The effect of consolidated periods in heterogeneous lot-sizing games

We consider a cooperative game defined by an economic lot-sizing problem with heterogeneous costs over a finite time horizon, in which each firm faces demand for a single product in each period and coalitions can pool orders. The model of cooperation works as follows: ordering channels and holding and backlogging technologies are shared among the members of the coalitions. This implies that each firm uses the best ordering channel and holding technology provided by the participants in the consortium. That is, they produce, hold inventory, pay backlogged demand and make orders at the minimum cost of the coalition members. Thus, firms aim at satisfying their demand over the planing horizon with minimal operation cost. Our contribution is to show that there exist fair allocations of the overall operation cost among the firms so that no group of agents profit from leaving the consortium. Then we propose a parametric family of cost allocations and provide sufficient conditions for this to be a stable family against coalitional defections of firms. Finally, we focus on those periods of the time horizon that are consolidated and we analyze their effect on the stability of cost allocations.


Introduction
Lot-sizing is one of the most important and also one of the most difficult problems in production planning.Although lot sizing and scheduling problems, and their variants, have been studied by many authors, providing different solution approaches, looking for more efficient solution approaches is still a challenging subject.Variants of the lot-sizing problem (henceforth, LSP) with complex setup and other variants which are more realistic and practical have received less attention in the literature.There has been little literature regarding problems such as LSP with backlogging or with setup times and setup carry-over.Since these problems are NP-hard, fast and efficient heuristics are required.A good survey on the subject is Karimi et al (2003).
Also there is little literature for cost-sharing in LSP.Among the pioneers we mention Van Den Heuvel et al. (2007) which focuses on the cooperation in economic lot-sizing situations (henceforth ELS-situations) with homogeneous costs.They consider a homogeneous finite horizon model where cost are all equal for all the players in each period.Each player must satisfy its demand in each period by producing or carrying inventory from previous stages but backlogging is not permitted.The main result in that paper is that the cooperative games induced by ELS-situations enjoy a nonempty core.Subsequently, Guardiola et al. (2008Guardiola et al. ( , 2009) ) present a new class of totally balanced combinatorial optimization games: production-inventory games (henceforth, PI-games).PI-games bring the essentials of inventory and combinatorial optimization games.They provide a cooperative approach to analyze the production and storage of indivisible items being their characteristic function given as the optimal objective function of a combinatorial optimization problem.PI-games can be seen as ELS games without setup costs but with backlogging.Guardiola et al. (2008Guardiola et al. ( , 2009) ) prove that the Owen set, the set of allocations which are achievable through dual solutions [see Owen, 1975 andGellekom et al., 2000] reduces to a singleton.In addition, that allocation is always in the core and it defines a population monotonic allocation scheme.This fact motivates the name of Owen point for this core-allocation on PI-games.The main difference between the ELS-games by Van Den Heuvel et al. (2007) and PI-games by Guardiola et al. (2009) is that the former considers set up costs but assume that costs are the homogeneous for all players in each period.Hence, both situations (ELS and PI) are, in general, different.
On the other hand, Xu and Yang (2009) present a cost-sharing method that is competitive, cross-monotonic and approximate cost recovering for an ELS-game under a weak triangle inequality assumption, along with numerical results showing the effectiveness of the proposed method.Li et al. (2014) present a cost-sharing method that is cross-monotonic, competitive, and approximate cost recovery, for the multi-level ELS-game, under a mild condition.This result extends that of the recent 1-level ELS-game of Xu and Yang (2009).
Dreschel (2010) focusses on cooperative lot-sizing games in supply chains.Several models of cooperation in lot-sizing problems of different complexity that are analyzed regarding theoretical properties like monotonicity or concavity and solved with the proposed row generation algorithm to compute core elements; i.e. determining stable and fair cost allocations.
