Jointed pricing decision of remanufacturing system under uncertain demand

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1 õ V } 33 8 Vol33 No Systems Engineering heory & Practice Aug (013) F4!"#$%&')(*)))-) A / ; (1 <=>@?A>BC@DFEGHIJ@DK <= 10016; LMNOC@DPEGIJ@DK <@ ; 3 =S@DKPSU@DK = ) WYX[Z\[^_à bcd[è f ghi[jklmǹ o pqrstu[vw[xỳ z è ~ ^_ vw[} vẁ [ l ƒ cd ˆ Š [pqœžu[vw[l z [ g t[p š i[œ Karush-Kuhn-ucker žÿ ; e z i œ ª ª«~ ^ª_ v wª ª ª ª ª±ª²ª³ eª^ª_ªa lª µ vª ªƒª ª}ªŒª ª¹ ºª»ª¼ ½[¾ ~ ^_ vw[ ª ª±ª²³ À vw[ ÁẦ ÃoÄ ~ ^_ vw[ ÁÅÆ ªÀ ~ ^_ vw vw[v ÇẦ ÃoÄ ^_a Œ ½[È É Ề ËÌ jklhi ; WP ; u[vw[xỳ z ; l Jointed pricing decision of remanufacturing system under uncertain demand GUO Jun-hua 1 YANG Li 3 LI Bang-yi 1 NI Ming (1 School of Economics and Management Nanjing University of Aeronautics and Astronautics Nanjing China; School of Economics and Management East China Jiaotong University Nanchang China; 3 Logistics Department Beijing Materials Institute Beijing China) Abstract Based on the two-item newsvendor model this paper researches on the jointed pricing decision of remanufacturing system under uncertain demand from the perspective of solo manufacturer Firstly the two-item newsvendor model is built based on profit maximization hen the concavity of objective function is proved and the Karush-Kuhn-ucker optimal conditions are given Moreover the solutions of the model and the influence of remanufacturing products WP to pricing productivity and profit are analyzed numerically he results show that the new products pricing and two products productivity will decrease while the new products pricing and manufacturer s profit will increase with the increasing of remanufacturing products WP Keywords uncertain demand; WP differentiation; two-item newsvendor model; jointed pricing 1 ÏÐ ÒÓỒ Ö ÙÚÛǛ ß ýªþ ö ªÿªîªï ªà àáầ ãäåæçèéề ëìíîïðñòóồ õö[ øùúûü ªÙ æ ªå 1991 (ªªû ) (Xerox)! " # $ % & ' è * ªÿªîªï ªá ' - / 0 ' [1 ; 1 (Kodak) 3343#353637èö[ ÿîï [ ồ õ 3ø 33?3@ 3G3H3I3J 3;3<3=3> îïð 3å A3B3C3D3E3>3F 3K3L 3M [3ÓN P3 [5 M [4Ś3 M U3V 3W3X3Ỳ 3Z3[3\3536ồ ø õö[ ; ^3_33a3b3c3d3eö[ 3536ô ªø o p ÿªîªï f g h i j k l \ m nªôªõªÿªîªï q c r U t u v ÿªîªïªôªõª sªôªõ [6 wyxzo }y~y y ƒo y y y yˆ ( ); y y yšy y y yˆ (1BGL104 10BGL010); yœyƒo y y yˆ (01BAB11011); yœyyžy y y (O y ) y yy y y (01BCB3009) ỹ yšy œyyž (1976 ) Ÿ y y y y y y ÿ yªy«y ªy«y ȳ y±y²y³yýµy y y y¹yº ;»y¼y½ (1963 ) Ÿ ¾y yàyá y ÿ y yâyã ªy«y ȳ y±y²y³yýµy y y y¹yº ; ÄyÅ (1974 ) Ÿ ÆyÇyÈyÉy y ÿ ªy«y ȳ y±y²y³yýµy y y y¹yº

