Choosing an optimal return rate and a reverse logistics policy from the solution of a constrained LQG problem.

Transcription

1 Preprints of the 9th Worl Congress The International Feeration of Atomatic Control Cape Town, Soth Africa. Agst 4-9, 04 Choosing an optimal retrn rate an a reverse logistics policy from the soltion of a constraine LQG problem. O. S. Silva Filho I. R. Salviano Center for Information Technology Renato Archer - CTI Campinas SP Brazil ( oscar.salviano@cti.gov.br) Abstract: Deman for final procts has grown significantly worlwie, particlarly, e to the increase of emerging contries on the consmer market. The big problem is that consmer of energy an se of raw material increase in parallel. Conseqently, an nsstainable environmental sitation will be observe on the net ftre as arging some researchers. In orer to mitigate sch a sitation, the rese of resorces can be an interesting initial soltion. However, rese procts or materials reqire a somewhat treatment or remanfactring process that together with the collection process can be highly epensive. The paper tries to ientify a retrn rate that leas to a minimm lower cost for rnning a reverse logistics. A chance-constraine LQG problem is se to provie both an optimal retrn rate an optimal proction policy for retrnable procts. A fictitios company is se as an eample of application.. INTRODUCTION De to eaggerate se of raw material, which is etracte from the earth, an the amont of energy spent to proce an istribte procts aron the worl, reverse logistics has become an essential part of an integrate spply chain. The main objective of a reverse system is to shift procts from market collectors to places where they can be treate or can be properly isposal. The prpose is that some procts can get vale along the reverse chain by mean of recycling or remanfactring processes. Ths, it is possible to contribte positively to improving the environment, recing waste by mean of remanfactring or recycling. A reverse logistics system can be formlate throgh two integrate mathematical moels: one represents the forwar channel of the spply chain, where procts are manfactre, store in serviceable inventory nits, an then move to the marketplace in orer to meet eman. The other enotes the backwar channel, where se-procts are recovere or iscare. Athors, like Fleischmann et al. (997), have provie a typology of qantitative moels for reverse logistics. Usally, in the literatre, three kins of problems involving sch moels are formlate as follows: (i) collecting, packaging, an istribting se procts. The collecting process starts at the marketplace, from where cstomers leave their ot-of-se procts; (ii) scheling of remanfactre an recycle process, which allows se procts retrning to the serviceable inventory of the company; an (iii) planning an controlling of items an procts for resing, withot any aitional process of remanfactring. The forwar channel moel is a ynamic system that is affecte by the stochastic flctation of eman. It is assme that the eman follows a istribtion of probability, which can be approimate by a normal istribtion, see Graves (999). As a main conseqence, the forwar channel system becomes a stochastic process. The problem formlate from this system belongs to the class of stochastic mathematical programming problem, which means that it is more comple to be optimally solve. It is worth mentioning that stochastic proction-inventory systems are sally fon in reverse logistics problems, particlarly in reason of the retrn rate not known precisely over the ftre perios. Fleischmann et al. (997) have shown that the traitional classification of proction an inventory stochastic problems, base on iscrete or continos-time moels, can be also applie to moel an solve problems of procts recovering, see for instance Oaret, et al. (0) an Kenné et al. (0) In this paper, a iscrete-time Linear Qaratic Gassian (LQG) moel with chance-constraints is consiere to represent a recovery problem. A two stages procere is provie to allow managers etermining an optimal retrn rate an optimal manfactring-remanfactring policy. In the first stage, ifferent costs are combine in orer to ientify the optimal retrn rate. In the seqence, the secon stage consists in fining a manfactring-remanfactring optimal policy. To solve the chance-constraine LQG problem, an eqivalent, bt eterministic problem is consiere. An Open-Loop Feeback Controller is se as a soltion approach. Sensibility analysis, provie from simple variation of some parameters as retrn rate of se procts, elay of retrn, or both, allows creating proction scenarios that help managers to make prospere ecisions. The paper is istribte as follows: section iscsses a constraine LQG moel for reverse logistics problem; section introces a two-stage procere, which allows managers eciing on the optimal retrn rate an proviing an optimal proction plan; an the section 4 presents a nmerical eample to illstrate the application of this procere Copyright 04 IFAC 897

