Production, Manufacturing and Logistics
Location with acceleration–deceleration distance

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Abstract

In this paper we investigate a model where travel time is not necessarily proportional to the distance. Every trip starts at speed zero, then the vehicle accelerates to a cruising speed, stays at the cruising speed for a portion of the trip and then decelerates back to a speed of zero. We define a time equivalent distance which is equal to the travel time multiplied by the cruising speed. This time equivalent distance is referred to as the acceleration–deceleration (A–D) distance. We prove that every demand point is a local minimum for the Weber problem defined by travel time rather than distance. We propose a heuristic approach employing the generalized Weiszfeld algorithm and an optimal approach applying the Big Triangle Small Triangle global optimization method. These two approaches are very efficient and problems of 10,000 demand points are solved in about 0.015 seconds by the generalized Weiszfeld algorithm and in about 1 minute by the BTST technique. When the generalized Weiszfeld algorithm was repeated 1000 times, the optimal solution was found at least once for all test problems.

Introduction

An important component of location modeling is distance. Most location models attempt to minimize distance, average distance (p-median), or maximum distance (p-center) (Current et al., 2002, Daskin, 1995, Love et al., 1988).

A common practice is to use distance and travel time interchangeably. The belief that the two are substitutable rests on the underlying assumption that speed of travel is constant, and therefore travel time is proportional to distance. In reality, however, travel speed is not constant. A person traveling by car leaves the origin at speed zero, accelerates to a cruising speed, and starts decelerating when approaching the destination. If traveling on a highway, travel speed on on- and off-ramps is slower than the cruising speed. Travel on the highway itself is not devoid of interruptions and variations in speed. On surface streets, frequent stops at traffic signals entail multiple events of being idle in addition to acceleration and deceleration. Similarly, when traveling by air, there is a period of take-off and acceleration until the plane reaches cruising speed, then, when approaching the destination and landing, a period of deceleration. This is true for other modes of transportation such as trains, boats, ships, barges, helicopters, and others. Regardless of the mode of transportation, these variations in speed, and thereby travel time, are exhibited.

The issue of acceleration/deceleration was first introduced by Kolesar (1975) for estimating the travel time of fire engines in the city of New York. However, he did not investigate the properties of the distance function, and did not apply his estimate to any location model. It is also discussed in various manufacturing systems. The movement of a robot’s arm starts at a stationary point and ends at a stationary point. Arm movements follow an acceleration and deceleration pattern. Typically, the acceleration and deceleration times are a significant portion of the travel distance. Acceleration/deceleration in storage and retrieval systems is discussed by Chang et al. (1995) and Wen et al. (2001), by Hark et al. (2004) for carousel systems, and in a servo system by Guo et al., 2002, Guo et al., 2003.

Acceleration and deceleration and their impact on travel speed, travel time and representation of distance come to bear mainly when one deals with relatively short distances. In relatively long distances, such as coast-to-coast travel, when the total distance significantly exceeds the combined acceleration plus deceleration distance, the impact of the variable speed on travel time is negligible.

In this paper we investigate the Weber location problem of finding the optimal location for a facility that minimizes the total weighted travel time for all travelers. The acceleration–deceleration distance can be defined for any distance measure (such as Euclidean, rectilinear, p, spherical, on network links) but in this paper we investigate mainly the Euclidean distance case. One may wish to incorporate in the model the direction of the runway at the airports which may further extend the travel distance. We also restrict the analysis to one acceleration and one deceleration during the trip. However, when traveling on a network there may be several acceleration and deceleration instances during the trip (e.g. in some intersections) which lead to more contrived formulations. The analysis of other distance measures and multiple stops is left for future research.

A special case of the model happens when the acceleration and deceleration distances are relatively large and none of the trips reach the cruising speed. In such circumstances travel time is proportional to the square root of the distance (see the following section). There are a few papers investigating similar models. Cooper (1968) analyzed the Weber problem based on the square root of the Euclidean distance. Morris (1981) analyzed the Weber problem using p distances raised to a power Morris (1981) also investigated the case 0<p<1 which exhibits properties similar to our objective. Brimberg and Love (1992) and Brimberg et al. (1994) investigated related distance measures such as the weighted sum of rectilinear and Euclidean distances.

Section snippets

Revisiting the definition of distance

In standard location problems a common assumption is that travel time is proportional to the distance d between origin and destination. This assumption is reasonable when the speed of travel is constant. However, in reality travel starts at speed zero, the speed increases to a certain value, and towards the end of the trip the speed goes down to a stop. Suppose that the acceleration is a1, the cruising speed is v, and the deceleration at the end of the trip is -a2. The distance traveled in time

Formulation of the location problem

The Weber (1909) location problem is formulated in terms of minimizing total travel time between all demand points and the facility which is equivalent to minimizing the total A–D distance.

Let n be the number of demand points; (ai,bi) be the location of demand point i; (x,y) be the unknown location of the new facility; d0 be the total acceleration plus deceleration distances by Eq. (1); di(x,y) be the Euclidean distance between demand point i and the new facility; ϕ(di(x,y)) be the A–D distance

Solution methods

In this section, we apply the heuristic approach “the generalized Weiszfeld algorithm” (Drezner and Drezner, 1998; Drezner, in press) based on the classical Weiszfeld (1937) algorithm, and a global optimization technique (the Big Triangle Small Triangle algorithm, Drezner and Suzuki, 2004) for solving the Weber problem based on A–D distances (9).

Computational experiments

Three programs were coded:

  • 1.

    one evaluates the value of the objective function at all demand points,

  • 2.

    one applies the generalized Weiszfeld algorithm, and

  • 3.

    one applies the “Big Triangle Small Triangle” algorithm.

We thank Atsuo Suzuki for his Fortran program that finds the triangulation based on Sugihara and Iri (1994) subroutines first developed in Ohya et al. (1984). The programs were coded in Fortran, using double precision arithmetic, compiled by Microsoft Fortran PowerStation 4.0 and ran on a

Conclusions

In most location models travel time is assumed to be proportional to the distance. In this paper we suggest that travel time is not an accurate representation of distance. We assume that every trip starts at speed zero, accelerates to a cruising speed and decelerates to a complete stop. This leads to an A–D distance which is larger than the distance when a constant cruising speed throughout the trip is assumed. The model is analyzed and solved assuming Euclidean distances in the plane. The

Acknowledgment

This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.

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