ïd3 k áýüs K, ó Œ±É Sü3Åìþ?1\ó, ½öáý G ½ v ^. Œ±rÏáýó G v ^wš 1n è ïü G ^, ½öòó 1n G ^., ïä káýüåüs KÚÓ.ÅüS K. K ãxe: 3üÅ üs K, k n ó J 1, J 2,, J n, ó J j ˆž r j, \óž p j j = 1, 2,, n), ó J j Œ±É Sü3Åìþ?1\ó; ½öáý G ½ v ^, Ù v ^ \óž α. P A ÅìÉó 8Ü, R Åìáý ó 8Ü. éu8i zžl v ^ƒú K, Äó kˆž œ/, ^nëêl«{œ±p 1 r j C ax + α J p j R j, Ù C ax = ax{c j J j A. éu8i zoóž v ^ƒú K, Ä kó 3"ž ˆ œ/, ^nëêl«{œ±p 1 J C j A j + α J p j R j. 3Ó.ÅüS K, k Ó.Å M 1, M 2,, M, n ó J 1, J 2,, J n ü13 žˆ, = õkü ˆž, O "ž r ž. ó J j \ó ž p j, v ^ e j j = 1, 2,, n). ó J j 3ˆƒ, Œ±É Sü3Å ìþ?1\ó, ½öáý G ½ v ^. P A ÅìÉó 8Ü, R Åìáýó 8Ü. 8I zžl v ^ƒú. ^nëêl«{œ±p P r j {0, r, online C ax + J e j R j. e 0 káýüåüs KÚ²1ÅüS K kïäj. 3 káýüåüs K, Zhang [1] ÄüÅìþ káý!ó kˆž üs K 1 r j C ax + J e j R j. Ñ 2-CqŽ{, Où K FPTAS. éù Kó 3 žˆœ/ 1 r j, online C ax + J e j R j, Lu [2] Ñ ' 2 `3 Ž{; kó õkü ˆžž, Cao Ú Zhang [3] Ñ ' `3 Ž{. Engels [4] ïäl œ/e káýüåü S K, 8I z\oóž v ^ƒú. y²ù K ÊÏ Â e NP-, Ñ FPTAS. Epstein [5] Ä káý!ü žó 3 LˆüÅüS K, 8I zoóž v ^ƒú. O3 Ž{ ' 2+ 3)/ , Ñ e.. ó \óž? êž, y² KØ3~ê '3 Ž{. Ma [6] ïä káý!ó 3 žˆ äk 5üÅüS K 1 r j, online, rej, agreeable J C j A j + J e j R j, ó äk 5 éu? ü ó J i J j, XJ p i < p j Kk e i e j ; XJ p i = p j r i < r j Kk e i e j. Ñ U? DSPT Ž{, ù Ž{ ' 2 `Ž{. Lu [7,8] ïä káýœ/e, ó kˆž Œ± 1\óüÅ üs K 1 p batch, r j, B C ax + J e j R j. éu1nþ B k ÚÃ œ/, O Ñ 2-CqŽ{. éu káý²1åüs K, Bartal [9] ïäl Ú3 œ/e káý Ó.ÅüS K, 8I zžl v ^ƒú. éuó l ˆœ /, Åìê? êž, Ñ5U' 2 1/ Ž{9 PTAS; Åì

