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A Model Predictive Control and Time Series Forecasting Framework for Supply Chain Management

A Model Predictive Cotrol ad Time Series Forecastig Framework for Supply Chai Maagemet Philip Dogais, Elei Aggelogiaaki, ad Haralambos Sarimveis Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig
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A Model Predictive Cotrol ad Time Series Forecastig Framework for Supply Chai Maagemet Philip Dogais, Elei Aggelogiaaki, ad Haralambos Sarimveis Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig Vol:, No:3, 8 Abstract Model Predictive Cotrol has bee previously applied to supply chai problems with promisig results; however hitherto proposed systems possessed o iformatio o future demad. A forecastig methodology will surely promote the efficiecy of cotrol actios by providig isight o the future. A complete supply chai maagemet framework that is based o Model Predictive Cotrol (MPC) ad Time Series Forecastig will be preseted i this paper. The proposed framework will be tested o idustrial data i order to assess the efficiecy of the method ad the impact of forecast accuracy o overall cotrol performace of the supply chai. To this ed, forecastig methodologies with differet characteristics will be implemeted o test data to geerate forecasts that will serve as iput to the Model Predictive Cotrol module. Keywords Forecastig, Model predictive cotrol, productio plaig. I. INTRODUCTION URING the last decades, idustrial goods productio has D shifted from the local or atioal level to facilities with global outreach that serve may atioal markets. This developmet has put substatial stress o the supply chais of compaies, demadig a better orgaizatio of performed actios ad thus callig for supply chai maagemet methodologies that ca improve customer service while reducig cost. This paper will demostrate the couplig of Model Predictive Cotrol with a forecastig system ad ivestigate its effectiveess for the problem of supply chai maagemet. A. Model Predictive Cotrol Model predictive cotrol (MPC) is owadays recogized as a stadard methodology for the cotrol of idustrial ad Mauscript received August 9, 6. Fiacial support by FAGE S.A. ad the Geeral Secretariat of Research ad Techology i Greece uder the PENED research program (ΕΔ38) is gratefully ackowledged. Η. Sarimveis is with the School of Chemical Egieerig of the Natioal Techical Uiversity of Athes, Zografou, Athes, 578 Greece (Correspodig author: ; fax: ; P. Dogais is with the School of Chemical Egieerig of the Natioal Techical Uiversity of Athes, Zografou, Athes, 578 Greece ( E. Aggelogiaaki is with the School of Chemical Egieerig of the Natioal Techical Uiversity of Athes, Zografou, Athes, 578 Greece ( process systems []. This is because of its capability to icorporate costraits for the maipulated ad/or the cotrolled variables, to hadle the oliearities ofte preset i dyamical systems ad to overcome modelig mismatch. The idea of model predictive cotrol is simple: A process model is used to predict the effect of a fiite umber of future moves o the cotrolled variables. This model is icorporated i a o-lie ope loop optimizatio problem, which determies the optimal cotrol sequece for a give performace criterio. The simplest MPC objective fuctio is the weighted sum of the two basic cotrol targets, amely the sum of squared differeces betwee the predicted outputs ad their set poits over the future predictio horizo ad the sum of squares of the cotrol moves over the cotrol horizo. After the solutio of the miimizatio problem is foud, oly the first of the future cotrol actios is implemeted to the system. The same procedure is performed repetitively at each time step. t- Past Iv(t) OR(t) t Future Iv t ( + j t) OR(t+j t) t+j Cotrol horizo Predictio horizo Set poit, TIv Fig. MPC basic cocept Future outputs Future iputs t+ch t+ph MPC was first applied to ivetory maagemet by Kapsiotis ad Tzafestas [], who studied a sigle maufacturig site problem ad icluded a pealty term i the objective fuctio for deviatios from a referece path for ivetory i order to compesate for productio lead times. 