INVENTORY ALLOCATION IN MULTI-PERIOD MULTI-ECHELON LOGISTICS NETWORKS OF MODULAR PRODUCTS

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1 8 th International Conference of Modeling and Simulation - MOSIM 10 - May 10-12, Hammamet - Tunisia Evaluation and optimization of innovative production systems of goods and services INVENTORY ALLOCATION IN MULTI-PERIOD MULTI-ECHELON LOGISTICS NETWORKS OF MODULAR PRODUCTS Ufuk Bahçeci, Orhan Feyzioğlu Galatasaray University Department of Industrial Engineering Ortaköy-İstanbul, Turkey ubahceci@gsu.edu.tr, ofeyzioglu@gsu.edu.tr ABSTRACT: In this study, inventory planning in logistics networks of modular products is investigated in a multi-period and multi-echelon setting. It is shown that this optimization problem can be converted to another problem which is briefly called modular proportional flow problem. This later problem is a minimum cost network flow problem with side constraints. The whole network consists of several disjoint subnetworks requiring the flows in some given sets of arcs to be proportional. Building efficient solution methods to this problem is important since it can be addressed several times as a subproblem. First, we show that the bases of this problem are good forests, and then design a network simplex algorithm that exploits this special structure. Finally, an illustrative example is supplied to demonstrate the algorithm. KEYWORDS: Modular Products, Logistics Networks, Side constraints, Proportional flow, Network Simplex 1 INTRODUCTION To increase product variety and achieve quick response with low cost, assemble-toorder(ato)systems have been widely adopted by contract manufacturers, including for example, the computer and telecommunication equipment, automobile, consumer electronics and many other industries. Further, many of them have multi-echelon inventory systems. Particularly, in production, stocks of raw materials, semi-finished products and finished products have to cross many stages before reaching the end state. During the past two decades, there has been an extensive literature that aims to study the related issues of ATO and multi-echelon inventory systems (see the excellent survey by Song and Zipkin 2003 and Axsater 2003). And, for a detailed description of the network models and their applications in practice, we refer reader to (Glover et al.; 1992). On the other hand, modular product design is also an important topic which drives to success in global competition (Antonio et al.; 2007). However, today s products have more and more complex structure, so the planning of logistics networks concerning the modularity is getting more complicated by the increasing number of modules. This type of logistics networks generally consist of several subnetworks, each of them responsible to supply some specific module. These subnetworks are not completely independent in the sense that to assemble a single product, several modules are required in proportional or equal amounts. Otherwise, unnecessary component stocks are held and transferred, or final products could not be build due to the shortage of components. Component commonality and substitution are another important concerns which enable to reduce costs associated with the mentioned problems. Based on this motivation, we present the following modular production planning problem. This problem can be encountered in practice, both stand alone or as a part of a more complex supply chain planning problem. Without loss of generality, an instance of the model having only three stages will be discussed to preserve the notational simplicity. Q : min t T s.t. ( w W k z Z ( ( k K x kt sw e kt 1 s w W k ( o kt swx kt s S k w W k sw + h kt s e kt s )+ p kt wzywz kt + vw kt gw kt )) + lzu t t z) (1) z Z + e kt s = r kt s s S k ; k K; t T (2)

2 z Z ywz kt x kt sw gw kt 1 + gw kt = s kt w s S k w W k ; k K; t T (3) wz u kt z + b t 1 z u kt 1 z = d kt z w W k y kt z Z; k K; t T (4) u t z = u1t z a 1 = u2t z z a 2 = = umt z z a m z z Z; t T (5) x kt sw, y kt wz, e kt s, g kt w, u kt z, u t z, 0 s S k ; w W k ; z Z; k K; t T (6) In model Q, the superscripts k K and t T denote module and period indices, respectively. The subscripts s S k, w W k and z Z are used to represent different elements in the three stages of the logistics network. The variables x kt sw are for the allocated quantities of inventory from the first stage to the second stage and ywz kt are for the inventory flows from the second stage to the third stage. In the first stage, e kt s is the unused