## 3 A Formal Description of the Multi-Period Product Display Problem
These intra-period dependencies account for the fact that some items might be positively or negatively correlated with others (i.e., showing them together might generate synergies or, on the contrary, might reduce their aggregated attractiveness). Apart from this intra-period dependencies between pairs of items, inter-period dependencies are also considered to account for the product's novelty (or the lack of it). Accordingly, the attractiveness value of every item is a dynamic input, i.e., it is reduced or increased by a certain percentage factor depending on whether the item was displayed or not in the previous period. Thus, \(\forall h\in\{1,2,\ldots,|H|\}\), the attractiveness value of item \(i\) in period \(h\), \(v_{ih}\), is recursively defined as:\[v_{ih}=f(v_{i(h-1)},x_{it(h-1)},a,b,u,w)=\left\{ \begin{array}{ll}Max\{a,(1-u)v_{ i(h-1)}\}&\sum_{t\in T}x_{it(h-1)}=1\\ Min\{b,(1+w)v_{i(h-1)}\}&\sum_{t\in T}x_{it(h-1)}=0\end{array} \right.,\]  
where \(0\leq a<b\) are bounds for the attractiveness values and \(u,w\geq 0\) are decreasing or increasing percentage factors, respectively. With this notation, the addressed problem can be formulated as follows:  
\[Max\sum_{h\in H}\sum_{t\in T}\sum_{i\in I}v_{ih}x_{ith}+\sum_{h\in H}\sum_{t \in T}\sum_{i\in I}\sum_{j\in I/i<j}d_{ij}x_{ith}x_{jth}. \tag{1}\]  
Subject to:  
\[\sum_{i\in I}x_{ith}=n\qquad\forall t\in T,\forall h\in H \tag{2}\] \[\sum_{t\in T}x_{ith}\leq 1\qquad\forall i\in I,\forall h\in H\] (3) \[\sum_{i\in c_{k}}x_{ith}\geq l_{c}\qquad\forall c_{k}\in C,\forall t \in T,\forall h\in H\] (4) \[\sum_{i\in I_{p}}x_{ith}\geq l_{p}\qquad\forall t\in T,\forall h\in H\] (5) \[\sum_{i\in I}m_{ih}x_{ith}\geq l_{m}\qquad\forall t\in T,\forall h\in H\] (6) \[v_{ih}=f(v_{i(h-1)},x_{it(h-1)},a,b,u,w)\qquad\forall i\in I, \forall h\in I\setminus\{0\}\] (7) \[x_{ith}\in\{0,1\}\qquad\forall i\in I,t\in T,h\in H\] (8) \[v_{ih}\in[a,b]\qquad\qquad\forall i\in I,h\in H \tag{9}\]  
The objective function (1) maximizes the total attractiveness of the planning horizon by considering the individual attractiveness of the items and the intra-period dependencies between each pair of selected items in each displaying table and period. Equation (2) ensures that the number of items on each table \(t\in T\) does not exceed a pre-defined value \(n\). Equation (3) guarantees that each item \(i\) cannot be selected more than once in a given period \(h\in H\). Equation (4) confirms that, inside each period, each table covers at least \(l_{c}\) items from each collection \(c_{k}\). Equation (5) guarantees that, inside each period, each table contains at least \(l_{p}\) expensive items. Equation (6) ensures, for each period, that the profit margin of each table should be greater than \(l_{m}\). Equation (7) introduces the inter-periods dynamic component in the attractiveness value of the items. Notice that this equation transforms the objective function into a non-smooth one due to the definition of the \(v_{ih}\) values. Equation (8) states that all decision variables are binary. Finally, Equation (9) bounds the values that variable \(v_{ih}\) can take. A'relaxed' version of this problem can be obtained when no bounds are imposed on the attractiveness values of each item. In that case, the objective function becomes quadratic since Equation (10) can be rewritten as:  
\[v_{ih}=v_{i(h-1)}(1-u)\sum_{t\in T}x_{it(h-1)}+v_{i-1}(1+w)\sum_{t\in T}(1-x_{ it(h-1)})\qquad\forall i\in I,\forall h\in H\setminus\{0\} \tag{10}\]