Implementation of Distributed Model Predictive Controllers Based on Orthonormal Basis Functions to Increase Supply Chain Robustness

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1 International Journal of Industrial Engineering & Production Research 313 pissn: Deceber 2014, Volue 25, Nuber 4 pp Ipleentation of Distributed Model Predictive Controllers Based on Orthonoral Basis Functions to Increase Supply Chain Robustness R.Madani& A.Raezani* & M.Beheshti Roina Madani, Master of science student, Departant of Electrical Engineering, Faculty of Electrical and Coputer Engineering, arbiat Modares University,ehran, Iran. roina.adani@odares.ac.ir Ain Raezani, Asistant Professor of Electrical Engineering, Departant of Electrical Engineering, Faculty of Electrical and Coputer Engineering, arbiat Modares University, ehran, Iran. raezani@odares.ac.ir Mohaad aghi Beheshti, Associate Professor of Electrical Engineering, Departant of Electrical Engineering, Faculty of Electrical and Coputer Engineering, arbiat Modares University, ehran, Iran. behesht@odares.ac.ir KEYWORDS Delay; Deand; Model Predictive Control; Orthonoral functions; Robustness; Supply chain; ABSRAC oday, copanies need to ake use of appropriate patterns such as supply chain anageent syste to gain and preserve their positions in copetitive world-wide arket. Supply chain is a large scale network consists of suppliers, anufacturers, warehouses, wholesalers, retailers and final custoers which are in coordination with each other in order to transfor products fro raw aterials into finished goods with optial placeent of inventory within the supply chain and iniizing operating costs in the face of deand fluctuations. Model Predictive Control (MPC) is a widely used eans on supply chains, but in presence of long delays and sudden disturbance changes (custoer deand changes), tuning of MPCs requires a tie consuing trial-and-error procedure and syste robustness will be highly decreased. In this paper, a control schee is proposed to increase supply chain robustness. Due to the large scale characteristic of supply chain, the odel is divided into different subsystes and is controlled by distributed odel predictive controllers. Each subsyste odel has been changed in which an integrated is ibedded, input and output changes are highly penalized in cost functions and Laguerre orthonoral basis functions are added in MPC's structures and it will be shown that the supply chain robustness will be increased toward high changes in custoer deand and toward long constant delays in distribution centres, also Results will be copared to previous conventional MPCs applied on supply chains by other authors IUS Publication, IJIEPR, Vol. 25, No. 4, All Rights Reserved Introduction Historically copanies leveraged a variety of factors to differentiate theselves fro their copetition, including product features, price, quality, product Corresponding author: Ain Raezani Eail: raezani@odares.ac.ir Paper first received Apr. 07, 2014, and in accepted for July, 27, availability and custoer service. In today s dynaic arket, copanies can no longer exploit the sae drivers, or ust exploit the differently, in order to reain copetitive. he supply chain is the best answer to above probles. A supply chain consists of all parties involved, directly or indirectly in fulfilling a custoer request. he supply chain not only includes the anufacturer and suppliers, but also contains transporters, warehouses, wholesalers, retailers and

