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Optimal Control of Distribution Chains for Perishable Goods
Proceedigs of the 15th IFAC Symposium on Information Control Problems in Manufacturing Proceedigs of the 15th IFAC Symposium on Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada Proceedigs of the 15th IFAC Symposium on Available online at www.sciencedirect.com Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada
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IFAC-PapersOnLine 48-3 (2015) 1049–1054 Optimal Control of Distribution Optimal Control of Distribution Optimal of Distribution Perishable Goods Optimal Control Control of Distribution Perishable Goods Perishable Goods Perishable Goods ∗ M. Gaggero F. Tonelli ∗∗
Chains Chains Chains Chains
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∗ ∗∗ M. M. Gaggero Gaggero ∗∗ F. F. Tonelli Tonelli ∗∗ ∗ M. Gaggero Tonelli ∗∗ National Research Institute of Intelligent Systems forF.Automation, ∗ of Intelligent for Automation, National Research ∗ Institute Council of Italy, I-16149 Systems Genoa, Italy (e-mail: [email protected]) ∗∗∗Institute of Intelligent Systems for Automation, National Research Institute of Intelligent Systems for Automation, National Research Council of Italy, I-16149 Genoa, Italy (e-mail: [email protected]) Department of Mechanical Engineering, Energetics, Management Council of Italy, I-16149 Genoa, Italy (e-mail: [email protected]) ∗∗ Council of Italy,ofI-16149 Genoa, Italy (e-mail: [email protected]) Department Mechanical Engineering, Energetics, Management ∗∗ and Transportation, University of Genoa, I-16145 Genoa, Italy ∗∗ Department of Mechanical Engineering, Energetics, Management Department of Mechanical Engineering, Energetics,Genoa, Management and University of (e-mail: [email protected]) and Transportation, Transportation, University of Genoa, Genoa, I-16145 I-16145 Genoa, Italy Italy and Transportation, University of Genoa, I-16145 Genoa, Italy (e-mail: (e-mail: [email protected]) [email protected]) (e-mail: [email protected]) Abstract: A discrete-time dynamic model of distribution chains for perishable goods is Abstract: A dynamic model of chains for perishable goods presented with an approach its optimal management based model predictive Abstract:together A discrete-time discrete-time dynamic for model of distribution distribution chains for on perishable goods is is Abstract: A discrete-time dynamic model ofwith distribution chains for the perishable goods is presented together with an approach for its optimal management based on model predictive control. The model is based on a directed graph, buffers representing amounts of goods presented together with an approach for its optimal management based on model predictive presented together with an approach for its optimal management based on model predictive control. The model is based on a directed graph, with buffers representing the amounts of goods for the various remaining lifetimes, whose graph, time evolution is obtained via balance equations. The control. The model is based on a directed with buffers representing the amounts of goods control. The model based on a directed with buffers representing the amounts of goods for the remaining lifetimes, whose time is obtained balance The amounts of goods toistransfer from node to graph, node evolution are chosen solving via a receding-horizon optimal for the various various remaining lifetimes, whose time evolution isby obtained via balance equations. equations. The for the various remaining lifetimes, whose time evolution is allows obtained balance equations. The amounts of transfer from to node chosen aato receding-horizon optimal control problem time step.node The proposed approach onevia trade among inventory amounts of goods goodsatto toeach transfer from node to node are are chosen by by solving solving receding-horizon optimal amounts of goodsattoeach transfer from node to node are chosendemand, by solving ato receding-horizon optimal control problem time step. The proposed approach allows one trade among inventory and transportation costs, satisfaction of the customers’ and reduction of the amount control problem at each time step. The proposed approach allows one to trade among inventory control problem each time step. The proposed approach allows one to trade inventory and transportation costs, satisfaction of the demand, and reduction the amount of wasted goods,atnamely with no lifetime and thus haveamong toof and transportation costs, goods satisfaction of remaining the customers’ customers’ demand, andthat reduction ofbe thediscarded amount and transportation costs, satisfaction of remaining the customers’ demand, andthat reduction ofbe the amount of wasted goods, namely goods with no lifetime and thus have to discarded from the distribution chain. Preliminary simulation results in three scenarios are reported to of wasted goods, namely goods with no remaining lifetime and thus that have to be discarded of wasted goods, namely goods withapproach. no remaining lifetime and thus that have are to be discarded from the distribution chain. Preliminary simulation results in three scenarios reported to show potential of the proposed from the distribution chain. Preliminary simulation results in three scenarios are reported to from the potential distribution the chain. Preliminary simulation results in three scenarios are reported to show show the potential of of the proposed proposed approach. approach. © 2015, IFAC (International Federation approach. of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. show the potential of the proposed Keywords: Model predictive control, optimization, distribution chains, perishable goods, Keywords: Model predictive control, optimization, distribution chains, perishable goods, inventory Keywords:control Model predictive control, optimization, distribution chains, perishable goods, Keywords:control Model predictive control, optimization, distribution chains, perishable goods, inventory inventory control inventory control 1. INTRODUCTION following, the term “perishable” has to be meant both in 1. INTRODUCTION following, the term hasobsolescence. to be meant both in the strict sense and “perishable” in the sense of 1. INTRODUCTION following, the term “perishable” has to be meant both in 1. INTRODUCTION following, the term hasobsolescence. to be meant both in strict sense and “perishable” in the sense of In this paper, the previous works (Alessandri et al., the the strict senseofand in the sense of obsolescence. The literature inventory and distribution management In this paper, the previous works (Alessandri et al., the strict sense and in the sense of obsolescence. 2011b,a, 2012) on strategic and tactical inventory control In this paper, the previous works (Alessandri et al., The literature of inventory and distribution management of perishable goods is less developed if compared with the In this paper, previous (Alessandri et al., The literature of inventory and distribution management 2011b,a, 2012) onthe strategic andworks tactical inventory control for multi-echelon (DCs) are extended 2011b,a, 2012) on distribution strategic andchains tactical inventory control of The literature of inventory and distribution management perishable goods isgoods. less developed if compared with the one of nonperishable Most of works focus on pricing 2011b,a, 2012) on strategic and tactical inventory control for multi-echelon distribution chains (DCs) are extended of perishable goods is less developed if compared with the to case of perishable goods. We (DCs) focus on for the multi-echelon distribution chains areperishable extended one of perishable goods is less developed if compared with the of nonperishable goods. Most of works focus on pricing models and on the goods. searchMost for of heuristics for on inventory for multi-echelon distribution chains areperishable extended to the case fixed of perishable goods. We (DCs) focus on one of nonperishable works focus pricing goods lifetime (Nahmias, all the goods one to the with case of perishable goods. We 1982): focus on perishable of nonperishable goods. works pricing models and stock on the searchMost for of heuristics for on inventory control replenishment e.g.,focus Liu Lian, to the case fixed oftoperishable goods. We 1982): focus on perishable goods with (Nahmias, all the goods models and and on the search for (see, heuristics for and inventory are assigned alifetime given remaining measured in models goods with fixed lifetime (Nahmias,lifetime, 1982): all the goods and on the search for heuristics for inventory control and stock replenishment (see, e.g., Liu and Lian, 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; goods with fixed (Nahmias, 1982):on. all the goods are assigned toreduces alifetime given remaining measured in control and stock replenishment (see, e.g., Liu and Lian, buckets, that the time lifetime, goes When the are assigned to a givenasremaining lifetime, measured in control and stock replenishment (see, e.g., Liu Lian, Goyal and Giri, 2001; Hsu, 2003; Chu et and al., 2005; Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo are assigned toreduces a reaches givenasremaining lifetime, measured in 1999; buckets, that thezero time goes on. When the 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; remaining lifetime the level or a level for which buckets, that reduces as the time goes on. When the Perakis 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo al., 2008). buckets, reduces as the thezero time the remaining lifetime reaches level or aon. levelWhen for which Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo the goodsthat are unmarketable, aregoes discarded from the et remaining lifetime reaches the they zero level or a level for which Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo al., 2008). remaining lifetime reaches the they zero level orofa level for which the goods are DCs unmarketable, are discarded from the et et al., 2008). DC. Clearly, involving such type goods require In al., this2008). work, the discrete-time dynamic model of DCs the goods are unmarketable, they are discarded from the et the goods are DCs unmarketable, they are from the In DC. Clearly, involving such typediscarded of require special attention reduce the amount of goods wasted goods thisonwork, the discrete-time dynamic model of DCs based a directed graph originally proposed in AlessanDC. Clearly, DCstoinvolving such type of goods require In this work, the discrete-time dynamic model of DCs DC. Clearly, DCs such type of require special attention toinvolving reducesuch the amount of goods wasted goods based In this work, the discrete-time dynamic model of DCs together with other goals as customers’ satisfaction on a directed graph originally