2 9th IFAC Worl Congress Cape Town, Soth Africa. Agst 4-9, 04 Manfactring X S Remanfactring X μ. Collection. Fig.. A forwar-backwar system ner ncertain eman S n. THE REVERSE LOGISTICS MODEL Figre illstrates the forwar an backwar channels of a niqe proct spply chain. Note that there are two stores in this figre: the first one (Store ) stocks manfactre an remanfactre procts to meet the eman, an the secon store (Store ) collects the retrne procts. These retrne procts can be remanfactre or isposal properly. It is worth emphasizing that the eman for procts mst be flfille by the combination between new procts (manfactre) an remanfactre procts (i.e., se procts that are collecte from the marketplace an, if possible overhale). Other featres an properties consiere for the system ehibite by figre are: a) eman is a ranom variable that follows a stationary stochastic process while the retrn process is assme essentially eterministic; b) both manfactring an remanfactring process has infinite capacity; c) similarly, the maimm physical storage capacity of inventory for warehoses an are assme nlimite; ) there is a constant time-elay associate with retrn procts from the market; an e) se-procts may be ispose of after being collecte. There are two main reasons to iscar seprocts: the first has a technical reason, which is relate to inappropriately retrne procts for remanfactring activities, an the secon has a financial reason, in which remanfactring all procts can significantly raise the inventories, an, as a reslt, sch strategy can increase the overall proction cost... Stochastic inventory-proction system The integrate forwar an reverse inventory-proction system illstrate in figre can mathematically be escribe by a iscrete-time stochastic control moel with two state variables that are relate to inventory levels on serviceable an remanfactrable nits; an three control variables that are relate to manfactring, remanfactring, an iscar rates. Note that, sch featres like ynamic of reverse logistics an ncertainties associate with the process of collecting an, simltaneosly, attening the market eman are sally reasonable consierations to jstify the se of stochastic control moels. In the literatre, it is possible to fin innmeros contribtions relate with this kin of moel, the majority of them consier the formlation in time-continos pattern; see, for instance, Dobos (00) an Miner an Kleber (00). For reaer interest in more generic themes of reverse logistics, it is recommene to visit the site The iscrete-time stochastic control moel is escribe by the following two ifference eqations, which represent, respectively, the inventory balance systems relate to forwar an reverse channel of the spply chain: (k+) = () (k+) = - - +r () where, for each perio k, the notation is given as follows: = inventory level of serviceable procts (store ); = inventory level of se proct (store ); = proction rate of manfactre procts; = proction rate of treate se-procts; = iscar rate of nserviceable procts; = eman level for serviceable procts; r = = level of se-procts The eman is a ranom variable that follows a normal istribtion with mean an variance given by ˆ an D finite, respectively. Since is an inepenent ranom variables, the inventory system () is a stochastic process. As a conseqence, the inventory level is a epenent ranom variable with mean ˆ an variance V 0. On the other han, the retrn rate r is eterministic variable given by r ˆ(k ), with 0 enoting a percentage of the total nmber of retrne nits. As a reslt, the inventory system escribe by the backwar process () shol be seen as a eterministic process. 898

3 9th IFAC Worl Congress Cape Town, Soth Africa. Agst 4-9, 04.. The stochastic reverse logistics problem Base on system ()-(), a chance-constraine Linear Qaratic Gassian (LQG) moel is formlate in orer to represent a logistic reverse problem as follows: Min,, s.t. E T h(t) h(t) Eh h c c c (k ) (k ) r Prob.( ) 0 k0 0 ; 0 ; 0 It is worth mentioning some characteristics of sing a qaratic cost: a) it penalizes eqally, both positive (i.e., ecess) an negative (i.e., backlogging), variations of ecision variables; an b) it inces high penalties for large eviations of the ecision variables from the origin, bt relatively small penalties for small eviation, see Bertesekas (000). Base on problem (), a two-stage procere that provies an optimal retrn rate (), an as well an optimal forwar an reverse policy for systems () an () are escribe net.. SOLVING (6) WITH OPTIMAL RETURN RATE () This section presents a strctre procere that allows the ecision maker efining an optimal retrn rate for a given proct. The procere takes into accont the qaratic programming moel, which is presente in the formlation (6), to choose the best retrn rate an to provie an optimal inventory-proction for the system given in (). It is known that the strategy se for selecting moels that help managers to implement a reverse logistics in a given spply chain epens on the costs relate to the logistics of retrning, which incles