2 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive 298 custoers to transfor products fro raw aterials into finished goods and deliver those goods into the hands of the end custoers as soon as possible. So the supply chain is a large scale syste with different subsystes that are highly coupled with each other. here are soe iportant issues that should be considered in the control of supply chains, such as safety stock control, inventory control, and bullwhip effect control. Safety stock is a ter used by supply chain anagers to describe a level of extra stock or inventory which should be aintained in warehouses to prevent risk of stock-outs due to uncertainties in supply and deand. Each stage or subsyste in a supply chain has specific safety storage level. Copanies today ust be fast and nible enough to react quickly to changes in custoer deand and do it with little inventory. Inventory control is concerned with iniizing the total cost of inventory. he purpose of inventory control is full filling any custoer deand with lowest level of inventory (safety stock levels) and reducing stocks-out without excessive inventories.another ter in the supply chains is bullwhip effect. he bullwhip effect is an observed phenoenon in forecast-driven distribution channels. It refers to a trend of larger and larger swings in inventory in response to changes in custoer deand (Fig.1).Our goal is to keep it as iniu as possible. Fig.1. Bullwhip effect rate According to [1], the bullwhip effect in each subsyste is obtained by following forula Var(inventory level) (1) BE(t) Var(custoer deand) Where Var(.) is the variance in tie t. Different ethods and control approaches are applies on supply chains. For exaple, reference nuber [2] has proposed a atheatical ethod for anaging inventories in a dual channel supply chain, a fuzzy approach is proposed for ulti-objective supplier selection in [3]. Model Predictive controller (MPC) is a widely used eans to deal with large ultivariable constraint control issues in industry. In supply chain studies there are a lot of papers using MPC to iprove supply chain perforance.kapsiotis was the first to apply MPC to an inventory anageent proble [4]; zafestas et al., considered a generalized production planning proble that includes both production/inventory and arketing decisions [5]; Perea-Lopez et al. eployed MPC to anage a ultiproduct, ulti-echelon production and distribution network with lead ties [6]; Dunbar and Desa applied a recently developed distributed /decentralized ipleentation of MPC to the proble of dynaic supply chain anageent proble, reiniscent of the classic MI Beer Gae [7]; Sarveis has written a review for dynaic odeling and control of supply chain systes[6]; Miranbeigi uses a ove suppression ter in supply chain cost functions in order to increase robustness toward changes in custoer deands [9]; Jie Li and Mian Peng have written a paper to solve inventory fluctuation caused by uncertain tie delay in a ulti-level supply chain [10]; Grasso presents a MPC strategy enriched with soft control techniques as neural networks and fuzzy logic to MPC self-tuning capability in a water drinks supply chain [11]. As it is clear, MPC is a widely used eans on supply chains, but in presence of long delays and sudden disturbance changes (custoer deand changes), tuning of MPCs requires a tie consuing trial-anderror procedure and syste robustness will be highly decreased. In this paper, a control schee is proposed to increase supply chain robustness. Due to the large scale characteristic of supply chain, the odel is divided into different subsystes and is controlled by distributed odel predictive controllers. Each subsyste odel has been changed in which an integrated is ibedded, input and output changes are highly penalized in cost functions and Leaguered orthonoral basis functions are added in MPC's structures and it will be shown that the supply chain robustness will be increased toward high changes in custoer deand and toward long constant delays in distribution centers, also results will be copared to previous conventional MPCs applied on supply chains by other authors. 2. Supply Chain Modeling and Control Strategy 2-1. Modeling Assue a supply chain with suppliers(s), anufactures (M), Wholesalers (W), Retailers (R) and final custoers(c) as the nodes of the syste. For each node k, there is an upstrea node denoted by k' which can supply node k and a downstrea node denoted by k" which can be supplied by k. Inventory control strategies are ade to avoid uncertain fluctuations. Each node has its own desired inventory level and inventory fluctuations should perfor in certain sall doain around their desired levels even though deands don t obey any specific structure. Each subsyste odel is as follows. d( invk(t)) p (t k ',k k ',k) d k,k" (t) (2) dt d(bo k(t)) p dt k R,W,M k,k" (t) d (t) k,k"

3 299 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive Where P k ',k (t) is the transferred product through (k',k) channel and is considered as anipulated variable, k ',k is transportation delay tie through channel k' and k, dk,k" is actual nuber of delivered products fro node k to node k" and is considered as a anipulated variable, pk,k" is deand rate sent fro node k" to node k and is considered as disturbance, inv k (t) and BO (t) are inventory level and backorder rate in node k k which are considered as syste outputs [9]." p k,k" and k ',k are the arbitrary paraeters in our odel. he control goal in our paper is to increase supply chain robustness through any kind of delay ( k ',k ) and any kind of deand ( p k,k" ). In order to use MPC, the above continues odel should be discretized as follows: invk(t 1) inv k(t) p (t-1- k ',k Lk ',k) d k,k" (t) BO k (t 1) BO k (t) p (3) (t) d k,k" (t) k,k" 2-2. Control Strategy In large scale applications, such as power systes, water distribution systes, traffic systes, anufacturing systes, and econoic systes, such a centralized control schee ay not suitable or even possible for technical or coercial reasons, it is useful to have distributed or decentralized control schees. Each controller will be optiized by its own policy and its own cost function and will sent its optial signal controls as disturbances (deands) to the next coupled controllers. he structure of distributed controllers is illustrated in Fig.2. Fig.2.Method 1, Decentralized conventional MPCs. In this paper, conventional MPCs in Fig.2 are replaced with orthonoral based MPCs in order to increase supply chain robustness Fig.3. Fig.3. Method 2, Decentralized orthonoral based MPCs. Soe advantages of ethod 2 than ethod 1 are: 1-Siple tuning. 2- Using orthonoral functions in ethod 2 reduces nuber of paraeters used for description of future control trajectory, so coputational volue decreases. 3- Using augented odel in ethod 2 and penalizing input and output changes causes soother responses Cost Functions In this paper decentralized forulation will be used. So instead of a centralized cost function, three individual cost functions will be used. Each cost function will do its own optiization in order to achieve optial orders for its coupled controllers. he cost functions are in quadratic fors. Each of the includes two least square parts. First part is to penalize outputs and the second part is to penalize inputs. k R,W,M he cost functions and constraints are obtained by inv (t) inv P (t) k k 2 k ',k 2 in W out ( ) W in( ) P k ',k,dk ',k BO k(t) 0 d k,k" (t) Subject to ax 0inv t in v, t 0 k Where Win>0 and W out > 0 are weighting atrices. 3. Discrete-tie MPC using Laguerre functions 3-1. Control Signal rajectory and Prediction he z-transfer function of Leaguers function is given as [12] 1 z (z) (z) 1 az k k1 1 2 (5) 1 With 1.where 0 < 1 is called the scaling 1 1 z factor and is selected by the user. Letting l 1(k) to l N(k) denote the inverse z-transfors of 1 (z) to 2 (z). his set of discrete-tie Laguerre functions are expressed in a vector for as L(k) l 1(k) l 2(k) l N(k) (6) According to (5), the set of discrete-tie Laguerre functions in vector (6) satisfies the following equation L(k1) AlL(k) (7) Where A is ( N N ) and can be expressed as l N 2 N 2 N 3 N 3 ( 1) ( 1) 2 Where 1 and the initial condition is given by N 1 N 1 L(0) 1 ( 1) (9) Assue that a discrete tie odel has the for as x (k 1) Ax (k) Bu(k) (10) y(k) Cx (k) We need to change the odel to suit our design (8) (4)