35 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig Vol:, No:3, 8 Perea-Lopez et al [3] employed MPC to the maagemet of a multi-level supply chai with multiple products where demad was determiistic, so the eed for a ivetory cotrol mechaism was reduced. Brau et al [] preseted a liear MPC methodology for large scale supply chai problems ad showed that MPC ca hadle ucertaities due to model mismatch ad usuccessful forecasts. Fially, Li et al [5] preseted a Miimum Variace Cotrol system with a set poit ot oly for the actual ivetory level, but also for the WIP (Work-I-Process) level, while customer demad was expressed by a ARIMA model. Their formulatio maitaied ivetory levels at a desired level avoidig the bullwhip effect ad whe compared, it proved superior to other frameworks. B. Forecastig Forecastig plays a cetral role i the efficiet operatio of a supply chai, as it provides valuable iformatio o the expected future directio of importat factors, thus eablig plaers to act preemptively ad more effectively. Various methodologies have bee proposed for forecastig ad they are typically time series algorithms that, depedig o the ature of the model they are based o, ca be classified as liear o oliear. The simplest method of all is aïve forecastig, where the forecast is assumed to be equal to the previous value, a method that is ofte used as a basis for compariso. Liear models are the most popular, partly due to their simplicity ad ease of use. Examples of widely used liear methodologies are the autoregressive movig average (ARMA) ad autoregressive itegrated movig average (ARIMA) [6], which is a geeralizatio of the former. The forecast i these two methodologies is produced after differecig the time series at a appropriate order (if ecessary) usig weighted past values of the time series ad past forecast errors. A geeral form of the ARIMA(p,d,q) model is the followig: p q = + i d i ϕil ( L) Xt θil εt () where L is the lag operator, φ i are the parameters of the autoregressive part of the model, θ i are the parameters of the movig average part, p is the order of autoregressio, d is the order of differecig, q is the order of the movig average process ad ε t are error terms. Depedig o the values of the parameters i the geeral form depicted i Eq. (), there are may types of ARIMA models, like the Autoregressive (AR) model, which is a ARIMA(p,,) model where oly past values of the fuctio are used to produce a forecast. Aother method widely used is the Holt-Witers, which is a expoetial smoothig methodology. As such, it uses weighted values of past time series occurreces, where the coefficiets decay expoetially with each period, thus givig more weight to recet values ad less to more distat oes. Its structure ca capture treds ad seasoality i data, makig it suitable for various types of time series data. The additive form of the Holt-Witers method is: Y = μ + bt+ S + e () t+ h t t t p+ h t where Y t+h is the predicted value of the h-th period ahead i time; μ t is a mea value of the series, which is updated as i Eq. (3), where p the periodicity of the seasoality; b t is a tred parameter of the series, updated as i Eq. () ad S t is the seasoal compoet of the series, updated as i Eq. (5). ( Y S ) ( )( b ) μ = α + α μ + (3) b t t t p t t ( ) ( ) ( ) = γ μ μ + γ b () t t t t St = δ Yt μt + δ St p (5) I most methods, icludig the two metioed above, the critical parameters of the equatios that describe the behavior of the time series are ot kow ad have to be established through a time-cosumig procedure of trial-ad error ad applicatio of statistical tests. Furthermore, the liear structure of the model is ot able to represet oliearities possibly preset i the time series. Artificial Neural Networks, ANN, ad more specifically Radial Basis Fuctio (RBF) eural etworks, is a oliear methodology that addresses the weakesses that were metioed above ad are preset i may models. Its iheret sophisticated structure allows it to capture the complexity i the behavior of series with oliearity, while at the same time model parameters ca be determied with algorithms that require o trial-ad error procedure. The RBF eural etworks cosists of three layers, as show i Fig.. The iput layer is used to feed the iput variables ito the model. The hidde layer cotais a umber of odes, which apply a oliear trasformatio to the iput variables, usig a radial basis fuctio. The output layer serves as a liear summatio uit. The eural etwork depicted i Fig. is a typical RBF eural etwork with oly oe output ode. Each hidde ode is associated with a vector c with dimesio equal to the umber of iputs to the ode, called a ceter. The activity ν of a hidde ode is the Euclidea distace betwee the iput vector ad the ode ceter. The hidde ode output is the value of the radial basis fuctio whe the activity ν is its iput variable. I the preset work, the thi-plate-splie radial basis fuctio is employed: f ( ν) = ν log( ν). A traiig algorithm for RBF etworks is based o usig a set of iput-output data (x i, y i ),,,, K i order to determie of the structure ad the parameters of the etwork that lead to a miimum error betwee the predicted output ad the actual values. This defies a Mixed Iteger Noliear Programmig (MINLP) optimizatio problem, which is solved usig special traiig algorithms. 