Im pact of capac ity a lloca tion on bullwh ip effect in supply cha in

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1 JOU RNAL O F SYST EM S EN G IN EER IN G V o l 7 N o 4 A ug 00 ( 30007) : : ; ; ; ; : F5 : A : (00) Im pact of capac ity a lloca tio o bullwh ip effect i supply cha i W AN J ie L IM i-q iag KOU J i- sog (Schoo l of M aagem e t T ia ji U iversity T ia ji Ch ia) Abstract: Co sider a supp lier sellig to m u lt ip le reta ilers w ho se dem ad s exceed the supp lier s capacity T he supp lier m u st im p lem e t a allocatio m echa ism to do le ou t p roductio W e so rt these allocatio m echa ism s i to truetellig direct m echa ism s ad iducigiflatio m echa ism ṡ A d w e dem o strate that the bu llw h ip effect w ill be expaded i iducigiflatio m echa ism ṡ W e also aalyze ad qua tify the bu llw h ip effect i liear allocatio m echa ism Key words: allocatio m echa ism ; the bu llw h ip effect; supp ly chai; gam e theo ry; liear allocatio m echa ism 0 [ ] 0 80 : ; ; 0 90 [ ] [3 ] [ 4 ] 5 : : ; : : ( ) Tsighua Togfag Optical Disc Co Ltd All rights reserved

2 00 8 : 34 ; ; 3 k ; 4 ; 5 [5 ] a i = k - i- X j j= [6 ] j = - i [ 7 8 ] 3 : k k ; [9 ] a i = kx ig X j j = j= k ; - - ; 4 : {X X X } m (m k ) m X i - Z i (i = ) X i - m X j - k a i = j= gm i m X i i 0 i > m a i j = : : 5 ) : {X X X } a i = k - i- X j j= a i = j = - i m k - m X i 0 : 6 : Tsighua Togfag Optical Disc Co Ltd All rights reserved m i= k 0 X i i m - m k - X j i= m j= i > m

3 : {S t- S t- S i t- S t- a i t = X i t m i t + m ax {0 k - m i t i= X i t} m ax {0 k - m i t m i t = S i t- - i= X i t X i t} i= S j t- j= g i= i i > t t - S i t- Z i {Η Α} i m i t 0 X i k i= w a i X i : a i X i - - a i ( - ) i X i B i = [0M ] M i q 3 q 3 B i A i = [0 + ) B i < A i : - - k q 3 S 3 i (q 3 q 3 q 3 ) - q 3 [q 3 + ) Tsighua Togfag Optical Disc Co Ltd All rights reserved

4 00 8 : 343 X Y ; X + Y > k gx - Y g < k X Y q 3 (X - Y + k) g Y k S 3 i (q 3 q 3 q 3 ) - f X = (Z X - w - X )X X + Y < k X - Y + k Z X - w - g X + Y > k - () gx - Y g < k (Z X - w - k) k X > k + Y 0 Y > k + X 3 5f X 5X = p = Κ(Z - q) p Z i - w - X X + Y < k q Κ (Z i - w - X + Y - k) X + Y > k () gx - Y g < k : 0 ; 0 X > k + Y Y > k + X (Z ) q Η f i (i = (Y h) ; Α ) (Y l) f i = p a i - w a i 5f [Y l Y i h ] = ΚZ i - w - Κa i 5a i Κ= a i = Z i - w Y l Θ (0 < Θ< ) Y h f X = ( ) X Y Θ(Z X - w - ax (Y l) ) ax (Y l) ax (Y h) = X k - Θ ( - Θ) (Z X - w - ax (Y h) ) ax (Y h) + X X + Y < k X X + Y h < k X + Y - k X - X + Y > k X + Y h - k X - X + Y h > k ax (Y ) = gx - Y g < k gx - Y hg < k k X > k + Y k X > k + Y h 0 Y > k + X X + Y < k 0 Y h > k + X Tsighua Togfag Optical Disc Co Ltd All rights reserved (3) (3a)

