A Primal-Dual Algorithm for Computing a Cost Allocation in the. Core of Economic Lot-Sizing Games

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1 1 2 A Primal-Dual Algorithm for Computing a Cost Allocation in the Core of Economic Lot-Sizing Games 3 Mohan Gopaladesikan Nelson A. Uhan Jikai Zou 4 October Abstract We consider the economic lot-sizing game with general concave ordering cost functions. It is well-known that the core of this game is nonempty when the inventory holding costs are linear. The main contribution of this work is a combinatorial, primal-dual algorithm that computes a cost allocation in the core of these games in polynomial time. We also show that this algorithm can be used to compute a cost allocation in the core of economic lot-sizing games with remanufacturing under certain assumptions. Keywords: cooperative games; cost allocation; economic lot-sizing; primal-dual algorithm 13 1 Introduction In this work, we study a class of cooperative games known as economic lot-sizing (ELS) games. In an ELS game, we have multiple retailers who each face a deterministic demand for the same product from a single manufacturer, over a finite discrete time horizon. Each retailer s demand in a given time period can be satisfied by product ordered in that time period, or any of the previous time periods. The retailers may place orders individually, or the retailers can form coalitions and place joint orders. We assume that the ordering and inventory costs are independent of the retailers. The cost to any coalition of retailers is their minimum total ordering and inventory cost. We formally define this setting as follows. Let Department of Statistics, Purdue University, West Lafayette, IN, USA. mohang@purdue.edu School of Industrial Engineering, Purdue University, West Lafayette, IN, USA. nuhan@purdue.edu School of Industrial Engineering, Purdue University, West Lafayette, IN, USA. zou7@purdue.edu 1

2 22 N = {1,..., n} be the set of retailers, or players; 23 T be the number of discrete time periods; 24 D i t be the demand faced by player i in period t, for all i N and t = 1,..., T ; 25 c t (x) be the cost of ordering x units in period t, for all t = 1,..., T ; 26 h t be the unit holding cost in period t, for t = 1,..., T. 27 We assume that c t ( ) is concave and nondecreasing, and that c t (0) = 0 for all t = 1,..., T. In 28 addition, let D S t = i S Di t for t = 1,..., T. For any S 2 N, let D S = (D S t ),...,T be the vector of demands for all time periods. The cost v(s) of coalition S 2 N is defined as the minimum total ordering and inventory cost to satisfy demands D S. In other words, v(s) is the optimal value of the following mathematical program: [A S ] min x,i (c t (x t ) + h t I t ) (1.1a) s.t. x t + I t 1 = D S t + I t for all t = 1,..., T, (1.1b) I 0 = I T = 0, (1.1c) x t 0, I t 0 for all t = 1,..., T, (1.1d) where x t is the amount ordered in period t, and I t is the amount of inventory at the end of each period t, for all t = 1,..., T. Here constraints (1.1b)-(1.1c) regulate the flow of product between inventory, ordering, and demand. The cooperative game (N, v) is an ELS game. Suppose that χ R N is a cost allocation: for each player i N, χ i is the cost allocated to player i N. The core (Gillies 1959) of a cooperative game (N, v) is the set of all cost allocations χ that satisfy the following constraints: χ i = v(n), (1.2a) i N χ i v(s) for all S 2 N. (1.2b) i S 38 The constraint (1.2a) ensures that the cost allocation χ is budget-balanced; that is, the sum of the costs 2

