EMSS International Mediterranean Modeling Multiconference. European Modeling Simulation Symposium. October 20-22,2005 Marseille, France

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1 International Meiterranean Moeling Multiconference EMSS European Moeling Simulation Symposium October -, Marseille, France EDITED BY Chiara BRIANO Clauia FRYDMAN Antonio GUASCH Miquel Angel PIERA Laboratoire es Sciences e l'information et es Systèmes DIPTEM University of Genoa International Meiterranean & Latin American Council of Simulation McLeo Institute of Simulation Science Liophant Simulation ISBN -97--

2 LOAD AND INVENTORY FLUCTUATIONS IN SUPPLY CHAINS J.-C. Hennet LISA, 6 avenue Notre-Dame-u-Lac, 9 Angers, France hennet@laas.fr KEYWORDS Orering policies, bullwhip effect, ranomness, multistage prouction. ABSTRACT The paper analyzes multi-stage supply chains subject to ranom variations of emans for final proucts. It is shown by analytical simulations that uner classical istribute prouction an orering policies, reactivity to eman changes generates important loa an inventory fluctuations. This phenomenon, commonly calle the bullwhip effect appears to be highly epenent on these policies. A new policy, calle the mean-eman riven policy is propose to attenuate this phenomenon. INTRODUCTION During the last fifteen years, supply chain analysis has become a major concern both in manufacturing theory an in inustrial practice. In the extene sense, which is now prevailing in the literature, a supply chain associates all the enterprises that contribute to prouction an sale of a family of proucts (goos or services). In this view, an ieal supply chain can be seen as a virtual enterprise esigne to satisfy some consumers nees in the most efficient an profitable way. Several performance inices have been propose to evaluate the quality of a supply chain, particularly in terms of costs an value, ecisional integration, agility, reactivity an reliability. Some obstacles to performance maximization have then appeare in the very nature of supply chains. In particular, the ecisional autonomy of enterprises sets some limits to communication, coorination an integration between the interacting entities. Responsiveness an reactivity to eman changes may generate fluctuations in work loa an inventory levels that may be amplifie as they propagate upwar along a supply chain. Since the work of H. Lee [Lee et al. 997], such a phenomenon has been referre to as the Bullwhip Effect. Observation of real inustrial cases an simulation stuies have shown examples of bullwhip effects. The bullwhip effect has been observe in many real supply chains, ranging from mechanical inustries to several foo sectors (see [Miragliotta, ]). A case reporte by the Supply On leave from LAAS-CNRS, 7 avenue u Colonel Roche, 77 Toulouse Ceex, France Chain Simulation Workgroup of Santa Fe Institute Business Network concerns Proctor an Gamble's Pampers ivision, who foun huge swings in weekly eman an orers within their supply chain. Managers, researchers, engineers an stuents all over the worl have been taught the bullwhip effect by playing the popular Beer Game, a simulation evelope by John Sterman s group at MIT's Sloan School of Management [Sterman 99]. The players of the game represent four noes in an iealize supply chain for a Beer company: retailer, wholesaler, istributor, factory. Play of the game prouce volatility much le that observe in real supply chains. As the backlog for orers increases, players orer too much input proucts, forcing their suppliers into severe backlogs. Conversely, excessive prouction propagates ownwar along the supply chain an the ecrease of orers amplifies. Key factors for generating bullwhip effects are eman uncertainty an istribute elays, but it has been shown that the loaing an orering policies themselves ten to amplify the phenomenon, either by lack of global information or by not using properly the global information available. In particular, [Sterman, 99] has stresse the estabilizing influence of misperception of the multiple feeback loops in the case of local base-stock policies. In terms of control system theory, loa an inventory variables can be viewe as the outputs of the supply chain system, the final emans being the external inputs subject to isturbances, an replenishment policies proucing the controlle inputs. From this approach, several authors [Dejonckheere et al. ] have stuie the stability property of the system uner classical base-stock policies an they have propose in particular to reuce the nervousness of this policy by using a proportional controller. The bounary of the control action set for a supply chain system is etermine by its istribute structure. [Miragliotta, ] has propose a partitioning of the causes of the bullwhip effect into structural eterminants an external triggers. More generically, this paper istinguishes internal an external factors an evaluates the possible improvements that can be expecte from using relevant external ata as parameters in local orering an loaing policies. In spite of their will to cooperate, the partners of a supply chain generally have ifferent constraints, objectives an practices. Local stock levels an prouction loas are 7