4 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive 300 purpose in which an integrator is ebedded. he augented odel can be expressed as follows according to [13] x (k 1) A 0 x (k) y(k 1) C y(k) A I (11) B u(k) CB x (k) y(k) 0 1 y(k) Where u u(k) u(k1) and x (k) x (k) x (k 1) denote the difference of the control input and the state variable respectively. By using discrete-tie Laguerre functions, u(ki k) can be approxiated by following equation N i j j j1 u(k k) c (k)l (k) L(k) (12) Where c 1 c2 cn. he state prediction has the for as follows: 1 i0 i i i x(k k ) A x(k ) A i 1 BL(i) Where (13) is obtained by replacing L(i) instead of u(ki k). Siilarly the prediction for the plant output can be written as 1 i0 i i i y(k k ) CA x(k ) CA i 1 BL(i) he cost function is in the following quadratic for N p J (r(k k ) y(k k )) Np Q (r(k k ) y(k k )) U(k k ) R U(k k ) he weighting atrices are Q >0 and R> 0. By substituting (12) into the cost function (15) we obtain N p J (r(k k ) y(k k )) 1 Q (r(k k ) y(k k )) (13) (14) (15) (16) R he objective is to iniize the cost function (16) to find the optial coefficient vector in the presence of input and states constraints 3-2. he Unconstrained Solution Without constraints, by substituting (14) into (16), according to [12] the optial solution of the cost function (16), is found as following equation 1 x(k ) (17) i 1 i 1 Where () A BL(i), Np 1 i0 ( ()Q () R) N 1 p and ( ()QA ) 3-3. he Inequality Constraints Model predictive control has the ability to handle hard constraints in the design. With paraeterization of the control signal trajectory, we can choose the locations of the future constraints. his could potentially reduce the nuber of constraints within the prediction horizon [12] Constraints on the Aplitudes of the Control Signal Suppose that the liits on the control signals are and u high ulow. Noting that the increent of the control k1 signal is u(k) u(i), then the inequality constraint i0 for the future tie k, k=1, 2... is expressed in following inequality as [12] k1 (18) u ( L(i)) u(k 1) u low i high i Constraints on the Output Variables Suppose that the constraints on the process output variables are given by ylow and y high. hen the inequality constraint for the future tie instant is in following inequality as [12] y CA x(k ) () y (19) low i high 4. Siulation A ulti echelon supply chain is used in siulation exaples. he supply chain network consists of supplier, anufacturer, wholesaler, retailer and final custoer. Only a single type of product will be distributed along the chain and the purpose of our inventory control isn't zero safety stock; safety storage is to keep custoer deand satisfaction, set as 100 units. Whole siulation tie is 80 tie unit (day). Prediction horizon and control horizon of 30 tie periods are selected. Inventory set points, initial inventory levels, axiu storage capacity at every node are reported in able 1. ab.1. Supply Chain Data Data Retaile r Warehous e Manufactur e Initial Inventory level Desired Inventory Maxiu storage capacity As it said before, custoer deand and transportation delays are the arbitrary paraeters in our supply chain odel and the control goal is to increase supply chain robustness through a sort of delay and any kind of custoer deand, so two kind of custoer deand is considered, first type of deand is a rando pulsatory deand with sall doain changes between 0 and 50

