Investment decision for supply chain resilience based on Evolutionary Game theory

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1 Invetent deciion for upply chain reilience baed on Evolutionary Gae theory Xiaowei Haijun Wang Manageent School Huazhong Univerity of Science and Technology Wuhan, Hubei, 4374, China Abtract We tudy the factor and iproveent ethod of upply chain reilience, then analyze the invetent deciion between upply chain eber baed on the Evolutionary Gae theory. We deduce that the governent hould ue punitive and ubidy ethod to control upply chain eber behavior and enhance the reilience of upply chain. Key Word: upply chain reilience, upply chain diruption, Evolutionary Gae theory Introduction Recently, upply chain diruption ha been tudied by expert and cholar in every vocation. Due to upply chain diruption, operating cot riing and brand influence declining often occur. When oe unexpected diater event jut like 9 happen, ieaurable econoic lo and caualtie will occur. In 2, the Philip chip factory in US fired, cauing the procureent of Ericon interrupted, reulting in a direct lo of $4 illion, and it ipact force Ericon' arket hare dropped fro 2% to 9%. It i increaingly iportant to iprove the upply chain reilience to reduce diruption loe. In thi paper, we firt tudy the factor and iproveent ethod of upply chain reilience baed on previou tudie by other cholar. In order to iprove upply chain reilience, copanie need to invet in any apect and only all eber of upply chain invet, the reilience of the whole upply chain can be enhanced. Then we build the odel of invetent deciion for reilience to analyze invetent deciion between upplier and anufacturer baed on the evolutionary gae. After that, we ue regulatory echani to propt all eber of upply chain to invet for upply chain reilience. For factor of upply chain reilience, Petti et al. (2) conidered that the factor of reilience can be claified a capability and vulnerability factor. If capabilitie increae and vulnerabilitie decreae, upply chain reilience will be iproved. For the iproveent ethod of reilience, Lee et al. (25) learned fro total quality anageent, found that copanie can achieve a high level of afety through proper anageent eaure and proce reengineering. Chritopher and Peck propoed a few iportant principle which can help copanie to enhance reilience. Sheffi (2) found that there are 3 way to enhance

2 upply chain reilience: increae redundancy, enhance agility, and change corporate culture. Evolutionary gae theory cobine gae theory and dynaic evolution, tudie the trend and tability of the fraction of eber in the gae. Recent year, any cholar ue evolutionary gae to tudy upply chain proble. Xiao et al. (26) tudied upply chain diruption and coordination baed on evolutionary gae. Zhu et al. (27) tudied the evolutionary gae between core copanie and governent in green upply chain. Becaue upply chain eber have bounded rationality, they can not necearily ake the optial deciion. So it i ore practical to ue evolutionary gae to analyze thi topic. The factor and iproveent ethod of Supply chain reilience A entioned above, after getting elicitation fro Petti tudy, we conider that fir hould conider the vulnerability and capability factor of upply chain, and build the ore balanced reilience. In our tudy, via reearching previou tudie of other cholar, we propoe the pecific ethod to enhance reilience in three apect. The claification i internal to the fir, external to the fir and virtual upply chain, hown a Table. It hould be noted that, the virtual upply chain refer to inforation yte and inforation traniion yte on the Internet which the actual upply chain relie on. Claification Internal to the fir External to the fir Virtual upply chain Table -The iproveent ethod of upply chain reilience The iproveent ethod Appropriate redundancy Supply chain flexibility Supply chain agility Viibility of the fir Diperibility of the fir Rik analyi and prevention Diruption repone eaure Rik anageent culture Deign reilience in upply chain Collaboration between fir Fitne to the environent Deign reilience in upply chain Meaning Build the afety tock of raw aterial and final product; Keep extra production ability and worker etc. Redeign and tandardize product and procedure; Reduce the type of coponent; Make flexible contract etc. Reduce reaction tie and total tie of aterial oving on the upply chain; Control of inventorie and production chedule. Buine intelligence; Inforation gathering; Technology upgrade and other ean. Decentralize deciion-aking, production capabilitie, peronnel and other critical reource. Etablih oe relevant departent; Develop diruption reaction echani; Reerve relevant reource etc. Etablih eergency repone departent; Develop interrupt repone echani; Deploy tuff and reource effectively etc. Rik aeent and continuou counication aong eployee; Knowledge haring and learning. Deign reilience in bottle neck and critical path; Balance the cot, efficiency and rik baed on copany trategy etc. Exchange inforation with upplier and cutoer; Collaborative planning, forecating and replenihent etc. Fitne to the external environent uch a natural, political, econoic, legal and cultural environent. Deign reilience in bottle neck and critical node of the oftware yte; Reerve interface to expand function and 2

