Jack A.A. van der Veen

Stability aspects of the traveling salesman problem based on k -best solutions

Abstract This paper discusses stability analysis for the Traveling Salesman Problem (TSP). For a traveling salesman tour which is known to be optimal with respect to a given instance (length vector) we are interested in determining the stability region, i.e. the set of all length vectors for which the tour is optimal. The following three subsets of the stability region are of special interest: 1. (1) tolerances, i.e. the maximum perturbations of single edges; 2. (2) tolerance regions which are subsets of the stability region that can be constructed from the tolerances; and 3. (3) the largest ball contained in the stability region centered at the given length vector (the corresponding radius is known as the stability radius). It is well known that the problems of determining tolerances and the stability radius for the TSP are NP -hard so that in general it is not possible to obtain the above-mentioned three subsets without spending a lot of computation time. The question addressed in this paper is the following: assume that not only an optimal tour is known, but also a set of k shortest tours (k ⩾2) is given. Then to which extent does this allow us to determine the three subsets in polynomial time? It will be shown in this paper that having k-best solutions can give the desired information only partially. More precisely, it will be shown that only some of the tolerances can be determined exactly and for the other ones as well as for the stability radius only lower and/or upper bounds can be derived. Since the amount of information that can be derived from the set of k-best solutions is dependent on both the value of k as well as on the specific length vector, we present numerical experiments on instances from the TSPLIB library to analyze the effectiveness of our approach.

Solving the k -best traveling salesman problem

Abstract Although k-best solutions for polynomial solvable problems are extensively studied in the literature not much is known for NP -hard problems. In this paper we design algorithms for finding sets of k-best solutions to the Traveling Salesman Problem (TSP) for some positive integer k. It will be shown that a set of k-best Hamiltonian tours in a weighted graph can be determined by applying the so-called partitioning algorithms and by algorithms based on modifications of solution methods like branch-and-bound. In order to study the effectiveness of these algorithms the time for determining a set of k-best solutions is investigated for a number of instances in Reinelt’s TSPLIB library. It appears that the time required to find a set of k-best tours grows rather slowly in k. Furthermore the results of numerical experiments show that the difference in length between a longest and a shortest tour in the set of k-best solutions grows only slowly in k and that also the ‘structure’ of the tours in the set of k-best tours is quite robust. Scope and purpose After having solved an optimization problem, it is often the case that one still must verify whether the optimal solution satisfies some additional restrictions that are not included in the original model. Such conditions are called “subtle conditions”. In the case of subtle conditions, it is particularly of interest to have a set of k-best solutions in order to see which of the solutions in that set satisfy the extra conditions and how high the “price” is for incorporating such conditions. Algorithms for finding sets of k-best solutions have mainly been studied in the literature for polynomially solvable combinatorial optimization problems. Little is known on algorithms for finding k-best solutions for NP -hard combinatorial optimization problems. In this paper, we study algorithms for finding k-best solutions for one of the most notorious NP-hard problems, namely the Traveling Salesman Problem (TSP). Properties of the sets of k-best solutions for the TSP, such as the increase in tour length and the number of edges that the k-best solutions have in common or differ in, are discussed in detail.

Sequencing jobs that require common resources on a single machine : a solvable case of the TSP

In this paper a one-machine scheduling model is analyzed wheren different jobs are classified intoK groups depending on which additional resource they require. The change-over time from one job to another consists of the removal time or of the set-up time of the two jobs. It is sequence-dependent in the sense that the change-over time is determined by whether or not the two jobs belong to the same group. The objective is to minimize the makespan. This problem can be modeled as an asymmetric Traveling Salesman Problem (TSP) with a specially structured distance matrix. For this problem we give a polynomial time solution algorithm that runs in O(n logn) time.

Low-complexity algorithms for sequencing jobs with a fixed number of job-classes

In this paper we consider the problem of scheduling n jobs such that makespan is minimized. It is assumed that the jobs can be divided into K job-classes and that the change-over time between two consecutive jobs depends on the job-classes to which the two jobs belong. In this setting, we discuss the one machine scheduling problem with arbitrary processing times and the parallel machines scheduling problem with identical processing times. In both cases it is assumed that the number of job-classes K is fixed. By using an appropriate integer programming formulation with a fixed number of variables and constraints, it is shown that these two problems are solvable in polynomial time. For the one machine scheduling case it is shown that the complexity of our algorithm is linear in the number of jobs n. Moreover, if the problem is encoded according to the high multiplicity model of Hochbaum and Shamir, the time complexity of the algorithm is shown to be a polynomial in log n. In the parallel machine scheduling case, it is shown that if the number of machines is fixed the same results hold.

Economic and Environmental Performance of the Firm: Synergy or Trade-Off? Insights from the EOQ Model

Over the last decades, corporations are increasingly expected to perform well on the triple bottom line: People, Planet and Profit. However, both in academia and in practice, there is no consensus on the feasibility of doing good and doing well simultaneously. The traditional view is that there is an unavoidable trade-off between the social and environmental performance of an organisation and its profitability. The other school of thought claimed that breaking the trade-off and creating a synergy, is not only desired but actually feasible. In this chapter, the validity of both views is tested by using a multi-objective approach to a variant of the well-known EOQ model. It is demonstrated that both views are not contradictory but valid under different conditions. As such this chapter helps to reach a better understanding of the factors that drive trade-offs and synergy behaviour of the triple bottom line measures.

An O(n) algorithm to solve the Bottleneck Traveling Salesman Problem restricted to ordered product matrices

The Bottleneck Traveling Salesman Problem (BTSP) is the problem of finding a Hamiltonian tour in a complete weighted digraph that minimizes the longest traveled distance between two successive vertices. The BTSP is studied in a graph where the distance matrix D=(di,j]) is given by di,j]=ai]·bj] with a1]?a2]???an] and b1]?b2]???bn]. It is observed that such so-called ordered product matrices (OPMs) have the following property. They are either “doubly graded matrices” or special “max-distribution matrices”. Using this characterization, it is shown that there is an O(n) algorithm to solve the BTSP restricted to OPMs.

On the equivalence of selected supply chain contract mechanism

This paper models a situation where a Supplier sells a fashion product to a Buyer who in turn sells the product to the consumers. Both the Supplier and the Buyer set their own selling price. For the above setting this paper designs different contract mechanisms such as Revenue sharing, Profit sharing, Quantity discounts, License fee and Mail-in-rebate contract mechanisms. The paper shows that the designed contract mechanism can provide both coordination and win-win. The paper also establishes the equivalence between the designed contract mechanisms and argues that industries can use one mechanism over the other in case of implementation problem.

Economic and Environmental Performance of the Firm: Synergy or Trade-Off?

Over the last decades, corporations are increasingly expected to fulfill the written and unwritten laws of doing business in a sustainable way. They are implicitly expected to perform well on the so-called triple bottom line; People, Planet and Profit. However, both in academia and in practice, there is no consensus on the feasibility of doing good (for society and environment) and doing well (economically) simultaneously. The traditional view is that there is an unavoidable trade-off between the social & environmental performance of an organization and its profitability. The other school of thought has challenged this view and claimed that doing good and doing well simultaneously, i.e., breaking the trade-off and creating a synergy, is not only desired but actually feasible. In this paper, the validity of both views (trade-off and synergy) is tested within the context of green sourcing, using a multi-objective approach to a variant of the well- known Economic Order Quantity (EOQ) model. It is demonstrated that both views are not contradictory but valid under different conditions and strategic focus. As such this paper helps to reach a better understanding of the factors and conditions that drive trade-offs and synergy behavior of the triple bottom line measures within the chosen problem setting.