Improved Approximation Algorithms for the Joint Replenishment Problem with Outliers, and with Fairness Constraints

The joint replenishment problem (JRP) is a classical inventory management problem. We consider a natural generalization with outliers, where we are allowed to reject (that is, not service) a subset of demand points. In this paper, we are motivated by issues of fairness -if we do not serve all of the demands, we wish to "spread out the pain" in a balanced way among customers, communities, or any specified market segmentation. One approach is to constrain the rejections allowed, and to have separate bounds for each given customer. In our most general setting, we consider a set of C features, where each demand point has an associated rejection cost for each feature, and we have a given bound on the allowed rejection cost incurred in total for each feature. This generalizes a model of fairness introduced in earlier work on the Colorful k-Center problem in which (analogously) each demand point has a given color, and we bound the number of rejections of each color class. In the JRP, we seek to balance the cost incurred by a fixed ordering overhead with the cost of maintaining on-hand inventory over a longer period in advance of when it is needed. More precisely, there a given set of item types, for which there is specified demand over a finite, discrete-time horizon, and placing any order at a given time incurs a general ordering cost and item-specific ordering costs (independent of the total demand serviced); in addition, for each unit of demand held in inventory for an interval of time, there is a corresponding item-specific holding cost incurred; the aim is to minimize the total cost.

We give the first constant approximation algorithms for the fairness-constrained JRP with a constant number of features; specifically, we give a 2.86-approximation algorithm in this case. Even for the special case in which we bound the total (weighted) number of outliers, this performance guarantee improves upon bounds previously known for this case. Our approach is an LP-based algorithm that splits the instance into two subinstances. One is solved by a novel iterative rounding approach and the other by pipage-based rounding. The standard LP relaxation has an unbounded integrality gap, and hence another key element of our algorithm is to strengthen the relaxation by correctly guessing key attributes of the optimal solution, which are sufficiently concise, so that we can enumerate over all possible guesses in polynomial timealbeit exponential in C, the number of features.