OR Spectrum (2009) 31:573–599 DOI 10.1007/s00291-008-0160-5 REGULAR ARTICLE A strategic and tactical model for closed-loop supply chains Maria Isabel Gomes Salema · Ana Paula Barbosa Póvoa · Augusto Q. Novais Published online: 31 December 2008 © Springer-Verlag 2008 Abstract In this paper, a strategic location-allocation model is developed for the simultaneous design of forward and reverse supply chains. Strategic decisions such as network design are accounted for together with tactical decisions, namely, production, storage and distribution planning. The integration between strategic and tactical decisions is achieved by considering two interconnected time scales: a macro and a micro time. At macro level, the supply chain is designed in order to account for the existing demands and returns, whose satisfaction is planned simultaneously at the micro level where tactical decisions are taken. A Mixed Integer Linear Programming formulation is obtained which is solved to optimality using standard Branch & Bound techniques. Finally, the model accuracy and applicability is illustrated through the resolution of a case study. Keywords Discrete time · Closed-/open-loop supply chains · Network design · Optimisation 1 Introduction Traditionally, supply chains start with raw materials and finish with products delivered to the final customer. Several intermediate levels may be present, associated with M. I. G. Salema (B) Centro de Matemática e Aplicações, FCT, UNL, Monte de Caparica, 2825-114 Caparica, Portugal e-mail: mirg@fct.unl.pt A. P. B. Póvoa Centro de Estudos de Gestão, IST, Av. Rovisco Pais, 1049-101 Lisboa, Portugal A. Q. Novais Dep. de Modelação e Simulação, INETI, Est. do Paço do Lumiar, 1649-038 Lisboa, Portugal 123 574 M. I. G. Salema et al. different kinds of facilities, products and transportation modes, but the end user is always assumed to represent the last level in the network. However, at some point in time, products leave the final customers and must be dealt with. Exception made to organic waste, most used products still have some intrinsic value that, in a sustainable environment, should be recovered. In the last decades, the question of how to deal with return products and how to collect and handle them in processing facilities became an important problem to both research community and society. However, most studies addressing the global supply chain only consider the traditional forward flow. The reverse supply chain that deals with the flow that leaves customers and goes back to factories or to proper disposal, has been mostly studied in an operational/tactical level. Strategic questions are yet to be answered by academia and industry (Guide et al. 2003). When dealing with the design and planning of global supply chains with reverse flows, most published papers either adopt a silo approach or are case oriented. As stated in the work of De Brito et al. (2004), a considerable number of case studies have been published in the last decade. Amongst the most generic models proposed in the literature, the work of Jayaraman et al. (1999) should be mentioned. The authors propose a mixed integer programming model that simultaneously considers the location of remanufacturing and distribution facilities, the transhipment, as well as the production and stocking of the optimal quantities of remanufactured products and cores. The model was tested on a set of problems based on the parameters of an existing electronic equipment remanufacture firm. The authors concluded that, if on one hand demand for remanufactured products is a key decision variable as in traditional distribution networks, on the other the optimal solution depends on having enough quantities of cores to remanufacture. Fleischmann et al. (2001) investigated whether or not to integrate collection and recovery within an existing forward distribution network. A model for the simultaneous design of forward and reverse supply chains is proposed. The chosen formulation uses a network balance constraint to integrate twowarehouse location models. The authors concluded this problem to be context dependent. Furthermore, they suggest that when the existing distribution network can no longer be used, a careful analysis of the total logistic chain should take place in order to decide whether a single network with two flows or two independent networks should be used. Jayaraman et al. (2003) propose both a strong and a weak formulation for the reverse distribution problem. Their research addresses the modelling of independent forward and reverse networks and considers product recall, product recycling and reuse, product disposal and hazardous product return. The weak formulation, in particular, was solved using an heuristic procedure, specially developed for this model. The authors stated that this heuristic found optimal solutions for a significant proportion of tested problems within a reasonable amount of time. Moreover, it also solved a very large number