3 oftware yte Redundancy of upply chain yte Protect virtual upply chain againt rik coverage; Conider fir trategie when deigning etc. Keep redundancy of hardware and oftware; Enure appropriate peed and bandwidth of network; Deploy relatively taff to adinitrate the virtual upply chain yte etc. Purchae anti-viru oftware to prevent coputer virue; Increae invetent to prevent network diruption, and inforation ditortion in traniion etc. Model Decription In order to enhance upply chain reilience, fir hould invet in any apect uch a increaing redundancy, iproving flexibility and agility of upply chain and o on, all of which require capital invetent. Thi paper differ fro prior paper, tudy upply chain eber invetent deciion for upply chain reilience baed on evolutionary gae (Webull 998), and propoe oe echani to control eber behavior. Auption In thi paper, we tudy the upply chain contituted by upplier and anufacturer (denoted by S and M). In our odel, an individual of upplier population (for hort individual S) randoly play a one-hot gae with a atched individual of anufacturer population (for hort individual M) every tie. Every individual ha two kind of invetent trategie for reilience: invet (for hort I) or not invet (for hort ). By invet for reilience, fir can reduce upply chain diruption loe, and iprove revenue of the. Auption are: () If both individual don t invet for reilience, their revenue when diruption occur are ( a ) R, ( b ) R ; R( R )and R( R )are repectively noral revenue of individual S and individual M when diruption do not occur. a( a ) and b( b ) are repectively revenue lo rate of individual S and individual M when diruption occur. (2) If both individual invet and iprove the reilience, their profitability can be iproved, then the diruption loe of the can be reduced. Their revenue are R ar a2r C and R br b2r C, repectively; a2r( a2r ) and b R ( b R ) are copenation value of reilience for individual S and individual M after 2 2 they invet, repectively. a2( a2 ) and b2( b2 ) are copenation rate of individual S and individual M, repectively. C( C ) and C( C ) are invetent cot of individual S and individual M, repectively. (3) If individual S invet and individual M doe not invet, the revenue of individual S i R ar a2r C when diruption occur. Becaue individual M can benefit fro individual S invet, for exaple anufacturer procureent can be guaranteed in quality, and ore price can be elected when purchaing, the cot of anufacturer can be reduced. Due to individual M free-riding, the revenue of individual M when diruption occur i T ( T R b R ). (4) Siilarly, if individual M invet and individual S doe not invet, the revenue of individual M i R br b2r C. Due to individual S free-riding, the revenue of it i T ( T R a R ). For the convenience of deontration, we denote a R and a R a individual S and ( a ) R with individual M revenue when all of the don t invet, in other word, replace 3

4 a R and ( b ) R with R a, repectively; denote R b and R b a individual S and individual M b revenue when all of the invet, in other word, replace ( a a2) R C with R and b b b2 R C with R, repectively. According to the above auption, we etablih the payoff atrix of individual S and individual M, which i hown in Table 2. Table 2- Payoff atrix of individual S and individual M M S I ( ) I 2 a a R C, b b2 R C 2 ( a a ) R C, T T, b b2 R C ( a ) R, ( b ) R Replicator dynaic yte According to the payoff atrix, we can get the replicator dynaic yte of upplier population (for hort population S) and anufacturer population (for hort population M). When the gae begin, we let the fraction of individual S uing trategy I be p, o the fraction of individual S uing trategy i -p; Siilarly we let the fraction of individual M uing trategy I be q, o the fraction of individual M uing trategy i -q. The revenue function when individual S chooe trategy I, trategy, and the average revenue function of population S repectively are: U q[( a a ) R C ] ( q)[( a a ) R C ] () 2 2 U qt ( q)( R a R ) (2) 2 U pu ( p) U 2 (3) Siilarly, the revenue function when individual M chooe trategy I, trategy, and the average revenue function of population M repectively are: V p[( b b ) R C ] ( p) [( b b ) R C ] (4) 2 2 V pt ( p)( R b R ) (5) 2 V qv ( q) V 2 (6) Becaue individual have bounded rationality, after a period of evolution they will re-chooe trategie baed on the revenue they get, the individual who have le revenue will change their trategie. Then the fraction p and q will change over tie. According to Malthuian dynaic yte (Friedan. 99), we can obtain that the replicator dynaic yte for population S and population M (denoted by yte) i 4

