THE BENEFIT OF ALTRUISTIC BEHAVIOUR ACHIEVED BY THE OUT POLICY WITH UNMATCHED PROPORTIONAL FEEDBACK GAINS IN A TWO- ECHELON SUPPLY CHAIN

Transcription

1 19 h Inernaional Conference on Proucion Research THE BENEFIT OF ALTRUISTIC BEHAVIOUR ACHIEVED BY THE OUT POLICY WITH UNMATCHED PROPORTIONAL FEEDBACK GAINS IN A TWO- ECHELON SUPPLY CHAIN Sephen M. Disney* an Takamichi Hosoa Logisics Sysems Dynamics Group, Cariff Business School, Cariff Universiy, Aberconway Builing, Colum Drive, Cariff, CF10 EU, UK. * Corresponing auhor Absrac We suy a wo echelon supply chain wih AR(1 eman an uni replenishmen lea-imes. Each echelon of he supply chain uses coniional expecaion o generae MMSE forecass. Boh echelons use hese forecass insie he Orer-Up-To policy o generae replenishmen orers. We invesigae ifferen scenarios: The firs is when each echelon aims o minimize heir own invenory holing an backlog coss. The secon scenario is concerne wih an alruisic reailer who is willing an able o sacrifice some of his own performance for he benefi of he oal supply chain. He oes his by smoohing he eman he places on he manufacurer by using a mache proporional conroller in he invenory an WIP feeback loops. The hir scenario is concerne wih an alruisic reailer wih wo unmache conrollers. The mache conroller case ouperforms he raiional case by 1.1%; he unmache conroller case ouperforms he mache conroller case by.9%. Keywors: Supply chains, muli-echelon invenory, orer-up-o policy, collaboraion, alruisic behavior 1 INTRODUCTION Assuming AR(1 eman, we suy a serially linke woechelon supply chain ha explois a generalize orer-upo OUT policy wih unmache (ha is, inepenen, feeback conrollers a he firs echelon (he reailer an a raiional OUT policy a he secon echelon (he manufacurer. We also assume ha minimum mean square error forecasing is use an uni lea-imes are presen a each echelon. The benefi from he reailer s alruisic behavior enable by he generalize OUT policy wih unmache feeback conrollers will be invesigae, 1]. Each player acs o minimize global invenory coss. To quanify is benefi, his sraegy will be compare wih oher wo sraegies; 1 a raiional sraegy where each player minimizes local invenory coss ], an an alruisic sraegy achieve by he generalize OUT policy wih mache feeback conrollers ]. As an inicaor of he supply chain performance, we will employ a meric ha consiss of he sum of he saionary sanar eviaion of he ne invenory levels a each echelon. This is a vali approach when safey sock have been opimize via he newsvenor principle as invenory coss are hen linearly relae o he sanar eviaion o he invenory levels, ]. We also quanify he bullwhip effec in he supply chain. We reveal he exac analyical expressions of he performance inicaors. We highligh ha he generalize OUT policy wih unmache conrollers enable us o manipulae he ynamics of a supply chain wih higher egree of freeom han he generalize OUT policy wih mache conrollers. Furhermore, we also iscuss wha kin of informaion shoul be share beween hese wo players o achieve he benefis from he alruisic sraegy we highligh herein. DIFFERENCE EQUATION REPRESENTATION OF THE TWO-ECHELON SUPPLY CHAIN We assume he eman face by he reailer is a