of other problems that conventional MIP tools could not handle. Later on, Fandel and Stammen (2004) propose a strategic model for the supply chain design. The authors extend the traditional supply chain to account for the recycling of products released by customers. Dynamic aspects of the network are modelled by the use of a two-step time structure. Although promising, this work did not present 123 A strategic and tactical model for closed-loop supply chains 575 any case or example and, therefore, neither the adequacy nor efficiency of the proposed approach to treat real world problems was tested, nor was the model proven as solvable. Finally, in the work developed by Salema et al. (2006), a multi-product model is proposed for the design of a reverse distribution network where both forward and reverse flows are considered simultaneously. These networks differ not only in terms of structure but also in the number of products involved. Based on a network design model, the propose MILP formulation considers binary variables to describe structural decisions (e.g. whether or not to open a factory) and continuous variables for the volume of products in transit along the network. One published case is modified and solved in order to provide a better insight into the model. Lu and Bostel (2007) proposed a closed-loop supply chain model for a remanufacturing network. The model considered producers, remanufacturing sites, intermediate centres and customers. The intermediate centres belong exclusively to the reverse network and send their products only to remanufacturing facilities. These, together with producers, directly supply customers’ demand, which means that remanufactured products are assumed as new. A lagrangean heuristic approach was developed and numerical experiments were performed on examples adapted from classical test problems. Within the works mentioned above, one important area of research that needs further study is the simultaneous design and planning of forward and reverse networks. The objective of this paper is to propose a multi-product and dynamic model for the design of supply chain network where production and storage planning are accounted for. A strategic locationallocation model is developed, where tactical decisions are explicitly modelled. Two interconnected time scales are considered, which allows some detail to be introduced in production, storage and distribution planning. All activities have capacity limits; maximum and minimum levels are imposed on the production on every time instance; storage maximum level is set; distribution flows, whenever they occur, must be within given lower and upper bounds. Traditionally, the strategic nature of location models considers instantaneous flows. The proposed model allows for the establishment of travel time between network levels. Another feature modelled is the time products are being used by customers, named as “usage time”. This introduces a lag time between the time that the product demands are satisfied and the availability of the products to be “remanufactured”. It may be questionable the relevance of introducing tactical planning decisions into a strategic model since uncertainty on data may exist. This option becomes however relevant since the decisions considered are taken based on a set of data aggregation with no high detail associated. In this way, the model becomes very useful to analyse the network robustness. As both kinds of decisions are integrated within a single formulation, analysis can be performed to observe the impact that changes in tactical parameters (such as products costs, distributions costs, storage costs, among others) may have on the network design. Once the locations are chosen, the tactical decisions should be reviewed considering updated information and should be analysed with the operational decisions (scheduling decisions, for instance). This paper is organised as follows. In the next section, a detailed description of the problem is given. Then, due to its significance in the model, the time structure 123 576 M. I. G. Salema et al. Cust. Cust. Warehouse Cust. Cust. Warehouse Dis. centre Factory Cust. Cust. Factory Cust. Dis. centre Warehouse Cust. Cust. Cust. Fig. 1 Supply chain with reverse flow is discussed in detail and two operators are defined. The mathematical formulation of the model follows, where variables, parameters and constraints are explained. The case study is then presented together with the results and a preliminary study on the model behaviour. Finally, some conclusions and future research directions are drawn. 