quantity of inventory in period t, which flows to the next period. In the second stage, gw kt has a similar interpretation. In the third stage, u t z is the unsatisfied demand of product z in period t, and u kt z s are auxiliary variables. Parameters o kt sw and p kt wz stand for the inventory allocation costs in the respective stages and they include purchasing and substitution costs. Similarly, h kt s and vw kt are for the holding costs. Further, lz t is for shortage cost and b t z is for the backorder ratio of product z in period t. rs kt, s kt w and d kt z denote the positive supply or negative demand amount of inventory for the elements in the respective stages. Finally, a k z is the component ratio from module k used in the assembly of product z. Note that, in model Q, constraints (2)-(4) are inventory balance constraints, each of them corresponds to one of the three stages. Constraint (5) is for the proportionality requirement between component flows belonging to different modules enabling the final assembly of product z in period t. If we fix the variables u kt z, this linear programming problem becomes a network problem. Indeed, constraint (5) is a side constraint of the network problem consisting of (1)-(6). Hence, we introduce the following notation. Let G (N, A) be a directed network where N = {1,, n} is the set of nodes and A = {(i, j) : i, j N } is the set of directed arcs. Let G k (N k, A k ) be a subnetwork of G where N k = {1,, n k } is the set of nodes and A k = {(i, j) : i, j N k } is the set of arcs corresponding to subnetwork k K = {1,, m}. In relation with the previously posed problem, each G k is associated with the supply network of k-module with the property that G k G k = for each k k and k, k K, and k K G k = G. We define u ij, c ij and p ij as being the flow upper bound, unit flow cost and proportionality coefficient related to arc (i, j) A respectively. Let b i > 0 if node i N is a supply node, and b i < 0 if node i N is a demand node. It is assumed that i N k b i = 0 for all k K. As the customer demand of a specific end-product is satisfied with the assembly of several specific modules in logistics networks of modular products, flows on several arcs belonging to separate subnetworks must be proportional to satisfy that demand. In a more general framework, let A sk be the subset of arcs in A k and A s = k A sk be the set of all arcs in A that should have proportional flow for requirement s respectively where s S = {1,, t}. In fact, the requirement s mentioned here corresponds to a particular end-product and period for problem Q given in (1)-(6). Hence, it is possible to formulate an equivalent problem for Q as presented below. In problem P1, which we briefly call modular proportional flow problem, we want to find the amount of flows f ij for each (i, j) A so that P1: min s.t. (i,j) A j:(i,j) A k f ij c ij f ij (7) j:(j,i) A k f ji = b i i N k, k K, (8) f ij /p ij are all equal (i, j) A s, s S, (9) 0 f ij u ij (i, j) A k, k K (10) In the network flow problem presented in (7)-(10), the constraints (8) are the flow conservation constraints corresponding to supply network of module k. The constraints (9) are the proportional flow constraints corresponding to the side constraints, and (10) impose lower and upper bounds on the arc flows. If the constraints (9) were not included in problem P1, then it would become a minimum cost network flow problem (MCNF) with disjoint subnetworks. MCNF is a well-known problem and there exist efficient algorithms to deal with it such as network simplex (Ahuja et al.; 1993). However, the addition of proportionality constraints complicates the problem and the available solution procedures developed for MCNF can not be applied directly. When the number of nodes is at least an order of magnitude larger than the number of side constraints, it becomes possible to exploit the special network structure (Helgason and Kennington; 1995). So far, several variants of the MCNF problem with side constraints have been studied. Based on the relaxation and decomposition techniques, Ali et al. (1988) solved the equal flow problem in which given pairs of arcs are required to have identical flow. Ahuja et al. (1999) introduced the simple equal flow prob-

