The balanced traveling salesmanproblem

In this paper we introduce the balanced traveling salesman problem (BTSP), which can be used to model optimization problems where equitable distribution of resources are important. BTSP is obviously NP-hard. Efficient heuristic algorithms are presented to solve the problem along with extensive computational results using benchmark problems from TSPLIB and random instances. Our algorithms produced provably optimal solutions for several test instances. Application of the model in solving the nozzle guide vane assembly problem related to the maintenance of aircraft engines is also discussed.     

TSP Heuristics: Domination Analysis and Complexity

   Abstract. We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph K _n produce a solution no worse than the average cost of a tour in K _n in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than

TSP Heuristics: Domination Analysis and Complexity

We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph Kn produce a solution no worse than the average cost of a tour in Kn in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than $\lceil{n}/{2}\rceil !$ , and for the Double Tree heuristic and a variation of the Christofides heuristic the domination numbers are shown to be one (even if the edge costs satisfy the triangle inequality). Further, unless P = NP, no polynomial time approximation algorithm exists for the TSP on the complete digraph $\vec{K}_n$ with domination number at least (n-1)!-k for any constant k or with domination number at least (n-1)! - (( k /(k+1))(n+r))!-1 for any non-negative constants r and k such that (n+r) $\equiv$ 0 mod (k+1). The complexities of finding the median value of costs of all the tours in $\vec{K}_n$ and of similar problems are also studied.     

Priority allocation rules for single machine total weighted linear and square tardiness problems

The single machine scheduling problems minimizing total weighted tardiness and square tardiness objectives have been studied in literature for many years. Applications of the model include sequencing problems in manufacture and logistics. This paper proposes two new priority allocation rules, PAR 1 and PAR 2, for solving these two problems. Unlike most known dispatch rules and constructive algorithms, our new rules take advantage of not only the jobs’ static characters values such as the process time, the due date and the weight, but also their dynamic characters values, i.e., the slack and the values of the objective function for different choices of some jobs. At any time when a job is being selected to process, some of the unprocessed jobs are delayed while the others are not. It means that the characters of these two sorts of jobs are different from each other. So, combining these characters with the objective function’s value can obtain effective dispatch rule. Experimental analysis based on the instances from the OR-Library discloses that our priority allocation rules, PAR 1 and PAR 2, are efficient and have significant advantages over traditional approaches.     

The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

In this paper we consider the quadratic minimum spanning tree problem (QMSTP) which is known to be NP-hard. Given a complete graph, the QMSTP consists of finding a minimum spanning tree (MST) where interaction costs between pairs of edges are prescribed. A Lagrangian relaxation procedure is devised and an efficient local search algorithm with tabu thresholding is developed. Computational experiments are reported on standard test instances, randomly generated test instances and quadratic assignment problem (QAP) instances from the QAPLIB by using a transformation scheme. The local search heuristic yields very good performance and the Lagrangian relaxation procedure gives the tightest lower bounds for all instances when compared to previous lower bounding approaches.     

The bottleneck k-MST

The k-MST is a well known NP-hard problem and several approximation algorithms exist to solve this problem with a guaranteed performance bound. A closely related problem, called the bottleneck k-MST (BMST(k)) can however be solved in O(mlogn) time on graph with n nodes and m edges. We propose two algorithms to solve BMST(k), one of complexity O(m+nlogn) and the other of O(m) time. We also consider a generalization of BMST(k) which subsumes many bottleneck problems studied in the literature and show that this generalized problem can also be solved in O(m) time.     

Learning multicriteria fuzzy classification method PROAFTN from data

In this paper, we present a new methodology for learning parameters of multiple criteria classification method PROAFTN from data. There are numerous representations and techniques available for data mining, for example decision trees, rule bases, artificial neural networks, density estimation, regression and clustering. The PROAFTN method constitutes another approach for data mining. It belongs to the class of supervised learning algorithms and assigns membership degree of the alternatives to the classes. The PROAFTN method requires the elicitation of its parameters for the purpose of classification. Therefore, we need an automatic method that helps us to establish these parameters from the given data with minimum classification errors. Here, we propose variable neighborhood search metaheuristic for getting these parameters. The performances of the newly proposed method were evaluated using 10 cross validation technique. The results are compared with those obtained by other classification methods previously reported on the same data. It appears that the solutions of substantially better quality are obtained with proposed method than with these former ones.     

A fast algorithm for a class of bottleneck problems

We show that if a bottleneck problem of sizem with an ordered list of element costs can be solved in O(ξ(m)) time, then the problem with an unordered list of element costs can be solved in O(ξ(m)) log^* m) time.