3 private ata that can only be integrate in local orering policies. However, with the progress of integrate information platforms, other ata can be share by all the partners of a supply chain. In this stuy, it is assume that emans for the final proucts of the supply chain are communicate in real time to all the partners. Then, three types of loaing an orering policies will be stuie: inventory-base local policies, internal orers-base policies an external orers base policies. This stuy will be conucte by analytical simulation with Scilab [Gomez, 999] on a proucts levels manufacturing network presente an moele in section. The ifferent orering policies will be escribe in section. Then, section will compare the simulation results an some conclusions will be finally presente. A MODEL OF A SUPPLY CHAIN SYSTEM Presentation of the ynamic moel The moel propose to escribe a supply chain is similar to the ynamic system presente in [Hennet, ]. It is base on the prouct structure of the final proucts with the bills of materials an lea-times associate to manufacturing stages. Transportation times are suppose inclue in leatimes. Classically (see e.g. [Muckstat an Rouny 99] an the references therein), a multistage prouction structure can be represente by an acyclic irecte graph with noes representing the prouction activities an arcs linking components to proucts. The total number of prouct types is n. In the consiere multistage prouction structures, there is a one-to-one corresponence between proucts an activities: each prouction activity has several input proucts an one output prouct. Prouction of one unit of prouct of type i (i=,,n) requires the assembly of components j=,...,n in quantities P ji. Proucts can be partitione into levels. Level is for en proucts, level l for proucts which are components of proucts of levels strictly less than l, an of at least of one level l- proucttype. If proucts are numbere in the increasing orer of their level, matrix P = (( P ij )) has a lower triangular structure with zeros on the iagonal. Each enterprise of the supply chain is suppose to perform one or several prouction activities using components provie by other enterprises upwar in the chain an/or proviing components to other enterprises ownwar along the chain. As in [Hennet, ], the planning horizon [,T ] is ivie in time perios, prouct lea-times are suppose constant an multiple of the elementary time-perio. The lea-time associate with prouct i is enote θ i. In the gozinto graph, constant urations θ i are thus associate with noes i (,..., n), an arcs (i,j) are value by Pij to represent the gozinto coefficients of the proucts. External emans for final proucts are suppose ranom with constant mean values an uniformly istribute boune variations. The quantities involve in the moel are escribe below for i (,..., n), k (,..., T ) : - s is the quantity of prouct i in stock at the beginning of perio k,, v is the quantity of prouct i whose prouction is - starte at perio k, z is the quantity of prouct i elivere at perio k- - is the eman for prouct i at perio k. The eman is external to the supply chain system for final proucts an internal for primary an intermeiate proucts, The system amits a ynamic representation in the form of a iscrete-time multivariable linear moel with istribute elays: si, k + = s + vi, k θ z i. () Demans for intermeiate proucts are consistent with the input (or gozinto) matrix, P, an with prouction levels: N = P v i= ij jk. Prouction levels, inventory levels an backorers are subject to pointwise-in-time constraints: positivity c is constraints an capacity constraints. In particular, if the prouction capacity for prouction of i at perio k, the prouction orer shoul satisfy: v. c Customers an proucers items being elivere on stock as much an as soon as possible, elivery variables are given by: = +. () z min(, s vi, k θ ) i Some properties of the moel The prouct structures consiere are characterize by an output matrix which is the ientity matrix (activity loas are measure in units of output proucts) an an input matrix P which is square, with nonnegative coefficients, lower-triangular with zeros on the main iagonal. Consier the vector of mean external emans for all the proucts = [ L ] T n. This vector being assume constant, stability requires that the mean prouction vector v = [ v L v ] T n is constant in steay state conitions an satisfies: = ( I P) v. Then, it is not ifficult to show that the inverse of matrix ( I P) exists an is non negative. It uniquely etermines the mean equilibrium prouction vector: ()