2 Problem definition A supply chain can be represented as a network where facilities act as nodes and links are related with direct flows between facilities. Such network may have several levels and each facility in each level can be connected to another facility in a different (not necessarily consecutive) level. In this work, three levels are considered: factories, intermediaries (warehouses and disassembly centres) and customers. Moreover, the supply chain is extended with the inclusion of reverse flows (see Fig. 1). The forward and the reverse chains can be regarded as having a two-echelon structure each one. In the forward network, factories are connected to customers through warehouses and in the reverse network customers send their products back to factories, through disassembly centres. It is assumed that no direct connection exists between factories and customers in either direction. The goal of the model is to maximise the global supply chain profit, assuming all prices, transfer-prices and costs to be known in advance. Several costs are considered: investment (whenever a facility is chosen), transportation, production, storage and penalty. Penalties apply to non-satisfied demands or returns. Forward and return products are treated as independent since it is assumed that the original products may loose their identity after use (e.g. paper recycling—paper prod- 123 A strategic and tactical model for closed-loop supply chains 577 ucts after use are simply classified as paper). However, if required, it is also possible to track the same product in both flows. Due to the unknown quality of return products, a disposal option is made available in the disassembly centres. Products that do not meet quality standards for remanufacturing are, thus, redirected to a different supply chain or simply to a depot. Furthermore, customers cannot introduce products coming from a different supply chain. Using the structural options defined above, the proposed model considers two levels of decisions, which correspond to two different time scales: a “macro time” scale, where the supply network is designed, and a “micro time” scale, where planning activities are set (e.g. global production and/or storage planning). These time scales can be years/months, years/trimesters, months/days or any combination that suits the problem. The chosen facilities will remain unchanged throughout the time horizon while the throughput may change over time. Traditionally, strategic supply chain models consider instantaneous flows, i.e. customers have their demand satisfied at the same instance in time at which products are manufactured. However, the proposed two-time scale overcomes this limitation and the formulation enforces the definition of “travel times”. Travel time is modelled as the number of micro time units that, for example, a product takes to flow from its origin to its destination. It is considered that products before entering the reverse network must be “used” by customers. Thus for each return product a “usage time” is considered, which is defined as the minimum number of micro time units that a supplied product remains in the customer before entering the supply chain as a return product. In short, the proposed model can be formulated as: Given • • • • • • • a possible superstructure for the location of the supply chain entities, the investment costs, the amount of returned product that will be added to the new products, the relation between forward and reverse products, the travel time between each pair of network agents, the minimum usage time for each return product, the minimum disposal fraction, and for each macro period and product • customers’ demand values, • the unit penalty costs for non-satisfied demand and return, and for each micro period • • • • • • the unit transportation cost between each pair of network facilities, the maximum and minimum flow capacities, the maximum and minimum production capacities, the maximum storage capacities, the initial stock levels, the factory production unit costs, 123 578 M. I. G. Salema et al. • the facilities unit storage costs, • the transfer prices between facilities, • customers’ purchase and selling prices. Determine • • • • the network structure, the production and storage levels, the flow amounts and the non-satisfied demand and return volumes. So as to • maximise the global supply chain profit. 3 Time modelling characterisation As referred above, the coexistence of two-time scales is one special feature of this model. These are closely interdependent and contemplate some aspects that should be analysed before entering the modelling detail. Consider T = {1, 2, . . . , t, . . . , T } as the set
of macro periods, where  the time horizon is divided in T equal size units, and T
= 0, 1, . . . , t
, . . . , n − 1 as the set of micro periods. Let the generic elements be referred respectively as t and t
. Note that T
is a set with n elements, i.e. |T
| = n and T is the time horizon. The interconnection between these two-time scales is depicted in Fig. 2, where each t ∈ T corresponds to n elements in T