5 dp p( U U ) p( p)[ q( a) R a2r C qt ] dt dq q( V V ) q( q)[ p( b ) R b2r C pt ] dt (7) Propoition. For the yte given by Eq. (7), we have the following reult: () (, ), (, ), (, ) and (, ) are it equilibriu. (2) If C a2r T a R C, C b2r T b R C, ( p, q) i b2 R C a2r C alo an equilibriu of yte, p, q. T b R T a R Proof. For yte, when dp / dt, dq / dt, obviouly we get that (,), (,), (,) and (,) are it equilibriu. If arand 2 2 br atify C a R T a R C, 2 C b2r T b R C, we can eaily know that p, q, thu ( p, q ) i alo an equilibriu. Evolutionarily table trategie The equilibriu of yte which we have got are not necearily ESS. So we hould ue Jacobian ethod (Hofbauer et al. 998) to judge the. The Jacobian atrix of yte i J [ q( a a2) R C qt ]( 2 p) p( p)[( a) R T ] q( q)[( b ) R T ] [ p( b b2 ) R C pt ]( 2 q) (8) The local tability of equilibriu i deterined by both deterinant and trace. The deterinant of the Jacobian atrix i detj ( 2 p)( [ q a a ) R C qt ]( 2 q)[ p( b b ) R C pt ] 2 2 p( p)[( a ) R T ] q( q)[( b ) R T ] (9) The trace of the Jacobian atrix i trj [ q( a a ) R C qt ]( 2 p) [ p( b b ) R C pt ]( 2) q () 2 2 Furtherore, we can derive the following. Propoition 2. () If a2r C, b2 R C, the equilibriu (, ) i the ESS of yte. (2)If C b2r T b R C, a2r C, (, ) i the ESS of yte. (3)If C a2r T a R C, b2 R C (4)If C a R T a R C, 5, (, ) i the ESS of yte. 2 C b2r T b R C, ( p, q) exit, (, ) and (, ) are the ESS of yte. b R T b R C, (, ) i the ESS. (5)If a R T a R C, 2 2 Proof. According to the Jacobian ethod, we know that if trj<, detj>, the equilibriu of the replicator yte i ESS. We can get the value of trj and det on the equilibriu of yte, then judge the tability of equilibriu and get the ESS. The

6 analyi of ESS in every cae i hown in Table 3. Cae Cae2 Cae3 Cae4 Cae5 Table 3-Stability analyi of equilibriu for yte Equilibriu TrJ DetJ Stability (,) + ESS (,) Saddle point (,) Saddle point (,) + + Untable (,) Saddle point (,) + ESS (,) Saddle point (,) + + Untable (,) Saddle point (,) Saddle point (,) + ESS (,) + + Untable (,) + + Untable (,) + ESS (,) + ESS (,) + + Untable ( p, q ) Saddle point (,) + + Untable (,) Saddle point (,) Saddle point (,) + ESS Evolutionary analyi reult Baed on the analyi above, we get the evolutionary gae proce of population S and population M in five cae, hown in Figure, repectively. Fro the phae diagra of dynaic evolution of yte, we can obtain the following reult. ( a ) ( b ) ( c ) 6