4 v = ( I P) () A vector of equilibrium inventory levels S = [ S L S ] T n can also be efine. Its computation epens not only on the probability istributions of external emans but also on the orering policy use for each activity. A necessary conition for the system stability is that the equilibrium state efine by the couple (v,s) satisfies all the capacity constraints. A -proucts example An example of a prouct structure for final proucts (numbere an ) is shown in Fig.. This is a three-level prouct structure with final proucts an at level, intermeiate prouct at level an primary proucts an at level. θ = θ Figure The gozinto graph of a prouct structure It is assume that the proucts are manufacture by ifferent enterprises. Prouction capacities at each perio are given by: C = [ ] T. The vector of external emans is stationary with mean value = [ ] T. The corresponing bill of material, given by equation (), gives the mean prouction vector: v = 7 9 () [ ] T k θ = θ = θ = At any perio k, is a uniformly istribute ranom variables in the interval [,] an is a uniformly k istribute ranom variables in the interval [,]. Accoringly, equilibrium inventory levels have been compute an use to initialize the inventory trajectories = when simulating the prouction network uner the ifferent policies. THE PRODUCTION AND ORDERING POLICIES An orering policy is a rule that etermines the amounts of components orere to suppliers as a function of the available knowlege on the current state of the system. In a similar manner, a prouction policy is a rule that etermines the amounts of proucts to be manufactures as a function of the available knowlege on the current state of the system. Classically, it can be assume that in a supply chain, each enterprise only knows its current an past manufacturing an supply orers, its current an past emans (from partners an from customers). Aitionally, some anticipate knowlege of future orers for each enterprise will be assume to construct the orer-base policy. In this stuy, it is also assume that current an past values of external emans for final proucts are available for all the partners of the supply chain. Such an assumption opens new possibilities for builing prouction an orering policies. The three types of policies that will be compare are the inventory-base policy, the orer-base policy an the eman-base policy. The inventory-base policy This policy acts more or less on the Kanban principle. At each prouction stage an at each perio, the quantity elivere is given by expression (). This elivery triggers prouction orer for prouct i: u = min( z, Ci ). Compoments availability conitions the prouction level for stage i through: s jk v = min ( u, ). (6) j ; P P ji ji What is typical of this policy is that internal orers for primary an intermeiate proucts j ; P ji are create from the current prouction level of proucts i through the following relation: = P v (7) jk i= : ji θ i Such a policy relies much on current inventory levels since they limit elivery an prouction at each noe of the prouction network. Such limitations propagate upwars with some elay through the Kanban mechanism escribe by expression (7). The orer-base policy The orer-base policy obeys a mechanism that combines lot for lot MRP an capacity constraints. Each enterprise provies proucts from its stock as long as the inventory level remains positive. Backorers are avoie by anticipating prouction for orers that cannot be satisfie 9

5 from the stock. The orering mechanism propagates upwar taking into account the bill of materials an lea times. The mean eman-riven policy The mean eman-riven policy is a very simple local policy in which prouction orers are compute from the bill of material () an from the estimation of mean emans for final proucts: vk = ( I P) ˆ k. () Backorers may be generate when prouction capacity an/or availability of input components are not sufficient to execute these orers. Recursive estimators of mean emans are constructe by taking the average of real eman ata on a given time winow note W: W ˆ = ˆ (9) + W W It can be note that ue to recursive eman estimation, the mean eman-riven policy is aaptive an can be applie to non stationary eman profiles. NUMERICAL RESULTS In orer to evaluate the bullwhip effects associate to the three policies, prouction an inventory s for the proucts have been estimate by series of runs over a perio time horizon. To each run correspons a ranom generation of uniformly istribute eman sequences for proucts an, respectively in the intervals [,] an [,]. Then, moel-base simulations have been performe using Scilab. [Gomez, 999]. The results are presente in table. Also, for comparison purposes, the same ranom eman trajectories of ( k ) an ( k ) have been use for the three policies. These eman trajectories are isplaye on Fig.. Then, figures, an show the prouction an inventory evolutions for the proucts uner the three policies for ata of Fig Fig. a Figure The inventory-base policy A first remark concerning this policy is that it oes not stabilize the system if initial inventory levels are not Fig. b sufficient. In that case, backorers ten to accumulate while prouction levels remain insufficient ue to the fact that they are conitione by inventory levels through equation (6). Important values of inventory levels have thus been require to obtain convergence for all the experiments. The selecte initial inventory values are given by the following vector: S=[ 7 ] 6 6 Prouction Inventory Figure Prouction an inventory levels From simulation results of Fig., it can be note that the inventory-base policy sustains important variations of prouct loa an prouct inventory at all the prouction stages. The orer-base policy As for the lot for lot MRP technique, the orer-base policy uses the initial stock in the transient stage an then, stocks are maintaine at the zero level, prouction anticipating eman. Better stability results have been obtaine for this policy in terms of eman satisfaction. Compare to the inventory-base policy, lower initial inventory levels have been neee: S=[ 7 ]. With this policy, inventory levels are completely stabilize in a few perios but prouction fluctuations remain important, of the same orer of magnitue as for the inventory-base policy.