. Two important entities must be formulated in this approach to model time: the “previous time” and the “forward time”. 3.1 Backward time operator Previous time appears associated with the concept of travel time, which is defined between two micro period instances: origin and destination. When dealing, in the same constraint, with these two instances, two different situations must be considered: first, the origin and destination belong to the same macro period (Fig. 3a) and second, they belong to different macro periods (Fig. 3b). In order to account for these situations an operator ϒ, called “backward time operator”, is defined. For t ∈ T and t
∈ T
, let (t, t
) be the current time instance and let τ be the number of micro time units associated with a previous event, which is to be accounted for at (t, t
). Consider ω1 ∈ Z the smallest integer greater or equal than ... t 0 t+1 1 ... t'-1 t' Fig. 2 Relation between macro and micro time scales 123 ... n-1 0 ... A strategic and tactical model for closed-loop supply chains 579 τ (a) ... t+1 t 0 τ 1 ... t'-τ ... t' ... n-1 0 (b) ... ... t'-τ ... t 0 t+1 1 ... ... t' ... n-1 0 Fig. 3 a Both micro periods belong to the same macro period, b Micro periods belong to different macro periods 
 = ω1 ). The operator ϒ gives the relevant time instance (“backward (i.e. τ −t n time”) for that previous event. This operator is defined as follows:  if t
− τ ≥ 0 (t, t
− τ ),
ϒ(t, t − τ ) =
(t − ω1 , ω1 n + t − τ ), if t
− τ < 0 ∧ t > ω1 τ −t
n To have a better insight into this operator let us consider an example where n = 12 and (t, t
) = (4, 6), i.e. there are 12 micro time instances and the current time instance is 4 on macro time and 6 on micro time. For a previous event characterised by τ = 5, the corresponding previous time instance is given by ϒ(t, t
− τ ) = ϒ(4, 1) = (4, 1) since t
− τ ≥ 0 (Fig. 3a). If, on the other hand, τ = 19 is assumed, then    τ −t
= ω1 ⇔ 19−6 = 2 and the previous time instance would be ϒ(t, t
− τ ) = n 12 (t − ω1 , ω1 n + t
− τ ) = (2, 11) since t
− τ < 0 (Fig. 3b). To simplify notation and unless there is ambiguity, the generic micro time element (t, t
) will be from now on denoted simply by t
. 3.2 Forward time operator Forward time appears whenever usage time is modelled. This is the case when customers inbound and outbound flows are considered in the same constraint. One must assure that a product remains in the customer at least the predefined usage time. As done for previous time, an operator is defined in order to allow the modelling of this other entity. The forward time operator  is defined as follows. For t ∈ T and t
∈ T
, let (t, t
) be the current time instance and let φ be the number of micro time units associated with an upcoming event, which is to be accounted
for at (t, t
). Consider ω2 ∈ Z the smallest integer greater or equal than t +φ−1 (i.e. n 
 t +φ−1 = ω2 ). The operator  gives the relevant time instance (“forward time”) n for that upcoming event. This operator is defined as follows:  if t
+ φ ≤ n − 1 (t, t
+ φ),
(t, t + φ) =
(t + ω2 , t + φ − ω2 n − 1), if t
+ φ > n ∧ t + ω2 ≤ card(T ) Consider where n = 12 and (t, t
) = (3, 8), i.e. there are 12 micro time instances and the current time instance is 3 on macro time and 8 on micro time. For a forward event characterised by φ = 2, the corresponding forward time instance is given by t
+ φ ≤ n − 1. If, on the other hand, φ = 16 (t, t
+ φ) = (3, 
10) = (3, 10) since   = ω2 ⇔ 8+16−1 = 1 and the forward time instance is assumed, then t +φ−1 n 12 would be (t, t
+ φ) = (t + ω2 , t
+ φ − ω2 n − 1) = (4, 11) since t
+ φ > n. 123 580 M. I. G. Salema et al. 4 Model formulation In this section, the proposed model is described in detail. Indices are presented first, then the definition of sets and variables are given, followed by the constraint definition and the objective function. In the description of variables and constraints, we start by defining those that are not time dependent, followed by the ones defined over the macro time and finally the ones over the micro time. 4.1 Indices Consider the following indices: i factories (i = 0, fictitious factory that allows products to leave the supply chain—disposal option), j warehouses, l disassembly centres, k customers, m f forward products, m r reverse products, t macro time and t
micro time. 4.2 Sets For the model formulation, several sets are needed, which can be divided into three groups: location, products and time. As location sets, we have: I potential location for factories (which can be generalised as I0 = I ∪ {0}, in order to allow return products to leave the network for a different type of processing), J potential location of warehouses, L potential location of disassembly centres, K potential location of customers’ sites. When dealing simultaneously with forward and reverse networks, one should be careful with the modelling of products, as they may undergo transformation while moving between networks. Thus, for each network different products were created, grouped into different sets: M f set of products in the forward network, Mr set of products in the reverse network. As mentioned above, in Sect. 3, two different time scales are defined over the time horizon: T = {1, time scale (referred as macro time), and
