Transportation Research Part B 50 (2013) 20–41
Contents lists available at SciVerse ScienceDirect
Transportation Research Part B
journal homepage: www.elsevier.com/locate/trb
A continuous approximation approach for assessment routing
in disaster relief
Michael Huang a, Karen R. Smilowitz a, Burcu Balcik b,⇑
a
b
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA
Industrial Engineering Department, Ozyegin University, Istanbul, Turkey
a r t i c l e
i n f o
Article history:
Received 21 September 2012
Received in revised form 24 January 2013
Accepted 25 January 2013
Keywords:
Vehicle routing
Continuous approximation
Disaster relief
Needs assessment
a b s t r a c t
In this paper, we focus on the assessment routing problem which routes teams to different
communities to assess damage and relief needs following a disaster. To address time-sensitivity, the routing problem is modeled with the objective of minimizing the sum of arrival
times to beneficiaries. We propose a continuous approximation approach which uses
aggregated instance data to develop routing policies and cost approximations. Numerical
tests are performed that demonstrate the effectiveness of the cost approximations at predicting the true implementation costs of the policies and compare the policies against more
complex solution approaches. The continuous approximation approach yields solutions
which can be easily implemented; further, this approach reduces the need for detailed data
and the computational requirements to solve the problem.
2013 Elsevier Ltd. All rights reserved.
1. Introduction
Vehicle routing problems in the humanitarian relief context often pose unique challenges not found in commercial settings. The typical vehicle routing model parameters, such as which communities to serve, the type of aid to deliver and the
amount to deliver, may all be unknown. Further, post-disaster environment is extremely time-sensitive. Not only should
routing decisions be made to ensure a quick response, the decisions themselves must also be made quickly, often with limited access to computing resources. Routing plans must also be easy to implement and communicate to local drivers.
These challenges are particularly acute in the needs assessment phase, which is the initial phase of the humanitarian relief cycle immediately following a disaster (Beamon, 2004). During the assessment phase, teams are dispatched to investigate the conditions of communities. The assessment reports may include the number of people affected by the disaster, the
magnitude of the need for food, water and other items, possible logistics problems such as road availability, and a status of
ongoing efforts to address pressing issues. Various manuals for assessment are available at the UN-HABITAT Disaster Risk
Assessment Portal (http://www.disastersassessment.org). Despite the importance of the assessment efforts during disaster
relief operations, the literature is scarce on this topic.
In this study, we study the Assessment Routing Problem (ARP), which focuses on routing of assessment teams in the
disaster area. The objective of the ARP is to minimize the sum of arrival times at communities, which reflects the time-sensitive nature of relief efforts. We develop a continuous approximation model for the ARP. The continuous approximation
model is a promising approach to complex humanitarian logistics problems for three reasons. First, the continuous approximation model aggregates discrete data parameters, which are often unknown or inaccurate in the immediate aftermath of a
disaster. Second, solutions can be obtained quickly, with limited computational resources. Third, solutions derived from the
⇑ Corresponding author. Tel.: +90 216 564 9390.
E-mail address: [email protected] (B. Balcik).
0191-2615/$ - see front matter 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.trb.2013.01.005
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
21
continuous approximation model are easy to implement in the field. In this paper, we compare the continuous approximation model with a discrete model (and corresponding tabu search solution approach) to analyze the performance of the continuous model. As shown in this paper, comparisons regarding data requirements and computational effort are
straightforward; however, implementation complexity can be difficult to quantify. Therefore, we explore several approaches
to analyze the concept of complexity. Additionally, we study the drawbacks of using an approximation model to obtain solutions, in terms of solution quality. To close the performance gap, we introduce a hybrid strategy that improves upon the solutions from the continuous approximation model by exploring alternate route sequencing with a tabu search.
The main contribution of the paper is to illustrate the use of continuous approximation in developing routing schedules
for humanitarian relief. The humanitarian relief supply chain is a new application for continuous approximation combining
alternative objectives and a limited number of vehicles. The ability to generate quick, understandable, non-complex and
competitive solutions is especially appropriate for assessment routing.
The paper is organized as follows: Section 2 reviews the literature in three related areas: humanitarian logistics, continuous approximation, and implementation complexity. Section 3 describes the ARP setting and proposes a discrete formulation and a continuous approximation model for the problem. Section 4 details the components of the continuous
approximation approach. Section 5 presents computational tests to evaluate the approximation and the performance of
the continuous approximation model. Section 6 concludes the paper and discusses future research.
