Sensitivity analysis in inventory models by means of ergodicity coefficients

We investigate the sensitivity of a class of (s,S) inventory models with respect to the perturbation of the demand distribution in terms of ergodicity coefficients. Ergodicity coefficients can be regarded as matrix norms that are useful to study qualitative as well as quantitative properties of some classes of stochastic systems. We obtain estimations of the absolute deviation of the stationary vector of the underlying Markov chain subject to perturbations. The particular structure of the transition matrix of this class of models allows us to derive simple closed form formulae for the computation of the ergodicity coefficients as well as the perturbation bounds. The perturbation bounds obtained can help us to decide whether or not the model remains an acceptable representation of the real system and thus decide whether or not it can be trusted for real life applications. Under perturbation, e.g. estimation errors or approximations, an optimal solution for the mathematical model may not be optimal for the real system and if implemented, the real performance measures may deteriorate considerably and deviate from the targeted values. Numerical examples are given to illustrate how the perturbation of the demand distribution may have a considerable impact on the optimal inventory policy and the performance measures in some cases. In a practical setting, the understanding of the sensitivity results for inventory models can help us to identify the parameters that have to be estimated with most attention and how to build models that are robust and as close as possible to the real systems they represent.