2 è v õ j Ë &? M N o Ú Û o Ú 1950 ±Ê²Ê³Ê ÊºÊËÌÊÊÏ 33 Ð3 Ferrer å Swaminathan 3Ò ÿîïồ õ sồ õ ầ ãªä Ó3Ô Ö (willing to pay WP) 3 3Ù t3u 3Û Ù 3Úîïð ô3ü [7 ß3à3M [8 án33â3m [9 áã3ä3m [10 áå3æ3ç3m [11 áq [1 M U3V m3ǹ ÿîïồ õ sồ õ WP t3u ø 3ì3í3 3;3*33\3Z3[ q3é3ê3d3è ồ õ3ëí [14 å3æ3ç3m [13 ïî M U3V3ð ì3ñ3ò3ó ì3í33 3ø3Û Z3[3\3ëí ÿîïồ õ q3b3c3d3è ÿî 3t3u ïồ õ3sồ õ û3û ÿîï õô ö3øîï 3ù3ú J3ü3ý Aõþ ǹ ÿîïồ õ3sồ õ WP t3u 3? E3ÿ J3ü3ý ì3í33v [15 n ồ õ ëí 333^3_33a m 3^ ñ 3d3è I 3? ÿîïồ õ3sô ì3í33v ëí ð 3ò3ó 3v o3p lîïð m3n ồ õ Ú3sồ õ ÿî ø3û ù o3p ïồ õ 3* 3k \í Karush-Kuhn-ucker ;!3" *3í U \ÿîïồ õ ầ ãä3ó3ô33ö3úîïð ø 33 Áô! "$#$%$&$'$(*)- 1 / ;<=>?@ABCDEFG?IHKJ 67SUVWXYZ[ LMNOP \ D E F G?H^J _? a b c d e 6 7? f g h i 6 7 l m n o p q 6 7 ρ (0 < ρ < 1) ρ j k l m? r [ s t u p w F G x 6 7 F G u y z } ~ v F G? 3 p F G? M ƒ J ˆ Œ 3 B C Ž Š 6 7 8? k 3 w F G x 6 7 F G š }? œ ž Ÿ B C F G? Ž 3 U ˆ a O ª «B C [ ± ² ž ³ \ 3 w F G F µ Z [ 6 7 F G F µ Z [ 3 1 c n c r L c d e 6 7? Z [ ¹ h º» 6 7 F G¾^w F G ¹ h? µ Z [ ; = c n c r ¼ ½ 3 À w F G 6 7 F G? ÁÂ Ã 3 ~ [ ± U Ä Å ~ Æ ~ 3 ~ Ç È É Ê À w F G À 6 7 F G À ¼ ½ D n = d n (p n p r )ε n o45wfgï 67FGÏ?Ê Ë ¼ ½ x š45 NIÌ ε n ε r p n p r Ð ε n ε r N _ Ò Ó Ô o Ô ³ f n ( ) f r ( ) Ö ¼ ½ o w F G 6 7 F F n ( ) F r ( ) p n p G? Ï x ÀÏ Ù?Ú 3 67FG?Û xwfgü r d n (p n p r ) d r (p n p r ) k àáâ?ãäåæ çèé ßkIv 67G?çêëxwFGzìí cdîï j ðñ kòó ±ö WP ôõ? ø àáâwfgx 67FG?ùêzìíú [7 9 3 À û w F G x 6 7 F G? o ü WP 1 ž } β d n (p n p r ) d r (p n p r ) ý þ ÿ o w F G 6 7FG? Ï d n (p n p r ) = pn pr d r(p n p r ) = βpn pr β() p n p r ; NÌ à á â w F G? ù ê ë Y 6 7 F G 0 < β < 1 [ ± Ä } > B C r À B C w F G 6 7 F G? F F? n r v 3 ~ w F G 6 7 F G F G W F? µ Z [ o Ä M Ü? µ F G ë o Ä h r (h r > 0) ¼ ½ ; s n s r ¼ ½ p q Š 0 < s n < p Ü? F G µ ë b À Ï n Ð 3 p à á â î ï p µ w F G W 6 7 F G W F G 4 [ H^J? D E F G F 6 7 F G? 5 þ ý þ [17 3! #"#$!%#&#'!(#)#*!##&!' c n > c r > 0 [16 D r = d r (p n p r )ε r h n (h n > 0) 0 < s r < p r 31 - / 1 Ï º3?3 Birge L89 [18 Petruzzi 0 67 B C e ; É } ~ F G B Ce <= Ï [19 ; Das ß >? 3 [ ±? [0 ; A ± ö B C 3? } ~ F G? Ž 3 D E < = B C 3 ú w F G 6 7 F G? À?Ê Ë ~ G = H w F G v F F G? F F x L? M O P Z [ W S S FG? I J O K N Š