proposed in Alessandri et al. (2011b) is extended to take into account the special attention to reduce the amount of wasted goods based on a directed graph originally proposed in Alessanspecial attention to goals reducesuch the transportation amount of wasted together with other as customers’ satisfaction directedof originally proposed ineach Alessanand reduction of inventory and costs.goods based dri et on al. a(2011b) isgraph extended to take into account the remaining lifetime goods. Toward this end, node together with other goals such as customers’ satisfaction dri et al. (2011b) is extended to take into account the together with other goals such as customers’ satisfaction and reduction of inventory and transportation costs. dri et al. (2011b) is extended to containing take the remaining lifetime goods. Toward thisinto end,account each node is equipped with aof of buffers with and reduction of inventory and transportation costs. remaining lifetime ofset goods. Toward this end,goods each node Classical examples of DCsand for transportation perishable goods can be remaining and reduction of inventory costs. lifetime of goods. Toward this end, each node is equipped with a set of buffers containing goods with remaining Then,containing the amounts of goods Classical examples DCs food for perishable goods need can be equipped with a lifetime. set of buffers goods with found in the field ofof products, which to ais certain Classical examples offresh DCs for perishable goods can be equipped withnode a lifetime. settoofnode buffers containing goods with certain Then, the amounts of goods to transferremaining from are computed by means of Classical examples DCsbefore for perishable goods need can be found in the field ofoffresh food products, which to ais a certain remaining lifetime. Then, the amounts of goods be delivered to customers their remaining lifetime found in the field of fresh food products, which need to ato certain remaining lifetime. Then, the amounts of goods transfer from node to node are computed by means of model predictive control (MPC) (Mayne et al., 2000) basfound in the field of fresh food products, which need to be delivered to level. customers before lifetime to transfer from node to node are computed by means of reach the zero In this case,their the remaining term “perishable” be delivered to customers before their remaining lifetime model to transfer from node to nodefuture are computed means of predictive control (MPC) (Mayne et al., 2000) basing on the information about requests ofbycustomers. be delivered tostrict customers before their lifetime reach the zero level. In this case, the remaining term “perishable” model predictive control (MPC) (Mayne et al., 2000) basis used in the sense. A perishable item is one that model reach the zero level. In this case, the term “perishable” predictive control (MPC) (Mayne et al., 2000) basing on the information about future requests of customers. The proposed approach is able to trade among the goals reach the level. sense. In thisA case, the term is used in zero the strict perishable item“perishable” is one after that ing on the information about future requests of customers. has constant expiration is used in theutility strict until sense.a Acertain perishable item isdate; one that The on the information about future requests of customers. proposed isofable to trade among the goals of theapproach amount wasted goods, satisfying the is used in theutility strict until sense. perishable item isdate; one after that ing has constant a Acertain expiration Thereducing proposed approach is able to trade among the goals it, its utility drops to zero (Nahmias, 2011). This may has constant utility until a certain expiration date; after of The proposed isofable to trade among the goals reducing theapproach amount wasted goods, satisfying the customers’ demand, as well as minimizing inventory and has constant utility until a certain expiration date; after it, its utility drops to zero (Nahmias, 2011). This may of reducing the amount of wasted goods, satisfying the have impacts in terms of sustainability the of it, itsnegative utility drops to zero (Nahmias, 2011). Thisof may reducingdemand, thecosts. amount of wasted goods, inventory satisfying and the as well as minimizing transportation it, itsnegative utility drops to zero (Nahmias, 2011). This2013a; have impacts in terms of sustainability of may the customers’ customers’ demand, as well as minimizing inventory and entire supply/distribution chain (Tonelli et al., have negative impacts in terms of sustainability of the transportation customers’ demand, as well as minimizing inventory and costs. have negative in terms of sustainability of the The entire supply/distribution chain (Tonelli al., dealing 2013a; transportation costs. Taticchi et al., impacts 2013). Similar issues occur et when first attempts entire supply/distribution chain (Tonelli et al., 2013a; transportation costs.of the use of MPC in industry are in entire supply/distribution chain (Tonelli et al., dealing 2013a; Taticchi etinvolving al., 2013).goods Similar issues occur when with DCs subject to obsolescence, such The first attempts the use MPC inplants industry areand in process of control forof (Qin Taticchi et al., 2013). Similar issues occur when dealing the The area first of attempts of the use ofchemical MPC in industry are in Taticchi etinvolving al., 2013). Similar issues when dealing with DCselectronic goods subject tooccur obsolescence, such The first attempts of the use of MPC in industry are in as, e.g., equipment. In this case, items do not the area of process control for chemical plants (Qin and Badgewell, 1997), with increasing diffusion in many other with DCs involving goods subject to obsolescence, such the area of process control for chemical plants (Qin and with DCs involving goods subject to case, obsolescence, as, e.g., electronic equipment. In this items dosuch not Badgewell, area of 1997), process control forofchemical plants (Qinother and loose their wealth properties with increasing diffusion in many fields, including management logistics operations (see, as, e.g., electronic equipment.asInfresh this food case, products, items do but not the Badgewell, 1997), with increasing diffusion in many other as, e.g., electronic equipment. Infresh this food case, items do but not Badgewell, loose their wealth properties products, 1997),management with increasing diffusion in many other become obsolete and then less as attractive for customers including logistics (see, e.g., et al., 2003;of andoperations Rivera, 2008; loose their wealth properties as fresh food products, due but fields, fields,Perea-Lopez including management of Wang logistics operations (see, loose their wealth properties as fresh products, food products, but become obsolete and then lessof attractive for customers due fields, including management of logistics operations (see, to the presence in the market newer which are e.g., Perea-Lopez et al., 2003; Wang and Rivera, 2008; et al., 2008, of the 2008; great become obsolete and then less attractive for customers due Alessandri e.g., Perea-Lopez et al.,2013). 2003; The Wangreasons and Rivera, become obsolete and then less attractive for customers due to the presence in the market of newer products, which are e.g., Perea-Lopez et the al.,capability 2003; The Wangreasons and Rivera, often advanced termsofofnewer performances. et al.,are 2008, 2013). of the 2008; great success of MPC exploit information to themore presence in the in market products,Thus, whichalso are Alessandri Alessandri et al., 2008, 2013). Thetoreasons of the great to the presence market products, whichalso are often advanced in termsofofnewer performances. Thus, Alessandri et al., 2008, 2013). Thetoreasons of the great in thismore case, it inis the important organize the transfer of success of MPC are the of capability exploit information on the future behavior the system, the possibility of often more advanced in terms to of performances. Thus, also success of MPC are the capability to exploit information often advanced in terms of performances. Thus, in thismore case, it important organize transfer of on of MPCbehavior are the of capability to exploit information goods theis of the to DC to avoidthe wastes. In also the the future the system, the possibility of in thisamong case, it isnodes important to organize the transfer of success in thisamong case, it important transfer of on the future behavior of the system, the possibility of goods theisnodes of the to DCorganize to avoidthe wastes. In the goods among the nodes of the DC to avoid wastes. In the on the future behavior of the system, the possibility of goods among the nodes of the DC to avoid wastes. In the
Copyright © 2015 IFAC 1099 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 1099Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 responsibility IFAC 1099Control. Peer review©under of International Federation of Automatic Copyright © 2015 IFAC 1099 10.1016/j.ifacol.2015.06.222
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S
node i goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
D
... ... goods of type 1
... goods of type 2
...
P
goods of type M
u01k
Fig. 1. Set of buffers at each node of the DC. u02k
dealing directly with constraints on state variables and control inputs, as well as the availability of a number of theoretical results about its properties. Here, the amounts of goods to transfer from node to node are obtained by solving optimal control problems involving the sliding-window optimization of a cost function. The system behavior within such a window is obtained by using the above-mentioned model of the DC, basing on future prediction of the customers’ requests. The idea of exploiting information about the future for the optimal ¨ management of DCs can be found also, e.g., in Ozer (2003); Robinson and Lawrence (2004); Wang and Toktay (2008); Tonelli et al. (2013b), although simpler heuristic methodologies are employed. The contribution of this paper is twofold: (i) a discretetime dynamic model to describe a generic DC for perishable goods is presented; (ii) the optimal amounts of goods to transfer from node to node are computed by means of MPC. The paper is organized as follows. Section 2 presents the model of the DC. Section 3 contains the MPC approach for the optimal DC management. Section 4 reports the adopted indexes for performance evaluation. Finally, numerical results are discussed in Section 5. 2. DYNAMIC MODEL OF DISTRIBUTION CHAINS FOR PERISHABLE GOODS Starting form the ideas presented in Alessandri et al. (2011b), we consider a prototype of a DC for perishable goods of various types made up of production units (PUs), distribution units (DUs), and sales units (SUs). A highlevel model of such a DC is a directed graph composed of nodes and arcs. Each node is equipped with sets of buffers that are filled up with quantities of goods of various types among M possible ones. To model the remaining lifetimes of goods, we consider a buffer for each remaining lifetime level and each type of goods (see Fig. 1). Given a graph with N − 1 nodes, let K(i) be the set of the indexes corresponding to the types of goods available at node i, i = 1, . . . , N − 1, and let Lk (i) be the set of the integers that correspond to the remaining lifetime levels of the goods of type k that are stored at the node i. In more detail, such a set is made up by the integers max max 1, . . . , lik , where lik is the maximum remaining lifetime level of goods of type k that can be stored at node i. Moreover, let I(i) and O(i) be the sets of the incoming and outgoing neighbors of node i, respectively. For the sake of brevity, the sets of the node indexes corresponding to PUs, DUs, and SUs are denoted by P, D, and S, respectively. Then, we introduce the fictitious nodes 0 and N as initial source for the PUs and final sink for the SUs to account for the flows of goods at the nodes belonging to P and