mean cenere auoregressive sochasic process of he firs orer. Thus, µ ρ µ, (1 ( 1, 1 an he eman face by he manufacurer in he secon echelon is he reailers orer,, o1,. ( In Eq (1, ρ is he auoregressive consan (-1< ρ <1, is he eman a ime an We assume µ is he average eman. µ >> so ha he possibiliy of negaive eman is negligible. is a sochasic whie noise process. In Eq (, o is he orers a ime. We will also assume ha here is a uni replenishmen lea-ime a each echelon. Aiionally, here is a one perio, orer of evens elay. Thus a boh echelons (where he firs par of he subscrip is use o inicae he echelon in quesion; x1 for he reailer an x for he manufacurer, he following invenory balance equaion hols, ns, ( x, nsx, 1 x, ox, where ns is he ne sock (invenory on han. In each orering policy we will also nee wo forecass of eman. One of hese forecass is he coniional expecaion of he eman in he nex perio an his is use o generae a esire WIP (or pipeline, orers place bu no ye receive arge. The oher forecas is he coniional expecaion of eman in he perio afer he replenishmen orer arrives, ha is, he forecas of eman in he nex, nex perio. For he reailer hese forecass are wip 1, E 1, 1] ρ1, ˆ 1, E 1, ] ρ 1, However, hese wo forecass are consierably more complex for he manufacurer. They are wip, E o ] 1 ρ ( ρ ρ ρ 1, o1, ns ns o 1 ρ ˆ E o, 1 ρ 1, o1, ns ns1, o1, 1 ρ1 ρ o, 1, 1, (7 We may use hese forecass in he following ifference equaion o generae orers (which hols a boh echelons, ( ( (6

2 1 1 o ˆ x, x, ( ns nsx, ( wipx, ox, 1 (8 These las few ifference equaions (Eqs 6-8 conain some new noaion. The firs is ns, he arge ne sock, a ime invarian arge safey sock ha is use o ensure a esire fill-rae or availabiliy of sock is achieve. The oher wo new erms are an. These are linear feeback gains in he ne sock an WIP feeback loops respecively. Feeback gains are a very simple an very well known echnique from he fiel of conrol heory for manipulaing he response of a ynamic sysem. When 1 we say hen he supply chain consiss of wo serially linke, raiional OUT policies; when, we say here are mache conrollers; when we say here are unmache conrollers. posiions urns ou o be ( ρ ρ (10 NS1] ( ρ( ρ( 6 ρ( 6 ρ( ρ( ρ NS (11 ( ρ 6ρ 6ρ ρ ρ ρ 6 These expressions for he variances may, in general, be obaine by a variey of ways, from sochasic analysis ], via he frequency omain ], conrol heory 6], or sae space mehos 7]. However we will no provie furher eails here ue o space requiremens. Figure 1 illusraes he invenory coss (via he sanar eviaions use in he objecive funcion, Eq (9 as a funcion of he auoregressive parameer, ρ. THE OBJECTIVE FUNCTION We will consier minimising he following objecive funcion J NS1] NS]. (9 This is an appropriae objecive funcion when here are invenory holing an backlog coss ha are linear in he invenory posiion in cases when ns has been se o he criical fracile o minimize he coss via he newsboy principle. The fac ha we have simply ae he wo sanar eviaions ogeher also implies ha he reailer s invenory holing an backlog coss are as imporan as he manufacurer s invenory holing an backlog coss. ] shows ha seing 1 a he manufacurer (he secon echelon yiels a minimum value of J in a given