The simple integer recourse (SIR) function of a decision variable is the expectation of the integer round-up of the shortage/surplus between a random variable with a known distribution and the decision variable. It is the integer analogue of the simple (continuous) recourse function in two stage stochastic linear programming. Structural properties and approximations of SIR functions have been extensively studied in the seminal works of van der Vlerk and coauthors. We study a distributionally robust SIR function (DR-SIR) that considers the worst-case expectation over a given family of distributions. Under the assumption that the distribution family is specified by its mean and support, we derive a closed form analytical expression for the DR-SIR function. We also show that this nonconvex DR-SIR function can be represented using a mixed integer second-order conic program.

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