N. G. F. Sancho

A new algorithm for the solution of multi-state dynamic programming problems

This paper presents a new algorithm for the solution of multi-state dynamic programming problems, referred to as the Progressive Optimality Algorithm. It is a method of successive approximation using a general two-stage solution. The algorithm is computationally efficient and has minimal storage requirements. A description of the algorithm is given including a proof of convergence. Performance characteristics for a trial problem are summarized.

A hybrid ‘dynamic programming/depth-first search’ algorithm, with an application to redundancy allocation

A hybrid ‘dynamic programming/depth-first search’ algorithm has been developed to solve non-linear integer programming problems arising in the reliability optimization of redundancy allocation. Initially, the technique solves the knapsack relaxation of the original mathematical programming problem using dynamic programming. Then, all solutions in some range of the relaxation problem are obtained via an enumerative depth-first search technique. The solutions are ranked and the optimal solution is given by the best one that satisfies the remaining constraints of the given problem. Computational complexity of the algorithm is also discussed. The salient features of our hybrid algorithm are its simplicity and ease of programming. Our algorithm also has an advantage over the traditional Lagrangian and surrogate dual approaches. It does not have to deal with the issue of ‘duality gap’ as in classical dual approaches, which is responsible for the failure to identify optimal solutions to the primal integer optimization problems. Of most importance, it guarantees to succeed in identifying an optimal solution.

A MULTI-OBJECTIVE ROUTING PROBLEM

A dynamic programming formulation is proposed for finding all Pareto optimal solutions for a routing problem within a specified overall cost, overall time and overall distance. A method is also proposed for finding other solutions which are not Pareto optimal solutions but which lie within the specified overall cost, overall time and overall distance.

A suboptimal solution to a hierarchial network design problem using dynamic programming

Abstract Each arc of a network has two nonnegative costs. The primary cost is larger than the secondary cost for each arc. Given a source node and a destination node, it is required to connect these in a path P , to which all other nodes are linked in one or more tree-like structures. The cost of the resulting tree is found from the primary costs of the arcs used in P and the secondary costs of all other arcs. A dynamic programming formulation is presented which finds a suboptimal solution to the problem, and the paper indicates the reasons for this formulation falling under certain circumstances.

The hierarchical network design problem with multiple primary paths

Abstract In this paper we develop a suboptimal solution for the hierarchical network design problem (HNDP) with multiple primary paths. In practice some networks may require solutions to the HNDP with more than one origin/destination pair to be connected by primary paths. This paper develops a dynamic programming solution to find a suboptimal solution to the problem.

A NEW TYPE OF MULTI-OBJECTIVE ROUTING PROBLEM

A dynamic programming model is proposed for finding all Pareto optimal solutions for a routing problem within a specified overall time, overall reliability and overall capacity. A method is also given for finding other solutions which are not Pareto optimal, but which satisfy the required constraints. This model uses the method proposed by Sniedovich2 which has certain significant advantages over a method previously examined by Sancho.1

Economic optimization in controlled fisheries

Abstract Dynamic Programming is used to optimize a formula for calculating the present value of all total profits in fisheries exploitation over time. This result may be calculated for each year, and economic comparisons can be made.

A new algorithm for solving certain variational problems

A technique for finding the solution of discrete, multistate dynamic programming problems is applied to solve certain variational problems. The algorithm is a method of successive approximations using a general two-stage solution. The advantage of the method is that it provides a means of reducing Bellmans “curse of dimensionality.” An example on the Plateau problem or the minimal surface area problem is considered, and the algorithm is found to be computationally efficient.

Technique for Finding the Moment Equations of a Nonlinear Stochastic System

A technique is described for deriving the moment equations of a nonlinear stochastic system with a random forcing term. The nonlinear term is then linearized by means of minimizing the mean‐square error between nonlinear and linear terms. The traditional derivation of the Fokker‐Planck equation for the conditional probability density, and hence the moment equations, is bypassed.

Regional surveillance of disjoint rectangles: a travelling salesman formulation

Mission planning for surveillance coverage is of both practical and theoretical interest. In brief, regional surveillance involves planning the search of certain given regions in the minimum possible time. The surveillance problem can therefore be described as a variant of the classical travelling salesman problem. The uniqueness of the problem lies in the different allowed entry and exit points. Additionally, the mission schedule has to ensure the probability of target detection must not be compromised. From the practical perspective, any reduction in travelling time provides immediate cost savings to the defence department. A dynamic programming formulation is derived for the regional surveillance problem. An example is included to illustrate the methodology.