Gröbner bases related to 3-dimensional transportation problems

This paper illustrates some work in progress on 3-dimensional transportation problems,of format r×s×t say. Following Conti and Traverso, a suitable Gröbner basis is sought for, which is hard to be calculated by means of Buchberger algorithm. A different approach involving graph theory makes the calculation tractable when r = s = t = 3 (and in fact whenever 3 ∈ {r, s, t}). Introduction The aim of this paper is to illustrate some work in progress on 3-dimensional transportation problems. A typical problem of this kind goes as follows. Given r production facilities F1 . . . Fr, let aik (i := 1 . . . r, k := 1 . . . t) denote the number of units of an indivisible good produced by Fi during the k-th month of a fixed period of t months. Assume that there are s outlets O1 . . . Os each one demanding a certain number of units per month, say bjk (j := 1 . . . s, k := 1 . . . t). If cijk stands for the cost associated with transporting one unit from Fi to Oj during the k-th month, one wishes to minimize the total cost of transportation during the whole period of t months. In mathematical terms, one wishes to solve the integer programming problem associated with the matrix A whose columns are {eij ⊕ e ′ ik ⊕ e ′′ jk | 1 ≤ i ≤ r, 1 ≤ j ≤ s, 1 ≤ k ≤ t}, where {eij} (resp., {e ′ ik} ; resp., {e ′′ jk} ) stands for the canonical basis of the Zmodule of r × s (resp., r × t ; resp., s × t) integer matrices. For 3-dimensional transportation problems see e.g. [Vl] and [St, Chapter 14]. It is well known that integer programming problems as above can be solved by a method first suggested by Conti and Traverso (cf. [CT]), which resorts to the calculation of suitable Gröbner bases (for more information about this method we refer the reader to the survey papers [HT] and [T]). More precisely, if IA denotes the toric ideal canonically associated with the matrix A described before, then one needs to find the reduced Gröbner basis of IA relative to some appropriate term order. 2000 Mathematics Subject Classification. Primary 13P10; Secondary 05C00, 90C10.