## 2 Conflicting Objectives and Non-Dominated Solutions of EMS Design Problem  
The emergency medical service system design is handled as a task to deploy \(p\) service centers in a set of \(m\) various candidates taking into account the demands of system users concentrated at \(n\) dwelling places of the serviced region. The basic decisions on the design can be modeled by a series of \(m\) zero-one decision variables connected to individual possible center locations. If there is a service center at location \(i\), the variable \(y_{i}\) has a value of one; otherwise, it takes a value of zero. Considering the given total number \(p\) of centers, the set of feasible solutions can be studied as a sub-set of vertices of the unit \(m\)-dimensional hypercube, where the Hamming distance of each feasible vertex from the origin of coordinates is exactly \(p\), and the shortest Hamming distance between two feasible solutions equals two. The set of all feasible solutions can be denoted by \(\mathbf{Y}\) and defined by (1). \[\mathbf{Y}=\left\{\mathbf{y}\in\left\{0,1\right\}^{m}:H(\mathbf{y},\mathbf{0})=p\right\} \tag{1}\]  
The above-mentioned Hamming distance of two vertices, \(\mathbf{y}\) and \(\mathbf{x}\), of the \(m\)-dimensional unit hypercube is defined by (2). \[H(\mathbf{x},\mathbf{y})=\sum_{i=1}^{m}\lvert x_{i}-y_{i}\rvert \tag{2}\]  
To describe the average system response time with respect to users, several formulations have been developed. The simplest one models the response time of the system to an individual user by the time-distance \(t_{ij}\) from the user \(j\) to the nearest center \(i\). In this case, the EMS system design problem collapses to a simple weighted \(p\)-median problem, which is able to be solved to optimality by tools of integer programming [13; 23; 24; 25; 26]. Nevertheless, this approach does not take into account the random occurrence in the demands for service, which is a natural characteristic of emergency medical service systems. Because of the random occurrence in the demands for service, the public service system design model must reflect the characteristics of a queueing system, where a temporary occupancy of the nearest center must be taken into account. This real system property is modeled by a series of \(r\) probability values. They express the probability of the case that the \(k\)-th nearest center to the current demand point is the first available one. Similar to [2], it is supposed that the probability values are approximately the same in all considered regions. The more accurate model of the response time for a system user \(j\) is expressed by the formula \(q_{1}t_{1(j)}+q_{2}t_{2(j)}+\ldots+q_{r}t_{ir(j)}\). The symbol \(ik(j)\) represents the \(k\)-th nearest service center to the user location \(j\), and \(q_{k}\) is the probability that the \(k\)-th nearest service center is the first one, which is available due to occupancy of the nearer centers. Thus, the formula expresses the expected response time of the system to the user \(j\). To obtain the value proportional to the average response time, it is necessary to multiply the expected response time to the user location \(j\) by the expected frequency \(b_{j}\) of the calls for demand satisfaction. After these preliminaries, the system objective function can be formulated by (3), where the result of the operator \(\min_{k}\{t_{ij}:\,i\in P\}\) returns the \(k\)-th minimal value of \(t_{ij}\) for \(i\in P\). \[f^{1}(\mathbf{y})=\sum_{j=1}^{n}b_{j}\sum_{k=1}^{r}q_{k}\min_{k}\left\{t_{ij}:\,i =1,\ldots,m;y_{i}=1\right\} \tag{3}\]  
Contrary to the system criterion, a fair criterion reflects disutility affecting the worst-situated minority of the concerned users. A fair criterion can be formulated in different ways, such as a min-max problem, a lexicographical min-max problem, or in many other forms. In this paper, we will consider the criterion of the number of demands (or calls) thatare located outside a radius \(T\) from the closest medical center.