3 lem in which only a single set of arcs is required to have identical flow and in particular, develop special purpose primal simplex algorithm. Calvete (2003) extended the simple equal flow problem by allowing multiple sets of arcs to have identical flow and so introduced the general equal flow problem. Mo et al. (2005a) considered an integrated manufacturing supply chain where multiple products are manufactured across multiple manufacturing plants by distilling from one raw material and following a similar way to that used in (Ahuja et al.; 1999) and (Calvete; 2003), a modified network simplex method which exploits the special structure of basis is presented. Recently, a more complicated model called manufacturing network flows allowing the synthesis of different materials to one product and the distilling of one material to many different products is introduced by Fang and Qi (2003). The authors modified the network simplex method according to the flow problem they defined and illustrated their method by solving a simplified version of their model. Mo et al. (2005b) expanded the study of manufacturing network flows by incorporating certain features of the ordinary multi-commodity network flow models. Lu et al. (2006) studied a manufacturing network model in which the assumption requiring the total flow in and out of a node to be equal is relaxed. Venkateshan et al. (2008) developed a network-simplex-based algorithm to solve a minimum cost flow problem formulated on such a generalized network. Lu et al. (2009) proposed a method to obtain an initial basic feasible solution for the same problem and initialize the existing network simplex algorithm, and also presented a network based approach to check the dual feasibility conditions. Since our primary interest is in dealing with modularity and proportionality requirements on some arc flows, the general equal flow problem is the closest model to our problem. Motivated by the work of Calvete (2003), we analyze the structure of the modular proportional flow problem and deliver an algorithm to benefit from the underlying network structure. Hence, the modular proportional flow problem can be viewed as a minimum cost network flow problem with side constraints plus a decomposable structure provided by modularity. Here, the logistics network consists of several disjoint subnetworks requiring the flows in some given sets of arcs to be proportional. This proportionality assumption between arc flows belonging to distinct subnetworks in a given set of arcs allows us to incorporate assembly or distillation process into the model. Nevertheless, it is possible to incorporate into the model other policy requirements on arc flows by this assumption. Since each subnetwork can be seen as a module of the whole system, our model extends the original general equal flow problem to the m-modules general proportional flow problem. First, we give a network formulation of this problem and show that its bases are good (r+m)- forests. Later, we describe in details an algorithm that exploits this structure. Finally, we illustrate the steps required in applying our algorithm. 2 REFORMULATION OF THE MODU- LAR PROPORTIONAL FLOW PROB- LEM In this section, we first reformulate the modular proportional flow problem given in (7)-(10) and then analyze the special structure of its coefficient matrix which allowed us to develop an efficient solution algorithm. Let c s = (i,j) A s p ijc ij and u s = min (i,j) A s u ij /p ij for all s S. Let a ij denotes the column associated to arc (i, j) in the node-arc incidence matrix of network G and a l ij be the l-th component of vector a ij. Let also a s = (i,j) A s a ij for all s S and a l s be the l- th component of vector a s. As a i ij = p ij, a j ij = p ij, a l ij for each l i, j N k and (i, j) A sk and all the subnetworks are disjoint from each other, we observe for each subnetwork k that l N k a l s = (i,j) A sk l N k a l ij = 0 s S. (11) Let G (N, Ã) be the network with à = A \ s SA s and G k (N k, Ãk) be the network with Ãk = A k \ t s=1a sk for all k K. Hence, problem P1 can be transformed into the problem P2 as follows: P2: min s.t. j:(i,j) à (i,j) à c ij f ij + f ij t c s f s (12) s=1 j:(j,i) à f ji + t a i sf s s=1 = b i i N (13) 0 f ij u ij (i, j) à (14) 0 f s u s s S (15) After this reformulation, we need to consider only P2. It must be noted that if t n, then there will be many basic feasible solutions of P2 involving only variables {f 1,, f t }, and the simplex algorithm will pivot between these basic feasible solutions. In turn, this will reduce the efficiency of the proposed algorithm since there will be no possibility to take the advantage of network structure. Therefore, we assume that t < n holds. Lemma 1 The rank of the matrix A corresponding to constraints (13) is equal to n m.