2. Literature review
Logistics problems occurring in humanitarian supply chains are being increasingly examined in the Operations Research
community. The studies generally consider one or more of the unique facets of humanitarian supply chain: multiple relief
commodities; heterogeneous vehicles; uncertainty in the supply, demand or traveling costs; and different objectives. In
many of these studies the problem is formulated as an integer programming problem and then solved through the use of
a commercial solver (e.g., Barbarosoglu and Arda, 2004; Tzeng et al., 2007; Balcik et al., 2008) or through the use of sophisticated heuristics (e.g., Haghani and Oh, 1996; Barbarosoglu et al., 2002; Ozdamar et al., 2004; Lin et al., 2009; Shen et al.,
2009; Van Hentenryck et al., 2010; Huang et al., 2012). Given the limited computational resources of many humanitarian
agencies, our paper investigates a less computational-intensive approach. Bartholdi et al. (1983) show, in the context of food
delivery for the elderly, that simple solution approaches can still yield highly competitive routing solutions.
Most studies in the humanitarian logistics literature address problems related to aid distribution during disaster relief
operations. The needs assessment phase has not received much attention in the literature, and existing studies are mostly
descriptive. Tatham (2009) investigates the use of unmanned aerial vehicles for needs assessment. Some studies discuss
sampling techniques used in the field by the survey teams to identify the needs of the affected communities. For instance,
Johnson and Wilfert (2008) describes a cluster sampling method, in which the disaster site is divided into mutually exclusive
clusters with well-defined boundaries. Daley et al. (2001) describes a geography-based sampling scheme implemented for
assessing the needs after the August 1999 earthquake in Turkey. To the best of our knowledge, there is no study that develops routing policies and models to support assessment efforts in relief operations.
Minimizing the sum of arrival times in the context of relief routing first appears in Campbell et al. (2008). They compare
the objectives of (1) minimizing the sum of arrival times and (2) minimizing the last arrival time against the traditional vehicle routing problem (VRP) objective of minimizing total travel time. They demonstrate that these objectives yield significant
differences in metric values. For example, they show when minimizing sum of arrival times, in the worst case, solutions generated by minimizing the total travel time differ from the optimal solution by a factor N, where N is the number of nodes. The
uncapacitated non-unit demand version of the problem with an objective of minimizing sum of arrival times is equivalent to
the traveling repairman problem (TRP), which is also called the minimum latency problem, shown by Blum et al. (1994) to be
NP-Hard. Most recent work focuses on developing polynomial time approximation algorithms for the TRP (e.g., Fakcharoenphol et al., 2007; Archer et al., 2008). Ngueveu et al. (2010) consider the objective with non-unit demand and develop a
metaheuristic based on genetic algorithms and local search procedures to solve the problem.
This paper approaches the problem of minimizing sum of arrival times using continuous approximation in humanitarian
relief. Continuous approximation adopts a ‘‘non-detailed’’ approach to solving logistic problems. The approach has been used
in facility location problems. Li and Ouyang (2010) and Cui et al. (2010) use continuous approximation to locate facilities that
may suffer disruptions. The approach has also been used in other logistics problems (see Langevin et al. (1996) for a review,
and Daganzo (2005), Francis and Smilowitz (2006), Smilowitz and Daganzo (2007), and Jabali et al. (2012) for more recent
work). In continuous approximation, detailed information is replaced with aggregate data. For example, the location of nodes
is replaced with a continuous choice set and density function. By working with aggregated data, insights into high level decisions (e.g., system configuration, route shape) can be made in constant time regardless of problem instance size. While discrete optimization techniques are best suited for small instances, approximations based on aggregated data are more
accurate as instances grow larger. The premium of time along with the lack of reliable information due to the chaotic environment following a disaster make continuous approximation an appealing approach for assessment in disaster relief.
