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Impact of display stock has been extensively discussed in literature in terms of stock dependent demand rate. In some recent papers a three-component inventory level dependent demand (as displayed stock level (DSL)) has been proposed. It has been assumed that demand is independent of instantaneous inventory level if inventory is above a certain level in a shelf. As the demand is independent of instantaneous inventory level, a retailer is always in dilemma whether inventory additional to that level will be kept in the shelf or to rent a new storage. In this paper we focus on this by developing three mathematical models. We show that the problem in choosing rented storage or shelf to keep additional inventory can always be eliminated (a) without the knowledge of optimal stocking points or (b) with the knowledge of any one optimal stocking point. Also, if there is enough opportunity to have an extra storage and a mild limit on inventory holding cost in the shelf is satisfied, then there is no need to display inventory in the shelf above the certain level. The results are illustrated by numerical examples and sensitivity analysis is carried out.

Mass display of items in departmental stores and busy marketplaces has psychological impact on customers and it stimulates sales of some retail items. Motivated by this, researchers and practitioners have proposed several demand states to explore demand variability with respect to display stock level (DSL). Mainly, two types of demand patterns, e.g. initial display level dependent and instantaneous display level dependent, are observed in the literature. Instantaneous DSL dependent demand is further classified into two types, e.g. single-state and two-state. With two-state, demand is independent of DSL below a certain inventory level but there is no upper limit of DSL dependency of demand. This limitation is treated in the literature by imposing a restriction on capacity of shelf space. Some researchers in recent years have treated this drawback alternatively by proposing a three-state DSL dependent demand. The three-state DSL dependent demand is structured by adding a state, as upper limit of DSL dependency, to two-state DSL dependent demand. Inventory displayed above that limit in the shelf has no impact on customers. Although it's possible to achieve utmost benefit of display level dependency of demand as long as display level is above that limit, still a retailer is always in indecision whether inventory should be piled up in the shelf above that level or not. In this paper, we discuss three models. Two models are based on the three-component DSL dependent demand (TCD from now on), where inventory is replenished above the limit of display dependency. In the first model inventory additional to that limit is stored in rented warehouse, whereas in the second model it is displayed in the shelf. The third model is based on the two-component DSL dependent demand (TWCD from now on). Concavities of all the models are verified. The key contribution of the paper is in responding to the following queries. First, under what circumstance the retailer would pile up inventory above the limit of DSL dependency of demand? That is, whether a retailer would choose two-state or three-state DSL dependent demand. Second, if inventory is replenished above the level of DSL dependency then would the inventory extra to it be stored in a rented warehouse or would it be displayed in the shelf? The rest of the paper is organized as follows. In Section 2 inventory literature is reviewed for the DSL dependent demand and two warehouse inventory problems. Mathematical models are developed and are analyzed in Section 3. Section 4 deals with some numerical analysis of the proposed model. Finally Section 5 deals with summary and concluding remarks.