A model of human performance on the traveling salesperson problem

Abstract

A computational model is proposed of how humans solve the traveling salesperson problem (TSP), Tests of the model are reported, using human performance measures from a variety of 10-,20-,40-, and 60-node problems, a single 48-node problem, and a single 100-node problem. The model provided a range of solutions that approximated the range of human solutions and conformed closely to quantitative and qualitative characteristics of human performance. The minimum path lengths of subjects and model deviated by average absolute values of O,OO!6, 0,9%, 2.4%, 1.4%,3.5%, and 0.02%for the 10-,20-, 40-, 48-, 60-, and 100-node problems, respectively. Because the model produces a range of solutions, rather than a single solution, it may find better solutions than some conventional heuristic algorithms for solving TSPs, and comparative results are reported that support this suggestion.The Euclidean form of the traveling salesperson problem (TSP) consists offinding the shortest path (tour) that passes through a set of points and returns to the origin. Any particular instance of a TSP obviously has a finite set of solutions, so there is a guaranteed method of solving any problem by exhaustive search of the solution space. However, exhaustive search becomes increasingly impractical as the number of points (n) increases. This is because the number ofpossible solutions increases as (n -I )!/2 (ignoring direction of travel), Thus, although a computer generating 1,000 solutions per second would require only 3 min to find all the solutions to a I O-node problem, a 20-node problem would occupy it for almost 4 million years. The increased capacity of computers over the past 10 years has made it possible to find guaranteed optimal solutions to fairly large TSPs within reasonable time periods. In 1986, a new record was set (since broken) when a 2,392-city problem was solved, breaking the previous record of 532 cities and requiring 27 h (Sangalli, 1992). There has been a long-term interest in the field of operations reseatch in devising economical procedures for finding solutions that approximate the optimal, and many such heuristic techniques have been developed and compared. To achieve "reasonable" approximations to an optimal solution to within a few percentage points above the shortest path, such procedures generally need to perform on the order of n 3 calculations (Golden, Bodin, Doyle, & Stewart, 1980). One approach to improving heuristic procedures has been to enlist the assistance of human operators (Krolak, Felts, & Marble, 1971;Michie, Fleming, & Oldfield, 1968). Krolak et al. compared computer-generated solutions with solutions produced by a human-computer interactive approach. A number of comparisons were reported that indicated that human-computer solutions were at least as short, and often shorter, than those generated by computer alone. The results imply that human operators respond to TSPs in a manner that is not only different from, but more effective than, the heuristics used by Krolak et al. Otherwise, the human-computer...

Individual Differences in Optimization Problem Solving: Reconciling Conflicting Results

“…A number of theoretical accounts have suggested that how people solve TSPs is, in part, based on perceptual processes (Graham et al, 2000;MacGregor et al, 2000). It seems plausible that, for very simple problems, perceptual processing predominates but, as problem diffi culty increases, more analytical processes come into play.…”

“…In other words, human solution times per problem increase in proportion to the number of nodes (Graham et al, 2000;Pizlo, Stefanov, Saalweachter, Li, Haxhimusa, & Kropatsch, 2006). Other generally agreed-upon fi ndings are that human solutions are typically close to optimal (Graham et al, 2000;MacGregor & Ormerod, 1996;van Rooij, Schactman, Kadlec, & Stege, 2006;Vickers, Butavicius, Lee, & Medvedev, 2001) and rarely self-intersect (MacGregor, Ormerod, & Chronicle, 2000;van Rooij, Stege, & Schactman, 2003;Vickers, Lee, Dry, & Hughes, 2003).…”

Some Tours are More Equal than Others: The Convex-Hull Model Revisited with Lessons for Testing Models of the Traveling Salesperson Problem

“…This observation has motivated researchers to set out to identify the human strategy for solving TSPs and implement it in a computational model. In this paper, we take a closer look at one such model, the Convex-hull (CH) algorithm model proposed by MacGregor, Ormerod, and Chronicle (2000), and its purported fi t to human performance on the TSP (for altogether different modeling attempts for TSP see Graham, Joshi, & Pizlo, 2000;and Pizlo et al, 2006). The CH model of MacGregor et al (2000) is a formal algorithmic elaboration on the convex-hull hypothesis put forth by MacGregor and Ormerod (1996;see van Rooij, Stege, & Schactman, 2003, for an overall assessment of the empirical support for this hypothesis) that proposes that people construct solutions to the TSP by fi rst perceiving (and mentally sketching) the boundary of the point set (called the convex hull) and then inserting one by one the interior points in the tour.…”

“…The CH model of MacGregor et al (2000) is a formal algorithmic elaboration on the convex-hull hypothesis put forth by MacGregor and Ormerod (1996;see van Rooij, Stege, & Schactman, 2003, for an overall assessment of the empirical support for this hypothesis) that proposes that people construct solutions to the TSP by fi rst perceiving (and mentally sketching) the boundary of the point set (called the convex hull) and then inserting one by one the interior points in the tour. To investigate the extent to which this CH algorithm captures the process underlying human performance on the TSP, MacGregor et al (2000) compared tours produced by the algorithm (for different starting points and directions of travel) with tours produced by humans. They studied in particular random or semirandom point sets, and found that tours produced by model and participants resembled each other in terms of tour length.…”

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AbstractTo explain human performance on the Traveling Salesperson problem (TSP), MacGregor, Ormerod, and Chronicle (2000) proposed that humans construct solutions according to the steps described by their convex-hull algorithm. Focusing on tour length as the dependent variable, and using only random or semirandom point sets, the authors claimed empirical support for their model.

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