Rectilinear minimax hub location problems

Abstract

The minimax hub location problem sites a facility to minimize the maximum weighted interaction cost between pairs of fixed nodes. In this paper, distances are represented by a rectilinear norm and may be suited to factory layout or street network problems. The problem is already well known (in 2-D) as the round trip location problem and is extended to 3-D in this paper. One rationale for the solution method is based on an extension of the geometric arguments used to solve the minimax single facility location problem. Suppose a budget is provided for interactions, and that each interaction must be accomplished for no more than this cost. The algorithm uses a bi-section search for the feasible budget until it finds the expenditure needed to provide for these flows. The extension in the present paper is that the nodes are permitted to be on different layers (levels). This 3-D version of the problem appears to be a new variant of the hub model. The models and solution techniques developed in the paper are illustrated using a small 55 node problem. Because of a relatively efficient implementation of the bi-section search, the algorithm in 2-D and 3-D is also applied successfully to a 550 node problem.

Appendix: finite linear program

The entire problem can also be solved in one pass by means of the following LP

$$ {\text{MIN}}\,{\text{G}},\quad {\text{s.t.}}\;W_{ij} \left[ {\left| {x_{i} - X} \right| + \left| {y_{i} - Y} \right| + \left| {z_{i} - Z} \right| + \left| {x_{j} - X} \right| + \left| {y_{j} - Y} \right| + \left| {z_{j} - Z} \right|} \right] \le G $$

by choosing $\left[ {X,Y,Z,G} \right]$

converting the absolute values into (2⁶) linear inequalities

$$ \pm (x_{i} - X) \pm (y_{i} - Y) \pm (z_{i} - Z) \pm (x_{j} - X) \pm (y_{j} - Y) \pm (z_{j} - Z) \le G/W_{ij} $$

Define a ^k, b ^k, c ^k, d ^k, e ^k, f ^k as a series of ±1 indicators varying systematically over all 64 combinations, and using the k superscript as a notation to indicate one particular combination, we see that

$$ a^{k} (x_{i} - X) + b^{k} (y_{i} - Y) + c^{k} (z_{i} - Z) + d^{k} (x_{j} - X) + {\text{e}}^{k} (y_{j} - Y) + f^{k} (z_{j} - Z) \le G/W_{ij}\quad {\text{ for}}\;k = 1,\; \ldots ,\;64 $$

and collecting all the coefficients of X,Y,Z, and G on the left hand side and the constants to the right hand side, after carefully reversing signs and inequality we get

$$ (a^{k} + d^{k} )X + (b^{k} + e^{k} )Y + (c^{k} + f^{k} )Z + (1/Wij)G \ge (a^{k} x_{i} + b^{k} y_{i} + c^{k} z_{i} + d^{k} x_{j} + {\text{e}}^{k} y_{j} + f^{k} z_{j} )\quad {\text{ for}}\,{\text{all}}\;i,j,k $$

which, with the objective, is a simple linear program in four variables. This problem is however likely to grow quite large in that these constraints exist for all the interacting pairs. While it was initially helpful to use this method to generate exact solutions, ultimately the paper uses a bi-section search, roughly comparable to that which was also used for the 2-D case. As shown in the main text, the intersection decision problem is solved using an imbedded LP.