## 3 A Formal Description of the Multi-Period Product Display Problem  
Consider a warehouse holding a set of products or items. It has to supply a retail store for different time periods, which define the planning horizon. Each item belongs to a certain collection (e.g., shirts or jeans in the case of clothes), has a selling price--which might vary with time--and a marginal profit--which is typically given as a percentage of the selling price. Depending on its selling price, an item is classified as 'expensive' or not. In any given period the retail store contains a set of tables, each of them displaying a subset of non-repeated items. Each item has an initial attractiveness value, estimated from experts' opinions and/or historical observations in an omnichannel environment--such as the number of times it has been selected and analyzed in the past, the feedback provided by the customers, etc. The attractiveness value can also depend upon other items currently being displayed in the table, since relations (or dependencies) between pairs of products may need to be considered. Among all the available products in the warehouse, a subset of different items should be selected to be displayed on the retail display tables. The dependency between each pair of items is registered in a dependency matrix. An inter-period dependency is also considered. The attractiveness value of each item is reduced by a known quantity (typically expressed as a percentage) every time the product is repeatedly shown in two (or more) consecutive periods. In other words, if an item is repeatedly exposed during several consecutive periods of time, its novelty disappears and, as a consequence, its attractiveness value is reduced. On the other hand, whenever an item has not been shown in the previous period, its attractiveness value is increased due to the novelty effect. The goal is then to solve a multi-period product display problem with dynamic attractiveness levels. In this problem, a subset of items has to be selected to be displayed at each table-period combination in order to maximize the aggregated attractiveness level over all periods. In order to make the problem more realistic, a number of additional constraints are also considered in this paper:  
1. Collection constraint: the subset of items assigned to each table should cover at least a minimum number of goods from each collection, \(l_{c}\geq 0\). 2. Price constraint: a minimum number of products at each table, \(l_{p}\geq 0\), should belong to the expensive category. 3. Profit constraint: the profit margin of each table should be greater than a threshold defined by the manager, \(l_{m}\geq 0\). A set of consecutive time periods \(H\) is considered, together with a finite set of items, \(I\), which are hosted in a warehouse. Each item \(i\in I\) is associated with a base price \(p_{i}>0\), a marginal benefit \(m_{ih}\geq 0\) (which might be different at each period \(h\in H\)), and an initial attractiveness value \(v_{i0}>0\). The final selling price of each item \(i\in I\) at period \(h\in H\) is given by \(p^{\prime}_{ih}\geq p_{i}\), i.e., \(m_{ih}=p^{\prime}_{ih}-p_{i}\). These items belong to a set of collections \(C=\{c_{1},c_{2},\cdots,c_{|C|}\}\), where \(I=\bigcup\limits_{k=1}^{|C|}c_{k}\). The subset of expensive items is given by \(I_{p}=\{i\in I/p_{i}\geq p_{0}\}\), where \(p_{0}\) is a threshold price value defined by the manager. At each period \(h\in H\), a subset \(S_{h}\) items must be exposed using a set of homogeneous tables \(T\), with each table containing a total of \(n>0\) items. The decision variable \(x_{ith}\) is equal to 1 if item \(i\) is selected for table \(t\) in period \(h\), and to 0, otherwise. Thus, the set of non-repeated selected items for each table \(t\in T\) in period \(h\in H\) is given by \(S_{th}=\{i\in I/x_{ith}=1\}\), being \(S_{h}=\bigcup\limits_{t\in T}S_{th}\). A matrix \(D=[d_{ij}]_{i,j\in I}\) provides the existing interaction value, \(d_{ij}\in\mathbb{R}\), between any pair of items \(i,j\in I\).