5 301 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive units and the second deand is a rando pulsatory deand with high and sudden doain changes between 0 and 600 units and Also two kind of short and long transportation tie delays are considered. In this part two control ethods is applied. In first ethod, distributed regular odel predictive controllers are applied and in second ethod the supply chain odel is changed in which an integrator is ebedded, input and output changes are highly penalized in cost functions and also Orthonoral basis functions are used in MPCs structure. Laguerre paraeters are chosen as a= [ ], N= [40 40]. It will be shown that if suddenly deand changed, the first ethod cannot predict this changes and has poor efficiency, Also in presence of long delays, tuning of conventional MPCs (first ethod) requires a tie consuing trial-and-error procedure and retuning is needed for any changes, but our proposed ethod (ethod 2) is robust through sudden deand changes and toward any short or long transportation delays and there is no need for retuning. he input and output tuning weight atrices in each echelon in both ethods are given in able 2 and table 3. Deand type Delay type ab.2. Input and output weights in first ethod Delay type 1: Short Delay type 2: Long transportation delays transportation delays LS,W LM,W LW,R LR,C LS,W LM,W LW,R LR,C Deand type 1: Deand type 2: W in 0.5 R 0.5, W out 0.3 R 0.09 W in 0.5 W 0.5, W out 0.3 W 0.09 W in 0.5 M 0.5, W out 0.3 M 0.09 W in 0.5 S 0.5, W out 0.3 S 1.5 W in 0.3 R 0.5, W out 1.5 R 1.2 W in 0.4 W 0.5, W out 1.1 W 0.8 W in 0.5 M 0.5, W out 1.3 M 0.9 W in 0.5 S 0.5, W out 1.3 S 1.5 W in 30 R 20, W out 10 R 5 W in 15 W 10, W out 20 W 15 W in 15 M 25, W out 20 M 15 W in 20 S 20, W out 30 S 20 W in 35 R 30, W out 15 R 10 - W in 25 20, W W out W W in 35 30, W M out M W in 25 20, W S out S ab.3. Input and output weights in second ethod Delay type Short and Long delay (Delay type 1 and Delay type 2) Deand type Deand type 1 and 2 W in , W R out , W R in , W W out 2 2, W W in , W M out , W M in , W S out 2 2 S In the following, proposed ethods should be ipleented to check whether they have an acceptable perforance in critical issues such as safety stock control, inventory control and bullwhip effect control. Also the approaches ust provide optial orders for upstrea nodes and optial nuber of delivered products (satisfied orders) to downstrea nodes. Siulation results for each cobination of custoer deand and delays {(deand 1,delay short), (deand 1,delay Long ), (deand 2,delay short), (deand 2,delay Long ) } are illustrated in figures 4-11 and both ethods are copared with each other. When deand is type 1 and distribution delay is type 1 (delay short), results are shown in Figs 4.a, 5.a and 6.a. However Method 2 perfors better than ethod 1, but both applied ethods have good perforances; It eans that, inventory levels in all echelons are alost near safety stock level (desired reference), (Fig4.a), Each upstrea node satisfies its downstrea node deand perfectly (fig6.a), and Also Bullwhip effect rate which is calculated according to forula (1), is under control, Fig (5.a). But the question is how is the perforance of ethod 1 in face of sudden deand fluctuations and in the presence of long distribution delays? However uch effort has been done to tune MPCs in ethod 1, but as illustrated in Fig (4.b), fig (5.b) and Fig (7), in the presence of long delays, ethod 1 loses its efficiency; It eans that there are lots of overshoots in output responses, each node cannot satisfy its downstrea node's deand precisely and bullwhip effect rates are increased. Also when there is sudden changes in custoer deand (deand type 3), ethod 1 cannot control the

6 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive 302 supply chain, Fig (8a, 9a, 10) and ethod 1 perforance becoes poorer when distribution delay tie increases, Fig (8b, 9b, 11) As it is clear, in all figures, ethod 2 plays uch better perforance than ethod 1 and ethod 2 is always robust toward any deand changes and any delay. All the results are suarized in able 4. Delay type Deand type ab.4. Results suarization Delay type 1 (Short Delay) Delay type 2 (Long Delay) Deand type 1 (pulsatory deand with Sall doain changes) Method 1 Method 2 Method 1 Method 2 Deand type 2 (pulsatory deand with high and sudden doain changes) Method 1 Method 2 Method 1 Method 2 Fig.4. Method 2 plays better perforance than ethod 1 on inventory control and safety stock control, Fig (a), especially when there is delay type 2, Fig (b). Fig.5. Method 2 plays better perforance than ethod 1 on bullwhip effect rate reduction, Fig (a), especially when there is delay type 2, Fig (b)

