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 $Ñ r. ùpbó køóônœ 8I žml kó xˆ r Åì žm Ñ ( 14 + ε)-cq 9 Ž{. ' c üs 1$Ñ CqŽ{ ã aò O 010 êæ aò 68M0 90B5 An improved algorithm for scheduling two identical machines with batch delivery consideration WANG Leiyang 1 LIU Zhaohui 1, Abstract In this paper we consider the scheduling problem on two identical (parallel) machines in which the finished jobs need to be delivered to a customer in batches by single vehicle. The goal is to minimize the makespan, i.e., the time by which the vehicle has delivered the last job and returned to the machines. We assume that the jobs have different sizes, and give an approximation algorithm with the worst-case performance ratio 14 + ε. 9 Keywords scheduling, batch delivery, approximation algorithm Chinese Library Classification O 010 Mathematics Subject Classification 68M0 90B5 0 Ú ó Ä 1$ÑüS K =ó ÄküÓ.Åþ\ó \óó d ýnþk x r. T K @dleeúchen [1] JÑ b kó ÔnŒƒÓ. ChangÚLee [] Äó äkøóônœœ¹. Äù «œ/ Œ±LãXeµ ½ n ó J 1, J,, J n z ó òäkdüó.å \ó, d ýnþ z x r. ó J i (i = 1,,, n) \óžm p i, kônœ ÂvFϵ01c60F * Ä7 8µI[g, ÆÄ7]Ï 8 (No. 11171106) 1. uànóœæêæx þ 007 Department of Mathematics, East China University of Science and Technology, Shanghai 007, China ÏÕŠö Corresponding author, Email: zhliu@ecust.edu.cn
1Ï 1$ÑüÓ.ÅüS KU?Ž{ 9 s i =xžó^ m s i. Ð ž ÊÅì Nþ z L«g xujøo m. Ó 1xó x1 Óž½Â1xžm x 1ó r Åì I žm. ùp Ä rœ/ xž m T. 8I žml kó x r Åì ªžm. éù K ChangÚLee [] @JÑ -CqŽ{ Zhong [] JÑU? 5/-CqŽ{ Su [4] TŽ{Ä:þ? Ú Ñ 8/5-CqŽ{. cnÿ z Ä:þ? ÚU? JÑ ( 14 9 + ε)-cqž{. 1 U?Ž{ Äk Ñ K U?Ž{ Ž{Ì Ú½ 1ÚSü\ó. duz ó køóônœ ± 1žÿÒ ¹C K. First Fit (FF) Ž{Ú First Fit Decreasing (FFD) Ž{ C K²;Ž{. éu ½ó L FF Ž{Sü có J j 1 (?Ò ) UNe f XJù fø Kr J j # f. FFDŽ{Äkró Uì s j lœ L, Uì FFŽ{?1Sü. y Ñü«1üÑ. üñ1 ó U FFD 5K?1 1. üñ ÄkE K ~µó J j (j = 1,,, n) dš p j kœ s j, Nþ z. 1 K, (1-9 ε)-cqž{ òc\ ó LmÞ Ù ó U s j lœ L, 1 FF Ž{?1 1. UüÑ1ÚüÑ1ê O b 1, b. éü«øó 1üÑ Ñü fž{ fž{süó \óž ^²;L(LS) 5K =r1(½ó ) L Åìk s rl Sü1(½ó ) SüÅìþ\ó. Ž{ H1 (H) Ú½ 1 ^üñ1 (üñ)?1 1 1 B 1, B,, B b1 (B 1, B,, B b ) 1 b1 ( 1 b ) ùp i ( i ) L«1 B i(b i ) ó \óž mú. Ú½ ÄkU1L {B 1, B,, B b1 1} ({B 1, B,, B b 1 }) ^ LS 5Kòù 1SüÅìþ(z 1Š N), ^ LS 5Kò1 B b1 (B b ) ó SüÅìþ. Ú½ @ó1@$ñ. -Ž{ H1 (H) žml max (C H max). Ž{ 1L Xe. Ž{ H 1. 1Ž{ H1.. e b 1, 4 Cmax H = Cmax. H1. e b 1 =, 4 1Ž{ H Cmax H = min{cmax, H1 Cmax}. H - Cmax K `Š ρ ó Åìþó `žm b éac K ` ê k max{ρ + T, b T }.
40 [ 4Šz 17ò Bå ^ ρ i (ρ i ) L«ŠâŽ{ H1 (H) 1 i 1ó Åìþ óžm KkXeÚn. Ún 1.1 ρ b1 ρ, ρ b ρ. y² ÄŽ{ H1 Ï B b1 \óžm 1 ŠâŽ{ Œ± óó ½ B b1. b J s óó KŠâŽ{ k ρ b1 p s Ù = n p j. ÓnŒ± ρ b ρ. j=1 Ún 1. + p s + p s ρ, max = max{ρ 1 + b 1 T, ρ + (b 1 1)T, ρ b1 + T, ρ b1 1 + T, ρ b1 + T }. y² XJ Äc b 1 1 1ó z [] ² ÑžmL max{ρ 1 + (b 1 1)T, ρ + (b 1 )T, ρ b1 + T, ρ b1 1 + T }, ±Ä k b 1 1ó Œ± max = max{ρ 1 + b 1 T, ρ + (b 1 1)T, ρ b1 + T, ρ b1 1 + T, ρ b1 + T }. Ún 1. XJ b 1, 4 K C H max 1 0 C max. y² dž{ H Œ b 1, 4ž C max H = Cmax ± H1 n 1. Œ I ÄŽ{ H1. dú max = max{ρ 1 + b 1 T, ρ + (b 1 1)T, ρ b1 + T, ρ b1 1 + T, ρ b1 + T }. e5?øü«œ/. œ/1 max = max{ρ b1 1 + T, ρ b1 + T }. 5 b 1 = 1ž I ÄCmax H1 = ρ b1 + T œ¹. Cmax H1 = ρ b1 1 + T ž Ï ρ b1 1 C max max{ ρ b1 1 + T + T + T + T T = ; qdún 1.1Œ ρ b1 ρ ρ + T ± + T, T } ± ρ b1 + T ρ + T ρ + T.