1
w1k 2
w2k
i
uijk
wik
τijk
j
wjk
dN −2 uN −2N k
N−2
dN −1 N−1
uN −1N k
wN −1k
Fig. 2. Graph representation of a DC for perishable goods. S, respectively. Fig. 2 shows a sketch of the DC graph, where the nodes 0 and N are omitted for the sake of simplicity. Let us introduce the following variables at each time bucket t = 0, 1, . . . , T , where T is a given horizon. 1) zik (t, l) ∈ R is the stock level of the buffer that contains quantities of goods of type k with remaining lifetime l at node i, i = 1, . . . , N − 1, k ∈ K(i), l ∈ Lk (i); 2) uijk (t, l) ∈ R is the amount of goods of type k with remaining lifetime l transferred from node i to node j, i ∈ I(j), j = 1, . . . , N , k ∈ K(i) ∩ K(j), l ∈ Lk (i); 3) τijk ∈ N is the delay required to transfer goods of type k from node i to node j, i ∈ I(j), j = 1, . . . , N − 1, k ∈ K(i) ∩ K(j); min 4) τik ∈ R is the minimum delay required to transfer goods of type k from node i to a SU node, i = P ∪ D, k ∈ K(i); it can be computed, for example, by means of a dynamic programming algorithm (Bellman, 1957) applied to the graph of the DC; 5) wik (t) ∈ R is the total amount of goods of type k at node i that are wasted, and thus have to be discarded from the DC, i = 1, . . . , N − 1, k ∈ K(i). For the SU nodes, the wasted goods are those with remaining lifetime l = 1 that are unsold/unused, whereas for PUs and DUs they are those that, because of the transfer delays, cannot reach the SU nodes with remaining lifetime greater than or equal to 1 (i.e., those for which min l ≤ τik ). In this case, the transfer of such goods from node to node is useless since they cannot be sold/used; thus, a convenient management choice is to discard them from the DC without carrying out additional transportation costs; 6) bik (t) ∈ R is the backlog level corresponding to goods of type k at node i, i ∈ S, k ∈ K(i); 7) dik (t) ∈ R is the customers’ demand of goods of type k at node i, i ∈ S, k ∈ K(i). We suppose that the customers of the DC are well-suited to buying goods with arbitrary remaining lifetime (but, of course, always greater than or equal to 1). In particular, without loss of generality, we assume that the demand
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dik (t) distributes uniformly in all the lifetime levels of goods, i.e., we define dik (t) d¯ik (t, l) = max , i ∈ S, k ∈ K(i), l ∈ Lk (i) lik The analysis of other demand distributions in the various lifetime levels, such as situations in which goods with closest expiration date are properly positioned in the shelves or reordered in order to be sold before the corresponding ones with greater remaining lifetimes, is out of the scope of this paper. A discrete-time dynamic model of the DC is then derived by using the stock and wasted levels of goods in all the buffers and the backlogs at the SU nodes as state variables, and is given by the following dynamic equations (for the sake of brevity, let Ik (i) {j ∈ I(i) : k ∈ K(i) ∩ K(j)}, and Ok (i) {j ∈ O(i) : k ∈ K(i) ∩ K(j)}): � zik (t, l + 1) + ujik (t − τjik , l + τjik ) j∈Ik (i) � min , zik (t + 1, l) = uijk (t, l + 1) if l ≥ τik − j∈Ok (i) min 0 if l < τik (1a) i ∈ P ∪ D, k ∈ K(i), l ∈ Lk (i) � zik (t + 1, l) = zik (t, l + 1) + ujik (t − τjik , l + τjik ) j∈Ik (i)
− uiN k (t, l + 1), wik (t + 1) = wik (t) +
�
i ∈ S, k ∈ K(i), l ∈ Lk (i) (1b) zik (t, l),
min l≤τik
i ∈ P ∪ D, k ∈ K(i), l ∈ Lk (i) (1c) wik (t + 1) = wik (t) + zik (t, 1) − uiN k (t, 1), i ∈ S, k ∈ K(i) (1d) � � bik (t + 1) = bik (t) − d¯ik (t, l), uiN k (t, l) + l∈Lk (i)
l∈Lk (i)
i ∈ S, k ∈ K(i) (1e) where t = 0, 1, . . . , T . Equations (1a) and (1b) model the balance of goods and the decrease of the remaining lifetime of goods at each time bucket. Equations (1c) and (1d) account for the quantities of goods that are wasted because they have no remaining lifetime or cannot reach the SU nodes with a remaining lifetime greater than or equal to 1. Finally, equation (1e) accounts for the dynamics of the backlogs at the SU nodes. To complete the model (1), we need to introduce various types of constraints. As to the state variables, the stock levels are required to be nonnegative and are subject to upper bounds representing the maximum amounts of goods that can be stored at the various nodes, i.e., we impose � max 0≤ zik (t, l) ≤ zik , i = 1, . . . , N − 1, k ∈ K(i) l∈Lk (i)
(2) is a positive constant where t = 0, 1, . . . , T and representing the maximum quantity of goods of type k that can be stored at node i. max zik
As regards the backlogs, we need to ensure only their positivity, as no upper bounds can be fixed a priori. Thus, we impose the following constraints for t = 0, 1, . . . , T :
bik (t) ≥ 0,
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i ∈ S, k ∈ K(i)
(3)
In order to prevent from the occurrence of stockouts, one can keep a safety stock to satisfy unexpected requests. Without loss of generality, we suppose to keep a unique safety stock for all the goods of the same type at each node, independently from the remaining lifetime. Thus, let sik ≥ 0 be the safety stock level at node i for goods of type k. From a “strategic” point of view, we constrain the stock levels to be higher than the corresponding safety stock, i.e., � l∈Lk (i) zik (t, l) ≥ sik , for i = 1, . . . , N − 1, k ∈ K(i), and t = 0, 1, . . . , T . As we shall see in the following, during online operations this constraint is relaxed, as such stocks are employed on demand to avoid stockouts. Concerning the control inputs, since the transportation of goods of type k from node i to node j can be made only with a delivery cycle time denoted by Ωijk ∈ N, the following constraints hold: � umin uijk (t, l) ≤ umax ijk (t), ijk (t) ≤ l∈Lk (i)
i ∈ I(j), j = 1, . . . , N − 1, k ∈ K(i) ∩ K(j) umin ijk (t)
(4a)
umax ijk (t)
and are poswhere t = rΩijk , r ∈ N; itive numbers representing the minimum and maximum amounts of goods of type k that can be transferred at time t from node i to node j, respectively. Note that we do not constrain the controls uiN k (t, l) at the SU nodes, as they model the quantities of goods that leave the SUs upon the demands dik (t), i.e., we choose umin iN k (t) = 0 and umax iN k (t) = +∞. To model the impossibility of transferring goods of type k from node i to node j at time instants that are not multiples of Ωijk , we impose the following: uijk (t, l) = 0, i ∈ I(j), j = 1, . . . , N − 1, (4b) k ∈ K(i) ∩ K(j), l ∈ Lk (i) where t �= rΩijk , r ∈ N. In other words, if t¯ = rΩijk , r ∈ N, we constrain uijk (t, l) = 0 for all t = t¯+ 1, . . . , t¯+ Ωijk − 1. As previously pointed out, a good management choice is to avoid to transfer goods that cannot reach the SU nodes with remaining lifetime greater than or equal to 1 since such transfers are useless and would only entail additional transportation costs. Thus, we constrain the control inputs as follows: uijk (t, l) = 0, i ∈ P ∪ D, j ∈ O(i), k ∈ K(i) ∩ K(j) (5) min . where t = 0, 1, . . . , T and l ≤ τik
3. OPTIMAL MANAGEMENT OF DISTRIBUTION CHAINS USING MPC In this section, we present the proposed approach for the optimal control of the DC. Specifically, we search for optimal policies to choose the amount of goods to transfer from node to node at the various time instants. When looking for such optimal policies, the following goals have to be pursued: (i) minimize inventory costs; (ii) minimize transportation costs; (iii) minimize the use of safety stocks; (iv) minimize the amount of wasted goods; (v) satisfy the customers’ demand as much as possible. Clearly, a tradeoff among these goals is needed since, in general, the customers’ demand is satisfied more promptly if a greater amount of transferred and stocked goods is
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guaranteed. However, the higher the amounts of goods stocked at the various nodes, the higher the storage costs. Furthermore, the higher the amounts of transferred goods, the higher the transportation costs. Such a tradeoff is searched for by means of an approach based on MPC, which consists in finding, at each time t, the control inputs that solve a finite-horizon optimal control problem of length Tp ≪ T and in applying only the first control action. The procedure is repeated at the next time instant, with an one-step-forward shift of the sliding window and by using new forecasts of the customers’ demand. The dynamics within the prediction horizon is governed by the state equation (1). In particular, we consider the following cost function to minimize: t+Tp N −1 � � � � Jt = zik (τ, l) c1 ατ −t−1 τ =t+1
+c3 γ τ −t−1
i=1 k∈K(i) l∈Lk (i)
N −1 �
�
�
pik [zik (τ, l)]
N −1 � � � � +c4 δ τ −t−1 wik (τ ) bi (τ ) + c5 ǫτ −t−1 i=1 k∈K(i)
i∈S k∈K(i)
+
�
c2 β τ −t
τ =t
N −1 �
�
�
�
gijk [uijk (τ, l)]
j=1 i∈I(j) k∈K(i)∩K(j) l∈Lk (i)
(6) where α ∈ (0, 1], β ∈ (0, 1], γ ∈ (0, 1], δ ∈ (0, 1], and ǫ ∈ (0, 1] are discount coefficients; c1 > 0, c2 > 0, c3 > 0, c4 > 0, and c5 > 0 are weight constants. The goal of satisfying the external demands is taken into account by the penalization of the backlogs. The function gijk : R → R accounts for transportation costs. Following Alessandri et al. (2011b), we focus on a piecewise-constant, strictly-increasing function, which accounts for the fact that transportation is usually accomplished by lot-sized carriers, and the larger the transferred goods, the higher the costs. Specifically, the following function is employed: 0 if uij = 0 1 vijk if 0 < uijk ≤ u1ijk 2 1 2 gijk (uijk ) = vijk if uijk < uijk ≤ uijk , .. . Q vijk if uijk > uQ−1 ijk i ∈ I(j),
j = 1, . . . , N − 1,
1 where 0 ≤ vijk ≤ ··· ≤
Q vijk
k ∈ K(i) ∩ K(j)
and 0 ≤ u1ijk ≤ · · · ≤
(7)
uQ ijk .
