A frontier-based facility location problem with a centralised view of measuring the performance of the network

Abstract

The p-median problem is among the most popular problem types in combinatorial optimisation with a broad range of application areas from facility location–allocation and network design to data mining and pattern recognition. This paper studies a variant of this problem, which incorporates data envelopment analysis into the location analysis. In the proposed problem, – in addition to minimising the spatial interaction among the facilities and the demand nodes – we aim at maximising the efficiency of the chosen p facilities. This view is in line with the centralised control mechanism which can often be seen in applications of the p-median problem: facilities are managed by a central authority who wishes to improve the efficiency of the whole system rather than maximising the individual efficiency of each facility. Real-world data from hospitals in Germany will be used to illustrate the proposed approach.

Keywords:

• Data envelopment analysis
• the p-median problem
• location analysis
• the capacitated clustering problem
• centralised control mechanism

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The process of determining individual DEA efficiency scores may mathematically be done by a single consolidated program (see, eg, program (8) in Beasley, 2003 or program (20) in Klimberg & Ratick, 2008). Hence, solving only a single DEA program does not always indicate that the overall efficiency of the whole network is optimized. See the discussion in Section 3.

2 Act on the Economic Security of Hospitals and the Regulation of Hospital Nursing Rates, see gesetze-im-internet.de/khg

3 Lower Saxony, situated in northwestern Germany, is the second-largest state by land area and fourth-largest in population.

4 These hospitals provide at least general medicine services to patients (Niedersächsischer Krankenhausplan, 2018).

5 Braunschweig is one of the fourth administrative regions (in German: Regierungsbezirk) of Lower Saxony and located in the south-east of the state.

6 All mathematical programming programs in this paper were encoded in AIMMS, version 4.14.

7 Where p = 1, in any feasible solution of (9), only one of the binary variables, corresponding to, eg, $j=j^{*},$ is equal to 1, while the remaining binary variables are equal to zero. This hospital $j=j^{*}$ is the one by which the objective function is maximized. This implies in this case that the program in (9) is equivalent to finding the maximum of individual efficiencies ${\theta }_j^{}$ (j = 1,…, 26), which could alternatively be computed by the conventional DEA program in (4). Working with program (4), remark 3 guarantees that we always have at least one hospital $j=j^{*}$ so that ${\theta }_{j^{*}}^{}=1.$ Hence, when p = 1, then ${\theta }_{\text{Overall}}^1=1.$

8 Applying the Explicit Exclusion (EE) method given in Section 3.2 did not results in another optimal solution in this case.

Additional information

Funding

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) in the context of the research fund AH 90/5-2.