In another paper, Gopaladesikan and Uhan (2011) consider a new class of cooperative cost-sharing games, associated with the ELS-problem with remanufacturing options (henceforth, ELSR).They investigate the relative strength and integrality gaps of various mathematical optimization formulations of ELSR.Using insights from these results, they study the core of the associated cost-sharing game and show it is empty, in general.However, for two special cases-zero setup costs, and large initial quantity of low cost returns-they find that the cost sharing game has a non-empty core, and that a cost allocation in the core can be computed in polynomial time.Zeng et al. (2011) consider the ELS-game with perishable inventory.In this cooperative game, a number of retailers that have a known demand through a fixed number of periods for a same kind of perishable goods collaborate to place joint orders to a single supplier.They first show that an ELS game with perishable inventory is subadditive, totally balanced and its core is non-empty.Then, they propose a core-allocation which allocates the unit cost to each period as equally as possible.Finally, a numerical example is given to illustrate the above results.Tsao et al. (2013) use the Nash game and the cooperation game in an imperfect production system to investigate the combined effects of lotsizing integration, learning effect, and an imperfect production process on a manufacturer-retailer channel.They also developpe a search procedure to solve the problem described, the optimal properties and a numerical study are conducted to seek for structural and quantitative insights into the relationship between the upstream and downstream entities of the supply chain.Numerical results indicated that the cooperation game policy created a higher cost reduction under a wide range of parameter settings.
Finally, Chen and Zhang (2015) consider the ELS-game with general concave ordering cost.In that paper, the dual variables are understood as the price of the demand per unit.They show that a core allocation can be computed in polynomial time under the assumption that all retailers have the same cost parameters (again homogeneous costs).Their approach is based on linear programming (LP) duality.Specifically, they prove that there exists an optimal dual solution that defines an allocation in the core and point out that it is not necessarily true that every optimal dual solution defines a core allocation.Toriello and Uhan (2014) also study ELS-games with general concave ordering costs and show how to compute a dynamic cost allocation in the strong sequential core of these games, i.e. an allocation over time that exactly distributes costs and is stable against coalitional defections at every period of the time horizon.
In this paper we study another class of totally balanced combinatorial optimization games called setup-inventory games (henceforth, SI-games) that arises from cooperation in lot-sizing problems with heterogeneous costs.Each firm faces demand for a single product in each period and coalitions can pool orders.Firms cooperate which implies that each firm uses the best ordering channel and holding technology provided by the participants in the consortium.That is, they purchase, hold inventory, pay backlogged demand and make orders at the minimum cost among the members of the coalition.Thus, firms aim at satisfying their demand over the planing horizon with minimal operation cost.
Therefore, each firm must solve the Wagner and Whitin extended model with backlogging costs, solved by Zangwill (1969) using dynamic programming techniques.Modeling cooperation in purchasing and holding costs has already appeared in literature.Additionally, our cooperation in backlogging is also natural, but new: all the members of a coalition pay backlogging cost (compensation for delayed demand) by the cheapest cost among those in the coalition.The larger the coalition the stronger so that it can "squeeze" a bit more their customers.SI-games are an extension of PI-games (Guardiola et al., 2009) since the latter do not include setup costs.The reader may note that whenever set up costs are zero in all periods SI-games become PI-games.Moreover, SI-games also generalize ELS-games in that all considered costs can be different for the different players in each period and, in addition, backorders are permitted.
The contribution of this paper is to prove that the above mentioned mode of cooperation is always stable in that there exist fair allocations of the overall cost of the system among the members of a coalition so that no subgroup of agents is better off by leaving the consortium (every SIgame has a nonempty core).Then we propose a parametric family of cost allocations for SI-games: the extended Owen points.We provide sufficient conditions for this to be a stable family against coalitional defections of firms, that is, every extended Owen point is a core-allocation.Finally, we focus on those periods of the time horizon that are consolidated and analyze their effect on the stability of cost allocations.Specifically we prove that for consolidated SI-situatuins, the single extended Owen point belongs to the core of the game.Our paper contributes as well to the emerging literature on the analysis of problems in Operations Research by means of cooperative games.The interested reader is referred to Borm et al. (2001) for further details on operations research games (including inventory games).