scenario. Therefore, in our wo-echelon moel, only he firs echelon (he reailer explois he feeback conrollers ( an o manipulae he ynamics of he supply chain. The manufacurer simply uses a raiional orer-up-o policy wih MMSE forecasing o minimise J. This is a naural consequence of our objecive funcion, (Eq 9. If he objecive funcion conains bullwhip relae coss hen his oes no hol an he manufacurer shoul incorporae feeback conroller(s ino his replenishmen rule. This is ousie he scope of his shor conference paper. However, we will quanify orer variance a boh echelons in our moel for compleeness. In he res of he paper we will compare hree scenarios: The raiional, local opimisaion. This scenario consiers he case when boh he reailer an he manufacurer are solely concerne wih minimising heir own, local invenory holing an backlog coss. We will suy his scenario in secion. The alruisic reailer, global opimisaion wih mache conrollers. This scenario consiers he case when he reailer is able an willing o aler his replenishmen rule (by uning in orer o minimise he oal supply chain coss. We assume in his case ha he reailer uses a generalise OUT policy wih mache conrollers,. We will suy his scenario in secion. The alruisic reailer, global opimisaion wih unmache conrollers. We will suy his scenario in secion 6 an is essenially he same as he previous sraegy bu wih inepenen, unmache conrollers in he reailer s replenishmen rule, ha is. ANALYSIS OF THE TRADITIONAL OUT POLICY SCENARIO; THE LOCAL OPTIMISATION Here he reailer uses 1 an hus he supply chain consiss of wo serially linke OUT policies wih MMSE forecasing. As 1 here are no sabiliy issues in he supply chain an he variance of he wo ne sock Figure 1: The invenory variances in he raiional supply chain The variance expressions for he eman an he wo orer raes are; 1 ( ρ 1] D 0 1 ρ ρ ( ρ ρ 1 ρ 1 O1 ρ O ρ 1 ( ρ 1( 1 ρ ( 1 ρ ( 1 ρ ρ ρ ρ ρ 1 1 (1 (1 (1 which have been ploe in Figure. Noe from Figure ha when ρ > 0 hen a bullwhip effec exiss as O 1 ]>D 1 ] an O ]>D 1 ]. Figure : Orer variances in he raiional supply chain ANALYSIS OF THE GENERALISED OUT POLICY SCENARIO WITH MATCHED CONTROLLERS By seing, a he reailer we have mache feeback conrollers. This yiels a new se of variance raio formulas. These are:

3 19 h Inernaional Conference on Proucion Research ( ( 1 p 1 NS1] 1 (1 ( 1 ρ ( 1 ρ ρ ( 1 ρ( ρ ( 1 ρ ( ρ ρ ( ρ( ρ NS ] (16 From Eq (1 we can see ha he vali range of o ensure sabiliy is 0. < <. We may use hese variance raios in he objecive funcion (Eq 9 an eermine he ha minimises he objecive funcion, *. Analyically his appears o be very ifficul o achieve. However, using numerical echniques is consierably less complex an resuls in he following graphical relaionship, see Figure. an shows ha he alruisic conribuion of he reailer resuls in a smoohing of he reailers orer variance. Thus, if he reailer incurs some bullwhip relae coss in his reail, warehousing or ransporaion aciviies, hen he may in fac, be even more willing o use he proporional feeback conroller o minimize coss a he supplier han his sylize analysis suggess. Figure. The eman an orer variances wih mache conrollers a * wih an alruisic reailer Figure. The opimal when