3 allocated to all of the players equals the joint cost that they incur together. The constraints (1.2b) ensure that the cost allocation χ is stable; that is, no subset of players can do better by leaving the coalition of all players and acting independently. The idea here is similar to a Nash equilibrium of a noncooperative game: an outcome is stable if no defection is profitable. The core is one of the most prominent solution concepts in cooperative game theory. In this work, we study the core of ELS games. In particular, we focus on how to efficiently compute a cost allocation in the core of these games Previous related work The economic lot-sizing problem was first studied by Wagner and Whitin (1958), and has since received an enormous amount of attention in the operations research literature. We refer the reader to the various surveys on this problem (e.g. Drexl and Kimms 1997; Brahimi et al. 2006) for details on its history. van den Heuvel et al. (2005) introduced ELS games. In their version of the game, backlogging is not allowed and the ordering cost in each period consists of a fixed setup cost and a linear cost. They showed that the core of such games is always nonempty using the Bondareva-Shapley theorem (see Bondareva 1963; Shapley 1967). They also showed that some special cases of these games are concave. Chen and Zhang (2006) studied ELS games with general concave ordering costs and backlogging. They proved that the core of these games is nonempty and showed how to compute a cost allocation in the core in polynomial time, by solving a modified dual linear program. Xu and Yang (2009) presented a so-called cross-monotonic cost-sharing method for ELS games that is approximately budget balanced. Guardiola et al. (2008, 2009) studied production-inventory games, which model a collaborative production and inventory setting similar in spirit to the ELS games of van den Heuvel et al. (2005) and Chen and Zhang (2006). In their setting, ordering, inventory, and backlogging costs are retailer-dependent, and all coalition members share the most advanced technology among them; that is, they each have access to the cheapest costs within the coalition Contributions of this work We begin in Section 2 by describing some useful properties of the economic lot-sizing problem and ELS games. In Section 3, we present the main contribution of this work: a combinatorial, 3

4 primal-dual algorithm that directly computes a cost allocation in the core of the ELS games in polynomial time. This is in contrast with the algorithm by Chen and Zhang (2006), which requires a linear programming subroutine. The algorithm of Chen and Zhang (2006), however, works on ELS games with backlogging, while our algorithm does not. Finally, in Section 4, we discuss how our algorithm can be used to compute a core cost allocation for a special case of the economic lot-sizing game with remanufacturing options (Richter and Sombrutzki 2000; Richter and Weber 2001) A review of some useful results on ELS games Wagner and Whitin (1958) made an important observation on ELS problems without backlogging: when ordering costs consist of a fixed setup cost and a linear variable cost, there exists an optimal ordering policy that is a zero-inventory policy; that is, a policy in which x t I t 1 = 0 for t = 1, 2,..., T Zangwill (1969) proved a similar property for ELS problems with backlogging and a general concave cost function. Instead of the mathematical program (1.1), we will use an alternate mathematical program that directly finds a minimum cost zero-inventory policy for the underlying ELS problem of v(s). Before presenting this mathematical program, we need the following well-known lemma, recast in the context of ELS games. Lemma 2.1. Any instance of an ELS game (N, v) as described above is equivalent to another instance of an ELS game (N, v) with zero holding costs, where c t (x) = c t (x) + x k=t h k 85 is the cost of ordering x units in period t, for each t = 1,..., T. In particular, for any S 2 N, v(s) = v(s) K S, where K S = i S K i and K i = h t k=1 t Dk i. (2.1) 86 Proof. By adding constraints (1.1b) and (1.1c), and substituting this in the objective function (1.1a), 4

5 87 we obtain the result Note that for each t = 1,..., T, we have that c t ( ) is nondecreasing and concave, and that c t (0) = 0. A zero-inventory policy can be viewed as a set of disjoint time intervals that cover the entire time horizon. With this in mind, consider the following formulation for v(s), based on finding the minimum cost zero-inventory policy of the underlying ELS problem: [B S ] min s.t. c S ijx ij K S (2.2a) i=1 j=i t i=1 j=t x ij = 1 for all t = 1,..., T, (2.2b) x ij {0, 1} for all 1 i j T, (2.2c) where for all 1 i j T, x ij is a binary variable indicating ordering product in period i to meet demands in periods i, i + 1,..., j, c S ij = c j ) i( DS t, and K S is defined in (2.1). Constraints (2.2b) ensure that the demand in every period t is satisfied by exactly one ordering interval. Chen and Zhang (2006) showed that the natural LP relaxation of this integer program obtained by replacing the binary constraints with nonnegativity constraints has the same optimal value. 97 Theorem 2.2 (Chen and Zhang 2006). The integrality gap of [B S ] is By multiplying both sides of constraints (2.2b) by D S t formulation that is equivalent to [B S ]: for each t = 1,..., T, we have the following [C S ] min s.t. c S ijx ij K S i=1 j=i t Dt S i=1 j=t x ij = Dt S for all t = 1,..., T, x ij {0, 1} for all 1 i j T. 100 The dual of its LP relaxation is: [D S ] max Dt S α t K S i=1 5