. . . , t, . . . , T } is the broader  T
= 0, 1, . . . , t
, . . . , n − 1 is the finer time scale (referred as micro time), noting that there are n micro intervals for each macro interval t. 4.3 Parameters 4.3.1 Time independent parameters γ is the minimal disposal fraction for sending some return products to an external processing facility; it takes values in the range [0, 1], 123 A strategic and tactical model for closed-loop supply chains 581 p r sm f i0 , sma f j0 , sm , smc r k0 is the initial stock of product m f in factory i and r l0 warehouse j, and of product m r in disassembly centre l and in the possession of customer k, respectively, p f i , f ja , flr fixed cost of opening factory i, warehouse j, and disassembly centre l, respectively, Upper and lower limits s gi p , g sja , glsr maximum storage capacity of factory i, warehouse j, and disassembly centre l, p p gi , h i maximum and minimum production of factory i, f1 f gi , h i 1 upper and lower bound values for flows leaving factory i, r2 r2 gi , h i upper and lower bound values for flows arriving at factory i, f g j 2 upper bound value for flows leaving warehouse j, glr1 upper bound value for flows arriving at disassembly centre l, Transfer parameters δab travel time between locations a and b, i.e. the number of micro time units required to travel from location a to location b, φm r usage time of product m r , i.e. number of micro time units the product m r spends at customers, before entering the reverse network. Product parameter Material balance constraints of factories and customers relate forward with reverse products. For factories, there is an inbound of return products that must accounted for together with the production of new ones. For customers, they receive forward products and use them. After use, the products are released onto the reverse chain and considered as return products. ηm f is the fraction of product m f that is returned by customers, εm f m r = 1 if, after use, product m f is considered as product m r ; zero otherwise. With these two parameters, another two, which are required in the model constraints, can be computed as follows: βm f m r = ηm f εm f m r fraction of forward product m f that once used is treated as return product m r , ηm f fraction of product m r that will be remanufactured as αm r m f =  product m f . m f ∈M f εm f m r 4.3.2 Macro time parameters dm f kt demand of product m f to be supplied to costumer k, over macro period t, u w ) unit variable cost of non-satisfied demand (return) of product m (m ) (cm cm f r r kt f kt for customer k, for macro period t. 123 582 M. I. G. Salema et al. 4.3.3 Micro time parameters Lower and upper bounds f h m2f kt
, h rm1r kt
lower bound value for flows reaching and leaving customer k, at time t
, respectively Selling and transfer prices πm1f i jt
unit transfer price of product m f from factory i to warehouse j, at time t
, f f πm2f jkt
unit price of product m f sold by warehouse j to customer k, at time t
, πmr1r klt
unit price of product m r bought by disassembly centre l to customer k, at time t
, πmr2r lit
unit transfer price of product m r from disassembly centre l to factory i, at time t
. Costs cm f it
unit production cost of product m f , manufactured in factory i, at time t
, p s sa cmp f it
cm
unit storage cost of product m f kept in factory i and in warehouse j, f jt
at time t , respectively sr
cm
unit storage cost of product m r kept in disassembly centre l, at time t . r lt f cm1f i jt
unit transportation cost of product m f from factory i to warehouse j, at time t
, f cm2f jkt
unit transportation cost of product m f from warehouse j to customer k, at time t
, r1 cm
unit transportation cost of product m r from customer k to disassembly r klt centre l, at time t
, r2 cm
unit transportation cost of product m r from disassembly centre l to factory r lit i(i ∈ I0 ), at time t
. 4.4 Variables The variables that are time independent will be described first. The macro time variables follow and finally the micro time variables will be defined. 4.4.1 Binary variables These variables describe the choice of sites in the supply chain and if a customer should or should not be supplied. p Yi Y ja Ylr Ykc = 1 if factory i is opened; zero otherwise, i ∈ I, = 1 if warehouse j is opened; zero otherwise, j ∈ J, = 1 if disassembly centre l is opened; zero otherwise, l ∈ L and = 1 if customer k integrates the supply chain; zero otherwise, k ∈ K . 123 A strategic and tactical model for closed-loop supply chains 583 4.4.2 Continuous variables Two sets of variables are defined over the macro time. These describe the amount of demand/return that is not satisfied: Um f kt non-satisfied demand of product m f of customer k, over macro period t, and Wm r kt non-satisfied return of product m r of customer k over macro period t. All other variables are defined for the micro time. These are the flow, stock and production variables. The flow variables are: X m1f i jt
