Expanding Neighborhood GRASP for the Traveling Salesman Problem

In this paper, we present the application of a modified version of the well known Greedy Randomized Adaptive Search Procedure (GRASP) to the TSP. The proposed GRASP algorithm has two phases: In the first phase the algorithm finds an initial solution of the problem and in the second phase a local search procedure is utilized for the improvement of the initial solution. The local search procedure employs two different local search strategies based on 2-opt and 3-opt methods. The algorithm was tested on numerous benchmark problems from TSPLIB. The results were very satisfactory and for the majority of the instances the results were equal to the best known solution. The algorithm is also compared to the algorithms presented and tested in the DIMACS Implementation Challenge that was organized by David Johnson.     

A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables

We show that the production-transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time. The algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem: monotonicity of the objective function along certain directions, small proportion of nonlinear variables and combinatorial properties implied by transportation constraints.     

A novel approach to Bilevel nonlinear programming

Recently developed methods of monotonic optimization have been applied successfully for studying a wide class of nonconvex optimization problems, that includes, among others, generalized polynomial programming, generalized multiplicative and fractional programming, discrete programming, optimization over the efficient set, complementarity problems. In the present paper the monotonic approach is extended to the General Bilevel Programming GBP Problem. It is shown that (GBP) can be transformed into a monotonic optimization problem which can then be solved by “polyblock” approximation or, more efficiently, by a branch-reduce-and-bound method using monotonicity cuts. The method is particularly suitable for Bilevel Convex Programming and Bilevel Linear Programming.      

The Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs

We prove that the Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs can be solved by an algorithm requiring a number of elementary operations and a number of evaluations of the nonlinear cost functions which are both bounded by polynomials inr, n, m, wherer is the number of nodes,n is the number of arcs andm the number of sinks in the network.On leave from Institute of Mathematics, P.O. Box 631, Bo Ho, Hanoi, Vietnam.     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

A global optimization approach for the linear two-level program

Linear two-level programming deals with optimization problems in which the constraint region is implicity determined by another optimization problem. Mathematical programs of this type arise in connection with policy problems to which the Stackelberg leader-follower game is applicable. In this paper, the linear two-level programming problem is restated as a global optimization problem and a new solution method based on this approach is developed. The most important feature of this new method is that it attempts to take full advantage of the structure in the constraints using some recent global optimization techniques. A small example is solved in order to illustrate the approach.The paper was completed while this author was visiting the Department of Mathematics of Linköping University.     

Serial-batching scheduling with time-dependent setup time and effects of deterioration and learning on a single-machine

This paper deals with serial-batching scheduling problems with the effects of deterioration and learning, where time-dependent setup time is also considered. In the proposed scheduling models, all jobs are first partitioned into serial batches, and then all batches are processed on a single serial-batching machine. The actual job processing time is a function of its starting time and position. In addition, a setup time is required when a new batch is processed, and the setup time of the batches is time-dependent, i.e., it is a linear function of its starting time. Structural properties are derived for the problems of minimizing the makespan, the number of tardy jobs, and the maximum earliness. Then, three optimization algorithms are developed to solve them, respectively.     

Single-machine serial-batching scheduling with a machine availability constraint,position-dependent processing time,and time-dependent set-up time

This article considers the single-machine serial-batching scheduling problem with a machine availability constraint, position-dependent processing time, and time-dependent set-up time. The objective of this problem is to make the decision of batching jobs and sequencing batches to minimize the makespan. To solve the problem, three cases of machine non-availability periods are considered, and the structural properties of the optimal solution are derived for each case. Based on these structural properties, an optimization algorithm is developed and an example is proposed to illustrate this algorithm.     

Expanding neighborhood search–GRASP for the probabilistic traveling salesman problem

The Probabilistic Traveling Salesman Problem is a variation of the classic traveling salesman problem and one of the most significant stochastic routing problems. In probabilistic traveling salesman problem only a subset of potential customers need to be visited on any given instance of the problem. The number of customers to be visited each time is a random variable. In this paper, a variant of the well-known Greedy Randomized Adaptive Search Procedure (GRASP), the Expanding Neighborhood Search–GRASP, is proposed for the solution of the probabilistic traveling salesman problem. expanding neighborhood search–GRASP has been proved to be a very efficient algorithm for the solution of the traveling salesman problem. The proposed algorithm is tested on a numerous benchmark problems from TSPLIB with very satisfactory results. Comparisons with the classic GRASP algorithm and with a Tabu Search algorithm are also presented. Also, a comparison is performed with the results of a number of implementations of the Ant Colony Optimization algorithm from the literature and in six out of ten cases the proposed algorithm gives a new best solution.