Solving A Two-Stage Problem with Scheduling and
Delivery in Supply Chain by Tabu Search Algorithm
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Solving A Two-Stage Problem with Scheduling and Delivery
in A Supply Chain by A Tabu Search Algorithm
Department of Industrial Engineering and Management National Chiao Tung University
StudentYu-Hao Chen AdvisorYung-Chia Chang Abstract
The concept of emphasizing on profit earned across an entire system in a supply chain has been an important subject discussed in supply chain management. Since different stages often have different, sometimes conflicting, objectives, globally optimal integrated solution for an entire system is often difficult to achieve.This research studies an integrated problem that jointly considers two important stages in a supply chain&product manufacturing and finished good delivery. An unrelated parallel machine scheduling problem is used to simulate product manufacturing and a vehicle routing problem is applied to represent delivery of finished goods. The objective is to find a system-wide solution to minimize the total cost across the entire system. Both unrelated parallel machine scheduling problem and vehicle routing problem are NP-Hard, the complexity of the studied two-stage problem is also NP-Hard. It means the solution time will grow exponentially when problem size goes large. Therefore, a tabu search algorithm, one of the meta heuristics, is presented to solve this problem. Computational analysis based on international studied problems and simulated data are presented to test the effectiveness and stability of the proposed tabu search algorithm.
key wordunrelated parallel machine scheduling vehicle routing problem
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wn ® (tabu)é m ^h 9 BF (tabu list)3 . $
BF r ª b à g x I · ; · 3 ê n BF S T
wÓ è Â Ç ó < \¿ 3 .9r ¿ I w I Z ^± ² q r ; 8 : | } 37'º % G H . 