Data Classification (Clustering) and Related Problems

Abstract

Numerous decision problems related, for example, to environmental monitoring, regional solid waste management, manufacturing systems, transportation services, and so forth, depend essentially on the choice of a relatively small number of ‘primary facilities’ (such as, for example, water quality analysis laboratories, waste disposal sites, warehouses, bus terminals or other facilities serving customers, or multipurpose machines in a workshop environment). Once the strategic facility design/location decision is made, the long-term operational characteristics of the system depend on the assignment of a—usually (much) larger—number of given entities (data to be grouped, water quality measurement sites in a region, population affected by the waste disposal sites, customers, users of public transport, jobs in a workshop, etc.) to the ‘primary facilities’ selected at the first stage. In many such cases, the primary (strategic) locational decisions are to be optimized, while the overall system performance (for example, total investment/setup costs plus long-term discounted operational costs) can be assessed—for each primary decision—via ‘expensive’ evaluation methods: algorithmic solution of embedded optimization problems, stochastic simulation, and so forth. For related discussions and examples, the reader is referred, for example, to Cormack (1971), Hartigan (1975), Jacobsen and Madsen (1980), Dempster, Fisher, Jansen, Lageweg, Lenstra and Rinnooy Kan (1981), Gordon (1981), Muroga (1982), Bodin, Golden, Assad and Ball (1983), Florian (1984), Golden and Assad (1986), Nitti and Speranza (1987), Pintér and Somlyódy (1987), Ermoliev (1988), Laporte (1988), Wallace (1988), Deininger and Lee (1989), Kaufman and Rousseeuw (1990), Galvão (1993), Ghost and Harche (1993). In a sense, the analysed system can be conceived as some ‘black box’ or ‘oracle’ that reacts in a complicated manner to decisions on the primary facility configuration setting. Naturally, there is no prior guarantee for unimodality of the overall performance indicator associated with such problems (since even its analytical form may be unknown). On the other hand, Lipschitz-continuity of the performance indicator—as depending on the basic design parameters—holds in many cases.