Model
We start by describing the basic form of lot sizing problems (see Johnson and Montgomery (1974) for further details).We focus here on periodic review inventory problems where a setup cost is incurred when placing an order, which in turns makes the cost structure non-linear.
A setup-inventory problem (hereafter SI-problem) can be described as follows.We consider T periods, numbered from 1 to T , where the demand for a single product occurs in each of them.This demand can be satisfied by purchasing or by own production, and can be done in three different periods: (i) the same period, (ii) an earlier period (as inventory), (iii) a later period (as backlogging).Every time an order is placed in a certain period, a fixed cost must be paid.Therefore, the model includes purchasing, inventory holding, backlogging and setup costs.We assume, without loss of generality, the initial and terminal inventories are set to zero.The objective is to find an optimal ordering plan, that is a feasible ordering plan that minimizes the sum of setup, purchasing, inventory holding and backlogging cost.
• h t = unit inventory carrying costs in period t and h = (h 1 , . . ., h T ).
• p t = unit purchasing costs in period t and p = (p 1 , . . ., p T ).
• q t = order size in period t.
• I t = ending inventory in period t.
• E t = ending backlogged demand in period t.
We consider that costs and demand can never be negative.Furthermore, the decision variables q t , I t and E t take integer values.The single-item formulation of the problem (T, d, k, h, b, p) is as follows: where, We define a feasible ordering plan for a SI-problem as σ ∈ R T where σ t ∈ T ∪ {0} denotes the period where demand of period t is ordered.We assume the convention that σ t = 0 if and only if d t = 0.It means that there is no order placed to satisfy a null demand at period t.Moreover, we define P (σ) ∈ R T as the cost vector associated to ordering plan σ (henceforth: cost-plan vector), where If σ * is an optimal ordering plan for a SI-problem, then the optimal cost is given by Notice that if all setup costs are zero, the problem we are dealing with is a PI-problem (see Guardiola et al. 2009).

