Composed Min-Max and Min-Sum Radial Approach to the Emergency System Design
This paper deals with the emergency service system design, where not only the disutility of an average user is minimized, but also the disutility of the worst situated user must be considered. Optimization of the average user disutility is often related t
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Abstract This paper deals with the emergency service system design, where not only the disutility of an average user is minimized, but also the disutility of the worst situated user must be considered. Optimization of the average user disutility is often related to the weighted p-median problem. To cope with both objectives, we have suggested a composed method. In the first phase, the disutility of the worst situated user is minimized. The second phase is based on the min-sum approach to optimize the average user’s disutility. The result of the first phase is used here to reduce the model size. We focus on effective usage of the reduction and explore the possibility of a trade-off between a little loss of optimality and computational time.
1 Introduction The emergency system design consists in locating a limited number of service centers at positions from a given finite set to optimize the service accessibility to an average user. This way the emergency system design can be tackled as the weighted p-median problem known also as min-sum optimization [1, 3, 4, 10]. Applying of the minsum objective may lead to such design, where some users are caught in inadmissibly distant locations from any service center, what is considered unfair. The fair designing emerges whenever limited resources are to be fairly distributed among participants. Fairness has been broadly studied in [2, 11]. We focus here on the simple min-max criterion used in the p-center problem. We study a composition of the min-sum and min-max approaches. It is based on the idea that the min-max optimization performed at the first stage of the composition considerably reduces the set of relevant distances considered at the following stage, where the min-sum location problem is solved. The extension of the original composition has been evoked by the finding that the M. Kvet (B) · J. Janáˇcek Faculty of Management Science and Informatics, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia e-mail: [email protected] J. Janáˇcek e-mail: [email protected] © Springer International Publishing Switzerland 2017 K.F. Dœrner et al. (eds.), Operations Research Proceedings 2015, Operations Research Proceedings, DOI 10.1007/978-3-319-42902-1_6
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M. Kvet and J. Janáˇcek
min-max stage puts on a very strong limit on the second stage of the composition and the price of fairness paid by the average user is too high. That is why the limit given by the min-max solution is relaxed taking some higher distance as the limit of the maximal distance. To be able to make fully use of the reduced set of relevant distances, the radial formulation [3, 5, 6, 8] has been employed for both stages. It was found that this way of problem solving is much more effective than the locationallocation formulation.
2 Composition of Min-Max and Min-Sum Optimization Methods The emergency system design problem is formulated as a task of location of at most p service centers so that the objective function value derived from distances between center locations and us
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