1Ï 1$ÑüÓ.ÅüS KU?Ž{ 41 œ/ max = max{ρ 1 + b 1 T, ρ + (b 1 1)T, ρ b1 + T }. b 1 = 1, ž ù«œ¹c² ¹œ/1 ± I Ä b 1 5 œ/. ÄkN Xe(Jµ ρ 1 = 1 b 1, ρ = b 1 1, ρ b 1. qï max{ + T, b T }? Œ± ρ 1 + b 1 T b 1 + b 1 T + T + (b 1 b 1 )T b T = b 1 + b 1 b 1 b, ρ + (b 1 1)T ρ b1 + T b 1 1 + b 1 1 b 1 1 b, + T + T + T b T = 1 + b. d b 1 11 9 b + 6 9 ( [5]) ±9 b 1!b Ñ ê N yþãnªñøœu 1 0. e5 IÄ b 1 =, 4 žœ¹. Äk ÄŽ{ H ŠâŽ{ Œ ±XeÚn. Ún 1.4 XJ b = K b XJ b = K b 4. y² Šâ FF5K5U'Œ b 1 7 b ( [6]) ± µxj b = K b XJ b = K b 5. e5 Iy²XJ b = K b 5. ^ y{y ² b b = b = 5 - Bin 1 Bin Bin Bin 4 Bin 5 üñ1(šâ f m^s?ò).?ø±eü«œ/µ œ/1 Bin Bin 4 Bin 5 k 1 ¹ü ±þó. Ø b Bin ¹kü ±þó Bin? ü ó O Bin 1 Bin ùñ L fnþ Bin 4 Ú Bin 5 1 L fnþ ± b > gñ. œ/ Bin Bin 4 Bin 5 Ñ ¹ ó. ù«œ/e Šâ FFD5K Bin ¹ ÔnŒ L Bin ó ó IÄù ó ÚBin, Bin 4, Bin 5 ó ÒŒ b 4 gñ. e Ún Ñ b 1 =, 4 žž{ H 5U'. Ún 1.5 XJ b 1 =, 4 K C H max ( 14 9 + ε)c max ùp ε 45. y² ÄkÚ? #ÎÒ - u v w OL«`) 1 1lmžm 11 ó \óžmú 1n1 ó \óžmú - L«éA K `dšþ. XJ b 1 =, 4 b. e5 n«œ/?ø. œ/1 b = ù«œ/?øž{ H. dún 1.4 Œ b.
4 [ 4Šz 17ò b ž aquún 1. œ/ 1 y² Œ± b = ž C H max C max. C H max = max{ 1 + T, + T, ρ + T }. C H max = max{ + T, ρ + T } aquún 1. œ/ 1 y² N e IÄ C H max = 1 + T œ¹. Šâ 1üÑŒ C H max C max. (1 9 ε) (1 9 ε)v, l qï Œ± 1 u + T v 1 + T (1 9 (1 9 ε)v. + T, + T, ε) v + (1 9 ε)t v + T + ( 9 ε (1 9 ε))t T = + 9ε 4 14 9 + ε. + 9ε 4 + 9 εt + T œ/ b = ù«œ/?øž{ H. aquœ/1 I Ä b = 4œ¹, dž C H max = max{ 1 + 4T, + T, ρ + T, ρ 4 + T }. C H max = max{ + T, ρ + T, ρ 4 + T } ž aquún 1. y² N e IÄ C H max = 1 + 4T œ¹. Šâ 1üÑŒ C H max C max. 4 (1 9 ε) (1 9 ε)w, l 1 4 (1 9 ε)w.
1Ï 1$ÑüÓ.ÅüS KU?Ž{ 4 qï Œ± w + T, T, + T, 1 + 4T (1 9 ε) w + 4 (1 9 ε)t w + T + (4 ε 4 (1 9 ε))t T = 14 9 + ε. + ε + εt + T œ/ b 4 ù«œ/?øž{ H1. Ï b 1 =, 4 aquún 1.y² N C H max max C max. ½n 1.1 ùp ε 45. C H max ( 14 9 + ε ), y² dún 1. Ú 1.5. ë z [1] Lee C Y, Chen Z L. Machine scheduling with transportation considerations [J]. Journal of Scheduling, 001, 4: -4. [] Chang Y C, Lee C Y. Machine scheduling with job delivery coordination [J]. European Journal of Operational Research, 004, 158: 470-487. [] Zhong W Y, Dosa G, Tan Z Y. On the machine scheduling problem with job delivery coordination [J]. European Journal of Operational Research, 007, 18: 1057-107. [4] Su C S, an J C H, Hsu T S. A new heuristic algorithm for the machine scheduling problem with job delivery coordination [J]. Theoretical Computer Science, 009, 410: 581-591. [5] Dosa G. The tight bound of first fit decreasing bin-packing algorithm is F F D(I) 11 O T (I)+ 6 9 [J]. Lecture Notes in Computer Science, 007, 4614: 1-11. [6] Xia B, Tan Z Y. Tighter bounds of the First Fit algorithm for the bin-packing problem [J]. Discrete Applied Mathematics, 010, 158: 1668-1675. 9