3 4Ï káýüåúó.åüs K 3 ê ½êž, Ñ FPTAS. éuó 3 Lˆœ/, Åìê? êž, Ñ ' `Ž{; Åìê = 2 ž, Ñ ' `Ž{. He Ú Min [10] ïäó 3 Lˆœ/e, káý ÓaÅüS K. Åìê = 2 ž, ÑCqŽ{, y² s žù Ž{ `, Ù ' s + 1)/s; Åìê = 3 ž ÑCqŽ{, y² s 2 žù Ž{ `, Ù ' s + 2)/s. Dósa Ú He [11] U?Å ìê 2 s < ž '. éuù KŒ äœ/, Ñ ' 1/2 + 1/4 + 1/s CqŽ{. 'uó 3 žˆ káý²1åüs K, 8 cïäj, Ä ù Kó ü13 žˆœ/. SNSüXe: 31 Ü, òy² 1 r j C ax +α J p j R j Ú 1 J C j A j+ α J p j R j ùü káýüåüs K Œ). 31Ü, éu káýå ìê 2 Ú Ó.Å ž3 üs K, O Ñ ' 2 Ú 4 2/ 3 Ž{. 3Ž{, P A ÅìÉó 8Ü, R Åìáýó 8Ü. ƒ A/3 `üs, P A ÅìÉó 8Ü, R Åìáýó 8Ü. 1 ü üåœ)œ/ Äk Ä8I zžl v ^ƒú K. kó 3"ž ˆž, K 1 C ax + α J p j R j ². XJó J j ó é8i¼êš z p j ; XJó J j ó é8i¼ê Š z αp j. Ïdéu K 1 C ax + α J p j R j, 0 < α 1 ž, áý kó Œ± `üs. α > 1 ž, É kó Sü3Åìþ\óŒ± `ü S. ó køóˆžž, y² K 1 r j C ax + α J p j R j Œ). éu 0 < α 1 Ú α > 1 ü«œ/, Ñ±eÚn. Ún1.1 1) 0 < α 1 ž, áý kó Œ± K 1 r j C ax + α J p j R j `üs. 2) α > 1 ž, 3 K 1 r j C ax + α J p j R j `üs, 3, ž t {r 1, r 2,, r n, 3 t ƒ ˆó þáý, Ù{ó UìERD5KSü3Åì þ?1\ó. y² 1) Ø w,, e y² 2). P t `üs, Sü3Åìþ\ óó Œˆž. w, k t {r 1, r 2,, r n t = ax{r j J j A. e3, ó J k áý Ùˆž r k òù ó SüÅìþ\ó, ž L õo\ p k, 8I¼êŠ ~ α 1)p k > 0, ó J k 3 `üs AT É, Sü\ó. Ïd α > 1 ž, 3 K 1 r j C ax + α J p j R j `üs, 3, ž t {r 1, r 2,, r n, 3 t ƒ ˆó þáý, Ù{ó UìERD5KS ü3åìþ?1\ó. Ïd α > 1 ž, epüs σt) áý t {r 1, r 2,, r n ž ƒ ˆó! ÉÙ{ó üs, 8I¼êŠ Zt) üs `üs, = `Š Z = in j). ŠâÚn 1.1, kxe½n. 1 j n ½n1.1 káýüåüs K 1 r j C ax + α J p j R j Œ). e5 Ä8I zoóž v ^ƒú K 1 J C j A j+

4 4 p r, S 18ò α J p j R j. ó køóˆžž, e α K 1 r j J C j A j +α J p j R j òz ²;üS K 1 r j C j. Lenstra [12] ²y² K 1 r j C j r NP-J. kó 3"ž ˆž, òy²ù K Œ). Ž{ H1 ò kó Uì SPT 5K, =Uì\óžš~^S #?Ò, É Œ α 1 ó, Uì SPT 5KSü3Åìþ\ó, áýù{ó. ½n1.2 Ž{ H1 K 1 J C j A j + α J p j R j `Ž{. y² du SPT Ž{Œ± )²;üS K 1 C j, Ïd K 1 J C j A j+ α J p j R j Éó Uì SPT 5KSü3Åìþ?1\ó. ò kó Uì SPT 5K #?Ò Sü\ó, 8I¼êŠ n j=1 C j = n j=1 n j + 1)p j. éuó J j, e n j + 1) α, áýù ó Œ é8i¼ê Š z~ [n j + 1) α]p j, Ïd áý v^ j n + 1 α ó. =3 `üs, Œ α 1 ó É, Ù{ n + 1 α ó áý. ÄKeù `üs, Éó Ø Œ α 1 J i Ú J k, p i < p k J i É!J k áý. Bå, P J i Sü3 k Éó SPTSê1 β, β α 1. æ^ K: áý J i, É J k, 8I¼êŠò ~ βp i + αp k βp k + αp i ) = β α)p i p k ) > 0. Ïdò Œ α 1 ó Uì SPT 5KSü3Åìþ\ó, áýù{ó, Œ± 8I¼ê J C j A j + α J p j R j. Ž{ H1 K 1 J C j A j + α J p j R j `Ž{. 2 Ó.Åœ/ 3!, ïä káýó.å3 üs K. kó ü13 žˆ, =ó õkü ˆž, O "ž r ž. 8I zžl v ^ ƒú, = P r j {0, r, online C ax + J j R e j, ùp C ax = ax{c j J j A. Cao Ú Zhang [3] ÑT KüÅœ/e., = 1 r j {0, r, online C ax + J j R e j e Œ±^Ó {y², éu K P r j {0, r, online C ax + J j R e j, 2 žùe e Ñ PÒ. éu? ó 8Ü S, P MS) = J p j S j/, RS) = J e j S j. S0) "ž ˆó 8Ü, Sr) r ž ˆó 8Ü. 3Ž{, P A 1 S0) Éó 8Ü, R 1 S0) áýó 8Ü; A 2 Sr) É ó 8Ü, R 2 Sr) áýó 8Ü. ƒq/, Œ±^ A 1, R1, A 2 Ú R2 5 ½Â `üs ƒéa8ü. P = ax{0, r MA 1 ), = ax{0, r MA 1), k±eún. Ún2.1 + MR 1 A 1). y² Øª3 = 0 ž w, á. 0 ž, d Ú ½ÂŒ, MA 1 ) + = r, MA 1) + r. Ïd MA 1 ) + MA 1) +, = + MA 1) MA 1 ) = + MR 1 A 1). kéuüåìœ/, Ñ3 Ž{ H2, y²ù Ž{ ' 2.