4Ü "d Ö [21,22]3 (1) t u w(start solution) r Ú9À I ® ) I ò ó { | e wö ÷ v 7- 9 } ~ . t u w3 (2) }w(neighborhood solution) / ø é ® ° sL ^M wo- t u wà á '}t u w^M w . }b é ® [8 : Z ± E F v; Þ b 37 8 ù r t u w,¬ °,© ç9i j h8 : Z ¢}wb é 9ú û ,¬ © çÈ ê }w j i hv® i h j; ¢ âÎ Ú1}w-9'¿ t ¥3 (3) b (move) s }w8â) 01 ´ ü Z^§[\01}ws BF 8íý a þ ¹ ô § 3 (aspiration criterion) §â) $ ´ ü 7l m ñ'}w8sâ) ¾0 ´ ü ' r <0 }w3 (4) BF s& . V8 J ´ ü ] ® ^M 39. / s w. V8e Ì c wf Ì ,$8 ì ¦ r BF ) I . + , N -+ , Í r ¿ ´ ü 9/ i ¡ $ ´ ü ; Ì BF íp . r b ¿ * º õ 7wÓ õ \¿ 3[$ ^ó BF S T ¿ ï y o r y }w; Ì B F Wm r BF w^ wÓ 3 (5) Ë g 2 (intensification) + , N -R ^¶ 9¼à + , ½R¼% + , ½Ë g 2
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z e { | e } ~ . ${ E F01wÀ À r Q sBattiti
and Tecchiolli[8]5 6 8w ^¢ Q 9 ¼û Î Ï t¼½8
J ¤ (1) 01wæ Q u 01w1(2) øÂ (limited
cycle)r ð s ½ æ Q w¢ "Â 1(3) ¸ (chaotic attractor)
n o | } Ú -. [ æ Q ¦ b 3,$s8 : } ~ .W
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NO°,» ª * W% b é 9Cr » "W% oi3\Cr » , 6»Up [rOS ¾8r OS T r D/ i ô & J OS | S W% 9 C D6»8Cr » "W% 0qZ \°& ñA B 8¹/ i ô & W% n 9 » "W% 7cjBI b é 96»ñr u °,ð D/ / i ô & 3¾8dij9/ i i/ i j C DW% 3E F 9 Min n j j j c w =1 ¾8wjBI 6»: 39º 2E F45 6 6b Â4 7 8 ' 1. 6»t u W% ] Æ 3 2. NO°,î M W% 3 3. 6»sLr NO8°,W% ò ó 3 4. X Y ,oNà á 3 5. r </ i 'r <6»L ; / i % Z [ ò ó 3 6. »r A B 8¹»r S S ¸ g ] r Ý í-S ÷ à ø3 7. S T r Dz p -ö ,- 3 8. r T S Dr ¿
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3.2.1
45 6 ® 7 .wÛ V@C D; E F7 "402 9 $; E F \Ü h012E F^I 'F º »Pe BI Min = f(x) s.t.x∈X xBI 6»» ,© çr Ü h XBI 6»» ,© çÜ h z h f(x)BI x 9 ( .Â4B V' Step 18 b }wb é Ë ã ¿ (iterations)Ë BF @ ô § 3 3 Step 2S Nâ) r t u w x$W BF 9ß z h * : ~ x9 (f(x)"9 Í 01wf(x)*3 Step 3 Í 01wx }w3 Step 4ñ }w8â) r <y w¾9 \ (9}w80 í s BF 8 * $r y w3 Step 5 y BF Ë ô § 3 Rà + , N -: ~ ¿ 3 Step 6O y w¾9 (\ Í 0 w# s BF 8Z[\þ ¹ ô § 3 $ $y w ¢ y " Í 0 wf(x)*3 Step 7o ã ¿ þ . 8 ¿ Wñ ß b é Í v w 90 wO ê õ Step 3â ã 3
3.3.2
? @7Ç ß Z I .wÞ z E FW^7¦ w .Q 8 b 9 \w> @r ~ Û q 7'945 6 9 \z Q 8 b 3 (1)t u w Í u Æ ½ G ] - I \45 6 ; E F'\ ( ,$45 6 ? @ Pinedo[45] w X Y þ ,Ö '0% °,W% 6
» · _ t.ê (weighted shortest processing time first rule, WSPT)
ê 8 : r { | e } ~ .7: ~ t u w - 9 Ã } ~ .Å ± @3WSPT Â4 ® 7°,W% @6»þ ( Þ b 6»© ç (Z · Û ç345 6 | Y { | e } ~ .wb o '3 Step 1· : ~ J r 6»þ @s NO'°,W% 0 (\ ± ² v (¢6»y Û F 7$y ¢6» T v ) ,...., , ( ... ) ,..., , ( ) ,... , ( 2 1 2 2 2 22 2 21 1 1 1 12 1 11 n nk n n n n k k w p w p w p Min w p w p w p Min w p w p w p Min ≤ ≤ ≤
Step 2Þ b 6»Û ç* ± ² NOW W% Þ b só r NO°,3 CkNOW W% 9lkk =1, 2,…,m¿ u (U8 9Y 3± ² s Step 18