6 Figure Prouction an inventory levels The mean eman-riven policy Prouction Prouction Inventory Inventory Figure Prouction an inventory levels Results show that uner the mean eman-riven policy, eman fluctuations are almost completely absorbe by the inventories of final proucts. Smooth prouction curves are obtaine for all the enterprises. Inventory variations are very limite for primary an intermeiate proucts. Important inventories are neee for final proucts but they can be kept very small for primary an intermeiate proucts. Variance analysis for the three policies The three policies have been simulate on series of ranomly generate eman trajectories for final proucts an. The mean eman values have been kept constant ( =, = ). Deman s associate to the uniform istributions have been evaluate an inicate in table. These s are compare with prouction an inventory s for the proucts. In interpreting the results, it is important to take into account the bill of materials, which naturally amplifies for components the means an s of prouction an inventory levels. Inventor y-base policy Orerbase policy Demanbase policy Proucts Deman 7 Prouction 6 Inventory 6 6 Prouction 7 Inventory Prouction 6 Inventory 6 Table Variance comparison for the three policies From this table, it is clear that uner the inventory-base policy, eman variations generate important prouction an inventory fluctuations along the supply chain. The orer-base policy amps oscillations of inventory levels but not those of prouction levels. The mean eman-base policy consierably attenuates the final eman isturbances at all the stages of the supply chain, except for the final stage, for which important inventories are neee>. CONCLUSION Fluctuations of prouction an inventory levels are typical of multi-level material replenishment processes with significant elays an fluctuating emans. In supply chains, this phenomenon, known as the bullwhip effect has been observe in many practical cases. By simulation of analytical moels, the stuy has shown that combination of classical local orering policies tens to sustain or even amplify this effect. Two classical local policies have been stuie: an inventory-base policy, inspire from the Kanban technique, an an orer-base policy, inspire

7 from MRP. In both case, combination of local feeback loops reinforces the oscillations an may generate instability. On the contrary, smoother prouction an inventory levels have been observe uner the so-calle mean eman-riven policy. Uner this policy, the partner enterprises o not irectly influence each other. They rather respon to integrate (or average) eman evolutions in parallel, global consistency in prouction an assembly being achieve through respect of the bills of materials. please consult the web site or write to hennet@laas.fr. REFERENCES [Dejonckheere et al. ] Dejonckheere J., Disney S.M., Lambrecht M.R. an Towill D.R., Measuring an avoiing the bullwhip effect: A control theoretic approach, European Journal of Operational Research (7,),, pp [Gomez, 999] Engineering an Scientific Computing with Scilab. Claue Gomez, eitor, Birkhaüser, 999. [Hennet, ] Hennet J.C. A bimoal scheme for multi-stage prouction an inventory control, Automatica (9),, pp [Lee et al. 997] Lee L.H., Pamanabhan, P. an Whang, S. Information Distortion In A Supply Chain: The Bullwhip Effect, Management Science (:), 997, pp. 6-. [Miragliotta, ] Miragliotta G. The bullwhip effect: a survey on available knowlege an a new taxonomy of inherent eterminants an external triggers, Preprints th WSPE, Vol., pp.69-7,. [Muckstat an Rouny 99] Muckstat, G.A. an R.O. Rouny (99). Analysis of multistage prouction systems. In: Hanbooks in Operations Research an Management Science, vol., S.C. Graves, A.H.G. Rinnooy Kan, P.H. Zipkin, Es.. North-Hollan. pp. 9, 99. [Sterman 99] Sterman, J., 99. Moeling Managerial Behavior: Misperceptions of Feeback in a Dynamic Decision Making Experiment. Management Science. (), -9. JEAN-CLAUDE HENNET is a Research Director of the French Research Center (CNRS). He grauate from the Ecole Nationale Supérieure e l Aéronautique et e l Espace, Toulouse, France, in 97, receive a MS egree in Inustrial Engineering from Stanfor University, California (97), a Docteur-Ingénieur egree from INSA-Toulouse in 97 an a egree of Docteur 'Etat from University Paul Sabatier, Toulouse in 9. He joine the Laboratory for Analysis an Architecture of Systems of CNRS (LAAS-CNRS) in 976, an since 979, he has hel a research position at CNRS. Since, he has been on leave from LAAS-CNRS an has joine the LISA lab of Angers University. His research interests are in the fiels of systems theory, optimization, control uner constraints, iscrete event ecision systems an manufacturing systems. For more information on his work,