3 q v M 8 Z } k N 8 U VXWXY Z [X\XX^X_XXaXbXcXdXeXfXgXhXXiXjXk 1951 ë J ˆ U V w F G U V 6 7 F G U V Ú v l w F G À m w Š ü D F G F ú S F G S F G? ë J ˆ J ˆ n n n D n ( n D n )s n v F Z [ ú U V À a w F G F D n c n n p n D n c n n ( n D n )s n ; ü ú w F G Z [ ú J ˆ F Z [ n D n n (D n n )h n p n n c n n ú U V ; ê w F G? n o U V p n n c n n (D n n )h n ¼ ½ p n D n p 6 7 F G? n o U V π n = pn D n c n n ( n D n )s n D n n p n n c n n (D n n )h n D n > n (1) z n = n d n (p n p r ) = r d r (p n p r ) max π = p np rz n pr D r c r r ( r D r )s r D r r π r = () p r r c r r (D r r )h r D r > r z n zr 6 7 8? [(p n c n )(d n (p n p r ) x) s n (z n x)f n (x)dx [(p n c n )(d n (p n p r ) z n ) h n (x z n )f n (x)dx [(p r c r )(d r (p n p r ) y) s r ( y)f r (y)dy [(p r c r )(d r (p n p r ) ) h r (x )f r (y)dy (3a) st d r (p n p r ) ρ(d n (p n p r ) r z n ) (3b) p n p r 0 (3c) 3 - r s t u 1 (3a) (3b) (3c) ¼ v? z º? O w x [1 F (z n ) < (p n h n s n c n )f n (z n ) (p n h n s n c n )f n (z n ) (1 β)[1 F n (z n ) (p r h r s r c r )f r ( ) 4(p n h n s n c n )f n (z n ) β(1 β)[1 F r ( ) < 1 β(1 β)[1 F n (z n ) [1 F r ( ) y z Ô œ d n (p n p r ) = pn pr d r (p n p r ) = βpn pr β() 5 (3a) } max π = (p n c n ) p ) p np rz n 1 β x s n (z n x) f n (x)dx (p n c n ) p ) z n 1 β z n h n (x z n ) f n (x)dx zr ) (4) βp (p r c r ) β(1 β) y s r ( y) f r (y)dy ) βp (p r c r ) β(1 β) h r (y ) f r (y)dy N ~ H = 1 F n (z n ) 0 β() 0 1 F r ( ) 1 F n (z n ) 0 (p n h n s n c n )f n (z n ) F r ( ) 0 (p r h r s r c r )f r ( ) p ƒ H 1 = < 0 ƒ 4 H = β() > ƒ H 3 = [1 F (z n) ƒ β() 4(pnhn sn cn)fn(zn) β() ˆ H 4 = (p r h r s r c r )f r ( ) H 3 (1 F r ( )) [ (pnhn sn cn)fn(zn) (1 F n (z n )) O F z º Š 3 (4) N ~ H 1 < 0 Z5 wx H > 0 Œ ýþ H 3 < 0 H 4 > 0 [1 F (z n ) < (p n h n s n c n )f n (z n ) (p nh n s n c n)f n(z n) ()[1 F n(z n) (prhr sr cr)fr(zr) β()[1 F r() < 1 4(pnhn sn cn)fn(zn) β()[1 F n(z n) [1 F r() 1 8 Ž