mache conrollers exis wih an alruisic reailer Using his opimal insie he objecive funcion resuls in Figure which escribes he minimise invenory coss in our supply chain. Figure. T he objecive funcion wih mache conrollers a * wih an alruisic reailer The general expressions for he orer variances are given by Eqs (17 an (18. ρ ρ ρ ρ ρ ( 1 ρ ( ( 1 ( ρ 1 O1 ( ρ( ρ 1 (17 ( ρ 1( 1 ρ ( 7ρ 1 ( ρ 1 ρ( 1 ρ 7 6 ( ρ 1( 1 ρ ( 6 0ρ ρ ρ ( 1 ρ ρ ρ ρ ( ρ 1( 1 ρ ( 1 ρ( ρ( ρ( ρ( ρ 0 1 ( ρ 1( 1 ρ ( 1 ρ( ρ( ρ( 9 ρ( 9 ρ ( 1 ρ( 1 ( ρ 1 ρ( 10 ρ( ρ( ρ ρ 7ρ ( ( ( ( ( ( ( O 1 ρ 1 ρ ρ ρ 6 ρ ρ ρ ρ 1ρ 8ρ 1 0 ( ( 1 ( ( ρ 1 ρ( ρ 1( 1 ρ (18 When has been se o * hen he orer variances can be ploe as shown in Figure. Comparison of Figures 6 ANALYSIS OF THE GENERALISED OUT POLICY SCENARIO WITH UN-MATCHED CONTROLLERS As here are wo unmache conrollers hen here is a nee o conuc a sabiliy analysis a he reailer. There is no nee o consier such issues a he manufacurer as here 1 which resuls in a sable sysem. Thus he analysis for he generalise OUT policy wih unmache conrollers consiss of a wo sage approach. 6.1 Sabiliy analysis Sabiliy can be reaily invesigae via ransfer funcions an we will exploi Jury s Inners approach o conuc he analysis, 8]. This ransfer funcion of he reailers orer is ( z ( ( ( z 1 ρ ( z ( z ρ ( ( z 1( z ( z O1 z 1. (19 z 1 ρ I is known ha sabiliy only epens upon feeback loops an hus we may ignore he fee-forwar auoregressive erm. Seing ρ 0 an simplifying resuls in ( z z ( z ( z 1( 1 z O1 (0 Jury s sabiliy es requires us o expan ou he enominaor an collec ogeher powers of z. ( ( 1 z A( z z (1 The firs par of Jury s sabiliy es is o ensure ha A(1>0 A 1 A z. Thus i follows ha >0. where ( ( z 1 The secon s This is rue iff, < ( 1 n. age of Jury s es is ha ( 1 A( 1 > 0. ( The hir an final sage of Jury s sabiliy es is ha cerain marices of he co-efficien of he enominaor of he sysems ransfer funcion are posiive innerwise. Because our ransfer is only of secon orer, his crieria easily reuces o he fac ha > ( 1 an >. ( 1

4 Numerical invesigaion reveals ha ( is non criical as i is enirely encompasse by (. I is ineresing o noe ha he sabiliy boun in firs sep of Jury s es resuls in >0, bu ( shows us ha can in fac be negaive. Careful invesigaion shows ha he unsable region of, becomes sable when <-1. For confirmaion a more irec sabiliy es is given in 9] an resuls in >0, Eq an Eq. The reunan coniion prouce by Jury s Inners Tes is no generae. Figure 6 illusraes he sabiliy region. Using hese values in he objecive funcion we may illusrae he invenory coss as shown in Figure 8. Here we can see he impac of he isconinuous * an *. Figure 8. The objecive funcion wih unmache conrollers wih an alruisic reailer Figure 6: The sabiliy bounary for he generalize OUT policy wih uni lea-imes 6. iance analysis The variance of he reailer s ne sock is given by, ( ( 1( 1 ρ ( ρ ρ NS1] ( ( 1 ( ( 1 ( The variance of he manufacurer s ne sock is given by, ρ ρ 1 ρ ρ 1 NS ρ ρ ρ (6 Using hese variance raios in he objecive funcion we may fin he opimal values of he unmache feeback conrollers. Again, analyically his is very ifficul, bu numerical echniques o exis an hey resul in values for an as shown in he Figure 7. Figure 7 conains some very remarkable feaures. For ρ < 0. 