Capacity management in public service facility networks: a model, computational tests and a case study

Abstract

In this work, we present a mathematical model to support location decisions oriented to rationalize facility systems in non-competitive contexts. In order to test the model, computational results are shown and an application to a real-world case study, concerning the Higher Education system in an Italian region, is discussed.

Appendix A

In the following, we prove that the combination of the three sets of constraints (2), (3), (13) guarantees that fractions of demand assume the same values defined by the expressions (12).

For each service k, consider the following subsets of J:

• $T_k$ subset of facilities that did not provide service $k( {T_k =\left\{ {j\in J:l_{kj} =0} \right\} });$

• $V_k$ subset of facilities at which service k has been closed $( V_k =\left\{ j\in J:l_{kj} =1,\right. \left. s_{kj} =1 \right\} )$;

• $W_k$ subset of facilities that still provide service $k( {W_k =\left\{ {j\in J:l_{kj} =1,s_{kj} =0} \right\} })$.

Note that the above introduced subsets form a partition of J; in fact, $V_k$ and $W_k$ form a partition of the set of facilities providing k($V_k \cup W_k =U_k , V_k \cap W_k =\emptyset )$ and $T_k$ is the complement set of $U_k$ to J ($T_k = J-U_k )$.

For each service k and user i, the equivalence between (2, 3, 13) and (12) is trivially proved for any facility j not providing k in the final configuration; i.e., $\forall j\in T_k \cup V_k .$

Indeed, conditions (3) impose:

$$\begin{aligned}&x_j^{ik} =0&\qquad \qquad \forall i\in I,\forall k\in K,\forall j\in T_k \cup V_k \end{aligned}$$

similarly to conditions (11), being respectively in $T_k$ and$V_k$ $\alpha _j^{ik} =0$ and $s_{kj} =1.$

Then, the equivalence has to be proved only for facilities j still providing k;  i.e., $j\in W_k .$

Conditions (12), for each service k,  define a proportional relationship between the fractions of demand assigned to each pair of facilities j and r belonging to $U_k =V_k \cup W_k .$

For each pair $( {j,r})\in U_k \times U_k$, one of the following conditions can occur:

• 1. $j\in W_k ,r\in V_k :$ facility j still provides service k ($s_{kj} =0)$ while r not anymore ($s_{kr} =1$,$x_r^{ik} =0)$;

• 2. $j\in V_k ,t\in W_k :$ facility r still provides service k ($s_{kr} =0)$ while j not anymore ($s_{kj} =1$, $x_j^{ik} =0)$;

• 3. $j,r\in V_k :$ service k has been closed at both facilities j and r ($s_{kr} =s_{kj} =1$, $x_r^{ik} =x_j^{ik} =0)$;

• 4. $j,r\in W_k : \quad :$ service k is still provided by both facilities j and r ($s_{kr} =s_{kj} =0)$.

We now demonstrate that conditions (13) become active only in the last case. With this aim, consider the paired conditions associated with ( j, r):

$$\begin{aligned} \left\{ {{\begin{array}{l} {x_j^{ik} \le \frac{\alpha _j^{ik} }{\alpha _r^{ik} }x_r^{ik} +s_{kr}} \\ {x_r^{ik} \le \frac{\alpha _r^{ik} }{\alpha _j^{ik} }x_j^{ik} +s_{kj}} \\ \end{array} }} \right. \quad \forall i\in I . \end{aligned}$$

From Table 5, in which the expressions of the above conditions in the single cases are reported, it is easy to understand that in the first three cases the constraints are trivially satisfied $\forall i\in I$.

Table 5 Possible expressions of conditions (12) for a generic pair $(j,r)\in U_k \times U_k$

In case d the two conditions become equivalent to the following one:

$$\begin{aligned} x_r^{ik} =\frac{\alpha _r^{ik} }{\alpha _j^{ik} }x_j^{ik} \quad \forall i\in I . \end{aligned}$$

Hence, for a particular user i and service k,  it is possible to express all the fractions of the demand assigned to the facilities in $W_k =\left\{ {r_1 ,\ldots ,r_w } \right\}$ as a function of the same variable $x_j^{ik}$ $( {j\in W_k })$. Therefore, replacing:

$$\begin{aligned} x_r^{ik} =\frac{\alpha _r^{ik} }{\alpha _j^{ik} }x_j^{ik} \quad \forall i\in I,\quad \forall r\in W_k . \end{aligned}$$

in (4), we have

$$\begin{aligned} \sum \nolimits _{j\in J} x_j^{ik}= & {} \sum \nolimits _{j\in T_k } x_j^{ik} +\sum \nolimits _{j\in V_k } x_j^{ik} + \sum \nolimits _{j\in W_k } x_j^{ik}\\= & {} \sum \nolimits _{j\in W_k } x_j^{ik} =x_{r_1 }^{ik} +\cdots +x_{r_w }^{ik} =\left( {\frac{\alpha _{t_1 }^{ik} }{\alpha _j^{ik} }+\cdots +\frac{\alpha _{r_w }^{ik} }{\alpha _j^{ik} }}\right) x_j^{ik} =1 . \end{aligned}$$

Hence:

$$\begin{aligned} x_j^{ik} =\frac{\alpha _j^{ik} }{\alpha _{r_1 }^{ik} +\cdots +\alpha _{r_w }^{ik} }=\frac{\alpha _j^{ik} }{\mathop \sum \nolimits _{r\in W_k } \alpha _r^{ik} } \end{aligned}$$

(14)

For a given service k, Eq. (14) holds for all the facilities in the set $W_k$ and each user i; then we can generally write:

$$\begin{aligned} x_j^{ik} =\frac{\alpha _j^{ik} }{\mathop \sum \nolimits _{r\in W_k } \alpha _r^{ik} } \quad \forall i\in I,\forall k\in K,\forall j\in W_k \end{aligned}$$

(15)

which is equivalent to (12) $\forall j\in W_k$.