Implementation complexity of the routes is important in humanitarian relief practice, and may often dominate concerns
for efficiency. More specifically, routes which traverse a large portion of the service region may not be desirable for drivers,
who would rather wish to visit communities in a compact area. In the literature, implementation complexity has been stud-
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M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
ied mostly in multi-period routing settings. In some industries such as parcel delivery, companies may wish to ensure service
regularity by assigning the drivers to the same customers and geographical areas over time to improve driver familiarity
with the customers and service territories. Zhong et al. (2009) utilize learning curves to model driver familiarity such that
a driver’s performance is a function of the number of times the driver visits a fixed service area over multiple periods. Francis
et al. (2007) use various metrics for measuring implementation complexity in a Periodic VRP (PVRP) setting. The PVRP, introduced in Beltrami and Bodin (1974) and Russell and Igo (1979), is a variation of the VRP in which driver routes are constructed for a period of time to minimize the travel cost while satisfying operational constraints such as vehicle capacity
and customer visit frequency. Francis et al. (2007) evaluate the complexity of routes by measuring the portion of the total
service region covered by a driver, the number of different drivers visiting a customer, and the variability in the customer
visiting times for a period of time. Groer et al. (2009) introduce the Consistent VRP (ConVRP), which require the same drivers
to serve the same customers at approximately the same time on each day. Smilowitz et al. (2013) consider customer and
region familiarity in the context of PVRP; specifically, they measure customer familiarity in terms of the number of times
that a driver visits a customer, and region familiarity in terms of the number of times that a driver repeatedly visits a region.
Different than these studies, ARP is a single period problem. Therefore, metrics from the multi-period settings may not be
directly applicable to measure implementation complexity in the context of ARP. In this study, we evaluate the implementation complexity of ARP routes in terms of the compactness of the service region for each driver. We evaluate different approaches to measure implementation complexity of ARP solutions obtained by discrete and continuous models.
3. Problem setting
The ARP determines the routes of a fixed number of vehicles to visit all communities in a disaster region such that the
sum of arrival times is minimized. We model the problem with a region with N community nodes. Unit demand is assumed
at each node, representing the need to visit the node; no relief items are distributed at this time. In addition to the N nodes, a
depot is located outside of the service region. Unlike commercial settings, where it is preferable to locate the depot centrally,
in disaster relief settings, vehicles are likely available outside the affected region. We denote the depot as node 0 and the
communities as nodes {1, . . . , N} and define tij as the travel time from node i to j; "i, j 2 {0, . . . , N}. A fleet of K uncapacitated
vehicles is initially located at the depot. The vehicles are considered uncapacitated because, in the assessment setting, there
is no limit on the number of points that can be visited. The objective is to visit all the nodes as quickly as possible minimizing
the sum of arrival times.
For ease of illustration, we introduce the following assumptions about the setting; these assumptions do not impact the
application of the discrete and continuous models described in this section, but create a more simplified setting to describe.
As shown in Fig. 1, the service region is rectangularly shaped with length of lr and width wr. The depot is located p units
below the midpoint of the service region’s lower border. Based on an average speed, travel times are set as distances measured using the Euclidean metric.
In the following subsections, we introduce two approaches to solve the ARP: Section 3.1 presents a discrete model of the
ARP and Section 3.2 introduces the continuous modeling approach in which some discrete parameters are approximated
with continuous functions. Fig. 2 displays the input/output relationships between the data, models and solutions for the
ARP solution approaches. The discrete model uses the full level of information provided by the instance data and is solved
using a tabu search heuristic; the objective value of the resulting solution is denoted as ZTabu. The continuous model takes as
Fig. 1. Stylized service region for the ARP.
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
23
Fig. 2. Input/output relationships between data, models and solutions for the ARP.
input an aggregated summary of the instance data and a parameterized policy. A policy is a set of rules that are followed to
generate a routing plan. An ideal policy should be simple enough so that a practitioner can implement the policy without
great computational resources while providing a solution which is reasonably close to the optimal solution. The continuous
model uses the aggregated information to construct analytical expressions which are used to determine the optimal parameter values for the inputted policy as well as provide a cost approximation ZCA of the policy solution. This approximation cost
can be used to evaluate a policy before implementation. The implementation of the resulting strategy generates a solution
whose objective is denoted as ZPolicy. In Section 4.3.2, we introduce a hybrid solution approach which generates solutions by
utilizing the full level of information provided by the instance data while using the continuous model to reduce the solution
space. Objective values of the hybrid solutions are denoted as ZHybrid.
3.1. Discrete model
The advantage of the discrete model is the use of detailed information regarding instances to obtain high-quality solutions. We present an integer program for the discrete model of the problem based on Campbell et al. (2008), with an added
constraint to guarantee K vehicles are used. We use the binary variables xij to determine the routing of the nodes; xij = 1 if
node j follows node i on a route, and 0 otherwise, " i, j 2{0, . . . , N}. The integer variables ai represent the arrival time at node i,
" i 2 {1, . . . , N}.