6 s.t. j Dt S α t c S ij for all 1 i j T Chen and Zhang (2006) showed the following lemma on the monotonicity of optimal solutions to [D S ]. Lemma 2.3 (Chen and Zhang 2006). Suppose we have α such that α t α t+1 for all t = 1,..., T If j ( j ) D t α t c i D t for all i j, 105 then for any D such that D t D t for all t = 1,..., T, we have that j D t α t c i ( j D t ) for all i j Computing a cost allocation in the core Now we present the main contribution of this work. The following algorithm finds a minimum cost zero-inventory policy by solving the formulations [C N ] and [D N ] simultaneously. In the following, we will refer to each of the constraints of [D N ] as an ordered pair (i, j). A set of ordered pairs E is said to be an exact cover if for each t = 1,..., T, there is exactly one (i, j) E such that i t j. Algorithm 3.1 (Primal-dual algorithm for ELS games). C. (set of tight constraints) τ 0. (continuous time counter) Set α 1,..., α T as active. while at least one α t is active do Simultaneously increase the value of all active α t s with the time counter until j DN t α t = c N ij for some i j. Make α i, α i+1,..., α j inactive (or leave them inactive, if already inactive). C C (i, j). end while Find a subset E C such that E is an exact cover. x ij 1 for all (i, j) E. 6

7 Compute the core cost allocation χ i = T Di tα t K i for all i N. Before proving the correctness of the above algorithm, we first show some properties of the solution α computed by the above algorithm. To simplify notation, in the following we denote D N t in Algorithm 3.1 as D t for t = 1,..., T, c N ij as c ij for 1 i j T, and K N as K. 127 Lemma (a) At any point in Algorithm 3.1, for some T 0 T, the variables α 1,..., α T0 variables α T0 +1,..., α T are inactive. are active, and the 130 (b) Let C be the set of tight constraints output by Algorithm 3.1. There exists an exact cover in C. 131 (c) Let α be a solution to [C N ] computed by Algorithm 3.1. Then αt αt+1 for all t = 1,..., T Proof. For all t = 1,..., T and any time in the algorithm τ, let α t (τ) be the value of α t at time τ. We show (a) by induction. The claim holds at the beginning of the algorithm, since α 1,..., α T are initialized to be active. Now consider the execution of Algorithm 3.1 at time τ, and suppose 135 that at any time before τ, the claim holds. Let α 1,..., α T1 be active and α T1 +1,..., α T be inactive 136 at time τ, where T 1 < T. The next time a constraint becomes tight is at time τ + τ, where τ = min i j, i T 1 { c ij } j D tα t (τ) j,α t is active at τ D. t 137 First, we claim that c it1 T 1 D tα t (τ) T1,α t is active at τ D t c ij j D tα t (τ) j,α t is active at τ D t for all 1 i j T 1. (3.1) 138 Note that for all 1 i j T 1, we have that T 1, α t is active at τ T 1 D t = D t and j D t =, α t is active at τ j D t. 139 In addition, for all t such that 1 t T 1, we have that α t (τ) = τ, since α t is active for all such t. 7