7 boh * an * are posiive. However as here are wo local opimums in he soluion space near ρ 0. 7 hen he opimal * an * is isconinuous in ρ. Ineresingly, he opimal * is negaive for ρ > Reurning now o he impac of alruisic reailer wih unmache conrollers on he variances of he orer raes we have; ( ( ρ ρ ρ 1 ρ ( ρ( ρ 1 ρ ( 1 ρ ( 1 ρ( ρ 1 ( ρ 1 ρ ρ ( ρ( ρ ( ρ 1( ρ 1 ( 1 ρ( ρ 1 ( 1 ρ ( ρ ( ρ 1 ( 1 ρ( ρ 1 1 1] O ( ( 1 ( ( 1 ( ( ρ ρ ρ ( 1 ( ( ρ ρ ρ ( 1 ( ρ 1 (7 6 αβ ( ρ 1 ρ ( 1 ρ( χ 16 ( ρ 1( φ ω τ ( ϕ υ η ( ξ λ ] O ( ( 1 ( ( 1 ψ ( ( 1 ψ ( ( ( ρ ρ ζ ( ( ( ρ ρ ζ ( 1 ρ (8 where we have use he following subsiuions as he formula is raher complex. ψ ( 1 ( 1, ζ ρ ( 1 (9 ( 1 ρ ( ( ρ 1 ( 1 ρ( ρ 1 ρ ( ρ ( ρ ρ 1( 1 7 α ρ ρ (0 ρ ρ ρ ( ( ( ( β (1 ρ 1 1 ρ ρ ρ 1 χ ( ( ( ( ( ( 1 ρ 1 ρ 1 ρ 1 ρ 1 ρ ρ ( 6 ( 1 ρ( 1 ρ ρ ρ ( ρ( ρ ( 1 ρ( ρ ( ( ( 1 ρ 1 ρ 1 ρ ( 1 ρ ρ( 1 ρ( ρ ( 1 ( 1 ρ ρ ρ( 1 ρ ρ( ρ( ρ 1 ( 1 ρ ( 1 ρ( 1 ρ ( 1 ρ ρ ρ ( φ ( ρ 1 ρ( 1 ρ ( ρ( 8 ρ( ρ 11 ( ρ 1( 1 ρ ( ρ( ρ ( 7 ρ( 10ρ 1 ϕ ( ( ( ( ( ( ρ ρ 1 ρ 1 ρ ρ ρ ( ρ( ρ( 0 ρ( 8 ρ( 1ρ 11ρ ρ ρ ρ 8 19ρ 16 ξ ρ λ ( ρ 1 ρ( 1 ρ ( 8 ρ( ρ( ρ ρ 10 ρ( ρ( ρ( ρ( 9 ρ( 1 ( ρ 1 ρ 9 ( ρ 1 ρ( 1 ρ( ρ ( ρ ( ρ ρ ρ 7ρ ρ 8ρ ρ ( ( ρ ρ 7 7ρ ρ ( ( 6 ( ρ 1 ( ρ( 9 ρ ( ρ( 1 ρ ( ρ( 7 ρ ρ ρ ( 1 ρ ( ρ ( ρ( ρ( ρ( ρ( 1 ρ ρ 1 (6 ( ( ( 9 ( ( ( 11 ( ( ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Figure 7. Tuning he unmache feeback conrollers o minimise supply chain invenory coss

5 19 h Inernaional Conference on Proucion Research ( ρ 1( 1 ρ ( ρ( 11 ρ( 1 ρ( ρ 16 ρ ρ 8 (7 ( ρ 1 ρ( 1 ρ( ( ρ 7( ρ 1 ρ ( ( ( ( ( ( ρ 1 1 ρ ρ 1 ρ ρ 11ρ 8 τ 6 7 ( ρ( ρ( ρ( 6 ρ( ρ ρ ρ ρ ρ 1 11ρ 1 ( ρ 1( 1 ρ ( 1 ρ( ρ( ρ( ρ( ρ( ρ( ρ 1 ( ρ 1( 1 ρ ( ρ( 7 ρ( 17 ρ( ρ( 1 ρ( ( ρ 1 ρ 11 ρ( ρ( ρ( ρ( 19 ρ ( ρ( 1ρ υ ( ρ 1( 1 ρ ( ρ( 8 ρ( ρ( ρ( ρ( ρ( ρ (8 ρ( 7 ρ ( 1 ρ( 6 ρ ( ρ( 1 ρ( ρ( ρ 1 ( ( ( ( ( ( ω (9 ρ 6 ρ 1 ρ 11 ρ 7 ρ ρ 6 η ( 1 ρ( ρ( ρ( ρ( 16 ρ( 1 ρ ( ρ( ρ 7 ρ 8 (0 These expressions are ploe in Figure 9. Figure 9. The orer variances wih unmache conrollers (of * an * a wih an alruisic reailer Table 1. Enumeraion of he hree supply chain scenarios Traiional Supply Chain, 1 Alruisic Reailer wih Mache Conrollers Alruisic Reailer wih Unmache Conrollers ρ Reailers Invenory Cos Man. Invenory Cos Toal Invenory Coss ** Reailers Invenory Cos Man. Invenory Cos Toal Invenory Coss % benefi above raiional supply chain * * Reailers Invenory Cos Man. Invenory Cos Toal Invenory Coss % benefi above mache supply chain NUMERICAL INVESTIGATIONS In orer o highligh he benefi of he unmache conrollers wih alruisic reailer we will now enumerae he invenory coss an orer variances for a range of values in he auoregressive eman parameer. This is shown in Table 1. We can see ha if he reailer is able o aler his replenishmen rule o incorporae mache feeback conrollers hen oal supply chain invenory coss may be reuce by as much as 1% when compare o a raiional supply chain. However, if he reailer is will o go even furher an use appropriaely une unmache conrollers, hen a furher.9% reucion in oal supply chain invenory coss may be gaine. 