4 ³ S N S v? Ì? Š e 7 3 ½ L ~ 195 ## # # #! # # Ô Ô h i w x Á Ù? Å ~ Ô š œ ^> U Ä (4) Ì } º w x Æ Ð p n p r Karush-Kuhn-ucker l N Ô Lagrangean L(p n p r z n ) = (p n c n ) p ) 1 β x s n (z n x) f n (x)dx (p n c n ) p ) z n 1 β z n h n (x z n ) f n (x)dx zr ) βp (p r c r ) β(1 β) y s r ( y) f r (y)dy (5) ) βp (p r c r ) β(1 β) h r (y ) f r (y)dy λ ρ p ) ( ) 1 β z βp n β(1 β) u n p n u r p r º w x ž Karush-Kuhn-ucker = (p ) (c n c r ) λ(1 ρ) p n 1 β = (βp ) (βc n c r ) λ(1 βρ) p r β(1 β) zr xf n (x)dx yf r (y)dy 33 z n z n f n (x)dx u n = 0 (6a) z n = (p n h n s n c n )F n (z n ) (p n h n c n ) λρ = 0 f r (y)dy u r = 0 (6b) = (p r h r s r c r )F r ( ) (p r h r c r ) λ = 0 (6d) λ ρ p ) ( ) 1 β z βp n β(1 β) = 0 (6e) u n p n = 0 (6c) (6f) u r p r = 0 (6g) Ž 5 (6a) (6h) 8 ƒ ž ž λ u n u r 0 (6h)? º š Ÿ Å ~ º ßv ë B C? º B C N 4 ª«[ ¹ U Ä ë B C 3 à á â 6 7 F G? œ± ë? º WP β 8 9 ² 3 J ˆ Ê β ³? > µ x Á º ¹ ε n ε r N U(0 )? Á» Ö N o ë ž ž = 60 β = 07 c n = 10 c r = 4 ρ = 06 h n = s n = 6 h r = s r = ¼ ž ë } 5 Karush-Kuhn-ucker º w x U Ä (6a) (6h) L B C º Matlab 8 p n ë º } = p r = 3535 zn = 413 zr = 957 π = M 5 1 BC½ Ž É? Nýþ 1 wx ë ÁÄ j¾à wxbc? Karush-Kuhn-ucker p~ BpÂ wfg º F 67FG º F 8 n = p n p r z n = r = βp n p r β() z r = Ã? ë Ç B p Â B C Ä A Å Æ? w F G 6 7 F G β? 3 F U V? o v 0 < β < 1 β ë } 5 B C (4) Ç ¼ 1 ½ ; È Ž ú Z 5 β 1 É ² Ê Ë 3 F U V Ê ½ β? > 6 Z Ï Ð B C 3 Ê M ¼ 1 Ì NÌ β? > 1 ½ F Ê β? > ½ U V Ê? β > 3 ½

5 ö 9 ú 9 Š S 8 Ò ñ Æ 8 U VXWXY Z [X\XX^X_XXaXbXcXdXeXfXgXhXXiXjXk ÓXÔ β XÖX XXÙXÚXÛXÜX β p n p r n r π ÓXÔ β ßXàXáXÚXâXã ÓXÔ β ßXäXåXÚXâXã 3 ÓXÔ β ßXæXçXèXéXêXÚXâXã ¼ 1 1 B C 3 ž ë ø 8 9 (1) v 1 ¼ 1? ì C 9 Ê î à á â 6 7 F G? ë? ï Æ î à á WP β ð ñ â 67FG?ùêòÓóm 67FG?Ïô ³ïõ?ö» wfgï L øùú wfg ;? Ï Ã û ð ô X? þ» w F G Ï Á m 6 7 F G Ï ß ü ý Ð ÿ ú 6 7 F G β = 09?3 m wfgï Êîàáâ 67FGùêòÓ?ïÆ 67FG Ð j¾à ùú w F G? Ï ; Š ÿ ú à á â 6 7 F G? Y ú 6 7 F G β = 01 k ¾ ð Y? Ï À ð () v ¼ 1? ì C í ˆ 9 Ê î ë? a w F G 6 7 F G? F β β = 0 ú Ã ï Æ Ê ³ Ê u Š ø β ë? a 6 7 F G? À Ï ï Æ À? F N Š I J 6 7 F G? Y Z [ S U ~ ; Šv ö^ 6 7 F G? F w F G? F ³ X F 8 (3) v 3 ¼ 1? ì C í ? U V Æ ú β = 0 ³ Ê ë? ï Æ ï Æ β Š v 6 7 F G? Y Z [ º» Ê î à á â 6 7 F G? ó m 6 7 F G? S U ~ ð F 8 Y? F m? À Ï m?u V ³ š 6 7B CaA j ¾œ ó m à á â 6 7 F G? ù ê p S m? U V p 6 7 f g? á D? F