4 Proof. The matrix A has n rows and one column for each arc in à and one column for each variable f s, A = [à a 1 a 2... a t ] where à is the node-arc incidence matrix of G. First, we observe that the maximum rank of A is n m because adding all the rows up yields the zero vector for each disjoint subnetwork k corresponding to node set N k. Furthermore, we assume without loss of generality that each network G k contains at least one spanning tree since otherwise we can add artificial arcs with sufficiently large costs. This implies that the rank of à is k (n k 1) = n m and thus rank(a) = n m. 3 STRUCTURE OF THE BASIS In the previous section, we have showed that basic feasible solutions of the linear problem P2 consist of n m basic variables whose corresponding vectors in A are linearly independent and the rest of the variables are fixed at their lower or upper bound. Following this fact, if none of the variables {f 1,, f t } are in the basis, then this basis can be represented by an m-spanning forest in G in order to get n m linearly independent vectors. Otherwise, if r of these variables {f 1,, f r } are basic, then we should select n r m variables {f ij, (i, j) Ã} whose associated vectors in the node-arc incidence matrix à are linearly independent and also independent of variables {f 1,..., f r }. This later case can be obtained by removing r arcs from an m-spanning forest in G, which will decompose it into r +m node-disjoint trees T 1 (N1 T, A T 1 ),..., T r+m (Nr+m, T A T r+m). This collection of trees will again span G and thus the resulting forest is a (r + m)-spanning forest in G which we will denote as F. We will now analyze the structure of the related bases. Let B denotes the submatrix of à associated with the (r + m)-spanning forest F and a 1,, a r be vectors associated with variables f 1,, f r. Given that the rank of B is equal to n r m, the rank of B = [ B a 1 a 2 a r ] is equal to n m if the vectors in B with vectors a 1,, a r are linearly independent. Accordingly, we provide in the following a suitable condition that guarantees B is a basis of the problem P2. But before going into the details, we need to introduce some additional notation. Let assume that T 1,, T z1 G 1, T z1+1,, T z2 G 2 and in general T z(k 1) +1,, T zk G k for all k K. Therefore the number of spanning trees in each subnetwork G k is equal to z k z (k 1) with z 0 = 0 and z m = r + m. In a similar fashion, let N1 T = {1,, n T 1 }, N2 T = {n T 1 + 1,, n T 2 }, and in general Nz T = {n T (z 1) + 1,, nt z } for all z Z = {1,, z 1, z 1 +1,, z 2,, z (m 1) +1,, z m } without loss of generality. Finally, let D be the matrix formed by the elements d z,s = l N a l z T s for all z Z = Z\{z 1, z 2,, z m } and s S = {1,, r}. Note that Z = r + m and Z = r by definition. Theorem 1 rank (B) = n m if and only if rank (D ) = r. Proof. Following necessary column arrangements, the matrix B can be reexpressed as in (16). T T z1 0 0 z1 0 0 T z(m 1) +1 0 z(m 1) T zm zm (16) where 0 are matrices of conformal dimensions with all entries equal to zero, T z is the node-arc incidence matrix of T z and z = 2 a n (z 1)+1 r 2 a n (z 1)+2 r a nz 1 a nz 2 a nz r a n (z 1)+1 1 a n (z 1)+1 a n (z 1)+2 1 a n (z 1)+2 for all z Z. Here, rank(t 1 ) = n T 1 1, rank(t 2 ) = n T 2 n T 1 1, and in general rank(t z ) = n T z n T (z 1) 1 for all z Z. As every non-singular square submatrix of the node-arc incidence matrix of a directed network is triangular, the matrix B can be rewritten as in B provided in (17). where T T z z T z (m 1) +1 0 z (m 1) T zm T z = ± ±1 z m (17)