8 CONCLUSIONS The unmache conroller generalise OUT policy ominaes he mache conroller case wih an alruisic reailer who is concerne wih minimising he global supply chain invenory coss. The benefi appears o be approximaely % reucion in he invenory holing an backlog coss. Closer inspecion reveals ha he alruisic conribuion of he reailer, in he unmache case, is even Average > Average > higher han in he mache case. However, he rewars are even higher when compare o he raiional supply chain where members are only concerne wih heir local invenory holing an backlog coss as he unmache conroller case is 18.% beer, on he average. In orer o gain his avanage he firs echelon nees o be able o unersan he manufacurer s cos srucure, he eman signal an he lea-imes in he supply chain an hen aler he srucure of his replenishmen rules. This is, inee, a very complex ask an we imagine ha i will ake consierable inusrial engineering effors o achieve. Even if his coul echnically be one hen he manufacurer has o unersan an use marke place informaion an be willing o share some of he economic benefi wih his cusomer. Oherwise, he reailer will have no incenive o make he alruisic conribuion an smooh his replenishmen orers. Of course, we have also assume a linear sysem exiss, an hus all unme eman has been backorere an he saisical properies of he eman signal are ime invarian. As a final poin, he analysis herein is very complex an raher ugly. Recen work in 10], suggess ha much more elegan resuls can be foun by exploiing he full-saefeeback conroller, a echnique avocae by moern

6 conrol heory. This will be explore in fuure research. 9 REFERENCES 1] Disney, S.M., Farasyn, I., Lambrech, M., Towill, D.R., an van e Vele, W., (006, "Dampening variabiliy by using smoohing replenishmen rules, Uner review a he European Journal of Inusrial Engineering. ] Hosoa, T. an Disney, S.M. (006a "On variance amplificaion in a hree-echelon supply chain wih minimum mean square error forecasing", OMEGA: The Inernaional Journal of Managemen Science (, -8. ] Hosoa, T. an Disney, S.M. (006b "The governing ynamics of supply chains: The impac of alruisic behaviour", Auomaica (8, ] Disney, S.M., Lambrech, M., Towill, D.R. an Van e Vele, W., (006, The Value of Coorinaion in a o Echelon Supply Chain: Sharing informaion, policies an parameers, Accepe for publicaion in IIE Transacions. ] Dejonckheere, J., Disney, S.M., Lambrech, M.R. an Towill, D.R., (00, The impac of informaion enrichmen on he bullwhip effec in supply chains: A conrol engineering perspecive, European Journal of Operaional Research 1(, ] Disney, S.M. an Towill, D.R., (00 On he bullwhip an invenory variance prouce by an orering policy, OMEGA: The Inernaional Journal of Managemen Science 1(, ] Gaalman, G. an Disney, S.M., (006 Sae space invesigaion of he bullwhip problem wih ARMA(1,1 eman processes, Inernaional Journal of Proucion Economics 10( ] Jury, E.I. (197 Inners an he sabiliy of ynamic sysems, John Wiley an Sons, New York. 9] Disney, S.M. (007 Supply chain aperioiciy, bullwhip an sabiliy analysis wih Jury s Inners, Uner review a IMA Journal of Managemen Mahemaics. 10] Gaalman, G. an Disney, S.M., (007, On bullwhip in a family of orer-up-o policies wih ARMA(, eman an arbirary lea-imes, Accepe for publicaion in he Inernaional Journal of Proucion Economics.