Þ b © ç¶ C 6» T · ¶ C 3s¶ C C i <6»W¶ ü : ~
$6»sJ r NÚ °,W% @¾NOW W% R7 ¾80(
¢$6»¶ C \ NO37 8 6» i ^ ¶ C NO q NOW
Ãlq : ^ y 9lq + piq$W% n 96»iCr » "W% oi37
Step 3± 6»Cr » "W% oi[Û q» "W% Z · C D7$© ç\y ¢6» T v o1 ≤ o2≤ …≤ on3¢ ¢ 6» L U¶ C ké C DrOS T & on/r9 WJ OS n C Dn/r<6 »Cr <C n/r<6»n 9Cr OS C D1on/r 9 WC r <C[n/r]<6»[Cr OS C D¾8[x]BI q\x0q C[n/r]+1<6»ê @Í * 6»([n/r]Ë[n/r]+2)» "W% Z Z Ú Z ê C[n/r]+1<6»n " Ì r T S ; M C D,- 7${ Þ b J O S é C D6»3J OS | S W% ® ± @ C D6»8sCr » "W% 0q(3 Step 4sÞ b J r OS C D6»* T : Þ b r OS 6» C D© ç[\S T C D® ñA B 8¹ | ,$Cr b ® : Cr <é ® ¯ / i ¥3 .9¢A B 8¹z </ i ¥ ; W% w 7 z </ i (6»)þ ) (09Cr <® ¯ / i 3 T \ C F <é C D/ i .9: ~ ñCr </ i ¥ ý a ® ¯ / i ¥W% w 7ý a ® ¯ / i þ ) (09CF <® ¯ / i 37 ${ ¢J OS é C D6»© çÛ 83 Step 5: ~ J r ,¬ » "W% ¢ý & þ 9"4¢ "4° n 9 3 7'7r º »¨¸ ` w. V¤ 5´ 6»Ë2ONOË2T S ¾Æ B3-1@B3-2 I 3
B3-1 6»þ @°,W% B 1 2 3 4 5 wi 3.21 1.49 1.07 5.59 6.58 pi1 68 72 84 56 89 pi2 34 104 101 48 59 B3-2 6» · Z [ dij 1 2 3 4 5 0 281 174 463 482 430 1 281 0 454 210 221 213 2 174 454 0 634 654 596 3 463 210 634 0 26 76 4 482 221 654 26 0 102 5 430 213 596 76 102 0 Step 1: ~ J r 6»þ @s NO'°,W% 0 ( * \± ² v (Þ b Û ç© ç(6» · Û çñB3-3^¹ Û ç© ç96»4Ë6» 5Ë6» 1Ë6»2@6»33 B3-3 6» · Û çB 1 2 3 4 5 pi1/wi 21.04784 48.44428 78.86291 9.93445 13.59334 pi2/wi 10.7215 69.92242 94.69498 8.509082 8.927996 3 4 5 1 2 Step 2Þ b 6»Û ç© ç* ± @ T NOW Ã Þ b 6»\ r NO°, B3-43 · ñ6»4 u ¶ C 6»4¶ C \NO1@N O2» "W% ¶ ü 956@ 48âÎ (0Z âÎ ¶ C \NO2
¶ ü 90+89@ 48+59 âÎ ¶ C \NO1$WNO 1@NO2W
W% ¶ ü 989 @486»1¶ C \NO1@NO2» "W% 989+68
@48+34 âÎ Ú( ¶ C \NO2$WNO1@NO2W
W% 989@8216»¶ C \NO1@NO2» "W% 989+72@
82+104âÎ Ú(¶ C \NO1$WNO1@NO2W W% 9161
@821¶ C 6»3\NO1@NO2» "W% 9 161+84@ 82+101â Î Ú( ¶ C \NO23¾6»Cr » "W% ¶ ü 948Ë89Ë 82Ë161@1833 B3-4 NOW W% B 4 5 1 2 3 1 l 0 89 89 161 161 2 l 48 48 82 82 183 oi 48 89 82 161 183 Step 3± @ oi( J r T S é C D6»[\[n/r]=2 ^ó 6 »4@6»1 · ^ ¶ C \S T 1\ Ú6» 5Í * 6»(6»1@6 »2)» "W% | ¤6»5@6»1 Z (7)Ú ¢6»5m ¶ C \ S T 10* N O9S T 1C D6»4Ë6»1@6»51S T 2C D6» 2@6» 3 B3-53 B3-5 Cr 6»» "W% 4 1 5 2 3 oi 48 82 89 161 183 Step 4ò ó Cr OS C D6»4Ë6»1@6»5 Cr <é C D / i ¥¶ ü : ~ A B 8¹/ i 4Ë/ i 1@/ i 5W% @¾6»þ (3: ~ N O¶ ü 986.19(/ i 4)Ë87.54(/ i 1)Ë65.27(/ i 5)