5 and z = for all z Z. Hence, rank(b) = d z,1 d z,2 d z,r 2 a n (z 1)+2 r a nz 1 a nz 2 a nz r a n (z 1)+2 1 a n (z 1)+2 m z k k=1 z=z k 1 +1 = n r m + rank(d) rank(t z ) + rank(d) where D is formed by the elements d z,s = l Nz T for all z Z and s S. As we have 0 = l N k a l s = l N T z (k 1) +1 a l s+...+ l N T z k a l s al s k K, s S from (11), rank(b) = n m holds if and only if rank(d ) = r. Therefore, a basic solution to the modular proportional flow problem consists of an (r + m)-spanning forest F in G where r = 0,..., t as well as variables {f 1,..., f r } verifying that rank(d ) = r. Note that for each (r + m)-spanning forest, there are ( t r) combinations of selecting r variables among {f 1,..., f t }. Definition 1 An (r+m)-spanning forest F in G is a good (r+m)-forest with respect to the variables {f s } s S with S S and S = r, if rank(d ) = r where D is formed by the elements d z,s = l Nz T al s for all z Z and s S and Nz T is the node set of tree T z in forest F. Theorem 2 A basic solution of the modular proportional flow problem is constituted by an (r + m)- spanning forest F in G where r = 0,..., t plus a set of r variables {f s } s S, S S, S = r verifying that F is a good (r + m)-forest with respect to {f s } s S. It is clear from the proceeding develop- Proof. ments. 4 NETWORK SIMPLEX ALGORITHM In this section, we give in details the main steps required for the network simplex algorithm developed to solve problem P Finding the initial basic feasible solution If none is conveniently available, the all artificial start method (Hillier and Lieberman; 2010) can be used to get a basic feasible solution with artificial variables in the network G. The initial basic feasible solution is constituted by the good m-forest defined by this feasible solution. All other variables are non-basic variables and are equal to their lower bound. 4.2 Computing the values of the basic variables From now on we assume that the basis is given by a good (r + m)-forest such that T z F for all z Z and the variables {f 1,, f r }. Let B be the set of arcs (i, j) Ã such that f ij is a basic variable, and B be the set of s S such that f s is a basic variable. Accordingly, we categorize non-basic variables such that L = {(i, j) Ã \ B : f ij = 0}, L = {s S \ B : f s = 0}, U = {(i, j) Ã \ B : f ij = u ij } and U = {s S \ B : f s = u s }. Finally, we let Vz 1 = {(i, j) U : i Nz T, j / Nz T } and Vz 2 = {(i, j) U : i / Nz T, j Nz T, (i, j)} for all z Z. Then, the following theorem guarantees that the value of variables {f 1,, f r } is solvable. Theorem 3 The value of the basic variables {f 1,, f r } is the solution of the following linear system: D f = b (18) where D is previously defined, f = (f 1,, f r ) t and b = (b 1,, b r) t with b z = b l u ij l N T z l N T z (i,j) V 1 z (i,j) V 2 z u ij s U a l su s z Z (19) Proof. After fixing the value of non-basic variables, each constraints in (13) can be reformulated as f ij f ji + a i sf s = ˆb i i N s B j:(i,j) B j:(j,i) B where ˆb i = b i j:(i,j) U u ij + j:(j,i) U u ij s U ai su s. Since u ij vanishes if i Nz T, j Nz T and (i, j) U, ˆbi = b i u ij + l Nz T l Nz T i Nz T,j / N z T,(i,j) U u ij a i su s i / Nz T,j N z T,(i,j) U l Nz T s U = b i u ij + u ij a i su s s U l N T z = b z (i,j) V 1 z (i,j) V 2 z l N T z

6 for all z Z. Similar to the proof of Theorem 1, we may solve the linear system (18) to obtain the value of variables {f 1,, f r }. Hence, the proof is complete. The values of basic variables {f 1,..., f r } affect the requirement of each supply and demand node. Then, the flow values of the remaining arcs in each tree T z z Z can be determined by applying the general procedure of the network simplex algorithm. 4.3 Computing the reduced costs Given a basic feasible solution, we have to verify if it is optimal by calculating node potentials π i for all i N and taking into account the fact that the reduced cost of each basic variable is zero. In other words, we should be able to find node potentials such that c π ij = 0 for all (i, j) B and cπ s = 0 for all s B where c π ij = c ij π i + π j for all (i, j) Ã and cπ s = c s i N ai sπ s for all s S. Since the basic variables can be easily derived from the spanning (r+m)-forest, the computation of the node potential π i of nodes i T z z Z is done by considering reduced costs c π ij = c ij π i +π j = 0 of arcs (i, j) T z z Z. We also need the reduced costs c π s = c s i N ai sπ i, s = 1,..., r of variables f 1,..., f r to be equal to zero. Hence, we update node potentials according to the following expression to conclude that the reduced costs of all basic variables are equal to zero: { } πi + σ π i = z if i T z z Z π i if i T z z {z 1,..., z m } (20) where σ z, z Z are obtained by solving the following linear system D t σ = c π (21) where D is previously defined, σ and c π are both column vectors formed of elements σ z, z Z and c π z, z Z, respectively. We will continue to denote the node potentials by π i, i N. The optimality condition corresponds to that c π ij 0 and c π s 0 for all f ij and f s non-basic variables at their lower bound and c π ij 0 and cπ s 0 for all f ij and f s non-basic variables at their upper bound. If the optimality condition is not satisfied the entering variable can be determined by following any of usual rules. 