Z IP ¼ min
N
X
ai
ð1aÞ
i¼1
subject to
X
8i 2 f1; . . . ; Ng
xij ¼ 1
ð1bÞ
j2f0;...;Ng
X
x0j 6 K
j2f1;...;Ng
X
xij j2f0;...;Ng
ð1cÞ
X
xji ¼ 0
8i; j 2 f1; . . . ; Ng
8i 2 f1; . . . ; Ng
8i; j 2 f0; . . . ; Ng
8i 2 f1; . . . ; Ng
tij þ ai 6 aj þ Mð1 xij Þ
ai P t 0i x0i
xij 2 f0; 1g
ai integer
8i 2 f0; . . . ; Ng
ð1dÞ
j2f0;...;Ng
ð1eÞ
ð1fÞ
ð1gÞ
ð1hÞ
The objective function (1a) minimizes the sum of arrival times at all nodes. Constraints (1b) ensure that each community
node is visited. Constraint (1c) guarantees only K vehicles are used by limiting the number of arcs leaving the depot. Constraints (1d) are flow balancing constraints. Constraints (1e) and (1f) ensure that the arrival times are properly assigned
and also act as subtour elimination constraints, with a sufficiently large constant, M. Finally Constraints (1g) and (1h) define
xij as binary and ai as integer, respectively.
Solving the discrete model can be challenging in the context of assessment routing. The ARP is NP-Hard and Campbell
et al. (2008) observe that standard commercial solvers have difficulty in solving the model. Thus, in our computational tests
in Section 5, we obtain an approximation of ZIP with a tabu search heuristic, denoted ZTabu. Appendix A describes the imple-
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M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
mentation of the tabu search. While heuristics such as tabu search reduce the amount of time needed to solve the discrete
model, such approaches still require a considerable amount of computation time and computing resources, as compared
with continuous approximation.
Further, the discrete model is sensitive to input parameter values. If the travel time parameters tij are incorrect, the level
of detail of the discrete model may become a disadvantage. This is especially true in the setting of assessment routing where
reliable information is limited. The detail-oriented solutions from the discrete models may lead to complex routes, in terms
of the geographic composition of routes. While more complex routes may reduce the objective value, they are undesirable
from a practical perspective, as drivers have a preference to serve familiar regions, which are compact and contiguous. The
desire for familiarity may offset gains in the objective value made by the complex routes.
3.2. Continuous approximation model
The continuous model aggregates discrete parameters regarding communities. The aggregation of parameters allows the
model to avoid operational details and focus on higher level decisions. Insight from the continuous model is useful in quickly
determining a routing policy for the ARP. The use of a policy is motivated by three factors: (1) the speed with which solutions
are generated; (2) a reduction in the need for precise information; and (3) the simplicity of the solutions (i.e., the ease to
implement the solutions generated from a policy).
In aggregating the discrete parameters, we replace the physical locations of the nodes with a density function d(x) defined
on a continuum of locations; d(x) represents the number of nodes per unit area about a point x in the service region. Travel
times are converted into distances. The travel times tij are replaced with two location-dependent types of distances: (1) a
linehaul distance L(x) which represents the distance from the depot to the location x and (2) a local distance d(x) which represents the expected distance between successive node visits centered at x and is a function of d(x). In the continuous
approximation literature (Daganzo, 2005), it is shown that if the density function d(x) varies slowly, without a significant
loss of accuracy, the problem may be solved by decomposition into multiple partitions of the service region each with a
near-constant density. For simplicity, we assume the community nodes are evenly distributed yielding a constant density
function d. Thus, the local distance does not change with location. Although the linehaul distance is still dependent on location, for presentation purposes, we omit x from the notation for L and d.
With this continuous approximation, a policy’s performance may be expressed as an analytical expression. This equation
acts as an approximation of the objective value of the policy solution. For a family of parameterized policies, by treating the
parameters as decision variables, optimal parameter values may be determined analytically. Hence families of parameterized
policies may be easily evaluated. Policies are constructed so that the service region is partitioned into contiguous blocks
called vehicle routing zones, each served by a single vehicle. In the practical setting, this allows for solutions that are easier
to implement, allowing drivers to be responsible for a particular area.