36 x x x3 xn 3 w w w3 w wl Σ ŷ Combiig Eqs. (6)-(8) we arrive at Eq. (9), which shows that ivetory at is related to order rate with a autoregressive with exogeous iput model (ARX) that also cosiders customer demad as a exteral measured disturbace. i (9) () = ( ) + ( ) () Iv t Iv t g Order t i Sales t Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig Vol:, No:3, 8 L Fig. A example of the RBF eural etworks architecture II. METHODOLOGY I this sectio the proposed cotrol methodology will be described i detail. Fig. 3 presets the block diagram of the cotrol scheme, cosistig of the process, the MPC law ad the forecastig policy of customers demad. I the followig, a brief descriptio of the process ad afterwards the MPC cofiguratio will be preseted. A. Process Model ad Material Balace I may productio-ivetory systems, the productio process is modelled by a pure delay uit, with a discrete T trasfer fuctio equal to z, where T is the lead time. However, such a assumptio is ot always realistic sice the productio rate may deped o orders give i differet times i the past. I this work, we assume that the process dyamic behavior is described by a Fiite Impulse Respose (FIR) model. I this case, the system output (productio rate R ( t ) ) will be give by the followig Eq.: i (6) () = ( ) R t g Order t i where Order ( t i), i =,..., is the order rate at -i, is the system order ad gi, i =,..., are the system parameters. Eq. (6) ca easily lead to the trasfer fuctio betwee productio rate ad order rate z-trasformed sigals: R z Order z = g z g z (7) which obviously is a geeralizatio of pure delay. From the block diagram of Fig. 3, ivetory Iv ( z ) is give by the followig equatio: Iv z R z Sales z z = where Sales ( z ) is the z-trasform of customers demad Sales ( t ) ad z (8) is the trasfer fuctio of the itegrator. B. Robust Model Predictive Cotrol Scheme I case of ivetory cotrol (Fig. ), maipulated variables of the proposed cotrol scheme are the future order rates Order ( t + j t ), j =,..., ch ad cotrolled variable is the predicted ivetory Iv t + j t, j =,..., ph. A predictor for ivetory is formulated based o the material balace represeted of Eq. (9). I order to test the robustess of the proposed cotrol scheme, we assume that the predictor is based o a approximatio of the process parameters g i, i =,..., ad ot their actual values (Eq. (6)). The ivetory predictor also uses a estimatio of ukow future sales ForSales ( t + j t), j =,..., ph. This estimatio ca be the simple projectio of curret sales over the predictio horizo, or ca be calculated from a forecastig policy, as is the case here. So, the optimizatio problem solved o lie is described by the set of Eqs. ()-(7). ph ch mi w Iv ( t + j t) TIv + r δorder ( t + j t ) ( + ) OR t i,..., ch j= j= ( + ) = ( + ) + i ( + ) ForSales( t+ j t) + e( t+ j t) Iv() t Iv t j t Iv t j t g Order t j i (5) () Iv t t = () ett, if j= et+ jt = (), else = () ( ) i ( ) + () e t t Iv t Iv t g Order t i Sales t (3) δ Order t + j t = Order t + j t Order t + j t, j =,..., ch () u mi Order( t + j t) u,,..., j = ch (5) δ u δ mi Order( t + j t) δ u,,..., j = ch (6) δ Order ( t + j t) =, j = ch,..., ph (7) where Iv t + j t, j =,..., ph is the j- step ahead predictio of ivetory, ph ad ch are the predictio ad the cotrol 37 Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig Vol:, No:3, 8 horizo respectively, TIv is the target ivetory value, δ Order ( t + j t), j =,..., ch are the future cotrol moves (Eq. ()), w, r are weight matrices ad et ( + jt), j=,..., ph is the predictor error (Eq. ()-(3)). Eq. () shows that the curret value of the predictor is equal to the actual. Eq. () deotes that the predictor error should correct oly the first predictio sice Eq.