Solving Dynamic Traveling Salesman Problem Using Dynamic Gaussian Process Regression

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 818529, 10 pages http://dx.doi.org/10.1155/2014/818529 Research Article Solving Dynamic Traveling Salesman Problem Using Dynamic Gaussian Process Regression Stephen M. Akandwanaho, Aderemi O. Adewumi, and Ayodele A. Adebiyi School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, University Road, Westville, Private Bag X 54001, Durban, 4000, South Africa Correspondence should be addressed to Stephen M. Akandwanaho; Received 4 January 2014; Accepted 11 February 2014; Published 7 April 2014 Academic Editor: M. Montaz Ali Copyright © 2014 Stephen M. Akandwanaho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper solves the dynamic traveling salesman problem (DTSP) using dynamic Gaussian Process Regression (DGPR) method. The problem of varying correlation tour is alleviated by the nonstationary covariance function interleaved with DGPR to generate a predictive distribution for DTSP tour. This approach is conjoined with Nearest Neighbor (NN) method and the iterated local search to track dynamic optima. Experimental results were obtained on DTSP instances. The comparisons were performed with Genetic Algorithm and Simulated Annealing. The proposed approach demonstrates superiority in finding good traveling salesman problem (TSP) tour and less computational time in nonstationary conditions. 1. Introduction A bulk of research in optimization has carved a niche in solving stationary optimization problems. As a corollary, a flagrant gap has hitherto been created in finding solutions to problems whose landscape is dynamic, to the core. In many real-world optimization problems a wide range of uncertainties have to be taken into account [1]. These uncertainties have engendered a recent avalanche of research in dynamic optimization. Optimization in stochastic dynamic environments continues to crave for trailblazing solutions to problems whose nature is intrinsically mutable. Several concepts and techniques have been proposed for addressing dynamic optimization problems in literature. Branke et al. [2] delineate them through different stratifications, for example, those that ensure heterogeneity, sustenance of heterogeneity in the course of iterations, techniques that store solutions for later retrieval and those that use different multiple populations. The ramp up in significance of DTSP in stochastic dynamic landscapes has, up to the hilt, in the past two decades attracted a raft of computational methods, congenial to address the floating optima (Figure 1). An indepth exposition is available in [3, 4]. The traveling salesman problem (TSP) [5], one of the most thoroughly studied NPhard theory in combinatorial optimization, arguably remains a main research experiment that has hitherto been cast as an academic guinea pig, most notably in computer science. It is also a research factotum that intersects with a wide expanse of research areas; for example, it is widely studied and applied by mathematicians and operation researchers on a grand scale. TSP’s prominence ascribe to its flexibility and amenability to a copious range of problems. Gaussian process regression is touted as a sterling model on account of its stellar capacity to interpolate the observations, its probabilistic nature, versatility, practical and theoretical simplicity. This research lays bare a dynamic Gaussian process regression (DGPR) with a nonstationary covariance function to give foreknowledge of the best tour in a landscape that is subject to change. The research is in concert with the argumentation that optima are innately fluid, cognizant that size, nature, and position are potentially volatile in the lifespan of the optima. This skittish landscape, most notably in optimization, is a cue for fine-grained research to track the moving and evolving optima and provide a framework for solving a cartload of pent-up problems that are intrinsically dynamic. We colligate DGPR with nearest neighbor (NN) algorithm and the iterated 2 Journal of Applied Mathematics Euclidean distance expression [11], we present the matrix 𝐶 between separate distances as 2 2 𝑐𝑖𝑗 = √ (𝑥𝑖 − 𝑥𝑗 ) + (𝑦𝑖 − 𝑦𝑗 ) . (3) Affixed to TSP are important aspects that we bring to the fore in this paper. We adumbrate a brief overview of symmetric traveling salesman problem (STSP) and asymmetric traveling salesman problem (ATSP) as follows. STSP, akin to its name, ensures symmetry in length. The distances between points are equal for all directions while ATSP typifies different distance sizes of points in both directions. Dissecting ATSP gives us a handle to hash out solutions. Let ATSP be expressed, subject to the distance matrix. In combinatorial optimization, an optimal value is sought, whereby in this case, we minimize using the following expression: Figure 1: Nonstationary optima [6]. 𝑛−1 𝑤𝜋(𝑛),𝜋(1) + ∑ 𝑤𝜋(𝑖),𝜋(𝑖+1) . local search, a medley whose purpose is to refine the solution. We have arranged the paper in four sections. Section 1 is limited to introduction, Section 2’s ambit includes review of all methods that form the mainspring of this work, which include Gaussian process, TSP, and DTSP. We elucidate DGPR for solving the TSP in Section 3. Section 4 discusses results obtained and draws conclusion. (4) 𝑖=1 Reference [12] formulates ATSP in integer programming 𝑛2 − 𝑛 zero-one variables 𝑥𝑖𝑗 or else it is defined as 𝑛 𝑛 𝑦 = ∑∑ 𝑤𝑖𝑗 𝑥𝑖𝑗 (5) 𝑖=1 𝑗=1 such that 𝑛 2. The Traveling Salesman Problem (TSP) ∑𝑥𝑖𝑗 = 1, Basic Definitions and Notations. It is imperative to note that in the gamut of TSP, both symmetric and asymmetric aspects are important threads in its fabric. We factor them into this work through the following expressions. Basically, a salesman traverses across an expanse of cities culminating into a tour. The distance in terms of cost between cities is computed by minimizing the path length: 𝑛−1 𝑓 (𝜋) = ∑ 𝑑𝜋(𝑖),𝜋(𝑖+1) + 𝑑𝜋(𝑛),𝜋(1) . 𝑛 ∑ 𝑥𝑖𝑗 = 1, 𝑖 [𝑛] , 𝑗=1 (6) ∑∑𝑥𝑖𝑗 ≤ |𝑆| − 1, ∀ |𝑆| < 𝑛, 𝑖∈𝑆 𝑗∈𝑆 𝑥𝑖𝑗 = 0 or 1, 𝑖 ≠ 𝑗 ∈ [𝑛] . There are different rules affixed to ATSP, inter alia, to ensure a tour does not overstay its one-off visit to each vertex. The rules also ensure that standards are defined for subtours. In the symmetry paradigm, the problem is postulated. For brevity, we present subsequent work with tautness: (1) 𝑦= 𝑖=1 We provide a momentary storage, 𝐷 for cost distance. The distances between 𝑛 cities are stored in a distance matrix 𝐷. For brevity, the problem can also be situated as an optimization problem. We minimize the tour length (Figure 5): 𝑗 [𝑛] , 𝑖=1 The first researcher, in 1932, considered the traveling salesman problem [7]. Menger gives interesting ways of solving TSP. He lays bare the first approac (...truncated)