The worst-case performance of the Cube per Order Index slotting strategy is infinitely bad - A technical note

A well-known and frequently applied policy to assign stock keeping units (SKUs) to (dedicated) storage locations in a warehouse is the Cube per Order Index (COI) slotting strategy. Basically, COI stores an SKU based on how frequently it is picked per unit of stock space required. Fast movers are located close to the Input-Output points. For single command order picking, COI slotting is well-known to minimize order picking travel time. For multi command this is no longer true. An interesting question is: how bad can it be? In this paper we show that there is no limit to this badness. Worst-case behavior of COI is infinitely bad. We construct a worst-case example that proves the following. Given an arbitrary positive integer p, there is (i) a warehouse configuration (ii) a set of SKUs (iii) a set of orders for these SKUs such that slotting these SKUs in the warehouse according to COI leads to an order picking travel time which is p times larger than the order picking travel time produced by an optimal slotting strategy.