4.4 Determining the exiting column and pivoting When we change the value of the entering variable, two possibilities exist. One is that the entering variable remains non-basic at its other bound, and other is that the non-basic variable enters the basis and one basic variable exit at one of its bounds. Without loss of generality, three cases can be identified assuming that the entering variable is at its lower bound. Case 1. The entering variable is f ij, i T z, j T z. In this case, only the tree T z is affected. When nonbasic arc f ij is added to tree T z, all variables along the cycle are adjusted accordingly. Case 2. The entering variable is f ij, i T h, j T q, h q. Since the addition of variable f ij with θ 0 changes the requirements of trees T h and T q by θ and +θ, respectively, the modification in (22) of the linear system (18) should be solved when calculating the new values of variables f 1,..., f r : b = ( b 1,, b h θ,, b q + θ,, b z m 1) t (22) Once the new values of variables f 1,..., f r are determined, it is possible to calculate the new value of the remaining basic variables by increasing θ until f ij or one of the basic variables reaches one of its bounds. Case 3. The entering variable is f s, for any s = r + 1,..., t. When the value of variable f s is increased by θ 0, the requirements of trees are changed accordingly. The new values of variables f 1,..., f r are determined by solving the modification in (23) of the linear system (18): b b 1 θ l T 1 a l s = b z m 1 θ (23) l T zm 1 a l s In the rest of the procedure, the new value of the remaining basic variables are calculated by adjusting θ accordingly until f s or one of the basic variables reaches one of its bounds. 5 NUMERICAL ILLUSTRATION 5.1 Network example In order to illustrate our algorithm, let us consider the example presented in Figure 1. In this example, m = 2 and t = 4. The number adjacent to any

7 node i denotes b i. The node sets N 1 = {1,..., 19} and N 2 = {20,..., 32} are defined for the two subnetworks. Table 1 further summarizes the data of the example. This example corresponds to a two-product two-period production/inventory model where each product consists of two modules. In the first period, demand nodes 15 and 28 are used to represent product 1 s demand corresponding to its first and second modules, respectively. In numerical terms, product 1 has 100 units of demand corresponding to 100 units of module 1 and 300 units of module 2, because it consists of one module 1 and three modules 2. In the second period, nodes 17 and 30 represent demand quantities for each of two modules. Following the same line of thinking, product 2 has nodes 16, 29 for the first period and nodes 18, 31 for the second period. Nodes 19 and 32 are dummy demand nodes for the first and second modules, respectively. In the first period, nodes 1, 2, 3, 7, 8, 9, 10 supply for the first module, and nodes 20, 21, 24, 25 supply for the second module. In the second period, nodes 4, 5, 6, 11, 12, 13, 14 supply for the first module, and nodes 22, 23, 26, 27 supply for the second module. Note that nodes 3, 21, 6 and 23 are dummy nodes. The flows through arcs (3, 15) and (6, 17) shows the unsatisfied module demand of product 1 in the first and second periods, respectively. The flow through arc (17, 3) corresponds to the back order quantity in terms of module 1. The interpretation of similar arcs belonging to other module and product is same. The proportionality coefficients of these arcs are given as follows: p 3 15 = 2, p 3 16 = 2, p 17 3 = 1, p 18 3 = 1, p 6 17 = 1, p 6 18 = 1, p = 6, p = 4, p = 3, p = 2, p = 3, p = 2. Then, it is possible to calculate the costs of the proportionality sets: c 1 = 48, c 2 = 36, c 3 = 60 and c 4 = 60. Further, the arcs from supply nodes to dummy demand nodes show the inventory hold at the end of period 2. Note that the flows between dummy nodes are used to balance each subnetwork. 