5 ax (Y l) = (X - Y h + k) g; X X + Y l < k (Y l) (X - Y l + k) g; X + Y l - k X - X + Y l > k (Y h) (X - Y h + (3b) k) g; (Y l) (X gx - Y lg < k - Y l + k) g; (Y h) ; k X > k + Y l 0 Y l > k + X (Y l) k ; (Y h) k 8 : (Y l) ; (Y l) ; X ; (Y h) (Y h) ; (Y l) X ; (Y l) k (Y h) X ; (Y h) ; 8 (Y l) X ; (Y h) 5f X 5X = Θ(Z X - w - X ) + ( - Θ) (Z X - w - X ) X + Y l < k X + Y h < k Θ(Z X - w - X ) + 0 X + Y l < k Y h > k + X Θ(Z X - w - X ) + X + Y l < k ( - Θ) (Z X - w - X + Y h - k) X + Y h > k gx - Y hg < k Θ(Z X - w - X + Y l - k) + X + Y l > k gx - Y lg < k ( - Θ) (Z X - w - X + Y h - k) X + Y h > k gx - Y hg < k Θ(Z X - w - X + Y l - k) + 0 X + Y l > k gx - Y lg < k Y h > k + X X > Y l + k X > Y h + k Y l > X + k Y h > X + k X > Y l + k Y h > X + k (4) X (Y ) X (Y ) = (Z X - w ) X < k - Y h (Z X - w ) X < m i (k - Y l Y h - k) ( + Θ) (Z X - w ) + ( - Θ) (Y h - k) m ax (k - Y h Y h - k) < X < k - Y l + 3Θ Z X - w + Y h - k + Θ(Y l - Y h) m ax (k - Y l Y l - k) < X < k + Y l Z X - w + Y l - k Y l - k < X < m i (k + Y l Y h - k) 0 X > Y h + k 0 X < m i (k + Y l Y h - k) 0 Y l + k < X < Y h - k X (Y ) = {X {X (Y ) X (Y ) X 3 (Y ) X 4 (Y ) X 5 (Y ) }} X m ax f X X (Y ) = X (Y ) = (Z X - w ) (5) Tsighua Togfag Optical Disc Co Ltd All rights reserved

6 00 8 : 345 ( X 3 + Θ) (Z X - w ) + ( - Θ) (Y h - k) (Y ) = + 3Θ X 4 (Y ) = Z X - w + Y h - k + Θ(Y l - Y h) X 5 (Y ) = Z X - w + Y l - k x 3 l = ( + Θ) (Α- w ) + ( - Θ) (X 3 h - k) + 3Θ X 3 h = Η- w - k + X 3 h + Θ(X 3 l - X 3 h ) X > Y l + k X > Y h + k Y > X l + k Y > X h + k X 3 X (Y ) X 5 l = (Y ) Θ( + Θ) (Α- w ) + ( - Θ) (Η- w ) + (Θ - ) k 4Θ X 3 h = X Y ( ( + 3Θ) (Η- w ) + Θ( + Θ) (Α- w ) + (Θ - - 4Θ) k ) X h = Y h X l = Y l k 4Θ : k > q 3 h q 3 h q 3 h ) ax = X a Y = Y (X Y ) k < q 3 l q 3 l q 3 l = Α- w q 3 h = Η- w X 3 l = ( + Θ) q3 l + ( - Θ) q3 h Θ Θ + (Θ - ) k 4Θ q 3 l q 3 h V ar X 3 ( + Θ) q 3 l < k < m i (q 3 l q 3 l = V ar q 3 l + h ) Θ ( - Θ) V ar q 3 h + Θ (Θ - ) k V ar 4Θ m i (q 3 l q 3 h ) < k < m ax (q 3 l q 3 h ) ( q 3 - Θ) h V ar q 3 h > 0 Θ : V ar X 3 l ( + Θ) ; X X 3 (Y ) 0 < Θ< V ar (q 3 > l ) ; Y X 4 (Y ) X 3 = (X 3 l X 3 h ) m i (q 3 l q 3 h ) < k < X 3 l = X 3 (X 3 ) X 3 h = X 4 (X 3 ) 3 m i (q 3 l q 3 h ) < k < m ax (q 3 l V ar q 3 l m ax (q 3 l q 3 h ) V ar X 3 ( + 3Θ) h = Θ Tsighua Togfag Optical Disc Co Ltd All rights reserved > Θ V ar (X 3 l ) V ar q 3 h +