7 ( d ) ( e ) Figure -The phae diagra of dynaic evolution of all cae for yte () If a2r C, b2 R C, ar 2 and br 2 are all le than the invet cot C, C. So the revenue after individual S and individual M invet are le than the revenue when they don t invet. See Fig. (a), (,) i untable, (,) and (,) are addle point, (,) i an ESS of yte. In other word, (, ) i an ESS of yte, population S and population M all don t invet. (2) If C b2r T b R C, a2r C, copenation value ar 2 i le than invet cot C, o individual S won t invet. br 2 i larger than invet cot C, b but R i le than free-riding revenuet. However, individual S doe not invet o that the free-riding behavior of individual M cannot be ipleented, inveting for reilience i optial for individual M. See Fig. (b), (,) i untable, (,) and (,) are addle point, (,)i an ESS of yte. Therefore population S doe not invet and population M invet. (3) Siilarly, if C a2r T a R C, b2 R C, br 2 i le thanc, b o individual M won t invet. ar 2 i larger than C, but R i le thant. See Fig. (c), (, ) i untable, (, ) and (, ) are addle point, (, ) i an ESS of yte. In other word, population M doe not invet, population S invet. C a R T a R C C b R T b R C, a 2 R and (4) If, b b b 2 R are larger than C and C. Becaue T and T are larger than R and R, population S and population M are ore inclined to ipleent free-riding behavior. But if they all don t invet, the free-riding behavior can t be ipleented. Therefore yte will eventually evolve to two ESS. See Fig. (d), (, ) and (, ) are untable point, ( p, q) i a addle point, (, ) and (, ) are ESS of yte. In other word, population S doe not invet and population M invet or population M doe not invet and population S invet. a R T a R C b R T b R C, R and R are larger (5) If, 2 2 than T andt, arand 2 br 2 are larger than C and C. See Fig. (e), (,) i untable, (,) and (,) are addle point, (,) i the ESS. In other word, population all invet. Evolutionary analyi under governent control In order to iprove the reilience of the whole upply chain and axiize the revenue, we hould let (, ) be the unique ESS, in other word, population all invet for reilience. Then we conider that the governent hould get involved in the anageent of upply chain, et up echani to regulate the behavior of population. Evolutionary analyi under punitive echani b b

8 If C a R T a R C, 2 C b R T b R C, the ESS of yte are 2 (,) and (,), the reilience of the whole upply chain cannot be fully iproved, o the upply chain cannot achieve optial operational efficiency. Therefore governent hould et up punitive echani to punih free-rider to propt the to invet. For cae 4 of yte, aue punihent for free-rider i P, the payoff atrix i hown a Table 4. Table 4-Payoff atrix of individual S and individual M under punitive echani S I 2 I ( a a ) R C, ( b b2) R C ( a a2) R C, T T P, 2 M ( b b ) R C ( a ) R, ( b ) R P The replicator dynaic yte under punitive echani (denoted by yte 2) i dp p p q a R a R C q T P dt dq q( q) p( b ) R b R C p T P dt ( )[ ( ) 2 ( )] [ 2 ( )] () p Propoition 3. For the yte 2 given by Eq. (), we derive the following: () (, ), (, ), (, ) and (, ) are it equilibriu. (2) If inequation (2) atifie, ( p, q) i alo an equilibriu of yte 2, b R C a2r C, q. T b R P T a R P 2 ax[ T ( a a ) R C, T ( b b ) R C ] P 2 2 in[ T ( a ) R, T ( b ) R ] (2) Proof. For yte 2, we let dp / dt, dq / dt, obviouly, we get that (, ),(, ),(, ) and (, ) are it equilibriu. If inequation (2) atifie, we can eaily know that p, q, then ( p, q ) i alo an equilibriu. Propoition 4. The neceary and ufficient condition under which (, ) i the unique ESS of yte 2 i: P ax[ T ( a a2) R C T ( b b2 ) R C, ] (3) Obviouly, inequation (3) i equivalent to P ax[ T R 2, T R 2]. Proof. Firt we know that the neceary and ufficient condition i trj<,detj>. [ a a R C ( T P )] [ b b R C ( T P )], and Eaily, we get 2 2 [ a a2 R C ( T P )][ b b2 R C ( T P )] a a R C ( T P ) and b b R C ( T P, then we can get 2 ). Therefore we ee 2 8