Section 4 describes a continuous approximation of the ARP in detail. In Section 4.3.2, we propose a hybrid approach that
combines the continuous and discrete approaches to the problem. Approximations yielded by the continuous model are used
to cluster the nodes into fixed vehicle routing zones. Once the routing zones are fixed, a routing schedule utilizing the location details from the discrete model is determined.
4. Assessment routing with continuous approximation
The continuous model gives insight into policy construction and provides means to approximate the performance of a
routing policy. By approximating policy performance with continuous approximation, one can quickly choose an appropriate
policy immediately after a disaster with limited data. Section 4.1 presents two routing policies and Section 4.2 develops
equations to evaluate these policies. Section 4.3 describes implementation of policy and hybrid solutions once a disaster
occurs.
4.1. Routing policies
This section outlines two routing policies considered for the ARP.
4.1.1. Swath heuristic for node sequencing
The ARP policies analyzed in this paper utilize the swath heuristic from Daganzo (1984b) for the Travelling Salesman
Problem (TSP) to sequence node visits. Fig. 3 illustrates the swath heuristic; a swath (a strip of width w) covers the entire
service area by following some path. Nodes in the service region are visited in order of appearance along the swath. Daganzo
(1984b) report that for the TSP on a square region, the swath heuristic returns a length about 20% higher than empirical solutions. When the swath heuristic is extended to problems with multiple vehicles such as the ARP, a partitioning of the service
region into vehicle routing zones must also be determined. Thus the policy must determine the following: (i) the manner in
which the swath covers the vehicle routing zone; (ii) the width of the swath; and (iii) the partitioning of the service region
into vehicle routing zones.
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
25
Fig. 3. Swath heuristic for the TSP: (a) Using a swath with width w, (b) cover the routing zone by following some path and (c) visit nodes in order of
appearance along the swath.
Fig. 4. (a) Swath for vehicle routing zone for the VRP (Daganzo, 1984b; Newell and Daganzo, 1986) and (b) swath for vehicle routing zone for delivery of
valuable resources (Newell, 1986).
Swath heuristic policies in Daganzo (1984a) and Newell and Daganzo (1986) are designed such that each vehicle routing
zone is an approximate rectangle elongated towards the depot. The vehicle initially travels away from the depot before turning around and traveling back towards the depot; see Fig. 4a. The vehicle routing zone has width 2w. An optimal swath width
w⁄ dependent on the density d is calculated. The optimal length of a routing zone can then be expressed in terms of the customer density and the capacity of the vehicles. Using the optimal width and length found, zones are sequentially inserted
into the service region, starting from the boundary farthest from the depot, making manual corrections to properly tessellate
the zones. Recently, Ouyang (2007) describes a systematic computer algorithm to automate the process of zone creation.
In Newell (1986), a similar analysis is performed for routing the delivery of valuable resources. In this context, the pipeline inventory cost of carrying items on a vehicle may dominate the travel cost; thus, the objective, as in the ARP, minimizes
the time between the start of the route and the delivery of all items. Newell (1986) prescribes a policy where every node is
visited on the outbound trip. Under this policy, each vehicle routing zone once again is approximately rectangular and elongated towards the depot. However, in this case, the swath does not turn within the vehicle routing zone; thus, the width of
the vehicle routing zone is the width of the swath; see Fig. 4b. Under this policy, the optimal width of the swath approaches
0; the vehicle routing zone should be as narrow as possible.
4.1.2. Routing policies with limited vehicles
The approaches in Newell and Daganzo (1986) and Newell (1986) first determine the size of each routing zone, before
partitioning the service region. However, if the number of vehicles is limited, as in the ARP, the partitioning of the service
region into vehicle routing zones is completed first. This ensures that the entire service region is served with the available
number of vehicles.
Ideally, the shape of the vehicle routing zones would mimic the valuable resource example with long narrow zones elongated towards the depot to minimize the linehaul distance. Due to the limit on the number of vehicles and the shape of the
service region, such a partitioning may not always be possible. For example, when applied on a wide service region, the policy used in Newell (1986) of visiting every node on the outbound trip results in very wide vehicle routing zones with swath
widths far greater than the optimal width. Therefore, we evaluate multiple policies based on different partitioning of the service region. The two most promising policies are presented below. Fig. 5 illustrates how the two policies partition their service region and how the swath for each policy covers their routing zones.
