Lou Caccetta

A Survey of Reachability and Controllability for Positive Linear Systems

This paper is a survey of reachability and controllability results for discrete-time positive linear systems. It presents a variety of criteria in both algebraic and digraph forms for recognising these fundamental system properties with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control, manpower planning, scheduling and other areas of operations research) but also in studying properties of reachable sets, in feedback control problems, and others. The paper highlights the intrinsic combinatorial structure of reachable/controllable positive linear systems and reveals the monomial components of such systems. The system matrix decomposition into monomial components is demonstrated by solving some illustrative examples.

An Improved Branch-and-Cut Algorithm for the Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) deals with the distribution of a single commodity from a centralized depot to a number of specified customer locations with known demands. The CVRP considered in this paper assumes common vehicle capacity, fixed or variable number of vehicles, and an objective to minimize the total distance traveled by all the vehicles. This paper develops several new cutting planes for this problem, and uses them in an exact branch-and-cut algorithm. Two of the new cutting planes are based on a specified structure of an optimal solution and its existence. Computational results are reported for 1,650 simulated Euclidean problems as well as 24 standard literature test problems; solved problems range in size from 15--100 customers. A comparative analysis demonstrates the significant computational benefit of the proposed method.

A branch and cut method for the degree-constrained minimum spanning tree problem

A problem of interest in network design is that of finding, in a given weighted graph, a minimum-weight spanning tree whose vertices satisfy specified degree restrictions. We present a branch and cut solution procedure for this NP-complete problem. Our algorithm is implemented and extensively tested. Computational results on 3150 random table problems ranging from 100 to 800 vertices are presented and discussed.

A new subtour elimination constraint for the vehicle routing problem

Abstract Vehicle Routing Problems (VRP) are concerned with the delivery of a single commodity from a centralized depot to a number of specified customer locations with known demands. In this paper we consider the VRP characterized by: fixed or variable number of vehicles, common vehicle capacity, distance restrictions, and minimization of total distance travelled by all vehicles as the objective. We develop an exact algorithm based on a new subtour elimination constraint. The algorithm is implemented using the CPLEX package for solving the relaxed subproblems. Computational results on 1590 simulated problems and 10 literature problems (without distance restrictions) are reported and a comparative analysis is carried out.

Optimal Design of All-Pass Variable Fractional-Delay Digital Filters

This paper presents a computational method for the optimal design of all-pass variable fractional-delay (VFD) filters aiming to minimize the squared error of the fractional group delay subject to a low level of squared error in the phase response. The constrained optimization problem thus formulated is converted to an unconstrained least-squares (LS) optimization problem which is highly nonlinear. However, it can be approximated by a linear LS optimization problem which in turn simply requires the solution of a linear system. The proposed method can efficiently minimize the total error energy of the fractional group delay while maintaining constraints on the level of the error energy of the phase response. To make the error distribution as flat as possible, a weighted LS (WLS) design method is also developed. An error weighting function is obtained according to the solution of the previous constrained LS design. The maximum peak error is then further reduced by an iterative updating of the error weighting function. Numerical examples are included in order to compare the performance of the filters designed using the proposed methods with those designed by several existing methods.

Optimal design of complex FIR filters with arbitrary magnitude and group delay responses

This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods.

Maximal cycles in graphs

Abstract Let G be a simple graph on n vertices and m edges having circumference (longest cycle length) t . Woodall determined some time ago the maximum possible value of m . The object of this paper is to give an alternative proof of Woodalls theorem. Our approach will, in addition, characterize the structure of the extremal graphs.

An application of discrete mathematics in the design of an open pit mine

Abstract The determination of the “optimum pit limit” of a mine is considered to be a fundamental problem in mine planning as it provides information which is essential in the evaluation of the economic potential of a mineral deposit, and in the formulation of long-, intermediate-, and short-range mine plans. A number of mathematical techniques have been proposed to solve this problem, some of the more elaborate ones posing considerable computational problems. In this paper we discuss the development and implementation of a graph-theoretic technique originally proposed by Lerchs and Grossman. Our implementation strategy involves the use of a dynamic programming technique to “bound” the optimum.

Equipment Selection for Surface Mining: A Review

One of the challenging problems for surface mining operation optimization is choosing the optimal truck and loader fleet. We refer to this problem as the equipment selection problem ESP. In this paper, we describe the ESP in the context of surface mining and discuss related problems and applications. Within the scope of both the ESP and related problems, we outline modeling and solution approaches. Using operations research literature as a guide, we conclude by pointing to future research directions to improve both the modeling and solution outcomes for practical applications of this problem.

Extremal solutions for p-Laplacian differential systems via iterative computation

Abstract In this paper, we study the extremal solutions of a fractional differential system involving the p -Laplacian operator and nonlocal boundary conditions. The conditions for the existence of the maximal and minimal solutions to the system are established. In addition, we also derive explicit formulae for the estimation of the lower and upper bounds of the extremal solutions, and establish a convergent iterative scheme for finding these solutions. We also give a special case in which the conditions for the existence of the extremal solutions can be easily verified.