6 A Ž» 3 j e Ÿ ³ ## # # #! # # B D N Š È ð É ø? Ó [ ± Ê 3 Ë! " # 5 ² ô õ β $ % & ' ( ë o 5 ì p (1) = 80 c n = 1 c r = 6 ρ = 07 h n = s n = 8 h r = s r = 4; () ì = 10 c n = 8 c r = ρ = 08 h n = s n = 6 h r = s r = 1; (3) = 90 c n = 10 c r = 4 ρ = 06 h n = 7 s n = 6 h r = 3 h r = ; (4) ì ˆ = 160 c n = 16 c r = 6 ρ = 05 h n = 9 s n = 8 h r = 5 h r = 3; (5) ì ð ½ ê = 00 c n = 13 c r = 5 ρ = 07 h n = 8 s n = 7 h r = 4 h r = ; È ) j ë º Á É w x \ * p ~ ý þ 1 ¾œ œ ð É ø e ø / ~ 3 É ² - j ¾œ Ä 0 A Å Æ π 7 < = 8 β 9 ; A B C 5 DFEFGFHFIFJFKFLFMFNFOFP β FFSFFUFVFW XFYFZ XF[FZ XF\FZ XFFZ XF^FZ β j 5 _a b c d e f g h i f g j k lm no p qj rs t e f g x i y f g z j } ~ 3 u v w fg ƒ efghiy fgj ˆq Š Œ wž Œ Œ w8 i y f g j WP β n š f g jq œžf Ÿh y j Ž w 8 7 ª j «β e f g t A i y f g ± «e f g h i y f g f Ÿ m t y ² 7 z «³ b c µ Œ ¹ º b c ¼ y j Š ¾» u v w ½ À d Å Æ j r s u v Á Â Ã Ä Ç È f g jë o b c» É Ê 5 Ì A Ï ρ o p q y j Š Œ 9 Ð ;» Ò j Ó Ô Ö z ; Ù Ú Û [1 Berko-Boateng V Azar J De Jong E et al Asset recycle management A total approach to product design for the environment[c// Proceedings of the 1993 IEEE International Symposium on Electronics and the Environment Arlington VA IEEE [ oktay L B Wein L M Zenios S A Inventory management of remanufacturable products[j Management Science (11) [3 ÜFF ßFàFá âfãfä åfæfçfèfféfêìëîífïfðfñfòfófôfõfö [J FøFùFúFûFFüFýFþ 005 5(3) 0 5 Gu L Gao G Shi L S Price decision analysis for reverse supply chain based on game theory[j Systems Engineering heory & practice 005 5(3) 0 5 [4 ÿ ífïfð êìëîífïfðf FøFéFñFòFóFô [J Fû (4) Wang Y Y Li B Y Shen L he price decision model for the system of supply chain and reverse supply chain[j Chinese Journal of Management Science (4) [5 åfæfçfèffé FíFïFðFñFòFóFôFõFö [J FøFùFú 008 3(1) Ge J Y Huang P Price decision analysis for closed-loop supply chain based on game theory[j Journal of Systems Engineering 008 3(1) [6 åfæ Fé FíFïFð! "FñFò # $ [J % & ' (Fü Fû (9) 17 1 Yi Y Y Pricing coordination in a closed-loop supply chain with remanufacturing[j Journal of Business Economics (9) 17 1 [7 Ferrer G Swaminathan J M Managing new and differentiated remanufactured products[j European Journal of Operational esearch () [8 ) * - / åfæ Fé 7 8FéFñFò ófô #$ [J Fû (6) Xu F Sheng Z H Chen G H he remanufactured products pricing strategy in a heterogeneous market[j Chinese Journal of Management Science (6) [9 ÿ 9 ; < = A BFõFé FíFïFð C 7FüFñFòFóFô [J D Fü EFó 009 4(5)