() is a autoregressive model. Eq. (3) gives the predictor error from curret sales ad ivetory value. Eqs. (5)-(6) are hard costraits that boud the maipulated variables ad the cotrol moves respectively. u mi, u, are the lower ad upper bouds for order rates ad δ u mi, δ u, are the lower ad upper bouds for cotrol moves. Eq. (7) esures that o cotrol moves are made after the cotrol horizo. Target Ivetory + MPC - Order Rate Forecastig policy Process Productio Rate - + Sales Fig. 3 Block diagram of MPC scheme for ivetory cotrol Ivetory III. RESULTS The proposed MPC-forecastig framework was tested i four test cases, where i each case a differet forecastig method was employed i order to produce sales forecasts. The forecastig methodologies used were aïve forecastig (for compariso purposes), Liear Autoregressio (Liear AR), Holt-Witers ad RBF eural etworks (RBF ANN). The test data was supplied by a leadig Greek dairy products maufacturer cocerig the sales of a fast movig product ad the results are show i Table I. The first colum cotais the average error for forecastig the sales time series ad the secod the sum of squared deviatios from the ivetory set poit. Method TABLE I IMPLEMENTATION RESULTS Average Forecastig Error Deviatio from ivetory set poit (SSQE) RBF ANN,53 7,33 Holt,958 8,3 Liear AR, 79,59 Naïve model,988,89 Demad Actual sales ANN forecast Fig. Compariso betwee Actual sales ad the ANN forecast These results idicate, first of all, the character of the time series, which is mostly oliear sice the oliear method used (RBF eural etworks) provides defiitively better forecasts tha the two other liear methods. Secodly, it becomes clear that employmet of a forecastig methodology leads to improved performace of the MPC module. I particular, the forecastig method that produced the best results, that is RBF eural etworks, led to a drastic reductio of the deviatio from the ivetory set poit, thus leadig to sigificatly less ivetory holdig costs. Figs. -6 show the results of the ANN forecastig case of framework implemetatio to the problem studied. As ca be observed i Fig., forecast values are close to the actual values, thus providig a advatageous isight for the future actios of the MPC module of the framework. Fig. 5 depicts productio orders, while Fig. 6 shows the course of product ivetory over the time period studied. It must be poited out that the ivetory gradually approaches the set poit over the course of time ad shows small ad decreasig variace from its set poit, especially towards the ed of the period studied. Order O(t) Fig. 5 Productio orders (ANN forecastig case) 38 Ivetory 8 6 Ivetory(t) Ivetory set poit 3 5 Fig. 6 Product ivetory (ANN forecastig case) Iteratioal Sciece Idex, Idustrial ad Maufacturig Egieerig Vol:, No:3, 8 IV. CONCLUSION A framework for supply chai maagemet based o Model Predictive Cotrol combied with a forecastig module was preseted. Various liear ad oliear forecastig methodologies were evaluated i order to ivestigate the existece of possible oliearity i the sales time series. The oliear method used, amely RBF eural etworks, exhibited superior forecastig performace, showig that the series had mostly oliear character. The simulatio results demostrated that forecast accuracy leads to improved cotrol performace, thus leadig to more efficiet maagemet of the supply chai. REFERENCES [] M. Morari,.JH. Lee, Model predictive cotrol: Past, preset, ad future, Computers & Chemical Egieerig, vol. 3, 999, pp [] G. Kapsiotis, S. Tzafestas, Decisio makig for ivetory/productio plaig usig model-based predictive cotrol, i: S. Tzafestas, P. Bore, L. Gradietti (Eds.), Parallel ad distributed computig i egieerig systems, Amsterdam: Elsevier, 99, pp [3] E. Perea Lopez, B. E. Ydstie, I. Grossma, A model predictive cotrol strategy for supply chai maagemet, i Computers & Chemical Egieerig, vol. 7, 3, pp. -8. [] M. W. Brau, D. E. Rivera, M. E. Flores, W. M. Carlyle, K. G. Kempf, A model predictive cotrol framework for robust maagemet of multiproduct, multi-echelo demad etworks, i Aual Reviews i Cotrol, vol. 7, pp [5] P. H. Li, S. S. Jag, D. S. H. Wog, Predictive cotrol of a decetralized supply chai uit. Idustrial Egieerig & Chemistry Research, vol., 5, pp [6] G. E. P. Box, G. M. Jekis & G. C. Reisel, Time series aalysis : forecastig ad cotrol, 3rd ed., Eglewood Cliffs, New Jersey, Pretice Hall,
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