An Application of Branch and Cut to Open Pit Mine Scheduling

The economic viability of the modern day mine is highly dependent upon careful planning and management. Declining trends in average ore grades, increasing mining costs and environmental considerations will ensure that this situation will remain in the foreseeable future. The operation and management of a large open pit mine having a life of several years is an enormous and complex task. Though a number of optimization techniques have been successfully applied to resolve some important problems, the problem of determining an optimal production schedule over the life of the deposit is still very much unresolved. In this paper we will critically examine the techniques that are being used in the mining industry for production scheduling indicating their limitations. In addition, we present a mixed integer linear programming model for the scheduling problems along with a Branch and Cut solution strategy. Computational results for practical sized problems are discussed.     

A modular integer GCD algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U−bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2,p/2) such that . When the algorithm is executed on a bit common CRCW PRAM with O(nlognlogloglogn) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/logn) time on the same number of processors.     

A theorem on integer flows on cartesian products of graphs

It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere‐zero 4‐flow. If both factors are bipartite, then the product admits a nowhere‐zero 3‐flow. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 93–98, 2003     

Sigma function solution of the initial value problem for Somos 5 sequences

The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

     

An improved linearization strategy for zero-one quadratic programming problems

We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed in [7], and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances.     

Liquids and liquid mixtures: Deduction of the BGY equation considering the angular distribution function

The Born→Green→Yvon equation for molecular fluid has been deduced considering the orientational distribution functions. The isotropic and anisotropic parts of the distribution function have been separated. The expressions deduced can be used in the case of mixtures and for the non-central type of intermolecular potential energy.     

Representations of unbounded optimization problems as integer programs

Any optimization problem in a finite structure can be represented as an integer or mixed-integer program in integral quantities.We show that, when an optimization problem on an unbounded structure has such a representation, it is very close to a linear programming problem, in the specific sense described in the following results. We also show that, if an optimization problem has such a representation, no more thann+2 equality constraints need be used, wheren is the number of variables of the problem.We obtain a necessary and sufficient condition for a functionf:SZ, withS Z ⁿ , to have a rational model in Meyer's sense, and show that Ibaraki models are a proper subset of Meyer models.This research was supported by NSF Grant No. GP-37510X1 and ONR Contract No. N00014-75-C0621, NR047-048.     

Metric inequalities for routings on direct connections with application to line planning

We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We derive a feasibility condition for path capacities supporting such direct connection flows similar to the well-known feasibility condition for arc capacities in ordinary multi-commodity flows. The condition can be expressed in terms of a class of metric inequalities for routings on direct connections. We illustrate the concept on the example of the line planning problem in public transport and present an application to large-scale real-world problems.