7 X 8 F G H / I J KFñ L M NFé 7 8 O PFñFò EFó 1955 Wang W B Da L Production and pricing strategy of closed-loop supply chain considering market segmentation[j Control and Decision 009 4(5) [10 ; < = åfæ 7 8 S 0Fé FíFïFðFñFòFü!" # $ [J Fû 010 7(5) Sun H Da L Pricing and coordination of remanufacturing closed-loop supply chain based on product differentiation[j Chinese Journal of Management 010 7(5) [11 U V W X V W Y Z åfæ\[ Fé FíFïFð S 0FñFòFóFô^! " [J Fø Fû (5) Bao X Y ang Z Y ang X W Coordination and differential price strategy of closed-loop supply chain with product remanufacturing[j Journal of Systems & Management (5) [1 _ a b * 8 c dfñfò A e A g h i ïfõfö [J FøFùFúFûFFüFýFþ (8) Dan B Ding X F Optimal pricing for the remanufactured products and analyzing the effect of cannibalization and market growth[j Systems Engineering heory & Practice (8) [13 U V j k l m n 7 8 c dfñfòfófô # $ [J o p q r k F Fø (1) Bao X Y Yang C H Pu Y Optimal pricing policy of the remanufactured products[j Computer Integrated Manufacturing Systems (1) [14 s t u v Y w x q 7 y z x q L M NFé F FøFé\[ FñFò EFó [J FøFùFúFûFFüFýFþ 009 9(8) 19 8 Li x Li Y J Cai X Collection pricing decision in a remanufacturing system considering random yield and random demand[j Systems Engineering heory & Practice 009 9(8) 19 8 [15 Guide V D Li J Y he potential for cannibalization of new products sales by remanufactured products[j Decision Sciences (3) [16 Savaskan C Bhattacharya S Van Wassenhove L N Closed-Loop supply chain models with product remanufacturing[j Management Science () 39 5 XF\ } [17 ÿ 9 ; < = %Fü [ N FíFïFð\[ FüFñFò #$ [J FûFùFú 010 4() Wang W B Da L he study on collection and pricing for closed-loop supply chain with retailer and the third party collecting[j Journal of Industrial Engineering and Engineering Management 010 4() [18 Birge J Drogosz J Duenyas I Setting single-period optimal capacity levels and prices for substitutable products[j he International Journal of Flexible Manufacturing Systyms [19 Petruzzi N C Dada M Pricing and the newsvendor problem A review with extensions[j Operations esearch [0 Das B Maiti M An application of bi-level newsboy problem in two substitutable items under capital cost[j Applied Mathematics and Computition (1) 410 4

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