To minimize the use of safety stocks, a penalization term based on the function pik : R → R is employed. Specifically, we focus on the following function: � −zik + sik if 0 ≤ zik ≤ sik (8) pik (zik ) = 0 if zik > sik According to the MPC paradigm, at each time t we need to solve the following finite-horizon optimal control problem. Problem MPC. Find min Jt
uijk (τ )
4. PERFORMANCE INDEXES We present in this section the indexes used to evaluate the performance of the proposed approach. First of all, concerning the inventory costs, let the total inventory level (TIL) be defined as the sum of the inventory levels for all t = 0, 1, . . . , T , i.e., −1 � T N � � � TIL zik (t, l) t=0 i=1 k∈K(i) l∈Lk (i)
As regards the transportation costs, let us consider the total transportation cost (TTC), given by the sum of the transportation costs over the considered time horizon: −1 � T −1 N � � � � gijk [uijk (t, l)] TTC t=0 j=1 i∈I(j) k∈K(i)∩K(j) l∈Lk (i)
i=1 k∈K(i) l∈Lk (i)
t+Tp −1
subject to (1), (2), (3), (4), and (5) from time τ = t to time τ = t + Tp − 1.
The capability of satisfying the customers’ demand is taken into account by the total backlog level (TBL), defined as the sum of the backlogs for all t = 0, 1, . . . , T , i.e., T � � � bik (t) TBL t=0 i∈S k∈K(i)
Concerning the amounts of wasted goods, let the total wasted goods (TWG) be defined as the sum of the levels of wasted goods at the various nodes for all t = 0, 1, . . . , T : −1 � T N � � wik (t) TWG t=0 i=1 k∈K(i)
To have a condensed indicator, we define the overall performance index (OPI) as the weighted sum of TIL, TTC, TBL, and TWG, i.e., OPI c1 TIL + c2 TTC + c4 TBL + c5 TWG where c1 , c2 , c4 , and c5 are the weight coefficients used in the cost function (6). 5. SIMULATION RESULTS In this section, we present the results obtained when applying the proposed approach for the management of the same single-item DC made up of 8 nodes considered in Alessandri et al. (2011b), adapted for perishable goods (see Fig. 3). Since such a DC involves only one type of goods, in the following we shall drop the subscript k from all the quantities. The DC was considered over one year (i.e., T = 52), where each bucket corresponds to one week. Three different patterns of the customers’ demand were considered, corresponding to situations that usually occur in practice. In Scenario A, the demand oscillates around an almost constant mean value. In Scenario B, the demand presents first a positive peak and then a negative one. In Scenario C, the demand is characterized by a seasonal pattern, with a peak in the central weeks of the year. The transportation delays, the delivery cycle times, the minimum and maximum amounts of goods that can be
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u13 u01
1 w1
u02
2 w2
3
u36 u37
w3
u14 u23 u24
4
u48
MPC Tp = 2
Tp = 3
A
OPI TIL TTC TBL TWG time [s]
5.33·1010 2.84·107 2.12·105 5.16·108 1.17·109 8.97
5.12·1010 2.85·107 4.09·105 4.95·108 1.68·109 119.2
5.11·1010 2.85·107 5.88·105 4.94·108 1.68·109 6420.1
2.24·1011 7.57·106 7.24·105 2.22·109 5.68·107 0.02
B
OPI TIL TTC TBL TWG time [s]
9.07·1010 2.82·107 1.81·105 8.93·108 1.33·109 7.08
8.69·1010 2.82·107 5.13·105 8.57·108 1.23·109 119.5
8.67·1010 2.82·107 6.21·105 8.55·108 1.23·109 716.6
3.00·107 11 7.55·106 7.24·105 3.00·109 6.43·107 0.02
C
OPI TIL TTC TBL TWG time [s]
8.02·1010 2.67·107 1.81·105 7.88·108 1.34·109 7.78
7.88·1010 2.64·107 6.87·105 7.75·108 1.30·109 173.7
7.86·1010 2.64·107 6.63·105 7.73·108 1.30·109 3802.4
2.95·1011 7.55·106 7.24·105 2.95·109 5.22·107 0.02
d7 u79
7 u57
w7
u25 5
Tp = 1
u46 w6
u47
w4
Table 2. Summary of the simulation results.
d6 u69
6
u58
d8 u89
8 w8
w5
1053
Fig. 3. DC for perishable goods used in the simulations. Table 1. Parameters of the considered DC. i
j
τij
Ωij
umin ij
umax ij
si
zi (0)
zimax
0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 7 8
1 2 3 4 3 4 5 6 7 6 7 8 7 8 9 9 9
2 3 1 1 2 2 1 3 1 1 2 1 1 2 -
4 4 2 2 2 2 2 1 1 1 1 1 1 1 -
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1400 3200 350 350 350 650 650 180 180 180 180 180 180 180 +∞ +∞ +∞
-
-
-
2500
40000
1.0·108
2100
50000
1.0·108
950
15000
1.0·108
960
15000
1.4·108
1450
15000
1.0·108
150 150 150
2000 2000 2000
1.0·108 1.2·108 2.0·108
transferred from node to node, the values of the safety stocks, as well as the initial and maximum buffer levels are reported in Table 1. The maximum remaining lifetime of goods limax was fixed equal to 10 for all i = 1, . . . , 8. The coefficients of the cost function (6) were chosen so as to satisfy more promptly the demands by penalizing much more the backlogs rather than the other terms, i.e., we chose c1 = 1, c2 = 0.1, c3 = 1, c4 = 100, c5 = 1 and α = 0.003, β = 0.003, γ = 0.003, δ = 0.005, ǫ = 0.003. Concerning the transportation cost function 1 2 20 gij (·), we chose (7) with vij = 10, vij = 20, . . . , vij = 200 1 2 19 and uij = 100, uij = 200, . . . , uij = 1900. The performances of the proposed approach were evaluated using different lengths of the prediction horizon Tp , i.e., we chose Tp equal to 1, 2, and 3. The results provided by MPC were then compared with an heuristic based on the classical lot-for-lot stock replenishment policy adapted to the setting of perishable goods (Feigin et al., 2002). All the simulations were performed in Matlab using a 3.0 GHz Pentium 4 PC with 2 GB of RAM. Table 2 summarizes the results and reports also the average time required to compute the optimal strategies. Figure 4 contains the plots of the demands, stock levels for three remaining lifetimes (l = 1, 3, and 5), and amount of wasted goods at node 6 obtained using MPC with Tp = 3. Looking at the simulation results, it can be noticed that, in all the scenarios, the MPC approach provides better results than the lot-for-lot heuristic in terms of the OPI.
lot-for-lot
Specifically, the former guarantees a more prompt satisfaction of the customers’ demand due to lower values of the TBL, whereas the latter provides reduced values of the TIL and of the TWG. Furthermore, the larger the prediction horizon Tp , the better the performances of MPC. In fact, the values of the OPI obtained with Tp = 3 are the lowest ones, as the optimization is performed using a larger amount of information about the future. Coherently, also the amount of wasted goods and inventory costs decrease as Tp increases. This is obtained at the price of a growth of the transportation costs, as a prompter demand satisfaction, provided by a reduction of the backlogs, is obtained with an increase of the amount of transferred goods. Concerning the computational effort, the MPC approach is more demanding than the lot-for-lot one since it needs to solve an optimization problem on line. By contrast, the lot-for-lot heuristic does not require such a step. Clearly, the larger the prediction horizon Tp for MPC, the larger the time required to compute the optimal policies, as there is the need of solving optimal control problems with an increasing number of unknowns. REFERENCES Alessandri, A., Cervellera, C., Cuneo, M., and Gaggero, M. (2008). Nonlinear predictive control for the management of container flows in maritime intermodal terminals. In Proc. IEEE Conf. on Decision and Control, 2800–2805. Alessandri, A., Cervellera, C., and Gaggero, M. (2013). Nonlinear predictive control of container flows in maritime intermodal terminals. IEEE Trans. Contr. Syst. Technol., 21(4), 1423–1431. Alessandri, A., Gaggero, M., and Tonelli, F. (2011a). Integer tree-based search and mixed-integer optimal control of distribution chains. In Proc. IEEE Conf. on Decision and Control, 489–494. Alessandri, A., Gaggero, M., and Tonelli, F. (2011b). Min-max and predictive control for the management of distribution in supply chains. IEEE Trans. Contr. Syst. Technol., 19(5), 1075–1089.
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Fig. 4. Plots of demands, inventory levels, and wasted goods at node 6 in three different simulation runs. Alessandri, A., Gaggero, M., and Tonelli, F. (2012). Optimal and predictive control of distribution chains by using integer tree-based search and mixed-integer programming. In 4th IFAC Nonlinear Model Predictive Control Conf., 238–244. Bellman, R. (1957). Dynamic Programming. Princeton University Press, Princeton, NJ. Chu, L., Hsu, V., and Shen, Z. (2005). An economic lotsizing problem with perishable inventory and economies of scale costs: Approximation solutions and worst case analysis. Naval Res. Logistics, 52, 536–548. Feigin, G., Katircioglu, K., and Yao, D. (2002). Distribution resource planning systems: a critique and enhancement. In International Series in Operations Research and Management Science, volume 60, 37–68. Springer. Goyal, S. and Giri, B. (2001). Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res., 134, 1–16. Gurler, U. and Ozkaya, Y. (2000). Analysis of the (s, S) policy for perishables with random shelf life. IEE Transactions, 40, 759–781. Hsu, V. (2003). An economic lot size model for perishable products with age dependent inventory and backorder costs. IEE Transactions, 35, 775–780. Liu, L. and Lian, Z. (1999). (s, S) continuous review models for products with fixed lifetimes. Oper. Res., 47, 150–158. Lonardo, P., Anghinolfi, D., Paolucci, M., and Tonelli, F. (2008). A stochastic linear programming approach for service parts optimization. CIRP Ann. Manuf. Technol., 57(1), 441–444. Mayne, D., Rawlings, J., Rao, C., and Scokaert, P. (2000). Constrained model predictive control: stability and optimality. Automatica, 36(6), 789–814. Nahmias, S. (1982). Perishable inventory theory: a review. Operations Research, 30(4), 680–708.