Cooperation in lot-sizing with heterogeneous costs
Next we address a variant of this model where several firms, facing each one a SI-problem, coordinate their actions to reduce costs.This coordination is driven by sharing ordering channels, backlogged and inventory carrying technologies.This means that cooperating firms make a joint order and pay backlogged and inventory carrying demand at the cheapest costs among the members of the coalition at each period.Formally, a setup-inventory situation (henceforth, SI-situation) is a tuple (N, D, K, H, B, P ) where: • N = {1, . . ., n} is the set of players.
′ is an integer demand matrix, where each row corresponds to the demand of a player, that is, ′ is an inventory carrying costs matrix, where • P = [p 1 , . . ., p n ] ′ is a purchasing costs matrix, where In order to simplify the notation we define Z as a matrix in which all costs are included, that is, Z := (K, H, B, P ).A cost TU-game is a pair (N, c), where N is the finite player set and c : P(N ) → R the characteristic function satisfying c(∅) = 0.The subgame related to coalition S, c S , is the restriction of the mapping c to the subcoalitions of S. A cost allocation will be x ∈ R n and, for every coalition S ⊆ N we shall write x(S) := i∈S x i the cost-sharing to coalition S (where x(∅) = 0).
For each SI-situation (N, D, Z) we associate a cost TU-game (N, c) where, for any nonempty coalition S ⊆ N, c(S) is the optimal value of the optimization problem SI(S), defined as: . ., T, q t ≥ 0, I t ≥ 0, E t ≥ 0, and integers, t = 1, . . ., T ; with Notice that for all nonempty S ⊆ N the characteristic function c can be rewritten as follows: where σ S , P S t (σ S ) ∈ R T are the optimal ordering plan and the cost-plan vector associated to SI(S), respectively.Every cost TU-game defined in this way is what we call a setup-inventory game (SI-game).
The reader may notice that every PI-game (as introduced by Guardiola et al., 2009) is a SI-game with K = 0. Hence the class of PI-games is a particular subclass of the SI-games.
First, we wonder whether the above model of cooperation is stable, i.e. whether there is a fair division of the total cost among the players such that no group of them has incentives to leave.As we had already announced the concept of core provides a direct answer to that question.The core of the game (N, c) consists of those cost allocations which divide the cost of the grand coalition in such a way that any other coalition pays at most its cost by the characteristic function.Formally, Core(N, c) = {x ∈ R n /x(N ) = c(N ) and x(S) ≤ c(S) for all S ⊂ N }.
In the following, fair allocations of the total cost will be called coreallocations.Bondareva (1963) and Shapley (1967) independently provide a general characterization of games with a non-empty core by means of balanceness.A collection of coalitions in N , B ⊆ P(N ) is a balanced collection if there exist nonnegative weights {λ S } S∈B such that S∈B:i∈S λ S = 1 for all i ∈ N .Those weight {λ S } S∈B are called balancing coefficient.A cost game (N, c) is balanced if for every balanced collection B with balancing coefficients {λ S } S∈B it holds that Then, Bondareva and Shapley prove that (N, c) has a nonempty core if and only if it is balanced.In addition, it is totally balanced game if the core of every subgame is nonempty.Totally balanced games were introduced by Shapley and Shubik in the study of market games (see Shapley and Shubik, 1969).
It is important to remark that even though the problems that define SI-games have totally unimodular constraint matrices, dual solutions do not induce core solution throughout the Owen construction.The reason is because the objective function is not linear.Therefore, the question whether the core of the corresponding game is empty or not is a challenging query and its study makes sense.
The main result of this section states that the cooperation in lot sizing problems with heterogeneous costs is always stable.In other words, SIgames are balanced.In what follows we include a technical lemma that helps in proving the following theorem.Actually, it provides a procedure for constructing new ordering plans out of existing ones.The rationale behind this construction is similar to the one used in Van den Heuvel et al. (2007).
Lemma 3.1 Let B be a balanced collection of coalitions with balancing coefficients {λ S } S∈B .Assume that for each coalition S ∈ B, π S is the optimal order plan for the problem (T, d S , k N , h N , b N , p N ).Let r be the smallest positive integer such that l S := rλ * S ∈ Z + for all S ∈ B where {λ * S } S∈B are rational numbers greater than or equal to {λ S } S∈B .Then, there exist {ψ j } j=1...,r feasible order plans for the problem (T, d N , k N , h N , b N , p N ) satisfying (i), (ii) and (iii): (i) For each t, there exists S(j) ∈ B such that (ii) S(j) is used at most rλ * S(j) times ∀j = 1, . . ., r.
Proof.The proof of this lemma is constructive so that at the end we will have a procedure to actually construct the corresponding order policies {ψ j } j=1...,r .
Let B be a balanced collection and {λ S } S∈B their corresponding balancing coefficients.Take λ * S as a rational number greater than or equal to λ S for all S ∈ B. There exists r ∈ N being the smallest positive integer such that l S = rλ * S ∈ Z + for all S ∈ B. Notice that this number r satisfies since S∈B λ * S ≥ S∈B λ S ≥ 1.Consider the following artificial set of coalitions, namely BA.For each coalition S ∈ B, consider l S = rλ * S replicas of S in the set BA. Therefore, we have in total m coalitions in the new collection (see (1)).We assume an arbitrary ordering of these coalitions so that we can refer, without confusion, to any coalition, say S k , by its index in such a sequence.Moreover, let π S k be the optimal order plan corresponding to the problem (T, Then we proceed to construct the feasible order plans {ψ j }.For each period t, 1 ≤ t ≤ T set: ii) If d N t > 0 do the following.Define Notice that N * t = ∅ and moreover {i} ⊂ N * t since there exists always at least one agent i ∈ N such that d i t > 0, otherwise d N t = 0. Therefore, Finally, we set ψ j t = π S (j) t for j = 1, . . ., r.Notice that since |C t | ≥ r this definition is well-stated.
The above construction satisfies the thesis of the lemma.
The following example illustrates how to implement the aforementioned procedure.
Example 3.2 Consider the following SI-situation with three players and four periods.Notice that all players have the same costs but different demands.
Next table shows the optimal ordering plans and the corresponding cost plan vectors for the problem (T, d S , k N , h N , b N , p N ) for each coalition S ∈ B.