,$Cr <® ¯ © ç9/ i 53 T \: ~ ñ/ i 5ý a ® ¯ / i W % @¾6»þ (3: ~ N O966.41(/ i 1)@ 18.19(/ i 4) 7 CF <® ¯ 9/ i 437${ v S T 16»C D© ç96»5Ë6»4 @6»117${ S T 26»C D© ç96»2@6»33 Step 5[B3-6^ó S T 1| S W% 989S T 2| S W% 9183: ~ z 6» » "W% 9B 3-6 I ¢z 6» » "W% ¶ ü ý & z 6 »þ * ° n 9 "4 B3-73 B 3-6 z 6» » "W% 5 4 1 1 89 519 621 842 2 3 2 183 357 991 B3-7 "4: ~ 1 2 3 4 5 cj 842 357 991 621 519 wi 3.21 1.49 1.07 5.59 6.58 2702.82 531.93 1060.37 3471.39 3415.02 11181.53 (2)}wb é À I }w ; ¶ ü 9; ; }õ .Ë! Ì õ .9¦ B µ r ~ W% 45 6 7; ; õ .- 9}w .3 ; ; }õ .9) I L Æ }; <6»p [È Æ õ . V b y w37A3-1RA3-29¨A3-1BI r t u wSiBI NO °,vS T C D6»© ç34¿ ¢; Æ }6» 5(J5)R6»1(J1) È Æ õ Ç y }wn o "A3-2 I S T 2o "· C D6» 10* \C D6» 53
a) t u w S1 S 2 S3 S4 S5 S6 S7 NO1 6»4 6» 5 NO2 6»3 6» 2 6»1 S T 1 6»3 6» 2 6»4 S T 2 6»5 6» 1 A3-1 ; ; õ .t u wI A b) }w S1 S2 S3 S4 S5 S6 S7 NO1 6»4 6» 5 NO2 6»3 6» 2 6»1 S T 1 6»3 6» 2 6»4 S T 2 6»1 6» 5 A3-2 ; ; õ .}wI A (3) BF à g 7Glover[22,23]Å Æ 8 b 9" n ® m p . " ¿ º §i . Û . w3 (4) ô § 3 45 6 8 b y w¤ : þ ¹ ô § 3 l m W9 h' F J à á (i) ø3 (ii) 9 ( \ Í v 01wf(x)*3 (iii) øu 01w3 (5)ñ ß 3 ê 45 6 8 o ã ¿ / P ´ ¿ Wn ñ ß 3
(6) = − ×100% (7) 45 6 .B VAA 3-3 I 3
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4.1
45 6 ) I C++# í < = Ve I 7 45 6 .wÅ 3[\45 6 ® H I C DE F,$âÎ C D¥¶ $ Ú9% zF 7Taillard w tai385& ' 8Í 120 ¥ PQ S T U VE F3
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6» 'à d¾8NO ¶ ü 9 2Ë4@86» ¶ ü 910Ë20Ë
40Ë80@ 1203S T S T 6» °o6» 910 WS T
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c d'; M 3
4.2
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4.2.1
D
p [Q o 8 b * Æ N OI B4-13ñB4-1 * o6» °WL M r ~ W% + , °7B4-19¨o6» [20 °40WwW% ³ 9 6» 2010 -16» [40 ° 80WwW% 9 6» 4010 -3 @a b c d e fg E Fp P o qWwW% ¤° "à ¦ Æ 9 ñA4-1^ . / ¹ $ r wW% + , °¦ 3í45 6 m . 6» 9160@ 200E F| ¤wW% ¶ ü 93W@11W3X Ãw ,Ö w${ E F 0 ` W% "4 q 45 6 E F6» 0q9 12039 \ qp PE FW45 6 Å Æ ¾ú Ú9ª .1 ã 3 w ¦ 6» ñB4-18m ^ * oNO °WL M r ~ W% m S °Z °2 g ¤ 6» ó 3 qq³ ¤µ NO 94r ~ W% ³ 9NO 2; -NO 8r ~ W% ³ 9NO
4; -[$^ó NO 9 r ~ W% ¤ 6» q3 sn v Ó ) ñA4-2^ * 6» g qW¾n v B¤ g 13^§ ,96» ÚWwß % Ú §n v q o6» 4 Wwß % g g q } ~ . §n v ß % m o q  c 37B4-19¨o6» 910@ 20Wn v q³ ´ ¶ m o6» °40@ 80W¾n v °´ ¶ F m| 7& k 0h ´ ¶ K m " ^ Æ og 83Zs6» 9120W n v 'Í c Þ ,^§9wß % ò p . q ¿ ^§ ¹ m \ ° ¿ vx I ¾ú Ê | } ^§v n v ; r b · w 2 3\[NO * ^| ¤o6» Æ W¾n v o q ¤ ` ¦ 45 6 } ~ . b g 8 ,9NO í= n v q2 5 ¤ 3