Nahmias, S. (2011). Perishable inventory systems. In International Series in Operations Research and Management Science, volume 160. Springer. ¨ Ozer, O. (2003). Replenishment strategies for distribution systems under advance demand information. Manage. Sci., 49(3), 255–272. Perakis, G. and Sood, A. (2006). Competitive multi-period pricing for perishable products: A robust optimization approach. Math. Program., 107, 295–335. Perea-Lopez, E., Ydstie, B., and Grossmann, I. (2003). A model predictive control strategy for supply chain optimization. Comput. Chem. Eng., 27(8), 1201–1218. Qin, S. and Badgewell, T. (1997). An overview of industrial model predictive control technology. In Chemical Process Control-V, 232–256. Robinson, E. and Lawrence, F. (2004). Coordinated capacitated lot-sizing problem with dynamic demand: a Lagrangian heuristic. Decision Sciences, 35(1), 25–53. Taticchi, P., Tonelli, F., and Pasqualino, R. (2013). Performance measurement of sustainable supply chains: A literature review and a research agenda. Int. J. Productivity Perf. Manag., 62, 702–804. Tonelli, F., Evans, S., and Taticchi, P. (2013a). Industrial sustainability: Challenges, perspectives, actions. Int. J. Business Innov. and Res., 2, 143–163. Tonelli, F., Paolucci, M., Anghinolfi, D., and Taticchi, P. (2013b). Production planning of mixed-model assembly lines: A heuristic mixed integer programming based approach. Prod. Planning Cont., 24(1), 110–127. Wang, T. and Toktay, B. (2008). Inventory management with advance demand information and flexible delivery. Manage. Sci., 54(4), 716–732. Wang, W. and Rivera, D. (2008). Model predictive control for tactical decision-making in semiconductor manufacturing supply chain management. IEEE Trans. on Control Systems Technology, 16(5), 841–855.
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IFAC-PapersOnLine 48-3 (2015) 1049–1054 Optimal Control of Distribution Optimal Control of Distribution Optimal of Distribution Perishable Goods Optimal Control Control of Distribution Perishable Goods Perishable Goods Perishable Goods ∗ M. Gaggero F. Tonelli ∗∗
Chains Chains Chains Chains
for for for for
∗ ∗∗ M. M. Gaggero Gaggero ∗∗ F. F. Tonelli Tonelli ∗∗ ∗ M. Gaggero Tonelli ∗∗ National Research Institute of Intelligent Systems forF.Automation, ∗ of Intelligent for Automation, National Research ∗ Institute Council of Italy, I-16149 Systems Genoa, Italy (e-mail: [email protected]) ∗∗∗Institute of Intelligent Systems for Automation, National Research Institute of Intelligent Systems for Automation, National Research Council of Italy, I-16149 Genoa, Italy (e-mail: [email protected]) Department of Mechanical Engineering, Energetics, Management Council of Italy, I-16149 Genoa, Italy (e-mail: [email protected]) ∗∗ Council of Italy,ofI-16149 Genoa, Italy (e-mail: [email protected]) Department Mechanical Engineering, Energetics, Management ∗∗ and Transportation, University of Genoa, I-16145 Genoa, Italy ∗∗ Department of Mechanical Engineering, Energetics, Management Department of Mechanical Engineering, Energetics,Genoa, Management and University of (e-mail: [email protected]) and Transportation, Transportation, University of Genoa, Genoa, I-16145 I-16145 Genoa, Italy Italy and Transportation, University of Genoa, I-16145 Genoa, Italy (e-mail: (e-mail: [email protected]) [email protected]) (e-mail: [email protected]) Abstract: A discrete-time dynamic model of distribution chains for perishable goods is Abstract: A dynamic model of chains for perishable goods presented with an approach its optimal management based model predictive Abstract:together A discrete-time discrete-time dynamic for model of distribution distribution chains for on perishable goods is is Abstract: A discrete-time dynamic model ofwith distribution chains for the perishable goods is presented together with an approach for its optimal management based on model predictive control. The model is based on a directed graph, buffers representing amounts of goods presented together with an approach for its optimal management based on model predictive presented together with an approach for its optimal management based on model predictive control. The model is based on a directed graph, with buffers representing the amounts of goods for the various remaining lifetimes, whose graph, time evolution is obtained via balance equations. The control. The model is based on a directed with buffers representing the amounts of goods control. The model based on a directed with buffers representing the amounts of goods for the remaining lifetimes, whose time is obtained balance The amounts of goods toistransfer from node to graph, node evolution are chosen solving via a receding-horizon optimal for the various various remaining lifetimes, whose time evolution isby obtained via balance equations. equations. The for the various remaining lifetimes, whose time evolution is allows obtained balance equations. The amounts of transfer from to node chosen aato receding-horizon optimal control problem time step.node The proposed approach onevia trade among inventory amounts of goods goodsatto toeach transfer from node to node are are chosen by by solving solving receding-horizon optimal amounts of goodsattoeach transfer from node to node are chosendemand, by solving ato receding-horizon optimal control problem time step. The proposed approach allows one trade among inventory and transportation costs, satisfaction of the customers’ and reduction of the amount control problem at each time step. The proposed approach allows one to trade among inventory control problem each time step. The proposed approach allows one to trade inventory and transportation costs, satisfaction of the demand, and reduction the amount of wasted goods,atnamely with no lifetime and thus haveamong toof and transportation costs, goods satisfaction of remaining the customers’ customers’ demand, andthat reduction ofbe thediscarded amount and transportation costs, satisfaction of remaining the customers’ demand, andthat reduction ofbe the amount of wasted goods, namely goods with no lifetime and thus have to discarded from the distribution chain. Preliminary simulation results in three scenarios are reported to of wasted goods, namely goods with no remaining lifetime and thus that have to be discarded of wasted goods, namely goods withapproach. no remaining lifetime and thus that have are to be discarded from the distribution chain. Preliminary simulation results in three scenarios reported to show potential of the proposed from the distribution chain. Preliminary simulation results in three scenarios are reported to from the potential distribution the chain. Preliminary simulation results in three scenarios are reported to show show the potential of of the proposed proposed approach. approach. © 2015, IFAC (International Federation approach. of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. show the potential of the proposed Keywords: Model predictive control, optimization, distribution chains, perishable goods, Keywords: Model predictive control, optimization, distribution chains, perishable goods, inventory Keywords:control Model predictive control, optimization, distribution chains, perishable goods, Keywords:control Model predictive control, optimization, distribution chains, perishable goods, inventory inventory control inventory control 1. INTRODUCTION following, the term “perishable” has to be meant both in 1. INTRODUCTION following, the term hasobsolescence. to be meant both in the strict sense and “perishable” in the sense of 1. INTRODUCTION following, the term “perishable” has to be meant both in 1. INTRODUCTION following, the term hasobsolescence. to be meant both in strict sense and “perishable” in the sense of In this paper, the previous works (Alessandri et al., the the strict senseofand in the sense of obsolescence. The literature inventory and distribution management In this paper, the previous works (Alessandri et al., the strict sense and in the sense of obsolescence. 2011b,a, 2012) on strategic and tactical inventory control In this paper, the previous works (Alessandri et al., The literature of inventory and distribution management of perishable goods is less developed if compared with the In this paper, previous (Alessandri et al., The literature of inventory and distribution management 2011b,a, 2012) onthe strategic andworks tactical inventory control for multi-echelon (DCs) are extended 2011b,a, 2012) on distribution strategic andchains tactical inventory control of The literature of inventory and distribution management perishable goods isgoods. less developed if compared with the one of nonperishable Most of works focus on pricing 2011b,a, 2012) on strategic and tactical inventory control for multi-echelon distribution chains (DCs) are extended of perishable goods is less developed if compared with the to case of perishable goods. We (DCs) focus on for the multi-echelon distribution chains areperishable extended one of perishable goods is less developed if compared with the of nonperishable goods. Most of works focus on pricing models and on the goods. searchMost for of heuristics for on inventory for multi-echelon distribution chains areperishable extended to the case fixed of perishable goods. We (DCs) focus on one of nonperishable works focus pricing goods lifetime (Nahmias, all the goods one to the with case of perishable goods. We 1982): focus on perishable of nonperishable goods. works pricing models and stock on the searchMost for of heuristics for on inventory control replenishment e.g.,focus Liu Lian, to the case fixed oftoperishable goods. We 1982): focus on perishable goods with (Nahmias, all the goods models and and on the search for (see, heuristics for and inventory are assigned alifetime given remaining measured in models goods with fixed lifetime (Nahmias,lifetime, 1982): all the goods and on the search for heuristics for inventory control and stock replenishment (see, e.g., Liu and Lian, 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; goods with fixed (Nahmias, 1982):on. all the goods are assigned toreduces alifetime given remaining measured in control and stock replenishment (see, e.g., Liu and Lian, buckets, that the time lifetime, goes When the are assigned to a givenasremaining lifetime, measured in control and stock replenishment (see, e.g., Liu Lian, Goyal and Giri, 2001; Hsu, 2003; Chu et and al., 2005; Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo are assigned toreduces a reaches givenasremaining lifetime, measured in 1999; buckets, that thezero time goes on. When the 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; remaining lifetime the level or a level for which buckets, that reduces as the