Leon Cooper

The Transportation-Location Problem

This paper defines a problem type, called the transportation-location problem, that can be considered a generalization of the Hitchcock transportation problem in which, in addition to seeking the amounts to be shipped from origins to destinations, it is also necessary to find, at the same time, the optimal locations of these sources with respect to a fixed and known set of destinations. This new problem is characterized mathematically, and exact and approximate methods are presented for its solution.

Facility location in the presence of forbidden regions, I: Formulation and the case of Euclidean distance with one forbidden circle

Abstract A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region R of the plane. Using the calculus of variations the shortest path between two points x , y ∉ R which does not intersect R is determined. If d( x , y ) is unconstrained distance, we denote the shortes distance along a feasible path by d ( x y ) . The constrained Weber problem is, then: given points x j ∉ R and positive weights wj, j = 1,2,…,n, find a point x ∉ R such that f( x )= Σ n j=1 d( x , x j ) is a minimum. An algorithm is formulated for the solution of this problem when d( x , y ) is Euclidean distance and R is a single circular region. Numerical results are presented.

Optimal location on a sphere

Abstract The problem of finding a point on the sphere S2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied. Three different metrics are considered as distances between points on S2: (A), square of Euclidean distance; (B), Euclidean distance; (C), great circle distance. Non uniqueness of minimizers is demonstrated and some pathological cases are studied. An algorithm, analogous to the Weiszfeld algorithm for the classical unconstrained Weber problem is formulated, and its convergence properties are investigated. A necessary and sufficient condition for a destination point to be a local minimizer is derived. Finally, a modified form of Steffensens acceleration is given and the results of numerical tests are presented. These results illustrate the predictions of the theory, and confirm the effectiveness of Steffensens acceleration.

An Always-Convergent Numerical Scheme for a Random Locational Equilibrium Problem

A probabilistic extension of the classical Weber problem is studied. N destinations in the plane,

The fixed charge problem—I: A new heuristic method

P_j ,j = 1, \cdots ,N

Solution of Integer Linear Programming Problems by Direct Search

, are given as random variables with specified probability density functions, and the problem is to find the location of the point P which minimizes the expected sum of the Euclidean distances

An extension of Fibonaccian search to several variables

\overline {PP_j }

The stochastic transportation-location problem

. Under mild assumptions on the density functions, the objective function is shown to be strictly convex and the minimum unique. An iterative scheme for finding P is shown to be a descent method which is globally convergent, and the iteration is shown to be locally linear. Finally, numerical examples using bivariate normal density functions are given.

The Weber problem revisited

Abstract A new approximate method for finding optimal or near optimal solutions to the fixed charge problem is described. It is very rapid, compared with previous methods and achieves results which are at least as good or better than previously published results. The method is useful in its own right. However, it will also form the basis for the development of an exact solution method to be described in subsequent work.

A study of the fixed charge transportation problem

An approximate but fairly rapid method for solving integer linear programming problems is presented, which utilizes, in part, some of the philosophy of “direct search” methods. The method is divided into phases which can be applied in order and has the desirable characteristic that a best feasible solution is always available. Numerical results are presented for a number of test problems. Some possible extensions and improvements are also presented.