5 4Ï káýüåúó.åüs K 5 Ž{ H2 Ú½ 1 éu S0) ó, É v^ p j 2e j ó UìLSŽ{?1\ó, áýù ó. Ú½ 2 éu Sr) ó, XJ + in{p j /2, e j e j, J j Sr) J j Sr) Üó. ÄKÉ v^ p j 2e j ó UìLSŽ{?1\ ó, áýù ó. ½n2.1 = 2 ž, 3 Ž{ H2 ' 2. y² Šâ Sr) Ä 8Ú Ä v3 Ž{Ú½ 2 5 œ/?ø. P Z H2 Ž{ H2 8IŠ, Z `Š. œ/ 1 Sr) = e S0) ó Z H2 = RS0)) = RA 1) + RR 1) MA 1) + RR 1) Z, 3 Ž{ `üs. e S0) kó É, P J l1 S0) Z H2 MA 1 ) + p l RR 1) = MA 1 A ) + MA 1 R ) + p l RR 1 A ) + RR 1 R ) MA 1 A ) + RA 1 R ) + p l MR 1 A ) + RR 1 R ) = MA 1) + RR 1) + p l 1 2 = MA ) + RR ) + p l 1 2. e J l1 A, þª Z H2 Z + Z /2 = 3Z /2; e J l1 R, þª Z H2 Z + e l1 2Z. œ/ 2 Sr) 3 Ž{áý Sr) Üó ŠâŽ{Ú½ 2, + in{p j /2, e j e j. J j Sr) J j Sr) œ/ 2.1 Sr) ó 3 `üs Z H2 MA 1 )+p l1 /2 + RR 1 ) + RSr)), Z MA 1) + RR 1) + RSr)). Œ±aq/ ^œ/ 1 {y². Z H2 MA 1 ) + p l1 /2 + RR 1 ) + RSr)) MA 1) + p l1 /2 + RR1) + RSr)) 2MA 1) + RR1) + RSr))) 2Z.

6 6 p r, S 18ò œ/ 2.2 Sr) kó 3 `üs É, dž dún2.1œ Z MA 1)+ + MA 2) + RR 1) + RR 2). Z H2 MA 1 ) + p l RR 1) + RSr)) MA 1 ) + p l RR 1) + + e J l1 A 1, Kk e J l1 R1, Kk J j A 2 J j Sr) { pj in MA 1 A 1) + MA 1 R1) + p l RR 1 A 1) + RR 1 R1) + + MR 1 A 1) + { pj in + { pj in J j R 2 MA 1 A 1) + RA 1 R 1) + p l MR 1 A 1) + RR 1 R 1) + + MR 1 A 1) + MA 2) + RR 2) =MA 1 A 1) + 2MR 1 A 1) + RR 1) + p l MA 2) + RR 2). Z H2 2MA 1 A 1) + 2MR 1 A 1) + RR 1) + + MA 2) + RR 2) = 2MA 1) + RR 1) + + MA 2) + RR 2) 2Z. Z H2 MA 1 A 1) + 2MR 1 A 1) + RR 1) + e l1 + + MA 2) + RR 2) 2MA 1) + 2RR 1) + + MA 2) + RR 2) 2Z. œ/ 3 Sr) 3 Ž{É Sr) v^ p j 2e j ó ŠâŽ{Ú½ 2, + J j Sr) in{p j /2, e j < P J l2 Sr) Sü\óó, k J j Sr) Z H2 MA 1 ) + + MA 2 ) + p l RR 1) + RR 2 ). œ/ 3.1 Sr) ó 3 `üs Üáý, dž Z MA 1) + RR 1) + RSr)). e j.