7 303 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive Fig.6. When deand is type 1 and delay is type 1, both ethods have good perforances in deand satisfaction, Fig (a), Fig (b); however ethod 2 has better perforance than ethod 1, Fig (b). Fig.7. When there is delay type 2, ethod 2 has poor perforance in deand satisfaction, Fig(a), but ethod 2 has good perforance and keeps its robustness through delay type 2, Fig(b). Fig.8. When deand is type 2, Method 2 plays uch better perforance than ethod 1 on inventory control and safety stock control (a), especially when there is delay type 2 (b).

8 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive 304 Fig.9. When deand is type 2, Method 2 plays uch better perforance than ethod 1 on bullwhip effect rate reduction (a), especially when there is delay type 2 (b) Fig.10. When deand is type 2, Method 2 in Fig (b), plays better perforance on downstrea deand satisfaction than ethod 1, Fig (a). Fig.11. When deand is type 2 and delay is type 2, Method 2, Fig (b), plays uch better perforance on downstrea deand satisfaction than ethod 1, Fig (a). 4. Conclusion A fraework for supply chain anageent based on distributed Model Predictive Controllers using orthonoral Laguerre functions was presented and also the supply chain odel changed to an augented odel in which an integrator was ibedded and input and output changes were highly penalized in cost functions. he siulation results, deonstrated that the proposed control approach is robust through any custoer deand and

9 305 R.Madani& A.Raezani & M.Beheshti Ipleentation of Distributed Model Predictive transportation tie delay and despite the deand fluctuations, in contrast to conventional MPCs without using orthonoral basis functions, the bullwhip effect rate decreased, better reference tracking and disturbance rejection achieved and soothness of the supply chain process in all echelons and subsystes kept. References [1] J, Li., Y. Chai.,Xianke Luo., "Cooperative Prediction for Multi-production Multi-Level Supply Chain Optiization,, Journal of Inforation & Coputational Science, pp ,2012. [2] eiory, H., Mirzahosseinian, H., Kaboli, A., "A Matheatical Method for Managing Inventories in a Dual Channel supply chain ". IJIEPR. 2008; 19 (4) : [10] Chai, J.Li.Y., Peng, M., ie-staping Predictive Strategy for Multi-echelon Inventory Control with Uncertain Lead ie, Int. J. Inf. Manag. Sci.,2012, pp [11] Grosso, J.M.C., Ocapo-Martínez, Puig, V., Learning-based tuning of supervisory odel predictive control for drinking water networks, Eng. Appl. Artif. Intell., vol. 26, no. 7, 2013, pp [12] Wang, L., "Discrete odel predictive controller design using Leaguered functions,", Journal of Process Control ELSEVIER, vol.14, 2004, pp [13] Wang, L.,"Model Predictive Control Syste Design and IpleentationUsing MALAB", Springer,2009. [3] Bagheri, F., arokh, M., "A Fuzzy Approach For Multi-Objective Supplier Selection." IJIEPR. 2010; 21 (1):1-9. [4] Kapsiotis., G., zafestas, S., Decision aking for inventory/production planning using odel-based predictive control, Parallel and distributed coputing in engineering systes. Asterda, Elsevier, 1992, pp [5] zafestas, S., Kapsiotis, G., Model-based predictive control for generalized production planning probles, Coputers in Industry, vol. 34,1997, pp [6] Lopez, P., Ydstie, Grossann, B., A odel predictive control strategy for supply chain anageent, Coputers & Cheical Engineering, vol. 27, 2003, pp [7] Dunbar, du W.B., Desa, S., Distributed MPC for dynaic supply chain anageent, in Assessent and Future Directions of Nonlinear Model Predictive Control, Springer, 2007, pp [8] Sariveis, H., Patrinos, P., arantilis, D., Kiranoudis, ". Dynaic odeling and control of supply chain systes," A review. Coputers and Research, Vol. 35, 2008, pp [9] Miranbeigi, M., Jalali, A., Miranbeigi, A., "Design of Distributed Optial Adaptive Receding Horizon Control for supply chain of Realistic Size under Deand Disturbances ". IJIEPR. 2011; 22 (3) :