The Dynamic Multi-Task Supply Chain Principal-Agent Analysis

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1 J. Servce Scence & Mngement 009 : 9- do:0.46/jssm Publshed Onlne December he Dynmc Mult-sk Supply Chn Prncpl-Agent Anlyss Shnlng LI Chunhu WANG Dol ZHU Mngement School Fudn Unversty Shngh Chn; Mngement School Soochow Unversty Suzhou Chn; Informton School Shngh Ocen Unversty Shngh Chn. Eml: Lsl@fudn.edu.cn eceved August 8 009; revsed September 5 009; ccepted November ABSAC In the supply chn by the composton of the suppler nd the retler the suppler offers products to the retler for sles whle the retler ffects the sles outcome by hs effort whch s dvded nto two dmensons. One s for the short-term sles tsk nd the other s for the long-term sles tsk. For the long-term development of the enterprse the suppler wnts to nspre the retler to mke more effort for the long-term tsk. However due to the symmetrc nformton the suppler cn t observe the retler s cton nd the morl hzrd wll come nto beng. o del wth ths problem we construct the dynmc mult-tsk supply chn prncpl-gent model by whch we nlyze the mpct of the nformton symmetry to the supply chn contrct. Furthermore by comprng the contrcts between the sngle-term mult-tsk nd two-term mult-tsk we hve nlyzed ther dfferent effect on the commsson rte. Keywords: Supply Chn Mngement Mult-tsk Prncpl-gent Dynmc Incentve Morl Hzrd. Introducton In the supply chn system there exsts the conflct between the locl nterests of the supply chn members nd the overll performnce of the supply chn whch leds to the system neffcency. At present one of the most mportnt reserch res of supply chn s to desgn the sutble coordnton mechnsm n order to obtn the globl optmzton of the supply chn performnce. In ths process the nformton plys very mportnt poston. As the supply chn members tend to hde ther prvte nformton to mntn nformton superorty ths wll led to Adverse Select nd Morl Hzrd n vrous felds []. In the recent decde scholrs hve studed on the ssue of the supply chn coordnton from vrous ngles. hese studes cn be roughly dvded nto two ctegores. One s to resolve the double mrgnlzton problem by contrct desgn n the stuton of the full nformton [ 4]. Such contrcts do not nvolve the nformton ncentve. he other s to study the supply chn ncentve problem n the stuton of the symmetrc nformton. Corbett etc. studed tht the optml quntty dscounts ncentve contrct between the suppler nd the retler [5]. Bsu etc. studed the ncentve ssues of the sles force under symmetrc nformton bsed on gency theory [6]. Ll etc. [78] nd Chen [9] extended the bove studes. Mny Chnese cdemcs re lso crred out reserch on ths ssue [0 4]. For the supply chn coordnton the uthor s reserch tem hd systemc reserch on the ssue erler. Some relevnt reserch results hve been publshed [5 ]. hs pper s the mportnt one of the systemc study. In the smple prncpl-gent model the gent s engged n one job nd the gent s effort s one-dmensonl. However n mny cses of the rel lfe gents re engged n the job of more thn one. Or even f there s only one job t nvolves more thn one dmenson. Furthermore t exts conflct n the dstrbuton of the sme gent s energy between the dfferent jobs. o del wth ths problem we construct the dynmc mult-tsk supply chn prncpl-gent model by whch we nlyze the mpct of the nformton symmetry to the supply chn contrct. Furthermore by comprng the contrcts between the sngle-term mult-tsk nd two-term mult-tsk we nlyzed ther dfferent effect on the commsson rte.. Assumptons nd Prmeters Set We mke the followng ssumptons for the trctble nlyss. Consderng Stckelberg model between suppler S who s the prncpl nd retler who s the gent the suppler offers the retler products to sle nd pys the retler ccordng to sles outcome whch s Copyrght 009 Sces