demand of product m f served by factory i to warehouse j, over period t
, f f X m2f jkt
demand of product m f served by warehouse j to customer k, over period t
, r1 Xm
return of product m r of customer k to the disassembly centre l over period r klt
t , and r2 Xm
return of product m r of disassembly centre l to factory i (i ∈ I0 ), over r lit period t
. In Fig. 4, the relation between flow variables and binary variables is represented, together with the different network echelons. The stock variables are: Sm f it
amount of product m f stocked in factory i, over period t
. Sma f jt
amount of product m f stocked in warehouse j, over period t
, r
Sm
amount of product m r stocked in disassembly centre l, over period t . r lt p Lastly, the production variable is Z m f it
amount of product m f produced by factory i, over period t
. All continuous variables are non-negative. 4.4.3 Auxiliary variables Auxiliary variables, which are not used as model decisions variables, are defined to ease the constraint formulation. These are the stock variables at customer sites and the dummy binary variables associated with flows. The former describe the amount of product m r kept by customer k, over micro period t
and is represented by Smc r kt
. The latter are used to define a set of disjunctive constraints. 4.5 Constraints As before, the constraints definition is made considering first those defined over the macro times and then those over the micro times. 123 584 M. I. G. Salema et al. X mf1f ijt ' Y ja Ykc Yi p X X mf 2f jkt ' Yl r r2 mr lit ' Intermediate facility Factory X mr1r klt ' Customer Fig. 4 Schematics of the interdependence between flow and binary variables 4.5.1 Macro time constraints These macro time constraints relate quantities aggregated in time such as demand, return and disposal with detailed ones such as flows. For each aggregate quantity, a constraint is defined. Demand constraint Each selected costumer k, in each macro period t, has a specific demand for product m f that needs to be satisfied. This demand can be (totally or partially) satisfied by all the inbound flows that reach customer k, during the time interval t. Assuming that customers may or may not integrate the supply chain, this constraint is only active if the customer is chosen (Ykc = 1) f j∈J t
∈T
X m2f jkϒ(t,t
−δ jk ) + Um f kt = dm f kt Ykc , ∀m f , k, t (1) Return constraint The return volume of each product m r for each customer k, in each macro period t is assured by constraint (2). It is considered that the total amount that is sent to any disassembly centre plus the nonsatisfied return amount (if any) must equal the total amount of used product that this customer has to send back. This total return volume is computed as the weighted sum of all forward product supplied to the customer. As above-mentioned, a usage time is taken into account by operator , which does not allow products to go into reverse network before staying a least some micro time periods in customers’ possession. 123 A strategic and tactical model for closed-loop supply chains l∈L t
∈T
585 r1 Xm + Wm r kt
r kl(t,t +φm ) r f = m f ∈M f j∈J t
∈T βm f m r X m2f jkϒ(t,t
−δ jk ) , ∀m r , k, t (2) Disposal fraction One of the features of the proposed network is that it allows products to leave the network. This is possible through the definition of a fraction of collected products that each disassembly centre sends either to recycling or to proper disposal. Thus, a part of the inbound flows of disassembly centre l can be sent to a fictitious factory, factory i = 0 that represents any facility outside the supply chain: γ k∈K t
∈T
r1 Xm ≤
r klϒ(t,t −δkl ) t
∈T
r2 Xm ∀m r , l, t
, r l0t (3) 4.5.2 Micro time constraints Having established, in the global chain, the relation between aggregated quantities and flows, it is important to relate the latter with the operational aspects, such as production, storage and transportation. As a result, the final network structure is also defined, namely sites and associated connections. Production constraints These constraints are established for two main purposes: to assure the connection between production factories and associated inbound and outbound flows and to guarantee factory capacity production limits. The production planning contemplated in this model is of a broad strategic nature, no consideration being given either to products’ bill-of-materials or to the difference between manufacturing and remanufacturing when outbound flows are considered. However, factories have specific inbound flows that need to be handled, which are the return products sent in by disassembly centres. • Material balance For each micro period t
, product m f and factory i, constraint (4) assures that the amount produced plus the used products (inbound flow) and the existing inventory (which was set in the previous micro period) equals the total product delivered by the factory (outbound flow) plus the remaining stock. If ϒ(t, t
− 1) = (1, 0), then p p Sm f iϒ(t,t