7 ( + Θ) Θ V ar q 3 l + (Θ - 4Θ- ) k V ar 4Θ ( + Θ) Θ V ar V ar q 3 l > 0 (Θ - 4Θ- ) k 4Θ = 0 0 < Θ< X l = Α- w ( + Θ) (Η- w ) + (Θ- ) k 4Θ 5m ax (q 3 l q 3 h ) < k < q 3 h q 3 l = Α- w q 3 h = Η- w V ar X 3 h ( + 3Θ) V ar q 3 > h Θ > 4 X h = ( + Θ) q3 h (Θ- ) k + Θ 4Θ Θ V ar X Θ ( + Θ) h = V ar q 3 h + Θ V ar X 3 l ( + Θ) ( - Θ) V ar q 3 h V ar q 3 = + l Θ Θ V ar q 3 = (Θ- ) k V ar l 4Θ ( - Θ) V ar q 3 h + V ar X Θ V ar q 3 h ( + Θ) l V ar q 3 = h 4Θ > Θ0 V ar X 3 l ( - Θ) V ar q 3 h V ar q 3 = + l Θ V ar q 3 > l ( - Θ) V ar q 3 h + 6 Θ V ar q 3 l Θ (Θ ) 7 m ax (q 3 l q 3 h ) < k < q 3 h : 4 Y Α> w X 3 (Y ) X = (X l X h) ( m ax (q 3 l q 3 h ) < ) Α{ } k < q 3 h Η= w { } Θ X l = q 3 l X h = X 3 (Y ) { } k X l = Α- w k m i (q 3 l q 3 h ) < k < m ax (q 3 l q 3 h ) m ax (q 3 l q 3 h ) < k < q 3 h X ( + Θ) (Η- w ) + ( - Θ) (X h - k) h = + 3Θ k { } k Tsighua Togfag Optical Disc Co Ltd All rights reserved X h =

8 00 8 : 347 k 3 : 3 m ax (q 3 l q 3 h ) < k < q 3 h Α= 0 m i (q 3 l q 3 h ) < k < m ax (q 3 l q 3 h ) Θ= 0 3w = 0 05 k = 0 47 Α { } Α w = 0 05 Θ= 0 k { } 3 5 Θ m i (q 3 l q 3 h ) < k < m ax (q 3 l q 3 h ) Α= 0 w = 0 05 Η= k = 0 Θ - ; - Θ : [ ]L ee H L Padm aabha V W hag S T he bullw h ip effect i supp ly chai [J ] Sloa M aagem et R eview (4): 93-0 [ ]Sterm a J D M odelig m aagerial behavio r: M ispercep tios of feedback i a dyam ic decisio m ak ig experim et [J ] M aagem et Sciece (3): [ 3 ]Tow ill D R Idustrial dyam ics modelig of supp ly chais[j ] Iteratioal Joural of Physical D istributio & L ogisticsm aagem e t (): 3-4 [ 4 ] L ee H L Padm aabha V W hag S Ifo rm atio disto rtio i a supp ly chai: T he bullw h ip effect [J ] M aagem et Sciece (4): [5 ]F rak C Zvi D Jeifer R K el al Q uatifyig the bullw h ip effect i a simp le supp ly chai: T he impact of Tsighua Togfag Optical Disc Co Ltd All rights reserved

9 fo recastig lead tim es ad ifo rm atio[j ] M aagem e t Sciece (3): [6 ]Cacho G P M arti L A Capacity cho ice ad allocatio: Strategic behavio r ad supp ly chai perfo rm ace [J ] M aagem et Sciece (8): [ 7 ]Ro seblatt M J Imp rovig p rofitability w ith quatity discouts uder fixed dem ad [J ] IIE T rasactios (4): [8 ]Kevi W Z Chael coo rdiatio ad quatity discouts[j ] M aagem et Sciece (9) : [9 ]Cacho G P M arti L A A equilibrium aalysis of liear p ropo rtioal ad uifo rm allocatio of scarce capacity [J ] IIE T ra sactio s 999 3: : (97- ) : (965- ) : (947- ) : ( 3 ) : [ ]Zeg J ili Physical ad hum a cap ital accum ulatio R &D ad ecoom ic grow th [J ] Souther Ecoom ic Joural (4): [ ] [J ] 000 (3): - 7 [3 ] R &D [D ] : 000 [4 ] [M ] : 00 [5 ] [J ] (4): [6 ]A gh io P How itt P A model of grow th th rough creative destructio[j ] Ecoom etrica (): [7 ]R eddig S T he low sk ill low quality trap: strategic comp lem etarities betw ee hum a cap ital ad R &D [J ] T he Ecoom ic Jou ral : [8 ]Cheg L K T ao Zh igag T he impact of public po licies o iovatio ad im itatio: T he ro le of R &D techo logy i grow th m odels[j ] I teratioal Ecoom ic R eview (): [9 ][] [M ] : 997 [ 0 ]Segerstrom P S Iovatio im itatio ad ecoom ic grow th [J ] Joural of Po litical Ecoom y 99 99: [ ] Stackelberg [M ] : 998 [ ] R &D [J ] 00 7 (): : (973- ) : (944- ) : Tsighua Togfag Optical Disc Co Ltd All rights reserved