time goes on. When the Perakis 1999; Goyal and Giri, 2001; Hsu, 2003; Chu et al., 2005; and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo al., 2008). buckets, reduces as the thezero time the remaining lifetime reaches level or aon. levelWhen for which Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo the goodsthat are unmarketable, aregoes discarded from the et remaining lifetime reaches the they zero level or a level for which Perakis and Sood, 2006; Gurler and Ozkaya, 2000; Lonardo al., 2008). remaining lifetime reaches the they zero level orofa level for which the goods are DCs unmarketable, are discarded from the et et al., 2008). DC. Clearly, involving such type goods require In al., this2008). work, the discrete-time dynamic model of DCs the goods are unmarketable, they are discarded from the et the goods are DCs unmarketable, they are from the In DC. Clearly, involving such typediscarded of require special attention reduce the amount of goods wasted goods thisonwork, the discrete-time dynamic model of DCs based a directed graph originally proposed in AlessanDC. Clearly, DCstoinvolving such type of goods require In this work, the discrete-time dynamic model of DCs DC. Clearly, DCs such type of require special attention toinvolving reducesuch the amount of goods wasted goods based In this work, the discrete-time dynamic model of DCs together with other goals as customers’ satisfaction on a directed graph originally proposed in Alessandri et al. (2011b) is extended to take into account the special attention to reduce the amount of wasted goods based on a directed graph originally proposed in Alessanspecial attention to goals reducesuch the transportation amount of wasted together with other as customers’ satisfaction directedof originally proposed ineach Alessanand reduction of inventory and costs.goods based dri et on al. a(2011b) isgraph extended to take into account the remaining lifetime goods. Toward this end, node together with other goals such as customers’ satisfaction dri et al. (2011b) is extended to take into account the together with other goals such as customers’ satisfaction and reduction of inventory and transportation costs. dri et al. (2011b) is extended to containing take the remaining lifetime goods. Toward thisinto end,account each node is equipped with aof of buffers with and reduction of inventory and transportation costs. remaining lifetime ofset goods. Toward this end,goods each node Classical examples of DCsand for transportation perishable goods can be remaining and reduction of inventory costs. lifetime of goods. Toward this end, each node is equipped with a set of buffers containing goods with remaining Then,containing the amounts of goods Classical examples DCs food for perishable goods need can be equipped with a lifetime. set of buffers goods with found in the field ofof products, which to ais certain Classical examples offresh DCs for perishable goods can be equipped withnode a lifetime. settoofnode buffers containing goods with certain Then, the amounts of goods to transferremaining from are computed by means of Classical examples DCsbefore for perishable goods need can be found in the field ofoffresh food products, which to ais a certain remaining lifetime. Then, the amounts of goods be delivered to customers their remaining lifetime found in the field of fresh food products, which need to ato certain remaining lifetime. Then, the amounts of goods transfer from node to node are computed by means of model predictive control (MPC) (Mayne et al., 2000) basfound in the field of fresh food products, which need to be delivered to level. customers before lifetime to transfer from node to node are computed by means of reach the zero In this case,their the remaining term “perishable” be delivered to customers before their remaining lifetime model to transfer from node to nodefuture are computed means of predictive control (MPC) (Mayne et al., 2000) basing on the information about requests ofbycustomers. be delivered tostrict customers before their lifetime reach the zero level. In this case, the remaining term “perishable” model predictive control (MPC) (Mayne et al., 2000) basis used in the sense. A perishable item is one that model reach the zero level. In this case, the term “perishable” predictive control (MPC) (Mayne et al., 2000) basing on the information about future requests of customers. The proposed approach is able to trade among the goals reach the level. sense. In thisA case, the term is used in zero the strict perishable item“perishable” is one after that ing on the information about future requests of customers. has constant expiration is used in theutility strict until sense.a Acertain perishable item isdate; one that The on the information about future requests of customers. proposed isofable to trade among the goals of theapproach amount wasted goods, satisfying the is used in theutility strict until sense. perishable item isdate; one after that ing has constant a Acertain expiration Thereducing proposed approach is able to trade among the goals it, its utility drops to zero (Nahmias, 2011). This may has constant utility until a certain expiration date; after of The proposed isofable to trade among the goals reducing theapproach amount wasted goods, satisfying the customers’ demand, as well as minimizing inventory and has constant utility until a certain expiration date; after it, its utility drops to zero (Nahmias, 2011). This may of reducing the amount of wasted goods, satisfying the have impacts in terms of sustainability the of it, itsnegative utility drops to zero (Nahmias, 2011). Thisof may reducingdemand, thecosts. amount of wasted goods, inventory satisfying and the as well as minimizing transportation it, itsnegative utility drops to zero (Nahmias, 2011). This2013a; have impacts in terms of sustainability of may the customers’ customers’ demand, as well as minimizing inventory and entire supply/distribution chain (Tonelli et al., have negative impacts in terms of sustainability of the transportation customers’ demand, as well as minimizing inventory and costs. have negative in terms of sustainability of the The entire supply/distribution chain (Tonelli al., dealing 2013a; transportation costs. Taticchi et al., impacts 2013). Similar issues occur et when first attempts entire supply/distribution chain (Tonelli et al., 2013a; transportation costs.of the use of MPC in industry are in entire supply/distribution chain (Tonelli et al., dealing 2013a; Taticchi etinvolving al., 2013).goods Similar issues occur when with DCs subject to obsolescence, such The first attempts the use MPC inplants industry areand in process of control forof (Qin Taticchi et al., 2013). Similar issues occur when dealing the The area first of attempts of the use ofchemical MPC in industry are in Taticchi etinvolving al., 2013). Similar issues when dealing with DCselectronic goods subject tooccur obsolescence, such The first attempts of the use of MPC in industry are in as, e.g., equipment. In this case, items do not the area of process control for chemical plants (Qin and Badgewell, 1997), with increasing diffusion in many other with DCs involving goods subject to obsolescence, such the area of process control for chemical plants (Qin and with DCs involving goods subject to case, obsolescence, as, e.g., electronic equipment. In this items dosuch not Badgewell, area of 1997), process control forofchemical plants (Qinother and loose their wealth properties with increasing diffusion in many fields, including management logistics operations (see, as, e.g., electronic equipment.asInfresh this food case, products, items do but not the Badgewell, 1997), with increasing diffusion in many other as, e.g., electronic equipment. Infresh this food case, items do but not Badgewell, loose their wealth properties products, 1997),management with increasing diffusion in many other become obsolete and then less as attractive for customers including logistics (see, e.g., et al., 2003;of andoperations Rivera, 2008; loose their wealth properties as fresh food products, due but fields, fields,Perea-Lopez including management of Wang logistics operations (see, loose their wealth properties as fresh products, food products, but become obsolete and then lessof attractive for customers due fields, including management of logistics operations (see, to the presence in the market newer which are e.g., Perea-Lopez et al., 2003; Wang and Rivera, 2008; et al., 2008, of the 2008; great become obsolete and then less attractive for customers due Alessandri e.g., Perea-Lopez et al.,2013). 2003; The Wangreasons and Rivera, become obsolete and then less attractive for customers due to the presence in the market of newer products, which are e.g., Perea-Lopez et the al.,capability 2003; The Wangreasons and Rivera, often advanced termsofofnewer performances. et al.,are 2008, 2013). of the 2008; great success of MPC exploit information to themore presence in the in market products,Thus, whichalso are Alessandri Alessandri et al., 2008, 2013). Thetoreasons of the great to the presence market products, whichalso are often advanced in termsofofnewer performances. Thus, Alessandri et al., 2008, 2013). Thetoreasons of the great in thismore case, it inis the important organize the transfer of success of MPC are the of capability exploit information on the future behavior the system, the possibility of often more advanced in terms to of performances. Thus, also success of MPC are the capability to exploit information often advanced in terms of performances. Thus, in thismore case, it important organize transfer of on of MPCbehavior are the of capability to exploit information goods theis of the to DC to avoidthe wastes. In also the the future the system, the possibility of in thisamong case, it isnodes important to organize the transfer of success in thisamong case, it important transfer of on the future behavior of the system, the possibility of goods theisnodes of the to DCorganize to avoidthe wastes. In the goods among the nodes of the DC to avoid wastes. In the on the future behavior of the system, the possibility of goods among the nodes of the DC to avoid wastes. In the
Copyright © 2015 IFAC 1099 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 1099Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 responsibility IFAC 1099Control. Peer review©under of International Federation of Automatic Copyright © 2015 IFAC 1099 10.1016/j.ifacol.2015.06.222
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S
node i goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
goods goods goods remaining remaining remaining lifetime 1 lifetime 2 lifetime 3
D
... ... goods of type 1
... goods of type 2
...