2 0 he Dynmc Mult-sk Supply Chn Prncpl-Agent Anlyss ffected by the retler s effort nd the rndom fctors. Set B ) s the retler s expected proft whose ownershp belongs to the suppler. denotes the retler s effort for the short sles gol. denotes the retler s effort for the long sles gol. C ) denotes the cost C C of the retler s effort stsfyng 0 0.e. the cost of the effort ncreses nd the mrgnl cost ncreses. For the smplcty Assume C ). he suppler cn t observe nd but cn observe nd verfy the sles outcome x whch s ffected by the retler s effort together wth the rndom vrbles denoted by x ) where ) s the output functon of the effort stsfyng 0 whch mens the mrgnl sles outcome of the effort s postve..e. more efforts men more sles; 0 whch me- ns mrgnl sles outcome decrese When the equl sgn s set up mrgnl sles unchnged). Set ) whch s the rndom vrble of Norml dstrbuton nd stsfy N 0 ;0 ; r) ; Set x x x) For the ske of smplfyng the clcultng ssume tht x ) ; x )..e. dfferent efforts result n dfferent nformton However dfferent nformton my be relevnt f nd re relevnt. ): x reflects x reflects. he ownershp of the sles profts belongs to the suppler nd the suppler offers the lner slry to py the retler. sx ) x x x ) where s x ) s the wge pd to the retler f the totl sles outcome s x. denotes the one-off welth trnsfer whch doesn t ffect the ncentve ntenson Clled Slry); ) whch denotes the ncentve ntensonclled Commsson te) tht mens f x ncrese by one unt the wge of the retler ncresed by unt.. he Sngle-Stge Mult-sk Model In the sngle-stge model the suppler offers one-tme wge schedule s x ) ccordng to whch the retler s rewrded. Assume the suppler s rsk-neurl the expected utlty functon s s follows: EU S B ) E ) ) B ) Assume the retler s rsk-verse nd the utlty s tht V x) e x where s the rsk verson coeffcent. When 0 the retler s rsk-neurl. When 0 the retler s rsk-verse. When 0 the retler s rsk preference. he retler s expected utlty s s follows: EU EV ) s x C )) ) o mke the nlyss smple we use the certnty equvlent CE) nsted of the expected utlty of the retler [8]. CE ) C ) 4) where ) s the expected wge s rsk verson coeffcent s the ncome vrnce / s the rsk cost. s the covrnce mtrx r of nd denoted by. r he suppler s the leder n the Stckelberg model who hs frst-step dvntge n the gme. However when he/she pursuts the proft mxmzton he/she must consder the ncentve comptblty constrnt nd prtcpton constrnt. hus the prncpl-gent model between the suppler nd the retler cn be rewrtten s the followng optmzton progrmmng. P) MxEU B ) 5) S s.. t I) CE r ) 0 r 6) IC) rg mx CE 7) where 6) s prtcpton constrnt I) nd 7) s ncentve comptblty constrnt IC).. he Full Informton Benchmrk In ths secton let s begn wth the full nformton cse where the retler s effort s observble nd verfble. hen the suppler cn ssgn n effort level to the retler by desgnng forcng contct. Under ths condton the ncentve comptblty 7) s nvld nd we only consder the prtcpton constrnt 6) whch s bndng. Nmely P) cn be rewrtten s follows: P) Mx US B ) 8) s.. t CE 0 Copyrght 009 Sces

3 he Dynmc Mult-sk Supply Chn Prncpl-Agent Anlyss Solvng P) we cn obtn tht: B ) C ) B ) C ) he Equton 9) s the clss condton of the Preto optmlty: the expected mrgnl proft of the effort s equl to the expected mrgnl cost. ht s smlr to the sngle-tsk cse. We hve the followng concluson. Proposton :Under the condton of full nformton by desgnng the lner ncentve contrct the gme between the suppler nd the retler cn cheve the Preto optmlty when the retler hs mult-dmensonl effort. 9). he Asymmetrc Informton Cse Generlly the suppler cn t observe the retler s cton nd only cn observe outcome x. In ths cse the ncentve comptblty constrnt 7) s vld. Substtutng 7) by the frst-order condton we cn obtn the equvlent progrmmng..e. 7) s equl to tht 0) Solvng the model P) r Mx B Substtutng by 0) we get ) ) r Solve the frst order dervtve of ) 4) bout obtn B r) ) 0 ) Solvng the bove equton we obtn tht B r ) Smlrly we get B r A) 4) By ) 4) we get the followng concluson. Proposton : When 0 the retler s rsk-neurl B C then )whch mens the gme cn get the Preto optmzton just s the full nformton cse. When 0 )s n nverse rto wth the rsk verson coeffcent wll reduce the ncentve ntensty ; )s n nverse rto wth vrnce ); n nverse rto wth the Covrnce r.e. the rndom fctors lso reduce the ncentve ntensty of. s lso n nverse rto wth. More mens less nd vce vers. 4. wo-stge Mult-sk Gme Model In the two-stge mult-tsk model suppose the retler s effort for the long tsk n the frst stge wll ffect the proft n the second stge of the supply chn. Set B ) denotes the expected effort proft of the frst Mx B ) ) r ) ) stge of the retler B ) denotes the expected proft of the second stge. Where nd denote the effort for the short tsk nd long tsk respectvely. Becuse the effort n the frst stge wll ffect the proft n the second stge t wll be the vrble of the output functon of the second stge. he ownershp of B ) nd B ) belongs to the suppler the suppler offers the wge schedule ccordng to the two-stge outcome. Smlrly to the ssumpton of onestge the observed outcome n the second stge s tht x x x x x x ) 4 5 5) where x x x x4 4 x5 5. Assume 4 5) whch s the rndom vrble n the second stge Independent wth ) the rndom vrble n the frst stge. he suppler offers the two-stge pyoff contrct ccordng the observed outcome s follows. s x) x x s x) x 4x4 5x5 he suppler s expected utlty s tht: EU S B ) Es x) B ) Es x) 7) 6) Now the certnty equvlent of the retler n the frst stge nd the second stge s tht CE 8) 4 5 9) CE Copyrght 009 Sces