9 that P ax[ T ( a a2) R C, T ( b b2 ) R C]. Secondly when inequation (3) atifie, baed on Table 5 we can obtain (, ) i an untable point of yte2, (, ) and (, ) are addle point, (, ) i the unique ESS of yte 2. Table 5-Stability analyi of equilibriu for yte 2 Equilibriu trj detj (, ) ( a2r C ) ( b2r C ) ( a2r C )( b2r C ) (, ) (, ) (, ) [( a a ) R C ( T P )] ( b R C ) 2 2 [( b b ) R C ( T P )] ( a R C ) 2 2 [( [( a a2) R C ( T P )] b b2) R C ( T P )] ) ( ] ( b2 R C )[( a a2 R C T P ) ( a R C )[ b b R C ( T P ] 2 ( 2) ) [( a a2) R C ( T P )] [ ( b b2) R C ( T P )] Thi illutrate that when P ax[ T R 2, T R 2], the punihent P i larger than the difference between free-rider revenue and the revenue when fir invet, the revenue when fir invet i larger than the revenue of free-riding, then upply chain eber all tend to invet for reilience, the reilience of the whole upply chain can be iproved. Evolutionary analyi under ubidy echani If a2r C, b2 R C, the ESS of yte i (,), the reilience cannot be iproved, and thi will for a viciou circle. So the governent hould et up ubidy echani to copleent the fir to propt the to invet. For cae of yte, aue ubidy for fir i S, then we get the payoff atrix hown a Table 6. I S Table-6 Payoff atrix of individual S and individual M under ubidy echani ( ) a a2 R C S, T, I 2 M ( b b ) R C S ( ) b b2 R C S ( ) T a a2 R C S, ( a ) R, ( b ) R The replicator dynaic yte under ubidy echani (denoted by yte 3) i dp p p q a R a R C S qt dt dq q q p b R b R C S pt dt ( )[ ( ) 2 ] ( )[ 2 ] (4) p 2 2 Propoition 5. For the yte 3 given by Eq. (4), we derive the following: () (, ), (, ), (, ) and (, ) are it equilibriu. 2 2 (2) If inequation (5) atifie, ( p, q ) i alo an equilibriu of yte 3. b R C S T b R, q 2 2 a R C S. T a R 9

10 ax[( C a R ),( C b R )] S 2 2 in[ T ( a a ) R C, T ( b b ) R C ] 2 2 (5) Proof. Siilar to Propoition 3. Propoition 6. The neceary and ufficient condition under which (, ) i the unique ESS of yte 3 i: S T R 2 T R 2 ax[, ] (6) Proof. Siilar to Propoition 4. Thi illutrate that when S ax[ T R 2, T R 2], the ubidy S i larger than the difference between free-rider revenue and the revenue when fir inveting. Supplier or anufacturer electing the invet trategy will get ore revenue than free-riding, then they all tend to invet. So when arand 2 br 2 are relatively all, governent hould ue ubidy ethod to control the behavior of fir to let the invet for reilience. Concluion Thi paper firt tudie the factor and iproveent ethod of upply chain reilience, then analyze the invetent deciion of upplier and anufacturer baed on Evolutionary Gae theory. We find that with the increae of arand 2 br 2, there appear different ESS uch a (, ), (, ), (, ), (, ). In order to iprove the reilience of the whole upply chain and axiize the revenue of upply chain eber, we conider that governent hould ue punitive and ubidy ethod to propt fir to invet. Thu the reilience can be iproved and the diruption lo can be reduced. There are everal area to extend thi reearch. Firt we can tudy reilience iproveent ethod of ulti-echelon upply chain. Secondly, we can conider coordination contract of upply chain in the invetent deciion baed on Evolutionary Gae theory. Reference H L. Lee, S Whang. 25. Higher upply chain ecurity with lower cot: Leon fro total quality anageent. International Journal of production econoic 96(3): Martin Chritopher, Helen Peck. 24. Building the reilient upply chain. International Journal of Logitic Manageent 5(2):-29. Sheffi Y. 2. Supply chain anageent under the threat of international terrori. International Journal of Logitic Manageent Review 2(2):-. Xiao T, Yu G. 26. Supply chain diruption anageent and evolutionarily table trategie of retailer in the quantity-etting duopoly ituation with hoogeneou good. European Journal of Operational Reearch 73(2): Zhu Q, Dou Y. 27.Evolutionary gae odel between governent and core enterprie in greening upply chain. Syte Engineering Theory and Practice 27(2): Pettit T J, Fikel J, Croxton K L. 2. Enuring upply chain reilience: developent of a conceptual fraework. Journal of Buine Logitic 3(): -2. Webull J.998. Evolutionary gae theory. Princeton Pre, Boton. Friedan D. 99. Evolutionary gae in econoic. Econoetrica 59: Hofbauer J, Sigund, K.998. Evolutionary Gae and Population Dynaic. Cabridge Univerity Pre, Cabridge.