8 140 Therefore (3.1) is equivalent to c it1 k=i j T 1 D k c ij D k for all 1 i j T 1. (3.2) k=i 141 Since c i (d) is nondecreasing and concave in d, and c i (0) = 0, we have ( c T1 i D ) t T1 D t c ( j i D t) j D t for all 1 i j T 1, and thus (3.2) holds. Now let τ be some time before τ when the algorithm made a constraint tight. Let α T +1,..., α T2 be the newly inactive variables at time τ, for some T, T 2 such that T 1 T < T 2 T. Note that if T 2 < T, then (T 2 + 1, T 3 ) C, for some T 3 such that T 2 < T 3 T. We have that j, α t is active at τ T 1 D t = D t for j = T,..., T We claim that if then c it T D tα t (τ) T,α t is active at τ D t c i,j 1 j 1 D tα t (τ) j 1,α t is active at τ D t > c ik k D tα t (τ) k,α t is active at τ D t > c ij j D tα t (τ) j,α t is active at τ D t for some k = T + 1,..., T 2, for all j = k + 1,..., T 2, (3.3) 147 or equivalently, k if c ik c it < D t α t (τ) for some k = T + 1,..., T 2, t=t +1 then c ij c i,j 1 < D j α j (τ) for all j = k + 1,..., T Note that α t (τ) = τ for t = T + 1,..., T 2. The claim holds, since the fact that c i (d) is concave 149 in d implies that c ij c i,j 1 c ik c it D k j t=t +1 D. t 150 By (3.1) and (3.3), we conclude that τ is achieved when one of the following constraints 8

9 151 becomes tight: (i, T 1 ) for some i = 1,..., T 1, or (i, T 2 ) for some i = 1,..., T 1 and T 2 = T 1 + 1,..., T such that (3.4) T 2 = T or (T 2 + 1, T 3 ) C for some T 3 = T 2 + 1,..., T. 152 Therefore, at time τ + τ, the variables α T0 +1,..., α T1 become inactive for some T 0 < T 1. So, the induction step is complete and (a) holds. Note that applying (3.4) recursively implies (b). Finally, since the value of the α t s are raised together until they become inactive, (a) implies (c). 155 The following theorem shows that Algorithm 3.1 computes optimal solutions to [C N ] and [D N ]. 156 Theorem 3.3. Algorithm 3.1 computes optimal solutions to [C N ] and [D N ] in polynomial time Proof. Suppose the algorithm computes x, α, and E. By construction, α is a feasible solution to [D N ]. By Lemma 3.2, E is an exact cover, and so x is a feasible solution to [C N ]. We also have that i=1 j=i c ij x ij K = c ij K = (i,j) E ( j ) D t αt K = (i,j) E D t αt K Hence, by strong duality, α is an optimal solution to [D N ], and x is an optimal solution to [C N ]. At every iteration, the algorithm spends O(T 2 ) time to determine the next tight constraint and which α t s become inactive. The algorithm runs through at most T iterations. Finally, an exact cover can be found in O(T ) time. Therefore, the algorithm runs in O(T 3 ) time Note that these results provide an alternate, algorithmic proof of Theorem 2.2. Finally, we put all these above results together to show that Algorithm 3.1 computes a cost allocation in the core of an ELS game Theorem 3.4. Suppose (N, v) is an ELS game. Then, the cost allocation χ computed by Algo- rithm 3.1 is in the core of (N, v). Furthermore, Algorithm 3.1 runs in polynomial time. 168 Proof. By Theorem 3.3, the solution α output by Algorithm 3.1 is an optimal solution to [D N ]. 9

10 169 Therefore, by Theorem 2.2, i N χ i = i N ( T ) Dtα i t K i = i N Dt N αt K N = v(n). 170 Hence χ is budget-balanced. Moreover, since α t α t+1 for all t = 1,..., T 1, by Lemma 2.3, for 171 any S 2 N, we have j ( j Dt S αt c i D S t ) = c S ij for all i j. 172 So, α is a feasible solution to [D S ] for any S 2 N. It follows that i S χ i = i S ( T ) Dtα i t K i = i S Dt S αt K S v(s). 173 Therefore, χ is stable, and so χ is in the core of (N, v) One might wonder if Algorithm 3.1 can be applied directly to an ELS game without the zero holding cost transformation described in Lemma 2.1. In particular, Chen and Zhang (2006) considered the formulation [B S ] (without constant K S ) for the ELS game with backlogging. In their { formulation, c ij is defined as min i l j cl ( j D t) + l 1 t h t k=i D k + j j t=l+1 h+ t 1 k=t D k}, where h t and h + t are unit backlogging and inventory costs in period t respectively. The following example shows that Algorithm 3.1 cannot be applied directly in this case Example 3.5. Consider the following instance of an ELS game with backlogging, with N = {1, 2} and T = 3. The ordering cost is of the form c t (x t ) = f t δ(x t ) + p t x t, where δ(x t ) = 1 if x t > 0, and 0 otherwise. The associated data is: D 1 = (3, 2, 5), D 2 = (1, 4, 4), f = (6, 7, 10), p = (1, 2, 1), h + = (1, 3, 3), h = (5, 1, 1) It is straightforward to show that v({1, 2}) = 41, while Algorithm 3.1 produces a cost allocation ( 65 3, 284 ) 15. Clearly, = <