−1) = sm f i0 which is the initial stock level at factory i. p Z m f it
+ m r ∈Mr l∈L r2 αm f m r X m + Sm f iϒ(t,t
−1)
r liϒ(t,t −δli ) X m1f i jt
+ Sm f it
, ∀m f , i ∈ I, (t, t
) f = p (4) j∈J • Capacity constraints In any factory i, the production level is double bounded. These bounds are assumed to be constant over the model horizon and depend only on the factory. Furthermore, 123 586 M. I. G. Salema et al. for a factory to be used, it must be installed. Constraints (6) and (7) model these cases: Z m f it
≤ gi Yi , ∀i ∈ I, (t, t
) (5) Z m f it
≥ h i Yi , ∀i ∈ I, (t, t
) (6) p p m f ∈M f p p m f ∈M f In addition, as storage is allowed in factories, an upper bound is set: Sm f it
≤ gi p Yi , ∀i ∈ I, (t, t
) s p m f ∈M f p (7) Warehouse storage constraints At the warehouses, a material balance is also established where the inbound flow plus the existing stock equals the outbound flow plus the remaining stock. This must be assured in every micro period, for every existing product. f X m1f i jϒ(t,t
−δi j ) + Sma f jϒ(t,t
−1) i∈I f = X m2f jkt
+ Sma f jt
, ∀m f , j, (t, t
) (8) k∈K Again, if ϒ(t, t
− 1) = (1, 0), then Sma f jϒ(t,t
−1) = sma f j0 which is the initial stock level at warehouse j. Furthermore, every warehouse j has a maximum storage capacity (g sja ), if chosen (i.e. Y ja = 1). m f ∈M f Sma f jt
≤ g sja Y ja , ∀ j, (t, t
) = (1, 1) (9) Disassembly centre storage constraints Disassembly centres can be viewed as reverse network warehouses. Therefore, constraints are similar to constraints (8) to (9). Equation (10) establishes the material balance, which relates storage volumes with the inbound/outbound flows. Thus, in each period t
, the inbound flow and the existing storage must equal the outbound flow plus the new storage volume. r2 r Xm ∀m r , l, (t, t
)
+ Sm r lt
, r lit r1 r Xm + Sm =

r lϒ(t,t −1) r klϒ(t,t −δkl ) k∈K (10) i∈I r r If ϒ(t, t
− 1) = (1, 0), then Sm = sm which is the initial stock level at
r l0 r lϒ(t,t −1) disassembly centres l. For each disassembly centre l, a maximum storage capacity (glsr ) is also considered if the facility is used (Ylr = 1). m r ∈Mr 123 sr r r Sm ∀l, (t, t
)
≤ gl Yl , r lt (11) A strategic and tactical model for closed-loop supply chains 587 Customer constraints Although customers are external to the company in charge of the supply chain, they are modelled as any other facility. Demand satisfaction is the criterion for customers to become part of the supply chain. In this model, we assume that there may be cases of customers, which are not chosen to be supplied on economic grounds. Thus, we introduced in the demand constraint (1), a binary decision variable that indicates whether or not a customer should be considered in this supply chain. Constraint (12) is, then, the material balance constraint for each customer. Similarly, to factories, customers have a transformation role: they “transform” forward products into “return” ones by using them. Another feature of this constraint is the modelling of “usage time” previously defined. Usage time is introduced in the outbound term, which makes the constraint to relate the inbound at time t
with the outbound at micro time t
+ φ. f m f ∈M f j∈J k∈K βm r m f X m2f jkϒ(t,t
−δ jk ) + Smc r kϒ(t,t
−1) r1 Xm + Smc r kt
, ∀m r , k, (t, t
)
r kl(t,t +φm ) = r (12) l∈L Once again, for the first micro period (t, t
) = (1, 1), the existing volume of products at customer site is the given parameter smc r k0 . Transportation flows constraints The transportation flows constraints have a double function. They assure that flows remain within certain pre-established limits and that they only occur between opened/existing facilities. All flows, in every micro time, must fall below a preset maximum level. The minimum level is modelled differently among flows. Factory inbound and outbound flows are not imposed on every micro time, but when they occur they must meet the minimum limit. This is modelled using auxiliary variables, which are described next. In terms of customers, it is assumed that a minimum flow amount must be sent and collected in every micro time, for every customer. • Factory outbound flow X m1f i jt
≤ gi 1 E i jt1
, ∀i, j, (t, t
) (13) X m1f i jt
≥ h i 1 E i jt1
, ∀i, j, (t, t
) (14) f m f ∈M f f f m f ∈M f f f f 2E i jt1
≤ Yi + Y ja , ∀i, j, (t, t
) f p (15) f where E i jt1
is an auxiliary binary variable for the flow between factory i and warehouse j, at micro time t
. • Customer inbound flow f m f ∈M f X m2f jkt
≤ g j 2 Y ja , ∀ j, k, (t, t
) f (16) 123 588 M. I. G. Salema et al. f m f ∈M f j∈J t∈T t
∈T
X m2f f X m2f jkt
≤ Big M1 Ykc , ∀k (17) jkt
≥ h m2f kt
Ykc , ∀m f , k, (t, t
) (18) f j∈J where Big M1 is given by Big M1 = g j 2 · |M f | · |J | · |T | · |T
| f • Customer outbound flow m r ∈Mr m r ∈Mr l∈L t∈T t
∈T
r1 r1 r Xm ∀l, k, (t, t
)
≤ gl Yl , r klt (19) r1 c Xm ∀k
≤ Big M2 Yk , r klt (20) r1 r1 c Xm ∀m r , k, (t, t
) :∼ (t=1 ∧ t
−φm r
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