S
π The reader may notice that there always exist 3 feasible order plans for the problem (T, These plans can be built by means of the following recursive procedure: for all t = 1, 2, 3, 4 such that S(1) = arg min P N t (π S ) , such that S(2) = arg min P N t (π S ) |S = S(1) , , In addition, for all S = S(1), S(2), S(3), P N t (π S ) ≥ P N t (π S(j) ) = P N t (ψ j ), for all t = 1, 2, 3, 4 and all j = 1, 2, 3. Proof.Let B ⊂ 2 N be a balanced collection and {λ S } S∈B their corresponding balancing coefficients.Then, where π S for each S ∈ B is an optimal order plan for the SI-problem (T, d S , k N , h N , b N , p N ).
Our goal is to prove that According to the construction in Lemma 3.1, take λ ε S S as a rational number greater than or equal to λ S such that There exists r ∈ N being the smallest positive integer for which rλ ε S S is integral for all S ∈ B. Then (4) can be rewritten as (6) Using the set BA defined in page 11, the first term in the left-hand side of ( 6) can be rewritten as Now, the expression in ( 8) is stated as: From the above formula and using the definition in (2) we get: Hence, since n i=1 d i t = d N t , the right-hand side of (10) equals the following: Thus, finally from ( 7) and ( 11) we get the following inequality: Hence, taking limit in (12) when λ ε S S → λ S for all S ∈ B we obtain: what concludes the proof.
We note in passing that every subgame of a SI-game is a new SI-game.Thus, Theorem 3.3 implies that every SI-game is totally balanced.
There exists an alternative proof of the balancedness of this class of games.Here we outline this proof for the sake of completeness.
From the central part of the proof of Theorem 3.3 we deduce that the balanced character is ensured for any balanced collection with rational balancing coefficients.Notice that balanced coefficients must be optimal solutions to the following linear problem (for any suitable choice of coefficients): Therefore, since the feasible region of the above problem has all its extreme points being rational numbers we deduce that balancedness holds for those choices.In addition, any non rational family of balanced coefficients must be a convex combination of extreme points in this polyhedron.Hence, we can apply the following construction.
Let B be a balanced collection with non-rational balancing coefficients {λ B S } S∈B .There exist B 1 , . . ., B k balanced collections with rational balancing coefficients {λ In fact, from the above argument we deduce something more general: proving balancedness for collections with rational balancing coefficients suffices.

Extended Owen points
We have just proven the stability of the grand coalition, in the sense of the core.We know that there always exists a core-allocation for SI-games but we do not know how to construct it.We propose to find suitable cost-allocations for SI-games which are easy to calculate and satisfy good properties.
The Owen point, introduced in Guardiola et al. (2009), is a coreallocation for PI-games which represents the cost that each player has to pay when producing at the minimum operational cost (see also Guardiola et al., 2008).If we consider a SI-situation (N, D, Z) with K = 0, that is a PI-situation, the Owen point, o = (o i ) i∈N , is given by In this section we introduce a parametric family of cost allocations with the flavor of the Owen point but appropriate to SI-games.We call it the family of extended Owen points.The interested reader is referred to Perea et al. (2009Perea et al. ( , 2012) ) for alternative extensions of the concept of Owen point.Before defining this new family of cost allocations, we need to introduce some previous concepts.
Let (N, D, Z) be a SI-situation and (N, c) the associated SI-game.We define the reduced SI-situation associated to (N, D, Z) as a SI-situation (N, D, Z) with Z = ( K, H, B, P ) where Note that reduced SI-situations are the simplest SI-situations in that all their costs are the same for all players in all periods.
We denote by (N, c) the cost game associated to the reduced SI-situation (N, D, Z).Notice that c(S) ≤ c(S) for all S ⊂ N and c(N ) = c(N ).Hence Core(N, c) ⊆ Core(N, c).Clearly, each ELS-situation corresponds with a reduced SI-situation for an appropriate choice of parameters since the costs involved in each period are the same for all the players (see Van Den Heuvel et al., 2007).Hence, ELS-situations are particular cases of SI-situations.
Next we define the following sets: • Set of ordering periods: T S := t ∈ T δ t (σ S ) = 1 for all S ⊆ N .
It is easy to check that, • Set of consolidated periods: Υ := {t ∈ T |∃i ∈ N such that δ t (σ S ) = 1 for all S ⊆ N with i ∈ S}.A period is consolidated if there exists at least one player such that he forces placing an order at this period to any coalition that he belongs to.
We can distinguish two classes of costs for every coalition S ⊆ N. Variable costs P S (σ S ) ′ d S , which depends on demands, and non-consolidated fixed costs t∈T S \Υ k S t .Next we define for each S ⊆ N N (S) : Notice that N (S) and M (S) represent the difference between the ordering plans σ S and σ N related to variable and non-consolidated fixed costs.
We are ready now to define the family of extender Owen points. where Notice that the above family of cost allocation is a parametric family depending on α ∈ R N + such that α(N ) > 0. Next proposition shows that, if the optimal ordering plan for the grand coalition reduces variable and non-consolidated fixed costs with respect to any coalition S ⊆ N , then the family of extended Owen points is a coreallocation family.+ such that α(N ) > 0, the allocation ω(α) = (ω 1 (α), . . ., ω n (α)) defined in ( 13) is a core-allocation.
Proof.Let (N, D, Z) be a SI-situation and (N, c) the corresponding SI-game.Then, It is easy to check that ω is efficient.Hence, ω ∈ Core(N, c).
The above result is illustrated in the next example.The corresponding SI-game is given by: Next table shows the optimal ordering plans, the corresponding cost-plan vectors, and the differences between the ordering plans: Since N (S), M (S) ≤ 0 for all S ⊆ N, we can conclude that (19, 13, 7) + 4 α(N ) (α 1 , α 2 , α 3 ) We note that ω(α) is not a game-theoretical solution since its definition only applies on SI-situations.A weaker sufficient condition to ensure the cost allocation ω(α) to be in the core is given by the following corollary.