7 4Ï káýüåúó.åüs K 7 œ/ 1 {aq, k k Z H2 MA 1 ) + + MA 2 ) + p l RR 1) + RR 2 ) = MA 1 ) + RR 1 ) + + p l { pj in + J j A 2 MA 1) + RR 1) MA 1) + RR1) < MA 1) + RR 1) + 2RSr)) 2Z. { pj in J j A 2 J j Sr) + { pj ) in œ/ 3.2 Sr) kó 3 `üs É, dž J j R 2 in J j R 2 in Z MA 1) + + MA 2) + RR 1) + RR 2). Z H2 MA 1 ) + + MA 2 ) + p l RR 1) + RR 2 ) MA 1 ) + RR 1 ) + + MR 1 A 1) + MA 2 ) + RR 2 ) + p l 2 2 { pj { pj MA 1 A 1) + 2MR 1 A 1) + RR 1) + + MA 2) + RR 2) + p l 2 2 2MA 1) + RR 1) + + MA 2) + RR 2) + p l 2 2 2MA 1) + + MA 2) + RR 1) + RR 2)) 2Z. Ïd3 Ž{ H2 ' 2. y3 Û káý Ó.ÅüS K, Ñ3 Ž{ H3, y²ù Ž { ' 4 2/. éu S0) ó, P S 1 0) = {J j p j e j /, J j S0), S 2 0) = S0)\S 1 0), ò S 1 0) ó U 1, 2,, S 1 0)?1?Ò. Ó/éu Sr) ó, P S 1 r) = {J j p j e j /, J j Sr), S 2 r) = Sr)\S 1 r), ò S 1 r) ó U 1, 2,, S 1 r)?1?ò. Ž{ H3 Ú½ 1 éu"ž ˆó, áý S 2 0) ó. P π i 0)i = 0, 1,, S 1 0) ) É S 1 0) c i ó UìLSŽ{?1\ó, áý S 1 0) Ù ó üs. l π i 0) À 8I¼ê üs, P σ 1. Ú½ 2 éu r ž ˆó, áýs 2 r) ó. Pσ 1, π j r))j =0, 1,, S 1 r) ) É S 1 r) c j ó UìLSŽ{?1\ó, áý S 1 r) Ù ó üs. l σ 1, π j r)) À 8I¼ê üs, P σ = σ 1, σ 2 ).

8 8 p r, S 18ò ½n2.2 3 Ž{ H3 ' 4 2/. y² P J k1 S 1 0) `üsé ó, J k2 S 1 r) `üs É ó. k 1 > 0 ž, P A 1 = {J 1, J 2,, J k1, R 1 = S0)\A 1; ƒa/, k 2 > 0 ž, P A 2 = {J 1, J 2,, J k2, R 2 = Sr)\A 2. P Z H3 Ž{ H3 8IŠ, Z `Š. œ/ 1 Sr) = œ/ 1.1 k 1 = 0. 3 `üs S 1 0) ó Üáý. d8ü S 2 0) ½ÂŒ S 2 0) ó 3 `üs Üáý. qï Sr) =, ±Ž{ H3 ü S `, = Z H3 = Z. œ/ 1.2 k 1 > 0. ŠâŽ{, k Z H3 MA 1) ) p k1 + RR 1) MA 1 A ) + MA 1 R ) + RR 1 A ) + RR 1 R ) ) Z. du A 1 R S 1 0), k MA 1 R ) RA 1 R ). Šâ k 1 ½ÂŒ R 1 A Ø ¹ S 1 0) ó, Ïd RR 1 A ) MR 1 A ). Z H3 MA 1 A ) + RA 1 R ) + MR 1 A ) + RR 1 R ) + = MA 1) + RR1) ) Z. 1 1 ) Z 1 1 œ/ 2 Sr) œ/ 2.1 k 1 = 0, k 2 = `üs A 1 S 2 0), RA 1) MA 1), A 2 S 2 r), RA 2) MA 2). Z H3 RS0)) + RSr)) = RA 1) + RR 1) + RA 2) + RR 2) MA 1) + RR 1) + MA 2) + RR 2) Z. ddœž{üs `. œ/ 2.2 k 1 = 0, k 2 > `üs A 1 S 2 0), RA 1) MA 1). Z H3 MA 1 ) + RR 1 ) + + MA 2) + RR 2) + RS0)) + + MA 2) + RR 2) ) 1 1 ) RA 1) + RR1) + + MR 1 A 1) + MA 2 A 2) + MA 2 R2) + RR 2 A 2) + RR 2 R2) ) ) Z