P
goods of type M
u01k
Fig. 1. Set of buffers at each node of the DC. u02k
dealing directly with constraints on state variables and control inputs, as well as the availability of a number of theoretical results about its properties. Here, the amounts of goods to transfer from node to node are obtained by solving optimal control problems involving the sliding-window optimization of a cost function. The system behavior within such a window is obtained by using the above-mentioned model of the DC, basing on future prediction of the customers’ requests. The idea of exploiting information about the future for the optimal ¨ management of DCs can be found also, e.g., in Ozer (2003); Robinson and Lawrence (2004); Wang and Toktay (2008); Tonelli et al. (2013b), although simpler heuristic methodologies are employed. The contribution of this paper is twofold: (i) a discretetime dynamic model to describe a generic DC for perishable goods is presented; (ii) the optimal amounts of goods to transfer from node to node are computed by means of MPC. The paper is organized as follows. Section 2 presents the model of the DC. Section 3 contains the MPC approach for the optimal DC management. Section 4 reports the adopted indexes for performance evaluation. Finally, numerical results are discussed in Section 5. 2. DYNAMIC MODEL OF DISTRIBUTION CHAINS FOR PERISHABLE GOODS Starting form the ideas presented in Alessandri et al. (2011b), we consider a prototype of a DC for perishable goods of various types made up of production units (PUs), distribution units (DUs), and sales units (SUs). A highlevel model of such a DC is a directed graph composed of nodes and arcs. Each node is equipped with sets of buffers that are filled up with quantities of goods of various types among M possible ones. To model the remaining lifetimes of goods, we consider a buffer for each remaining lifetime level and each type of goods (see Fig. 1). Given a graph with N − 1 nodes, let K(i) be the set of the indexes corresponding to the types of goods available at node i, i = 1, . . . , N − 1, and let Lk (i) be the set of the integers that correspond to the remaining lifetime levels of the goods of type k that are stored at the node i. In more detail, such a set is made up by the integers max max 1, . . . , lik , where lik is the maximum remaining lifetime level of goods of type k that can be stored at node i. Moreover, let I(i) and O(i) be the sets of the incoming and outgoing neighbors of node i, respectively. For the sake of brevity, the sets of the node indexes corresponding to PUs, DUs, and SUs are denoted by P, D, and S, respectively. Then, we introduce the fictitious nodes 0 and N as initial source for the PUs and final sink for the SUs to account for the flows of goods at the nodes belonging to P and
1
w1k 2
w2k
i
uijk
wik
τijk
j
wjk
dN −2 uN −2N k
N−2
dN −1 N−1
uN −1N k
wN −1k
Fig. 2. Graph representation of a DC for perishable goods. S, respectively. Fig. 2 shows a sketch of the DC graph, where the nodes 0 and N are omitted for the sake of simplicity. Let us introduce the following variables at each time bucket t = 0, 1, . . . , T , where T is a given horizon. 1) zik (t, l) ∈ R is the stock level of the buffer that contains quantities of goods of type k with remaining lifetime l at node i, i = 1, . . . , N − 1, k ∈ K(i), l ∈ Lk (i); 2) uijk (t, l) ∈ R is the amount of goods of type k with remaining lifetime l transferred from node i to node j, i ∈ I(j), j = 1, . . . , N , k ∈ K(i) ∩ K(j), l ∈ Lk (i); 3) τijk ∈ N is the delay required to transfer goods of type k from node i to node j, i ∈ I(j), j = 1, . . . , N − 1, k ∈ K(i) ∩ K(j); min 4) τik ∈ R is the minimum delay required to transfer goods of type k from node i to a SU node, i = P ∪ D, k ∈ K(i); it can be computed, for example, by means of a dynamic programming algorithm (Bellman, 1957) applied to the graph of the DC; 5) wik (t) ∈ R is the total amount of goods of type k at node i that are wasted, and thus have to be discarded from the DC, i = 1, . . . , N − 1, k ∈ K(i). For the SU nodes, the wasted goods are those with remaining lifetime l = 1 that are unsold/unused, whereas for PUs and DUs they are those that, because of the transfer delays, cannot reach the SU nodes with remaining lifetime greater than or equal to 1 (i.e., those for which min l ≤ τik ). In this case, the transfer of such goods from node to node is useless since they cannot be sold/used; thus, a convenient management choice is to discard them from the DC without carrying out additional transportation costs; 6) bik (t) ∈ R is the backlog level corresponding to goods of type k at node i, i ∈ S, k ∈ K(i); 7) dik (t) ∈ R is the customers’ demand of goods of type k at node i, i ∈ S, k ∈ K(i). We suppose that the customers of the DC are well-suited to buying goods with arbitrary remaining lifetime (but, of course, always greater than or equal to 1). In particular, without loss of generality, we assume that the demand
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dik (t) distributes uniformly in all the lifetime levels of goods, i.e., we define dik (t) d¯ik (t, l) = max , i ∈ S, k ∈ K(i), l ∈ Lk (i) lik The analysis of other demand distributions in the various lifetime levels, such as situations in which goods with closest expiration date are properly positioned in the shelves or reordered in order to be sold before the corresponding ones with greater remaining lifetimes, is out of the scope of this paper. A discrete-time dynamic model of the DC is then derived by using the stock and wasted levels of goods in all the buffers and the backlogs at the SU nodes as state variables, and is given by the following dynamic equations (for the sake of brevity, let Ik (i) {j ∈ I(i) : k ∈ K(i) ∩ K(j)}, and Ok (i) {j ∈ O(i) : k ∈ K(i) ∩ K(j)}): � zik (t, l + 1) + ujik (t − τjik , l + τjik ) j∈Ik (i) � min , zik (t + 1, l) = uijk (t, l + 1) if l ≥ τik − j∈Ok (i) min 0 if l < τik (1a) i ∈ P ∪ D, k ∈ K(i), l ∈ Lk (i) � zik (t + 1, l) = zik (t, l + 1) + ujik (t − τjik , l + τjik ) j∈Ik (i)
− uiN k (t, l + 1), wik (t + 1) = wik (t) +
�
i ∈ S, k ∈ K(i), l ∈ Lk (i) (1b) zik (t, l),
min l≤τik
i ∈ P ∪ D, k ∈ K(i), l ∈ Lk (i) (1c) wik (t + 1) = wik (t) + zik (t, 1) − uiN k (t, 1), i ∈ S, k ∈ K(i) (1d) � � bik (t + 1) = bik (t) − d¯ik (t, l), uiN k (t, l) + l∈Lk (i)
l∈Lk (i)
i ∈ S, k ∈ K(i) (1e) where t = 0, 1, . . . , T . Equations (1a) and (1b) model the balance of goods and the decrease of the remaining lifetime of goods at each time bucket. Equations (1c) and (1d) account for the quantities of goods that are wasted because they have no remaining lifetime or cannot reach the SU nodes with a remaining lifetime greater than or equal to 1. Finally, equation (1e) accounts for the dynamics of the backlogs at the SU nodes. To complete the model (1), we need to introduce various types of constraints. As to the state variables, the stock levels are required to be nonnegative and are subject to upper bounds representing the maximum amounts of goods that can be stored at the various nodes, i.e., we impose � max 0≤ zik (t, l) ≤ zik , i = 1, . . . , N − 1, k ∈ K(i) l∈Lk (i)
(2) is a positive constant where t = 0, 1, . . . , T and representing the maximum quantity of goods of type k that can be stored at node i. max zik
As regards the backlogs, we need to ensure only their positivity, as no upper bounds can be fixed a priori. Thus, we impose the following constraints for t = 0, 1, . . . , T :
bik (t) ≥ 0,
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i ∈ S, k ∈ K(i)
(3)
In order to prevent from the occurrence of stockouts, one can keep a safety stock to satisfy unexpected requests. Without loss of generality, we suppose to keep a unique safety stock for all the goods of the same type at each node, independently from the remaining lifetime. Thus, let sik ≥ 0 be the safety stock level at node i for goods of type k. From a “strategic” point of view, we constrain the stock levels to be higher than the corresponding safety stock, i.e., � l∈Lk (i) zik (t, l) ≥ sik , for i = 1, . . . , N − 1, k ∈ K(i), and t = 0, 1, . . . , T . As we shall see in the following, during online operations this constraint is relaxed, as such stocks are employed on demand to avoid stockouts. Concerning the control inputs, since the transportation of goods of type k from node i to node j can be made only with a delivery cycle time denoted by Ωijk ∈ N, the following constraints hold: � umin uijk (t, l) ≤ umax ijk (t), ijk (t) ≤ l∈Lk (i)
i ∈ I(j), j = 1, . . . , N − 1, k ∈ K(i) ∩ K(j) umin ijk (t)
(4a)
umax ijk (t)
and are poswhere t = rΩijk , r ∈ N; itive numbers representing the minimum and maximum amounts of goods of type k that can be transferred at time t from node i to node j, respectively. Note that we do not constrain the controls uiN k (t, l) at the SU nodes, as they model the quantities of goods that leave the SUs upon the demands dik (t), i.e., we choose umin iN k (t) = 0 and umax iN k (t) = +∞. To model the impossibility of transferring goods of type k from node i to node j at time instants that are not multiples of Ωijk , we impose the following: uijk (t, l) = 0, i ∈ I(j), j = 1, . . . , N − 1, (4b) k ∈ K(i) ∩ K(j), l ∈ Lk (i) where t �= rΩijk , r ∈ N. In other words, if t¯ = rΩijk , r ∈ N, we constrain uijk (t, l) = 0 for all t = t¯+ 1, . . . , t¯+ Ωijk − 1. As previously pointed out, a good management choice is to avoid to transfer goods that cannot reach the SU nodes with remaining lifetime greater than or equal to 1 since such transfers are useless and would only entail additional transportation costs. Thus, we constrain the control inputs as follows: uijk (t, l) = 0, i ∈ P ∪ D, j ∈ O(i), k ∈ K(i) ∩ K(j) (5) min . where t = 0, 1, . . . , T and l ≤ τik
3. OPTIMAL MANAGEMENT OF DISTRIBUTION CHAINS USING MPC In this section, we present the proposed approach for the optimal control of the DC. Specifically, we search for optimal policies to choose the amount of goods to transfer from node to node at the various time instants. When looking for such optimal policies, the following goals have to be pursued: (i) minimize inventory costs; (ii) minimize transportation costs; (iii) minimize the use of safety stocks; (iv) minimize the amount of wasted goods; (v) satisfy the customers’ demand as much as possible. Clearly, a tradeoff among these goals is needed since, in general, the customers’ demand is satisfied more promptly if a greater amount of transferred and stocked goods is
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guaranteed. However, the higher the amounts of goods stocked at the various nodes, the higher the storage costs. Furthermore, the higher the amounts of transferred goods, the higher the transportation costs. Such a tradeoff is searched for by means of an approach based on MPC, which consists in finding, at each time t, the control inputs that solve a finite-horizon optimal control problem of length Tp ≪ T and in applying only the first control action. The procedure is repeated at the next time instant, with an one-step-forward shift of the sliding window and by using new forecasts of the customers’ demand. The dynamics within the prediction horizon is governed by the state equation (1). In particular, we consider the following cost function to minimize: t+Tp N −1 � � � � Jt = zik (τ, l) c1 ατ −t−1 τ =t+1
+c3 γ τ −t−1
i=1 k∈K(i) l∈Lk (i)
N −1 �
�
�
pik [zik (τ, l)]
N −1 � � � � +c4 δ τ −t−1 wik (τ ) bi (τ ) + c5 ǫτ −t−1 i=1 k∈K(i)
i∈S k∈K(i)
+
�
c2 β τ −t
τ =t
N −1 �
�
�
�
gijk [uijk (τ, l)]
j=1 i∈I(j) k∈K(i)∩K(j) l∈Lk (i)
(6) where α ∈ (0, 1], β ∈ (0, 1], γ ∈ (0, 1], δ ∈ (0, 1], and ǫ ∈ (0, 1] are discount coefficients; c1 > 0, c2 > 0, c3 > 0, c4 > 0, and c5 > 0 are weight constants. The goal of satisfying the external demands is taken into account by the penalization of the backlogs. The function gijk : R → R accounts for transportation costs. Following Alessandri et al. (2011b), we focus on a piecewise-constant, strictly-increasing function, which accounts for the fact that transportation is usually accomplished by lot-sized carriers, and the larger the transferred goods, the higher the costs. Specifically, the following function is employed: 0 if uij = 0 1 vijk if 0 < uijk ≤ u1ijk 2 1 2 gijk (uijk ) = vijk if uijk < uijk ≤ uijk , .. . Q vijk if uijk > uQ−1 ijk i ∈ I(j),
j = 1, . . . , N − 1,
1 where 0 ≤ vijk ≤ ··· ≤
Q vijk
k ∈ K(i) ∩ K(j)
and 0 ≤ u1ijk ≤ · · · ≤
(7)
uQ ijk .
