Abstract
An example of design might be a warehouse floor (represented by a setS) of areaA, with unspecified shape. Givenm warehouse users, we suppose that useri has a known disutility functionf isuch thatH i(S), the integral off iover the setS (for example, total travel distance), defines the disutility of the designS to useri. For the vectorH(S) with entriesH i(S), we study the vector minimization problem over the set {H(S) :S a design} and call a design efficient if and only if it solves this problem. Assuming a mild regularity condition, we give necessary and sufficient conditions for a design to be efficient, as well as verifiable conditions for the regularity condition to hold. For the case wheref iis thel p-distance from warehouse docki, with 1<p<∞, a design is efficient if and only if it is essentially the same as a contour set of some Steiner-Weber functionf λ=λ1 f 1+⋯+λ m f m ,when the λ i are nonnegative constants, not all zero.
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Communicated by S. E. Dreyfus
This research was supported in part by the Interuniversity College for PhD Studies in Management Sciences (CIM), Brussels, Belgium; by the Army Research Office, Triangle Park, North Carolina; by a National Academy of Sciences-National Research Council Postdoctorate Associateship; and by the Operations Research Division, National Bureau of Standards, Washington, D.C. The authors would like to thank R. E. Wendell for calling Ref. 16 to their attention.
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Chalmet, L.G., Francis, R.L. & Lawrence, J.F. Efficiency in integral facility design problems. J Optim Theory Appl 32, 135–149 (1980). https://doi.org/10.1007/BF00934720
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DOI: https://doi.org/10.1007/BF00934720