9 4Ï káýüåúó.åüs K 9 MA 1) + RR1) + + MR 1 A 1) + MA 2 A 2) + RA 2 R2) + MR 2 A 2) + RR 2 R2) ) MA 1) + RR1) + + MR 1 A 1) + MA 2) + RR2) + 2Z ) Z = 3 1 ) Z. 1 1 ) Z œ/ 2.3 k 1 > 0, k 2 = `üs A 2 S 2 r), RA 2) MA 2). Z H3 MA 1) + RR 1) ) p k1 + RSr)) = MA 1 A 1) + MA 1 R1) + RR 1 A 1) + RR 1 R1) ) p k1 + RA 2) + RR2) MA 1 A 1) + RA 1 R1) + MR 1 A 1) + RR 1 R1) ) p k1 + MA 2) + RR2) MA 1) + RR1) + MA 2) + RR2) ) Z. œ/ 2.4 k 1 > 0, k 2 > ) Z Z H3 MA 1 ) + RR 1 ) + + MA 2) + RR 2) ) MA 1) + RR 1) ) p k1 + + MR 1 A 1) + MA 2) + RR 2) ) MA 1 A 1) + MA 1 R1) + RR 1 A 1) + RR 1 R1) ) p k1 + + MR 1 A 1) + MA 2 A 2) + MA 2 R2) + RR 2 A 2) + RR 2 R2) ) MA 1 A 1) + RA 1 R1) + MR 1 A 1) + RR 1 R1) ) p k1 + + MR 1 A 1) + MA 2 A 2) + RA 2 R2) + MR 2 A 2) + RR 2 R2) MA 1) + RR1) ) Z 4 2 ) Z. 1 1 ) ) Z + + Z + MA 2) + RR 2)

10 10 p r, S 18ò Ïd3 Ž{ H3 ' 4 2/. ë z [1] Zhang L, Lu L, Yuan J. Single achine scheduling with release dates and rejection [J]. European Journal of Operational Research, 2009, 198: [2] Lu L, Ng C T, Zhang L. Optial algoriths for single-achine scheduling with rejection to iniize the akespan [J]. International Journal of Production Econoics, 2011, 130: [3] Cao Z, Zhang Y. Scheduling with rejection and non-identical job arrivals [J]. Journal of Systes Science and Coplexity, 2007, 20: [4] Engels D W, Karger D R, Kolliopoulos S G, et al. Techniques for scheduling with rejection [J]. Journal of Algoriths, 2003, 49: [5] Epstein L, Noga J, Woeginger G J. On-line scheduling of unit tie jobs with rejection: iniizing the total copletion tie [J]. Operations Research Letters, 2002, 30: [6] Ma R, Yuan J. Online scheduling on a single achine with rejection under an agreeable condition to iniize the total copletion tie plus the total rejection cost [J]. Inforation Processing Letters, 2013, 113: [7] Lu L, Zhang L, Yuan J. The unbounded parallel batch achine scheduling with release dates and rejection to iniize akespan [J]. Theoretical Coputer Science, 2008, 396: [8] Lu L, Cheng T C E, Yuan J, et al. Bounded single-achine parallel-batch scheduling with release dates and rejection [J]. Coputers & Operations Research, 2009, 36: [9] Bartal Y, Leonardi S, Marchetti-Spaccaela A, et al. Multiprocessor scheduling with rejection [J]. SIAM Journal on Discrete Matheatics, 2000, 13: [10] He Y, Min X. On-line unifor achine scheduling with rejection [J]. Coputing, 2000, 65: [11] Dósa G, He Y. Preeptive and non-preeptive on-line algoriths for scheduling with rejection on two unifor achines [J]. Coputing, 2006, 76: [12] Lenstra J K, Rinnooy Kan A H G, Brucker P. Coplexity of achine scheduling probles [J]. Annals of Discrete Matheatics, 1977, 1:

An improved algorithm for scheduling two identical machines with batch delivery consideration

An improved algorithm for scheduling two identical machines with batch delivery consideration 01c $ Ê Æ Æ 117ò 11Ï March, 01 Operations Research Transactions Vol.17 No.1 1$ÑüÓ.ÅüS KU?Ž{ [ 1 4Šz 1, Á ïä 1$ÑüÓ.ÅüS K. T K ó üó.åþ\ó óó d ýnþ z $Ñ r. ùpbó køóônœ 8I žml kó xˆ r Åì žm Ñ ( 14 + ε)-cq 9