costs for collecting, transporting, storing, treating an isposing. Besies there are also inirect costs like energy costs for treating of retrnable proct an no-measres costs relate to polltion in part e to an increase of transportation in rban centers. In fact, these costs are often high an can rece the profitability of the spply chain. However, the most part of these costs is tangible. This means that the spply chain can obtain intangible gains by associating its image to an environmentally sstainable spply chain, which reces the etraction of raw material of the earth. The procere propose below takes into accont the efinition of scenarios to choose the optimal rate of retrn for a given proct, consiering the costs in forwar an backwar (reverse) channels of the spply chain. After k () setting the best rate, the net step is to ientify an optimal proction-inventory schele that consiers the projecte levels of eman. It is assme here that the spply chain rns in a make-to-stock pattern. The procere is given in two stages: Stage : Determining an optimal retrn rate First, it is important to set some basic assmption relate to problem (6): Collection an transportation costs are incle in the inventory cost of the proct retrnable. Deman is assme to be eqal to the monthly average eman (i.e., = ˆ ). Note that sch average eman ˆ is compte from historical ata that contain monthly recor of sales. Then, some scenarios can be constrcte by relating costs between the forwar an reverse channels of the spply chain. This allows comparison of ifferent scenarios that involve costs with inventory an manfactring of serviceable procts (i.e., c an h, respectively) an the costs of inventory, treatment an isposal of collecte procts (i.e., h, c an c, respectively). There are several possibilities of combinations of these costs. Lastly, the problem (6) is solve, bt consiering the retrn rate "" as being the ecision variable that mst be fon. The follow two steps provie the scheme: Step : Setting eman eqal to its average monthly eman, (i.e., = ˆ for k =,,, T), the problem () is transforme to a Mean Vale Problem, which is given as: Min,, s.t. h ˆ c ˆ (k ) ˆ ˆ ˆ (T) h c (k ) ˆ ˆ ˆ ; (T) 0 ˆ c 0 ; 0 ; 0 Step : Changing the prices in the criterion of 4 (i.e., h, h, c, c an c ), it is possible to create ifferent scenarios of retning by solving the problem (4) when varies in the range (0, ]. The best vale of (i.e., = * ) is the one that provies the smaller criterion of (4). Stage : Determining optimm proction an inventory levels for serviceable an retrnable procts, with = *. Compting a tre optimal soltion (i.e. an optimal closeloop soltion) for the stochastic problems, like problem (9), is not a simple task. Ths, in the literatre, it is possible to h ˆ(k ) T k0 ˆ h ˆ (4) 899

4 9th IFAC Worl Congress Cape Town, Soth Africa. Agst 4-9, 04 fin alternative approaches that provie near-optimal soltions. The seqential sboptimal techniqes (Bertesekas, 000) can be interesting approaches to proviing revise policies. In this contet, Open-Loop Feeback Controller (OLFC) is a sboptimal alternative propose here for investigation. This procere permits pating the optimal policy perioically, taking into accont the crrent state of systems ()-(). Note that the feeback characteristic of the OLFC makes it an aaptive procere. This means that the soltion provie by the procere is better than that one given by an open-loop procere. Note that an open-loop soltion is obtaine from the classical mean optimal problem. Note that the problem () is a stochastic optimal control problem with perfect state information. So, it is perfectly possible to measre the inventory level (i.e. the state) at the beginning of each new perio k. This characteristic allows immeiate application of the OLFC procere since it oes not nee se state estimators. Basically, the steps of the OLFC procere are (Bertesekas, 000): Step. In the beginning of each perio k, the eact position of the serviceable ( (t)) an retrnable ( (t)) inventory levels are measre. Step. With these information (i.e., ( = ( (t)) an ( (t) = ( (t))), an optimal proction policy { } k= t, t+,, T- is compte from solving the following eqivalent problem (Silva Filho, 0): Min,, s.t. h c (T) c (k ) (k ) (k ) ; 0, 0 ; 0 ; 0 h (T) where, the integration constant of the criterion K t, when it takes in probability, epens of the variance of eman an the perio t, it gives by K t (5) (h h ) (T t) (6) c, The physical constraints of serviceable an collecte procts are transforme into eqivalents, bt eterministic ineqalities given by: T kt h t h Step As a reslt, it can be establishe a proction policy to the problem (0) that ses the reslt provie eactly ring perio t. This means that the rest of policy is completely ignore. Some comments: The main characteristics abot this procere are: (a) it reqires the soltion of N problems in a rolling horizon scheme. This is eqivalent to maintain the proction plan constantly revise; (b) only the ecision of the perio k=t (i.e, i (t), i=,,) is effectively applie. The rest of the policy (i.e, { i (t+), i (t+),..., i (T-); i=,,}) is ignore since it oes not take into accont the actal level of inventory; (c) there is easy comptational implementation of mltiimensional problems. The procere permits the application of ifferent optimisation techniqes, an then, ser can choose the one that reqires less comptational effort; () fiing t= an = ˆ ( =0, k), the problem (5) becomes a static D optimisation problem, also known as mean problem; (Silva Filho, 009) an (e) being aaptive, the OLFC provies a better soltion than the one provie by the mean optimal soltion. 