26
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
Fig. 5. Example of partitioning of service region into routing zones along with their swath paths for (a) the VST policy and (b) the HDT policy.
The vertically-oriented single tier (VST) policy is an extension of the policy in Newell and Daganzo (1986). The service region is divided into routing zones of equal area by splitting the service region along its length. The resulting routing zones
have width wz = wr/K and length lz = lr. When the service region is wide, the VST policy results in wide vehicle routing zones
and long linehauls. For this reason, a policy which divides the service region horizontally is considered. The horizontally-oriented double tier (HDT) policy partitions the service region by making a single length-wise division along the center of the
service region and subsequent cuts along the width of the service region. The vehicle routing zones are oriented perpendicular to the service region with width wz = 2lr/K and length lz = wr/2.
Once the service region is partitioned into routing zones, the swath heuristic is employed within each routing zone. With
long, narrow vehicle routing zones, the swath should be a single outbound trip. However, if wz exceeds a certain bound, a
single outbound trip results in excessive traveling along the width of the routing zone. Therefore, a narrower swath is used
which makes multiple trips along the length of the routing zone. In the example of Fig. 5, vehicles in the VST policy use three
trips and in the HDT policy use two trips.
4.1.3. Routing policy parameter settings
To determine the optimal swath width w, the average local distance d between consecutive points is decomposed into its
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
transverse and longitudinal components: d ¼ EXY ð X 2 þ Y 2 Þ, where X and Y are random variables which respectively represent the transverse and longitudinal distance between two consecutive nodes. The swath width is optimized when the local
distance is minimized.
As Daganzo (1984a) shows, the random transverse distance X has the same distribution as the random variable representing the distance between two uniformly random points on a line segment of width w; the cumulative distribution function
(cdf) of X is Pr{X > x} = (1 x/w)2, 0 6 x 6 w. The random longitudinal distance Y has the distribution of a Poisson process
with rate dw has cdf Pr{Y > y} = exp(dwy), y P 0. Thus,
EX ¼ w=3 and EY ¼ 1=dw
ð2Þ
It is difficult to express d in terms of w analytically; Daganzo (1984a) gives the following tractable approximation for d:
d¼
w
1
þ
wðdw2 Þ;
3 dw
ð3Þ
where wðxÞ ¼ x22 ðð1 þ
lnð1 þ xÞ xÞ. By minimizing Eq. (3), Daganzo (1984a) reports the optimal value of w for the EuclidpxÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ean metric is w ¼ 2:95=d.
The actual width differs from w⁄, since the width w must accommodate the need to traverse the length of the routing zone
an integer number of times:
w ¼ Min wz ;
wz
roundðwz =w Þ
ð4Þ
The first argument restricts the swath width to the zone width wz when w⁄ P wz. When w⁄ 6 wz, the second argument
applies the constraining correction. Eq. (4) is used in both the VST and HDT policies to determine the swath width.
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
27
4.2. Approximation of policy performance
In this section, we derive equations for ZCA, a cost approximation of policy solutions.
Consider the arrival times for a single vehicle serving n nodes. Because the continuous model uses a constant local distance d between nodes, the arrival times form the terms of an arithmetic sequence with an initial term, L, the linehaul distance for this vehicle, and a common difference of d. The sum of arrival times can be represented as the sum of the first n
terms of an arithmetic sequence:
nL þ
nðn 1Þ
d
2
ð5Þ
The sum of all arrival times, ZCA, is the summation of Eq. (5) for the K vehicles. Note that given constant customer density
and equal vehicle routing zone areas, each vehicle serves the same number of nodes. Thus the value of n is the same for each
vehicle and is equal to the total number of nodes divided by the number of vehicles: N/K. Similarly, the local distance d
(which may be obtained through Eq. (3)) is constant throughout the service region and thus is constant for each vehicle.
Therefore, for the calculation of ZCA, the second term in Eq. (5) can be simply multiplied by K. However, the linehaul distance
L is dependent on location and varies from vehicle to vehicle. Thus,
Z CA ¼ nLT þ K
nðn 1Þ
d
2
ð6Þ
where LT is the total linehaul which can be calculated as a separate sum. The linehaul L for each routing zone is modeled as
the distance from the depot to the first node visited in the routing zone, which is the node closest to the depot. The linehaul
value depends largely on how the service region is partitioned into vehicle routing zones and to a lesser degree on the width
of the swath. The calculation of LT is illustrated with an example for calculation under the VST policy in Fig. 6 for two instances with different number of vehicles.