4 he Dynmc Mult-sk Supply Chn Prncpl-Agent Anlyss where s covrnce mtrx 4 nd 5. Denoted by r r r r 0) r r For obtnng the retler s optml effort of the second stge solve rg mx CE nd get 4 ) Consderng the prtcpton constrnt nd ncentve comptblty constrnt the suppler need solve the followng progrmmng: P) MxE U B ) Es B ) Es P s.. t IC) rg mx CE CE I) CE CE 0 ) Insted IC) n ) by the frst-order condton nd substtute by 0) ) Solve the progrmmng P nd get B ) r ) B ) r4[ r5 + r r + r r 4 5 ) + 5 Solve by dervte ) bout nd get ] ) B B ) r 0 4) B B r Solve by dervte ) bout nd get B B ) r 0 B B r B) 5) 6) 7) Becuse B ) doesn t nvolve the vr- B ble 0. he Equton ) s the sme to the Equton 5). Comprng 4) wth 7) becuse B 0 t s evdent B>A. hus we hve the followng concluson. Proposton : By desgnng dynmc mult-tsk contrct the suppler cn nspre the retler to py more effort for the long-term gol wthout the premse of chngng the retler s effort for the short gol. It shows tht the dynmc contrct desgn s conducve to mntn the long-term supply chn prtnershp. 5. Conclusons he supply chn contrct desgn s the mportnt mens of the supply chn coordnton. For dfferent envronment t wll gretly mprove the level of supply chn collborton by the desgn of pproprte contrct. In ths pper we hve studed the ncentve contrct between the suppler nd the retler. Becuse of symmetrcl nformton the suppler cn t observe the effort level of the retler. herefore the suppler cn only nspre the retler s dfferent effort level by the ncentve mechnsms desgn. he mjor study of the pper s on how to desgn the dynmc ncentve contrct to stmulte retlers to py more efforts for the long-term under symmetrc nformton nd mult-tsk envronment whch hs the gudng role for estblshng the supply chn dynmc llnce. At the sme tme our study extends the exstng reserch results of the prncpl-gent. In our reserch work for the ske of smplfyng the techncl nlyss nd the clcultng we focused on the second-term mult-tsk gme. In the future we wll extend our reserch to mult-term mult-tsk model whch would be chllengng nd menngful. 6. Acknowledgements he uthors would lke to thnk the referees for ther helpful suggestons. he work s supported by Chn Postdoctorl Scence Foundton under Grnt No nd Specl Grde of Fnncl Support from Chn Postdoctorl Scence Foundton under Grnt No nd Ntonl Nturl Scence Foundton under Grnt No EFEENCES [] W. Y. Zhng Gme heory nd Informton economcs [M] Shngh People's Publshng House 997. [] G. Cchon Supply chn coordnton wth contrcts Hndbooks n opertons reserch nd mngement scence: Supply Chn Mngement North Hollnd 00. [] G. Cchon nd M. Lrvere Contrctng to ssure supply: Copyrght 009 Sces

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