B4-1 E FD N OB . / C =10 20 40 80 120 Z [ C =2 31722.88 86582.19 307057.31 1459690.88 3513643.25 4 N/A 69920.24 232804.34 1059734.00 2298402.92 8 N/A N/A 211210.89 767060.31 1689519.87 \ ] ^ 10 20 40 80 120 2 10% 12% 27% 33% 32% 4 N/A 12% 34% 39% 34% 8 N/A N/A 26% 42% 43% _ # (`) 10 20 40 80 120 2 1.13 6.66 54.72 744.97 3428.38 4 N/A 9.75 90.11 1065.57 4316.18 8 N/A N/A 197.82 2145.92 6353.30 A4-1 6» @r ~ W% A( E FD)
! ! ! A 4-2 6» @n v A( E FD)
4.2.2
B
CF Ó ) 9= > °,W% @C DW% ] _ sÆ o % p [ * N OB4-2 I ñB8^| ¤G : ¦ @E F D{ è 3ñA4-3
^ . / ¹ o6» °WwW% m + , °n 6» o ; -WwW% ³ 9 m- K· 6»/ 160@200WwW% o v Æ o` W 9 p 6 ê 3¢ S T 6» °ñA4-4^¹ n v T w 2 ñ6»(6» 910 @20)´ ¶ mú û w 7 q6»(6» 940Ë80@ 120)´ ¶ J mú û 3 } ~ . §¨ n v $; E F"4$ } ~ . b s^[ NO à d' * v ó n v ® Æ o b q2 5 ¤ 45 6 } ~ .Ê h g b 3
B 4-2 E F B N OB . / C =10 20 40 80 120 Z [ C =2 69324.22 215805.77 864704.14 4638726.00 11034931.33 4 N/A 137390.36 563811.34 2668395.50 6113460.33 8 N/A N/A 368612.04 1716420.83 4034696.75 \ ] ^ 10 20 40 80 120 2 10% 10% 15% 27% 26% 4 N/A 13% 22% 32% 28% 8 N/A N/A 25% 36% 30% _ # (`) 10 20 40 80 120 2 0.82 7.48 56.08 738.17 3418.94 4 N/A 9.46 87.30 1046.72 4258.08 8 N/A N/A 194.12 2270.17 6224.74 A 4-3 6» @r ~ W% A( E F B)
! ! ! A4-4 6» @n v A( E F B)
4.2.3
P
CJ Ó ) ê ® PQ = > ? !°,W% q\C DW% Wà dPQ * ^ñB4-3 * 6» °WwW% q2 °ñA4-5^ . / * $¤ sNO °WwW% ¤2 °$¤ @Í ; < E Fî q 3sn v Ó ) Ú9 ® s 6» 40WÍ ; < E Fn v Ú ³ ^/ ´ ¶ J m4 E Fn v ´ ¶ mú û ís NO 'n v 5 t 8 m . q} ~ . ¤ b ¤ 3s¾ú Ó ¶ 6» (6» 910 @ 20)Wn v ^/ ´ ¶ mú û 6» q(6» 980)W^/ ´ ¶ J m# s6» 120Wn v } t Zn v ± 9 ´ ¶ F mJ § 3 Ê h g 3B 4-3 E FP N OB . / C =10 20 40 80 120 Z [ C =2 146013.26 476036.38 2024693.58 11108592.33 27343025.33 4 N/A 287108.78 1186850.87 6030547.67 14224423.00 8 N/A N/A 727180.77 3466723.92 8035616.00 \ ] ^ 10 20 40 80 120 2 6% 15% 13% 24% 24% 4 N/A 14% 15% 28% 23% 8 N/A N/A 6% 33% 25% _ # (`) 10 20 40 80 120 2 1.03 6.70 54.68 738.31 3460.98 4 N/A 10.27 86.77 1053.71 4192.03 8 N/A N/A 193.68 2306.47 5918.54 C _ # a`b =2 =4 =8 A 4-5 6» @r ~ W% A( E F P)
