## 5 Test Cases
### Artificial Test Instance Groups EVC and CSS  
Test instances from the groups EVC and CSS are generated with the same approach. The \(n\) possible locations for service stations are randomly distributed in the Euclidean plane with coordinates \(\mathrm{coord}(v)\), \(v\in V\), chosen uniformly from the grid \(\{0,\ldots,L-1\}^{2}\), with \(L=\lceil 10\sqrt{n}\rceil\). The fixed costs \(z_{v}^{\mathrm{fix}}\), as well as the variable costs \(z_{v}^{\mathrm{var}}\) for setting up a service station at each location \(v\in V\), are uniformly chosen at random from \(\{50,\ldots,100\}\). The budget is assumed to be \(B=\lceil 7.5\cdot n\rceil\) so that roughly \(10\%\) of the stations with average costs can be expected to be opened.  
The number of use cases for each user \(u\in U\) is chosen randomly according to a shifted Poisson distribution with offset one, expected value three, and a maximum value of five, i.e., the number of use cases never exceeds five. Each of these use cases \(c\in C_{u}\) is associated with an individual demand \(D_{u,c}\) randomly chosen from \(\{5,\ldots,50\}\) and, depending on the benchmark group, with one (EVC) or two (CSS) SPRs.  
Each SPR \(r\in R_{u,c}\) of a use case \(c\) also is associated with a particular geographic location \(q_{r}\in\{0,\ldots,L-1\}^{2}\). In order to model similarities in the users' SPRs, these locations are selected in the following correlated way. First, ten _attraction points_\(A\) with uniform random coordinates are selected from \(\{0,\ldots,L-1\}^{2}\). Then, each use case location is derived by randomly choosing one of these attraction points \((a_{x},a_{y})\in A\) and adding a small individual offset to the coordinates, i.e.,  
\[q_{r}=\left(\lfloor\mathcal{N}(a_{x},\sigma_{v})\rfloor,\lfloor\mathcal{N}(a_{ y},\sigma_{v})\rfloor\right), \tag{18}\]  
where \(\mathcal{N}(\cdot,\cdot)\) denotes a random value sampled from a normal distribution with the respectively given mean value and standard deviation \(\sigma_{v}\). If obtained coordinates are not in \(\{0,\ldots,L-1\}^{2}\), a new attraction point is chosen, and the deviation is re-sampled.  
A service point location \(v\in V\) receives a rating w.r.t. an SPR \(r\), according to a sigmoidal decay function applied to the Euclidean distance, and is also perturbed by a Gaussian noise with a standard deviation of \(\sigma_{r}\):  
\[w^{\prime}_{r,v}=\mathcal{N}\bigg{(}\frac{1}{1+6e^{0.5\left\lVert q_{r}- \mathrm{coord}(v)\right\rVert-6}},\sigma_{r}\bigg{)}. \tag{19}\]  
The parameters of the sigmoid function are chosen so that \(w^{\prime}_{r,v}\) decreases as the distance between \(v\) and \(q_{r}\) increases and becomes approximately zero at a distance larger than twelve. Additionally, as motivated in Section 3, we discretize the rating \(w^{\prime}_{r,v}\) by rounding to the closest value in \(\{0,0.25,0.5,0.75,1\}\), obtaining \(w_{r,v}\). Hence, \(w_{r,v}=\lfloor 4\cdot\min(1,\max(0,w^{\prime}_{r,v}))+0.5\rfloor/4\).  
Six sets of 30 benchmark instances were generated for EVC, as well as CSS. As detailed in Table 1, these sets consider \(n\in\{100,200,300\}\) potential service point locations in combination with different numbers of users and two different settings for the standard deviations of the Gaussian perturbations \(\sigma_{v}\) and \(\sigma_{r}\). Note that parameters \(\sigma_{v}\) and \(\sigma_{r}\) control how similar suitability values for locations of SPRs generated from the same attraction point are. The larger \(\sigma_{v}\) and \(\sigma_{r}\), the more different are the preferences of the users. Hence, we have chosen two different settings for \(\sigma_{v}\) and \(\sigma_{r}\) to show how COA behaves under conditions in which preferences of users towards locations for service points have a higher and lower similarity, respectively. Values for \(\sigma_{v}\) and \(\sigma_{r}\) have been determined experimentally in preliminary tests. In the following, we will denote instance sets primarily by the pair \((n,m)\)