5.2 Iteration 1 We have a 2 -spanning forest F = {T 1, T 2 } with z 1 = 1 and z 2 = 2. T 1 contains nodes {1,..., 19} and T 2 contains nodes {20,..., 32}. Since f s, s = 1,..., 4 are non-basic variables and we have a 2 -spanning forest in G, this is a basic solution to the modular proportional flow problem. Next we compute the node potentials and reduced costs for non-basic variables as given in Table 2. Since all non-basic variables are at their lower bound, f 1 is the entering variable and we increase its value by θ. The exiting variable is f 7 15, at its lower bound, and θ = Iteration 2 We have a 3 -spanning forest F = {T 1, T 2, T 3 } with z 1 = 2 and z 2 = 3. T 1 contains nodes {8, 15}, T 2 contains nodes {1,..., 19} \ {8, 15} and T 3 contains nodes {20,..., 32}. Since D = ( 2), this is a good 3 - forest. Next, we compute the node potentials. Since c π 1 = 9, we normalize them and find σ 1 = 4.5. At iteration 2, the reduced costs for non-basic variables are summarized in Table 2. Since all non-basic variables are at their lower bound, we choose f 2 9 as the entering variable and increase its value by θ. The exiting variable is f 1 7, at its lower bound, and θ = Iteration 3 We have a 3 -spanning forest F = {T 1, T 2, T 3 } with z 1 = 2 and z 2 = 3. T 1 contains nodes {8, 15}, T 2 contains nodes {1,..., 19} \ {8, 15} and T 3 contains nodes {20,..., 32}. Since D = ( 2), this is a good 3 -forest. The node potentials are calculated and normalized with σ 1 = 4.5. Next, the reduced costs for non-basic variables are calculated. Since all non-basic variables are at their lower bound, we choose f 7 15 as the entering variable and increase its value by θ. The new value of variable f 1 is calculated by solving the system D ( f1 ) = ( b 1 + θ ), and we find f 1 = 37.5 θ/2. The exiting variable is f 7 16, at its lower bound, and θ = Iteration 4 Figure 2 displays a 3 -spanning forest F = {T 1, T 2, T 3 } with z 1 = 2 and z 2 = 3. T 1 contains nodes {7, 8, 15}, T 2 contains nodes {1,..., 19} \ {7, 8, 15} and T 3 contains nodes {20,..., 32}. Since D = ( 2), this is a good 3 -forest. The node potentials are calculated and normalized with σ 1 = 4.5. Then, the reduced costs for non-basic variables are given in Table 2. Since all non-basic variables are at their lower bound, this basis is optimal. 6 CONCLUSION This study deals with inventory planning in logistics networks of modular products in a multi-period and multi-echelon setting. In order to benefit from the network stucture of this problem, we reformulated it as the modular proportional flow problem. This later problem is a minimum cost network flow problem with side constraints where the whole network consists of several disjoint subnetworks requiring the flows in some given sets of arcs to be proportional. Indeed, this study extends a previous work on a network model with side constraints, the general equal flow problem. This extension allows us to incorporate the modularity and the proportionality into the general equal flow problem. The bases of the mod-

8 Figure 1: Network example (i,j) (1,7) (1,8) (2,9) (2,10) (7,15) (8,15) (9,15) (10,15) (7,16) (8,16) (9,16) (10,16) (1,4) c ij u ij (i,j) (2,5) (3,19) (4,19) (5,19) (4,11) (4,12) (5,13) (5,14) (6,19) (11,17) (12,17) (13,17) (14,17) c ij u ij (i,j) (11,18) (12,18) (13,18) (14,18) (3,15) (3,16) (17,3) (18,3) (6,17) (6,18) (20,24) (20,25) (22,26) c ij u ij (i,j) (22,27) (24,28) (25,28) (24,29) (25,29) (26,30) (27,30) (26,31) (27,31) (20,22) (21,32) (22,32) (23,32) c ij u ij (i,j) (21,28) (21,29) (30,21) (31,21) (23,30) (23,31) c ij u ij Table 1: Illustrative example data (i,j) (1,7) (1,8) (2,9) (2,10) (7,15) (9,15) (10,15) (7,16) (8,16) (4,12) (5,14) (13,17) (14,17) It It It It (i,j) (11,18) (12,18) (25,28) (24,29) (27,30) (26,31) (1) (2) (3) (4) It It It It Table 2: Reduced costs of non-basic variables

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