To minimize the use of safety stocks, a penalization term based on the function pik : R → R is employed. Specifically, we focus on the following function: � −zik + sik if 0 ≤ zik ≤ sik (8) pik (zik ) = 0 if zik > sik According to the MPC paradigm, at each time t we need to solve the following finite-horizon optimal control problem. Problem MPC. Find min Jt
uijk (τ )
4. PERFORMANCE INDEXES We present in this section the indexes used to evaluate the performance of the proposed approach. First of all, concerning the inventory costs, let the total inventory level (TIL) be defined as the sum of the inventory levels for all t = 0, 1, . . . , T , i.e., −1 � T N � � � TIL zik (t, l) t=0 i=1 k∈K(i) l∈Lk (i)
As regards the transportation costs, let us consider the total transportation cost (TTC), given by the sum of the transportation costs over the considered time horizon: −1 � T −1 N � � � � gijk [uijk (t, l)] TTC t=0 j=1 i∈I(j) k∈K(i)∩K(j) l∈Lk (i)
i=1 k∈K(i) l∈Lk (i)
t+Tp −1
subject to (1), (2), (3), (4), and (5) from time τ = t to time τ = t + Tp − 1.
The capability of satisfying the customers’ demand is taken into account by the total backlog level (TBL), defined as the sum of the backlogs for all t = 0, 1, . . . , T , i.e., T � � � bik (t) TBL t=0 i∈S k∈K(i)
Concerning the amounts of wasted goods, let the total wasted goods (TWG) be defined as the sum of the levels of wasted goods at the various nodes for all t = 0, 1, . . . , T : −1 � T N � � wik (t) TWG t=0 i=1 k∈K(i)
To have a condensed indicator, we define the overall performance index (OPI) as the weighted sum of TIL, TTC, TBL, and TWG, i.e., OPI c1 TIL + c2 TTC + c4 TBL + c5 TWG where c1 , c2 , c4 , and c5 are the weight coefficients used in the cost function (6). 5. SIMULATION RESULTS In this section, we present the results obtained when applying the proposed approach for the management of the same single-item DC made up of 8 nodes considered in Alessandri et al. (2011b), adapted for perishable goods (see Fig. 3). Since such a DC involves only one type of goods, in the following we shall drop the subscript k from all the quantities. The DC was considered over one year (i.e., T = 52), where each bucket corresponds to one week. Three different patterns of the customers’ demand were considered, corresponding to situations that usually occur in practice. In Scenario A, the demand oscillates around an almost constant mean value. In Scenario B, the demand presents first a positive peak and then a negative one. In Scenario C, the demand is characterized by a seasonal pattern, with a peak in the central weeks of the year. The transportation delays, the delivery cycle times, the minimum and maximum amounts of goods that can be
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u13 u01
1 w1
u02
2 w2
3
u36 u37
w3
u14 u23 u24
4
u48
MPC Tp = 2
Tp = 3
A
OPI TIL TTC TBL TWG time [s]
5.33·1010 2.84·107 2.12·105 5.16·108 1.17·109 8.97
5.12·1010 2.85·107 4.09·105 4.95·108 1.68·109 119.2
5.11·1010 2.85·107 5.88·105 4.94·108 1.68·109 6420.1
2.24·1011 7.57·106 7.24·105 2.22·109 5.68·107 0.02
B
OPI TIL TTC TBL TWG time [s]
9.07·1010 2.82·107 1.81·105 8.93·108 1.33·109 7.08
8.69·1010 2.82·107 5.13·105 8.57·108 1.23·109 119.5
8.67·1010 2.82·107 6.21·105 8.55·108 1.23·109 716.6
3.00·107 11 7.55·106 7.24·105 3.00·109 6.43·107 0.02
C
OPI TIL TTC TBL TWG time [s]
8.02·1010 2.67·107 1.81·105 7.88·108 1.34·109 7.78
7.88·1010 2.64·107 6.87·105 7.75·108 1.30·109 173.7
7.86·1010 2.64·107 6.63·105 7.73·108 1.30·109 3802.4
2.95·1011 7.55·106 7.24·105 2.95·109 5.22·107 0.02
d7 u79
7 u57
w7
u25 5
Tp = 1
u46 w6
u47
w4
Table 2. Summary of the simulation results.
d6 u69
6
u58
d8 u89
8 w8
w5
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Fig. 3. DC for perishable goods used in the simulations. Table 1. Parameters of the considered DC. i
j
τij
Ωij
umin ij
umax ij
si
zi (0)
zimax
0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 7 8
1 2 3 4 3 4 5 6 7 6 7 8 7 8 9 9 9
2 3 1 1 2 2 1 3 1 1 2 1 1 2 -
4 4 2 2 2 2 2 1 1 1 1 1 1 1 -
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1400 3200 350 350 350 650 650 180 180 180 180 180 180 180 +∞ +∞ +∞
-
-
-
2500
40000
1.0·108
2100
50000
1.0·108
950
15000
1.0·108
960
15000
1.4·108
1450
15000
1.0·108
150 150 150
2000 2000 2000
1.0·108 1.2·108 2.0·108
transferred from node to node, the values of the safety stocks, as well as the initial and maximum buffer levels are reported in Table 1. The maximum remaining lifetime of goods limax was fixed equal to 10 for all i = 1, . . . , 8. The coefficients of the cost function (6) were chosen so as to satisfy more promptly the demands by penalizing much more the backlogs rather than the other terms, i.e., we chose c1 = 1, c2 = 0.1, c3 = 1, c4 = 100, c5 = 1 and α = 0.003, β = 0.003, γ = 0.003, δ = 0.005, ǫ = 0.003. Concerning the transportation cost function 1 2 20 gij (·), we chose (7) with vij = 10, vij = 20, . . . , vij = 200 1 2 19 and uij = 100, uij = 200, . . . , uij = 1900. The performances of the proposed approach were evaluated using different lengths of the prediction horizon Tp , i.e., we chose Tp equal to 1, 2, and 3. The results provided by MPC were then compared with an heuristic based on the classical lot-for-lot stock replenishment policy adapted to the setting of perishable goods (Feigin et al., 2002). All the simulations were performed in Matlab using a 3.0 GHz Pentium 4 PC with 2 GB of RAM. Table 2 summarizes the results and reports also the average time required to compute the optimal strategies. Figure 4 contains the plots of the demands, stock levels for three remaining lifetimes (l = 1, 3, and 5), and amount of wasted goods at node 6 obtained using MPC with Tp = 3. Looking at the simulation results, it can be noticed that, in all the scenarios, the MPC approach provides better results than the lot-for-lot heuristic in terms of the OPI.
lot-for-lot
Specifically, the former guarantees a more prompt satisfaction of the customers’ demand due to lower values of the TBL, whereas the latter provides reduced values of the TIL and of the TWG. Furthermore, the larger the prediction horizon Tp , the better the performances of MPC. In fact, the values of the OPI obtained with Tp = 3 are the lowest ones, as the optimization is performed using a larger amount of information about the future. Coherently, also the amount of wasted goods and inventory costs decrease as Tp increases. This is obtained at the price of a growth of the transportation costs, as a prompter demand satisfaction, provided by a reduction of the backlogs, is obtained with an increase of the amount of transferred goods. Concerning the computational effort, the MPC approach is more demanding than the lot-for-lot one since it needs to solve an optimization problem on line. By contrast, the lot-for-lot heuristic does not require such a step. Clearly, the larger the prediction horizon Tp for MPC, the larger the time required to compute the optimal policies, as there is the need of solving optimal control problems with an increasing number of unknowns. REFERENCES Alessandri, A., Cervellera, C., Cuneo, M., and Gaggero, M. (2008). Nonlinear predictive control for the management of container flows in maritime intermodal terminals. In Proc. IEEE Conf. on Decision and Control, 2800–2805. Alessandri, A., Cervellera, C., and Gaggero, M. (2013). Nonlinear predictive control of container flows in maritime intermodal terminals. IEEE Trans. Contr. Syst. Technol., 21(4), 1423–1431. Alessandri, A., Gaggero, M., and Tonelli, F. (2011a). Integer tree-based search and mixed-integer optimal control of distribution chains. In Proc. IEEE Conf. on Decision and Control, 489–494. Alessandri, A., Gaggero, M., and Tonelli, F. (2011b). Min-max and predictive control for the management of distribution in supply chains. IEEE Trans. Contr. Syst. Technol., 19(5), 1075–1089.
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Scenario A 20000
Scenario B
d6
20000
Scenario C
d6
20000
15000
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10000
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0
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50 l=1 l=3 l=5
z6
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0 12000
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10000 8000 6000 4000 2000
0 0 15000
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0 15000
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0 0
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0
t 50
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