11 185 4 Extension to ELS games with remanufacturing Consider the following setting, which shares many similarities with an ELS game. We have a set of retailers who face demands for the same product, which they order from a single manufacturer. The retailers can either place an order for newly manufactured products, or send some of its returned used products to be remanufactured. We assume that demands can be satisfied by either newly manufactured products or remanufactured products in other words, serviceable products. The retailers maintain separate inventories for serviceable products and returned used products. We assume that the ordering and inventory costs are independent of the retailers. Every coalition of retailers incurs the minimum ordering and inventory cost required to satisfy their joint demands. We call this setting an economic lot-sizing game with remanufacturing, or ELSR game for short. We define an ELSR game (N, v) formally as follows. Let T be the number of discrete time periods. For t = 1,..., T, let 197 D i t be the demand for serviceable products faced by player i in period t, for all i N; 198 R i t be the amount of returned used product received by player i in period t, for all i N; 199 c m t (x) be the cost of ordering x units of newly manufactured product in period t; 200 c r t (z) be the cost of ordering z units of remanufactured product in period t; 201 h s t be the unit holding cost for serviceable products in period t; 202 h r t be the unit holding cost for returned products in period t We assume that c m t ( ) and c m t ( ) are both concave and nondecreasing, and that c m t (0) = c r t (0) = 0 for all t = 1,..., T. Let Dt S = i S Di t and Rt S = i S Ri t for t = 1,..., T. In addition, for any S 2 N, let D S = (Dt S ),...,T and R S = (Rt S ),...,T. The cost v(s) of coalition S 2 N is defined as the minimum total ordering and inventory cost with demands D S and returns R S. We can view v(s) as the optimal value of the following mathematical program: [E S ] min (c m t (x t ) + c r t (z t ) + h s tit s + h r t It r ) (4.1a) s.t. I s t = I s t 1 + x t + z t D S t for all t = 1,..., T, (4.1b) 11

12 I r t = I r t 1 + R S t z t for all t = 1,..., T, (4.1c) I s 0 = I r 0 = I s T = I r T = 0, (4.1d) x t, z t, I s t, I r t 0 for all t = 1,..., T, (4.1e) where x t and z t are the amount of newly manufactured and remanufactured products ordered in period t respectively, and It s and It r are the amount of serviceable and returned products in inventory at the end of period t respectively, for t = 1,..., T. Constraints (4.1b)-(4.1d) represent the balance of products in the serviceable and returns inventories respectively. Richter and Sombrutzki (2000) and Richter and Weber (2001) considered a special case of the ELSR problem in which the ordering costs consist of a fixed setup cost and a linear variable cost, the quantity of returns in the first period is large, and the unit holding cost of returned product is less than that of serviceable product. In particular, c m t (x) = f m t δ(x) + p t x for all t = 1,..., T, c r t (z) = ft r δ(z) + r t z for all t = 1,..., T, R1 S Dt S for all S 2 N, (4.2) h s t h r t for all t = 1,..., T Richter and Sombrutzki (2000) and Richter and Weber (2001) showed that in this case there is always an optimal solution that is a zero-inventory policy. A zero-inventory policy for an ELSR problem is a solution in which an order is placed only if the serviceable inventory goes to zero, and this order consists entirely of either newly manufactured or remanufactured products. To illustrate, in the decision variables of [E S ], a zero-inventory policy for ELSR satisfies x t z t = 0, I s t 1 (x t + z t ) = 0 for all t = 1,..., T. 221 In general, an ELSR game may have an empty core, as the following example shows. 222 Example 4.1. Consider the following ELSR game with N = {1, 2, 3} and T = 2. The ordering 223 cost functions are of the form c m t (x) = f m t δ(x) + p t x and c r t (z) = f r t δ(z) + r t z for t = 1,..., T. The 12