"#$ % & ' ( ) * =2 ( ) * =4 ( ) * =8 A 4-66» @n v A( E F P)
4.2.4
D
B
P
Ú& f J < E FN O^| ¤ ¼°,W% 9c l W¾ wW% ] t 8 Û qw ¦ ` 4} ~ b m ^ó : °,W % q r ~ W% 3sn v Ó ) J < E F] S 6» °w 2 ZO¢6» ª b ¹ ¾n v î q5 6» W(6» 910@ 20)n v ³ 9´ ¶ m6» qW (6» 940Ë80@120)n v ^w 7 k³ ´ ¶ J m3,$ Ú$J < E F* | ¤ ¼°,W% @C DW% q 9 wW% @n v 3
4.3
p . $J <E F(DËBËP)* ^Ð 9 7& N O" ; i d N ¼
1. 45 6 .sw$Û V@C D; E FWo6»
°wW% m S ª °Zn v ^x v q2 g w 2 s6
940Ë80)n v L U³ ^/ ´ ¶ J mð 6» / 120W n v º } 9'Í L Un v ^/ ´ ¶ F m0h 9´ ¶ J m< BI 45 6 } ~ .sw$Û V@C D; E FÊ h g 3 2. 45 6 .w ¦ s E F P8n v ¤Q  b s¾ E F8] B¤¦ h b $ ¼NO q 9 : 6» Æ ¾n v ] ¤ b l û q2 5 I 45 6 } ~ .Ê h g b 3 3. sNO °W¾wW% ³ 9µ "à NO 9 \wW% ¤ 6» ó 3 q,$NO Q 8 b Ú øZ[\Þ | Z Ú X ÃNO"4± @6» q Þ b NO I ,$45 6 z NO 92Ë4Ë8Wà dk « Q X Ú3 4. p [& J < E FN O^| ¤ ¼= > °,W% @C DW% 9 ¨ s6» þ . 120à ¢ '] §s W% u v 0 w í$0 w@¿ u w Ú^t u "4³ ´ ¶ F m3
5.1
45 6 w r } ~ .I 7wJ KL M NOÛ V@S T C D ; hE F 7 Í §ª ht¿ u wo- ð p PQ N O| ¤^t u "4³ ´ ¶ F m3G : å G ^¶ 9ß ¨ @Ð ¯ ; f 3sß ¨ f I .w$; E Fs6 » s1207'W§shtW% u v 0 w p [PQ v ó 45 6 . @ b g 81sÐ ¯ f ¤ù = « ` a c b [ E FPQ N Ov ó I 4} ~ .^ t u "4³ ´ ¶ F m^k « I n v , = > ? !@C Dr - ; t u ? !@C D "4¡ k « x v ) w 2 / & '( ] = N O3
5.2
4¼½ * ã Å Æ 5 6 A' 1. ^N h¾ú y z e { | } ~ .> ) ¾à _ 7 § v É 0 w3 2. ^Ûw c 7 8 øS ÷ à øËW øË°Ì M W% vp W% Ä ô § 9 hç Ð à d3 3. _ ` $; E Fht'\ w Ã} ~ .Å W§ Ê ® & 3 4. H I .Q 8 b BF à g o 2Ë}wb é v® P y | } v } ~ . O Ê @ É 3
4. D é Ë´ ´ E ¼ .sF 2Û VI -7BOPP FILM9
¨½d º qß ,« ,V5 6 ? @ ¼½ A B m< 3 5. Õ Ö × ¼õ / i ; hPe H I ½ ì G 2¶ ! qß « ² H ß Î ? @ ¼½ A B m< 3 6. Ç È É ËI º ËJ K L ¼M N .ËO Ë Ì .@ ß % L Í .sS T ´ µ E FI 5 6 @Æ Ú½r ï : £ P Q C9R CJ 113-144A B mP 3 [ ]
7. Baker, B.M. and M.A. Ayechew, ” A genetic algorithm for the vehicle
routing problem”, Computers & Operations Research, vol.30, pp.787-800, 2003.
8. Battiti, R. and G. Tecchiolli, ” The reactive tabu search”, ORSA Journal on
Computation, vol.6, pp.126-140, 1994.
9. Brandao, J., ” A tabu search algorithm for the open vehicle routing
problem”, European Journal of Operational Research, vol.157, pp.552-564, 2004.
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http://neo.lcc.uma.es/radi-aeb/WebVRP/
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