## Section: Application Domains

### Inventory control

In addition of inventory control combined with routing problems, as mentioned above, we are also interested in multi-echelon inventory control. We focus in particular on the simpler case of a network consisting of one warehouse supplying many retailers. In this setting, we want to understand, using game theory, what is the benefit or impact of coordination in such a network. We start with the simpler case where the retailers are facing continuous demands with constant rates. In this case, Roundy [144] could give a very good approximation to the global optimal (within 2% of optimality) and we are currently exploiting this result to show that coordination has a pretty high saving potential. Chen, Federgruen and Zhang [63] have proposed a sophisticated coordination mechanism that ensure global optimality in this context. Nevertheless, this coordination procedure seems to be unpractical and thus we are also studying simple coordination mechanisms that would yield better overall solutions. We also study the new dual-balancing technique proposed by Levi et al. [115] that has proven to be a powerful technique for approximation results in inventory control. We are implementing this technique for the simple case of one warehouse facing correlated and non stationary stochastic demand to compare theoretical guarantee and practical ones. We want to adapt those ideas to provide the first approximation algorithm for the one-warehouse multi-retailer problem under periodic review and holding and penalty costs. This work is a collaboration with IBM Research Zurich Lab (Dr. Richard Boedi, Dr. Eleni Pratsini) and University of Grenoble (Prof. Jean-Philippe Gayon, Prof. Christophe Rapine).