13 224 associated data is: D1 1 = 4, D2 1 = 5, D1 2 = 4, D2 2 = 4, D1 3 = 4, D2= 3 4, R1 1 = 8, R2 1 = 1, R1 2 = 7, R2 2 = 10, R1 3 = 9, R2 3 = 9, f1 m = 2, f2 m = 2, p 1 = 1, p 2 = 4, h s 1 = 1, h s 2 = 1, f1 r = 1, f2 r = 4, r 1 = 2, r 2 = 2, h r 1 = 2, h r 2 = The cost function of this game is v({1, 2, 3}) = 105, v({1, 2}) = 64, v({1, 3}) = 64, v({2, 3}) = 79, v({1}) = 27, v({2}) = 42, v({3}) = It is straightforward to show that this game has an empty core On the other hand, we can show that ELSR games under condition (4.2) have a non-empty core, and a cost allocation in the core can be computed by Algorithm 3.1. Using the same technique as the proof of Lemma 2.1, we can show the following lemma for ELSR games Lemma 4.2. Any instance of an ELSR game (N, v) as described above is equivalent to another instance of an ELSR game (N, v) with zero holding costs, where c m t (x) = c m t (x) + x h s k and c r t (x) = c r t (x) + x (h s k hr k ) k=t k=t are the cost of ordering x units of newly manufactured and remanufactured products in period t respectively, for each t = 1,..., T. In particular, for any S 2 N, v(s) = v(s) + K S, where K S = i S K i and K i = ( h r t t Rk i hs t k=1 t k=1 D i k ). (4.3) Note that for each t = 1,..., T, we have that c m t ( ) and c r t ( ) are nondecreasing and concave, and that c m t (0) = c r t (0) = 0. Under (4.2), there is always a sufficient number of returns to cover the demands of the entire time horizon. Therefore, in any zero-inventory policy, at each ordering point, a coalition of players 13

14 will place an order entirely of either newly manufactured or remanufactured products, depending on which type of product satisfies demand up to the next ordering point at a lower cost. We define a ij = min { c m i ( j D S t ) ( j, c r i D S t ) }. 240 Then, under (4.2), the following formulation gives v(s) for any S 2 N : [F S ] min s.t. a S ijx ij + K S i=1 j=i t Dt S i=1 j=t x ij = Dt S for all t = 1,..., T, x ij {0, 1} for all 1 i j T, where for all 1 i j T, x ij is a binary variable indicating ordering product in period i to meet demands in periods i, i + 1,..., j, and K S is defined in (4.3). Since c m (d) and c r (d) are both concave in d, min{ c m (d), c r (d)} is concave in d. As a result, Theorem 2.2, Lemma 2.3, Algorithm 3.1, Lemma 3.2 and Theorem 3.3 can be all applied to [F S ]. We extend Theorem 3.4 to the special case of ELSR games under (4.2): Theorem 4.3. Suppose (N, v) is an ELSR game such that (4.2) holds. Then, the cost allocation χ computed by Algorithm 3.1 (using [F S ] and the dual of its LP relaxation) is in the core of (N, v). 248 References O. N. Bondareva. Some application of linear programming methods to the theory of cooperative games. Problemi Kibernitiki, 10: , N. Brahimi, S. Dauzere-Peres, N. M. Najid, and A. Nordli. Single item lot sizing problems. European Journal of Operational Research, 168:1 16, X. Chen and J. Zhang. Duality approaches to economic lot sizing games. Working paper, A. Drexl and A. Kimms. Lot sizing and scheduling survey and extensions. European Journal of Operational Research, 99: ,

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