A multi-agent inventory problem is a situation in which several agents face individual inventory problems and can coordinate their orders to reduce costs. This paper analyses a multi-agent inventory problem in which each agent faces a continuous-review inventory problem, with a deterministic linear demand, no holding costs and a limited capacity warehouse. In the case under study, shortages are allowed as follows. Goods are bought from an external supplier and then they are stored in each agent’s warehouse. These stored goods usually satisfy demand. However, each agent may alternative produce their own goods, which are less costly but of lower quality. When a shortage of the purchased goods occurs, demand is satisfied with the produced goods until a new order arrives. The problem under study herein arises in a farming community, and is a variation of a problem addressed by Fiestras-Janeiro et al. (2015). However, the existence of two acquisition costs makes it substantially different from Fiestras-Janeiro et al.’s problem and significantly complicates its analysis since the resulting cost functions may now be non-convex. This paper establishes the optimal inventory policies for our problem and obtains a stable order structure when agents allocate the joint costs using a proportional rule. In addition, it illustrates the performance of our model and results in an example.

Authors acknowledge the financial support of Ministerio de Economía y Competitividad through projects MTM2011-27731-C03-01, MTM2011-27731-C03-03, MTM2014-53395-C3-1-P, MTM2014-53395-C3-2-P and MTM2014-53395-C3-3-P, and of Xunta de Galicia through the ERDF (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2016-040, and Centro Singular de Investigación de Galicia ED431G/01).

Authors and Affiliations

1. Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

   A. Saavedra-Nieves

2. Departamento de Matemáticas, Universidade da Coruña, A Coruña, Spain

   I. García-Jurado

3. Departamento de Estatística e Investigación Operativa, Universidade de Vigo, Vigo, Spain

   M. G. Fiestras-Janeiro

Corresponding author

Correspondence toA. Saavedra-Nieves.

Appendix

In this appendix, we detail the steps to obtain an optimization problem equivalent to Problem P:

Using the explicit expression for the cost of any coalition given in (5) and denoting\(W=1/x\), we rewrite problem P above as

In what follows, the most efficient coalition will be determined by the set of index\(i=1,\dots n\), such that\(X_i+Y_i=1\). For every\(i=1,\dots ,n\), we denote by\(Y_i\) the binary variable that represents the membership (or not) ofi toI(S), and by\(X_i\) the other binary variable associated to the membership (or not) ofi to\(S{\setminus } I(S)\), respectively.

Thus, we can transform this problem into a separable problem using a well-known variable transformation (see, for instance, Williams2013, p. 28). If we define

then we can rewrite the problem as

Constraints four and five of Problem P2 can be linearized using the variables\(Z_i^X=X_iW,\)\( Z_i^Y=Y_iW\) and the results in Tawarmalani et al. (2002). In addition, it is necessary to include the following set of constraints associated to this transformation:

withM being a sufficiently large constant. Then, we obtain the following problem P3:

withM being a sufficiently large constant.

In this formulation, the last two addends of the objective function can be linearized using the variables\(R_i^X=X_iU,\)\( R_i^Y=Y_iU\) and, again, the results in Tawarmalani et al. (2002). Then we obtain a new formulation by introducing the following constraints:

withM being a sufficiently large constant. Then we obtain the following problem P4:

withM being a sufficiently large constant. Finally, for every\(i=1,\ldots ,n\), we transform the terms\(Z_i^YU\) into a difference of two new terms as follows:

and, thus,\(Z_i^YU=\alpha _i^2-\beta _i^2\). Then, the final formulation of our problem is

Thus, the minimization problem defined in Step 3 of Procedure4.1 can be expressed as a quadratic problem with some binary variables.

Reprints and permissions