4. NUMERICAL EXAMPLE A fictitios company makes to stock an istribtes a proct to the marketplace. Uner the pressre of environmental movements, the spply chain epartment of this company has been seeking alternatives to implement a moel of reverse logistics that not cases a strong impact on its profitability. In short, crrently, the company is in charge of manfactring the final proct of this spply chain an istribting it into the marketplace. In the brief ftre, the company will have to collect it back an to treat or ispose of it. The operational ata of the company are given in the Table : Table. Company s ata Elements Description T= months Planning horizon α=0.9 Cstomer satisfaction level =50 Stanar eviation of eman 0 =00 Initial inventory level of serviceable 0 00 Lower an pper bons of serviceable 750 Upper proction bons h =; c = Crrent proction an inventory costs $ Total cost of the company per year The company aims to answer some qestions: a) assming ifferent costs, as shown in Table, what is the best rate of retrn for the company?; b) after selecting optimm rate of retrn, which will be the final cost of the optimal reverse policy?; an c) compare with the crrent policy moel, what ifference is verifie?,,,k,k k k k k (7) Initially, let s consier the first qestion: assming that inventory costs are h> h, it becomes possible to evalate the behavior of rates of retrn for ifferent sitations relate to the proction an isposal costs, see Table. 890

5 9th IFAC Worl Congress Cape Town, Soth Africa. Agst 4-9, 04 Table. Costs relations with h>h Sitations Proction an isposal costs c > c = c c < c = c c< c = c 4 c < c = c 5 c> c > c 6 c< c < c Setting the mean vale of eman to the problem () an solving the step of stage, the best vale for a retrn rate for the company is promptly ientifie. The relation among retrn rates an costs provie ifferent crves that are shown in the figre. Fig.. Retrn rate verss costs From figre, it is possible to ientify two optimal sitation for the company that is to consier c>c=c or c=c=c. The retrn rate for these two sitations is to ½, that is, =0, Scenario withot a reverse policy ( = 0%) Initially, let s assme that the company oes not se a reverse system as the one ehibite by figre. The operation policy of the company in terms of serviceable inventory level an manfactring rate is shown in figre Serviceable level Manfactring rate months Fig.. Inventory an proction levels for =0 It is worth mentioning that the company is rnning in its maimm loa, which means that 750. Abot the inventory levels, every proct manfactre by the company is practically se to meet the eman at once. However, a small part of procts is kept in stock to reayelivery. In fact, the company ses a performance measre for cstomer satisfaction that look for garanteeing 95% of reay-elivery stock to the market place. The ineqality given by (7) shows how to keep a save stock for reayelivery with k = 95% of chances of no violation of serviceable inventory constraint. Table provies the costs of the company to rn its proction operation withot a reverse policy (=0). Note that both holing serviceable an manfactring new proct costs are consiere in this Table. The iea is to compare these costs with those costs that will obtain with = 50%. 4., Scenario with the retrn rate = 50% Figres 4 an 5 illstrate optimal inventory an proction trajectories for forwar an backwar channels. In this scenario, 50% of collecte procts retrn from the marketplace. These procts are checke before to be sent to remanfactre or to be properly ispose. An interesting aspect to be note throgh this sitation is the rection of the proction of new procts. Practically, new proct proction rate was rece to 47% compare to the previos sitation (i.e., withot reverse policy; figre ). Note, from figre 5, that remanfactring level of se-procts was close to maimm capacity. Srely, this new featre reces costs, particlarly e to prchase of components to proce new procts. Note that this featre eplains the reason of the cost for remanfactring to be less than the cost for manfactring new procts Serviceable inventory Retrnable inventory months Fig. 4. Inventory-proction levels withot reverse policy Proction an isposal rates Manfactre Dispose efective Remanfatre months Fig. 5. Optimal proction an isposal rates 89