Demand P urchasing Inventory Backlogging
The corresponding SI-game is shown in the next table: The optimal ordering plans, the corresponding cost-plan vectors, and the differences between the ordering plans can be found in the last table: It can be easily checked that N (S) + α(S) α(N ) M (S) ≤ 0 for all α ∈ R N + such that α(N ) > 0 and all S ⊆ N. Hence We finish this section with a simpler sufficient condition to check core membership.where N (S), M (S) are the corresponding values for the reduced SI-situation for all S ⊆ N, then ω(α) is a core-allocation.

Consolidated situations and stability
We focus now on those SI-situations that are consolidated.Then we analyze their effect on the stability of the extended Owen points.
A consolidated SI-situation is described by means of a property of the ordering periods: whenever a player places an order in a period every coalition that contains that player places an order in the same period as well.This idea of consolidation is a refinement of the original scheme since it makes coalitions to perform as any of its individual members.The concept is formalized in the following definition.From the above definition, it is clear that in any consolidated SI-situation, M (S) = 0 for any coalition S ⊆ N , since t∈T S \Υ k S t = 0. Based in this fact, we can provide an extended Owen point in the core for consolidated SI-games.The following technical lemma is needed to prove this result.Lemma 5.2 Let (N, D, Z) be a consolidated SI-situation and (N, c) the corresponding SI-game.Then P S t (σ S ) ≥ P R t (σ R ) for all t ∈ T with d S t = 0 and for all S ⊆ R ⊆ N.
Proof.Suppose that ∃t ′ ∈ T with d S t ′ = 0 such that P S t ′ (σ S ) < P R t ′ (σ R ) then σ S t ′ = r and σ R t ′ = r ′ with r = r ′ .δ r (σ S ) = 1 therefore δ r (σ R ) = 1 since (N, D, Z) is a consolidated SI-situation.If we take the next feasible plan, and this is a contradiction because σ R is an optimal ordering plan of coalition R ⊆ N .Note that the above lemma exhibits a monotonicity property with respect to the ordering policies.We mean that, the smaller the coalition, the greater the cost of satisfying demand in each single period.Guardiola et al. 2008Guardiola et al. , 2009) and a subclass of games that enjoys a population monotonic allocation scheme.
The stability property of the above mentioned mode of coordination leads us to mention two related future research lines: (1) analyzing the cooperation aspects of broader subclasses of inventory situations for which it is possible to provide explicit solutions; and (2) studying the relationships between the cores that arise from situations with and without set up costs.

Theorem 3 . 3
Every SI-game (N, c) associated to a SI-situation (N, D, Z) is balanced.

Proposition 4 . 1
Let (N, D, Z) a SI-situation and (N, c) the corresponding SI-game.If N (S), M (S) ≤ 0 for all S ⊆ N, then for each α ∈ R N

Example 4 . 2
Consider the following SI-situation with three periods and three players:

Corollary 4 . 3 Example 4 . 4
Let (N, D, Z) a SI-situation and (N, c) the corresponding SI-game.If there exists α ∈ R N + such that α(N ) > 0 and N (S)+ α(S) α(N ) M (S) ≤ 0 for all S ⊆ N, then ω(α) is a core-allocation.The example below illustrates the above condition.Consider the SI-situation described by the following table.

Definition 5 . 1 A
SI-situation (N, D, Z) is consolidated if T S ⊆ Υ for all S ⊆ N .