Research: Shelf space allocation

Shelf space allocation

The retailing sector in the UK is an extremely competitive arena. We only need to consider some high profile companies to show that this is the case. A particular example is provided by the recent decline of Marks and Spencer, who were the leading high street retailer (and in recent years they are starting to show an improvement in their profitability). A further example is given by C&A's decision to close all of its high street outlets. Yet another example is the decline of J Sainsburys from its position as the leading food retailer in the UK in the 1990's (in 1996, Tesco opened up a 2% lead over their rivals and continue to maintain an advantage). Asda, after merging with Wal-Mart, increased its market share dramatically and overtook Sainsbury's as the second biggest supermarket in the UK. In July 2003, Asda had gained 17% of market share, while Sainsbury's had slipped from 17.1% to 16.2%. Tesco retains the top spot with 27% of the overall market share. Finally, there was a battle over Safeways, which was recently up for sale.

This level of competitiveness is unlikely to decline. On the contrary, the high street (or more likely, out of town shopping centres) is likely to become even more competitive.

Shelf space is a scarce resource for a retailer. Shelf space allocation involves the distribution of appropriate amount of shelf space among different products, together with their locations, in a supermarket in such a way that the total profits and/or customer satisfaction are maximised. A shelf space allocation is also called a planogram, which is used to show exactly where and how many facings of each item should physically be placed onto the store shelves. Figure 1 gives an example of a planogram.

An example of a simple planogam

Due to the limited shelf space, planograms are one of the most important aspects that are used to improve financial performance. Electronic planograms can be also used for inventory control and vendor relation improvement. However, generating planograms is a challenging and time-consuming process because the simplest form of planogram problem (ignoring all marketing and retailing variables) is already a bounded problem, a well-known NP-hard problem which is very difficult to solve. The difficulty is further increased when we consider other merchandise, such as fresh food, clothing and frozen food. This is due to their special display requirements and the fact that they do not use standard shelf fitments. Currently, producing planograms is largely a manual process (there is software assistance available (e.g. Galaxxi) but it involves significant human interaction and does not provide any guidance or suggestions in deciding a good quality layout) and the shelf space allocation is mainly based on historical market share. However, this approach may lose substantial sales as the display space may have different sales influence with respect to different items. Using the same display space, different items may obtain different sales and hence affect the profits of the organisation.

Planograms are a subset of the wider domain of space planning which includes more well-known research areas such as bin packing and knapsack problems. Some of techniques that have already been successfully applied to problems within this wider domain may also be promising to shelf space allocation problems.

An extension of this shelf space allocation problem is fresh food inventory and space allocation which was specipically investigated in the project. More information can be found from here.

Related problems

1. Bounded knapsack problem

Given a set of n items and each item i is associated with a size si and a profit pi. Give a knapsack with capacity C. The bounded knapsack problem is to decide how many of each item to be placed into the knapsack in a way such that the total profits of the selected items are maximised while their total size does not exceed the knapsack capacity. The problem is usually formulated as follows:


Subject to:

In a special case, when bj =1, the problem becomes a 0-1 knapsack problem. If , the problem degenerated as an unbounded knapsack problem.

2. 0-1 multi-knapsack problem

This problem distinguishes 0-1 knapsack problem in that there are several knapsacks available. Given m knapsacks and n items. Each knapsack has a capacity ci and each item has a size si and profit pi. The 0-1 multi-knapsack problem is formulated as follows:


Subject to:

Model formulation

Due to the scarcity of space within stores, several researchers have concentrated on studying the relationship between the space allocated to an item and the sales of that item. Most have reached a common conclusion that a weak link exists between them and the significance depended on the type of items (Kotzan and Evanson, 1969; Cox, 1970; Curhan, 1972; Dreze et al., 1994; Desmet and Renaudin, 1998) . Figure 2 gives an example of the relationship between the number of the facings of an item and its demand. We can see that with the increase of the facings, the item's demand is increasing while the increase magnitude is decreasing.

Figure 2: The relationship between the displayed space and the demand

Several space allocation models have been proposed by market researchers. Most formulated the demand rate of an item as a function of the space allocated to the item, of which a classic model appears as a polynomial form proposed by Baker and Urban (Baker and Urban, 1988) :


where  is the demand rate of the product, x is the number of facings or the displayed inventory.  is a scale parameter and  is the space elasticity of the product. The advantageous characteristics of this model included the diminishing returns (the increase in the demand rate decreased as the space allocated to this shelf increased), inventory-level elasticity (the shape parameter represents the sensitivity of the demand rate to the changes of the shelf space), intrinsic linearity (the model can be easily transformed to a linear function by a logarithmic transformation and parameters can then be estimated by a simple linear regression) and its richness.


Suppose we have n items to be allocated on m shelves. Each item is defined by a five-tuple (li, pi, ,Li, Ui) where li (respectively, pi, ,Li, Ui) is the length (respectively, profit, space elasticity, lower bounds, upper bounds) of item i. The length of shelf j is denoted by . Assume that the total profit of item i is proportional to its unit profit and physical constraints in the other two dimensions (height and depth) are ignored. We have following space allocation model

Maximise                                                                  (2)

subject to:







The decision variables are , representing the number of facings of item i on shelf j and is the total number of facings of item i. Constraint (3) ensures that the length of a shelf is greater than the total length of the facings assigned to this shelf. Constraint (4) ensures that the lower and upper bounds of the number of facings for each item are satisfied. Constraint (5) ensures that the number of facings for each item is an integer. The objective is to maximise the overall profit without violating the given constraints. The model is a non-linear, multi-constraints optimisation problem. If , the model degenerates into a multi-knapsack problem.


Baker, R. C. and Urban, T. L., A Deterministic Inventory System with an Inventory-Level-Dependent Demand Rate. Journal of the Operational Research Society, 39(9): 823-831, 1988.

Cox, K., The Effect of Shelf Space Upon Sales of Branded Products. Journal of Marketing Research, 7: 55-58, 1970.

Curhan, R., The Relationship Between Space and Unit Sales in Supermarkets. Journal of Marketing Research, 9: 406-412, 1972.

Desmet, P. and Renaudin, V., Estimation of Product Category Sales Responsiveness to Allocated Shelf Space. International Journal of Research in Marketing, 15: 443-457, 1998.

Dreze, X., Hoch, S. J. and Purk, M. E., Shelf Management and Space Elasticity. Journal of Retailing, 70(4): 301-326, 1994.

Kotzan, J. and Evanson, R., Responsiveness of Drug Store Sales to Shelf Space Allocations. Journal of Marketing Research, 6: 465-469, 1969.