6 9th IFAC Worl Congress Cape Town, Soth Africa. Agst 4-9, Comparing scenario s costs Table provies the costs relate to both scenarios. Note that the total cost of the scenarios is more than the oble of scenario, which means that the se of remanfactre procts can ecrease the overall costs of proction. It is important to nerstan here that sch a reslt is only possible becase it was assme that the costs incrre in the reverse channel are less than the costs of the forwar channel. Table. Costs of each scenario ($) Costs (K=000) Scenario (=0) Scenario (=0,5) Serviceable holing 4,70 K 5,45 K Retrnable holing 40 K Manfactring 6.8,0 K.500 K Remanfactring 909,9 K Disposing 6.959,0 Total cost 6.559,90K.808,80 K 4.4. Final comments abot scenarios The se of reverse logistics scheme joint to a goo retrn policy for se procts can rece the holing an proction costs of the company. However, it is reqire that the cost for remanfactring be at least slightly less than the cost for manfactring new procts. The two scenarios previosly analyze show sch a sitation. In fact, in scenario one (=0), the manfactring of new procts is intense an ses all proction capacity. Comparatively, in the scenario two (=0,5) retrnable procts are remanfactre an, as a conseqence, the manfactring process of new procts is rece aron 47%, which means less epensive cost for the company. This characteristic is observe in the Table. 5. CONCLUSION The se of raw material etracte from the earth an energy spent to proce an istribte procts aron the worl have one reverse logistics an essential part of an integrate spply chain. The main objective of a reverse system is to move procts from market collectors to places where they can be remanfactre or properly ispose. The prpose is that some se-procts can get vale along the reverse chain by mean of any retreatment. As a conseqence, it is possible to improve the environment, recing waste. In this paper, reverse logistics channel is ae to the forwar channel in orer to rece inventory an proction costs. A stochastic ynamic problem is introce to this proposal. The problem is, therefore, efine by two proctioninventory systems; one is a forwar stochastic system, an the other a reverse system. Dring the operation of these systems, manfactring an remanfactring procts are available in a serviceable inventory to meet the eman. After a perio, procts alreay se retrn from the market to the company sing a backwar channel, which incles a retrnable inventory an a remanfactring process. A two stage procere is consiere in this sty. In them, a stochastic linear qaratic Gassian (LQG) moel with constraints is formlate. In the first stage, the ranom natre of the problem is eliminate by changing the eman by its monthly average vale. Ths, in this first stage, the optimal retrn rate is provie, taking into accont ifferent schemes of costs (as shown in Table ). In the secon stage, the stochastic problem is solve by mean of an Open-Loop Feeback controller, where an estimate eman is se to provie an optimal annal plan for manfactring, remanfactring an isposal variables. Throgh scenarios analyses, it is possible to compare this optimal plan with a retrn rate obtaine from stage, with the scenario where a company oes not se a reverse channel. This procere can help managers to evelop ecisionmaking abot appropriate policies of retrning for the company. In fact, this simple eample of a make-to-stock company, it is possible to reach important conclsion regars to the se of retrne procts to rece costs for the company. Acknowlegment: The paper has been spporte by CNPq ner Process Nmber: 0606/00-. REFERENCES Bertesekas, D. P. (000). Dynamic programming an stochastic control, Athena scientific, Vol.. USA. Dobos, Imre (00) Optimal Proction-inventory strategies for HMMS-type reverse logistics system, Int. J. Proction Economics, 8-8, Fleischmann, M., J. M. Bloemhof-Rwaar, R. Dekker, E. Van er Laan, J. A. E. E. Van Nnen an L. N. Van Wassenhove (997). Qantitative moels for reverse logistics: A review, Ero. Jor. of Opera. Research, 0. Graves, S. C. (999). A single-item inventory moel for a non-stationary eman process, Manfactring & Service Operations Management, Vol., No. Kenné, J-P.; Deja, P.; Gharbi, A. (0). Proction planning of a hybri manfactring remanfactring system ner ncertainty within a close-loop spply chain, International Jornal of Proction Economics, Vol. 5, Isse. Minner, S. an Kleber R. (00) Optimal Control of Proction an Remanfactring in a Simple Recovery Moel with Linear Cost Fnctions, OR Spektrm, Springer-Verlag, : -4. Oaret, S.; Polotski, V.; Kenné, J-P.; Gharbi, A. (0). Optimal Proction Control of Hybri Manfactring/Remanfactring Failre-Prone Systems ner Diffsion-Type Deman, Applie Mathematics, 4, Silva Filho, O. S. (009), Optimal linear qaratic gassian problem applie to reverse logistics system, Proceeings of the th IFAC Symposim on Information Control Problems in Manfactring Moscow, Rssia, Jne -5. Silva Filho, O. S. (0), An open-loop soltion for stochastic proction remanfactring planning problem, Proceeing of 8 th ICINCO, Netherlans. 89