A primal–dual algorithm for computing a cost allocation in the core of economic lot-sizing games

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.

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

• $N=\{1,\dots ,n\}$ be the set of retailers, or players;

• $T$ be the number of discrete time periods;

• $D_t^i$ be the demand faced by player $i$ in period $t$, for all $i\in N$ and $t=1,\dots ,T$;

• $c_t\left(x\right)$ be the cost of ordering $x$ units in period $t$, for all $t=1,\dots ,T$;

• $h_t$ be the unit holding cost in period $t$, for $t=1,\dots ,T$.

We assume that $c_t(\cdot )$ is concave and nondecreasing, and that $c_t\left(0\right)=0$ for all $t=1,\dots ,T$. In addition, let $D_t^S={\sum }_{i\in S}{D}_t^i$ for $t=1,\dots ,T$. For any $S\in 2^N$, let $D^S={\left(D_t^S\right)}_{t=1,\dots ,T}$ be the vector of demands for all time periods. The cost $v\left(S\right)$ of coalition $S\in 2^N$ is defined as the minimum total ordering and inventory cost to satisfy demands $D^S$. In other words, $v\left(S\right)$ is the optimal value of the following mathematical program:

$$\left[A^S\right]\underset{x,I}{min}\sum _{t=1}^T(c_t\left(x_t\right)+h_t{I}_t)$$$$\mathrm{s.t.}{x}_t+I_{t-1}=D_t^S+I_t\text{for all }t=1,\dots ,T\text{,}$$$$I_0=I_T=0\text{,}$$$$x_t\ge 0\text{,}{I}_t\ge 0\text{for all }t=1,\dots ,T\text{,}$$

where $x_t$ is the amount ordered in period $t$, and $I_t$ is the amount of inventory at the end of period $t$, for all $t=1,\dots ,T$. 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 $\chi \in {\mathbb{R}}^N$ is a cost allocation: for each player $i\in N$, ${\chi }_i$ is the cost allocated to player $i\in N$. The core [5] of a cooperative game $(N,v)$ is the set of all cost allocations $\chi$ that satisfy the following constraints:

$$\sum _{i\in N}{\chi }_i=v\left(N\right)\text{,}$$$$\sum _{i\in S}{\chi }_i\le v\left(S\right)\text{for all }S\in 2^N\text{.}$$

The constraint (1.2a) ensures that the cost allocation $\chi$ is budget-balanced; that is, the sum of the costs allocated to all of the players equals the joint cost that they incur together. The constraints (1.2b) ensure that the cost allocation $\chi$ 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.

The economic lot-sizing problem was first studied by Wagner and Whitin [13], 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. [2], [4]) for details on its history.

van den Heuvel et al. [12] 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 [1], [11]. They also showed that some special cases of these games are concave. Chen and Zhang [3] 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 [14] presented a so-called cross-monotonic cost-sharing method for ELS games that is approximately budget balanced. Guardiola et al. [6], [7] studied production-inventory games, which model a collaborative production and inventory setting similar in spirit to the ELS games of [3], [12]. 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.

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, 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 [3], which requires a linear programming subroutine. The algorithm in [3], 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 [9], [10].

It was recently brought to our attention that the combinatorial primal–dual algorithm of [8] for the economic lot-sizing problem can also be used to compute a cost allocation in the core of ELS games. Their algorithm is based on a mathematical program different from the one used in this work.