The expected longitudinal component of the linehaul is the same for each vehicle; the transverse component varies from
vehicle to vehicle. The longitudinal component of the linehaul equals the distance from the depot to the service region plus
the longitudinal distance to the first node from the service region border. The expected longitudinal distance to the first node
li is the length of the routing zone divided by one plus the expected number of nodes visited in one traversal of the length of
the zone. Hence, li ¼ 1þllzzwz d. Thus the expected longitudinal component of the linehaul is p + li.
With an even number of vehicles, as in Fig. 6a, the depot aligns with the border of one of the routing zones. Hence,
the transverse linehaul distance for a vehicle is a multiple of the zone width plus the transverse distance to reach the
first point from the vehicle routing zone border which is half the swath width. Therefore, the expected transverse linehaul distance for a vehicle equals jwz + w/2, where j represents the number of zones (possibly 0) between the border of
the vehicle routing zone served and the center of the service region. With an odd number of vehicles, as in Fig. 6b, the
depot aligns with the middle of the center zone. For the other zones, the expected transverse distance is of the same
form as the even vehicle case with the addition of an extra wz/2 from the center zone. For the center zone, the vehicle
need not go all the way to the zone’s border; the expected transverse distance is half the zone width minus half the
swath width: (wz w)/2.
For simplicity, because the random portion of both the longitudinal and transverse portions of the linehaul are dominated
by their constant portions, we estimate the expected distance of the linehaul by taking the square root of the sum of the
square of the expected longitudinal distance and the square of the expected transverse distance. Utilizing the symmetry
of the solution, the total linehaul LT using the VST policy is approximately:
Fig. 6. Component-wise breakdown of the linehaul for a routing zone using the VST policy in (a) an instance with 4 routing zones (b) an instance with 5
routing zones.
28
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
LT ¼
8 K
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
2
X
2
>
>
>
2
jwz þ w2 þ ðp þ li Þ2
>
>
< j¼0
if K is even
K11
>
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
>
X
2
>
2
>
wz
w 2
>
þ
ðp
þ
l
Þ
þ
2
jwz þ w2z þ w2 þ ðp þ li Þ2 if K is odd
>
i
2
2
:
ð7Þ
j¼0
Note that Eq. (7) is only applicable to the VST policy; equations may be similarly developed for other policies by summing an
approximation for the linehaul for each routing zone in the policy.
With an approximation for LT, Eq. (6) provides an approximation for the policy’s performance that can be quickly calculated. The computational results in Section 5.3 show that the approximation is accurate and effective in predicting the objective value. Hence, different policies can be quickly and accurately compared.
4.3. Implementation of policy and hybrid solutions
This section describes the steps followed to implement policy and hybrid solutions for a given problem instance.
4.3.1. Policy implementation
As input, a policy uses the data from the problem instance and information from the continuous model. In a given instance, the service region dimensions, the location of nodes (represented in Cartesian coordinates), and the number of vehicles are given. From the continuous model the dimensions of the routing zone (wz and lz), the locations of the routing zones
and the swath width for each routing zone are obtained. From this information, it is determined which nodes lie in each vehicle routing zone. This can be easily accomplished by inspection of each location’s coordinates.
For each routing zone, the nodes in the routing zone are then sorted to simulate the swath traveling up and down the
length of the routing zone. The locations are first sorted in the direction of the width of the routing zone; in the VST policy,
this is the x-coordinate and for the HDT policy the y-coordinate. The locations are sorted in ascending or descending order
dependent on the location of the routing zone relative to the depot. The locations are separated into intervals of width w
representing the swath’s successive passes along the length of the routing zone; the nodes in each interval are sorted in
the direction of the length of the routing zone alternating between ascending and descending order for each interval. The
nodes are now sorted in the order they are visited.
4.3.2. Hybrid implementation
The policy solutions are generated with the presumption of limited computing resources. Once the vehicle routing zones
are found from the continuous model, the order in which the service nodes are visited can be found by inspection. However,
if modest computational resources are available, the solutions can be improved by utilizing discrete model methodologies to
determine the order in which the nodes are visited.
The resulting hybrid solutions have three advantages: (1) the routes benefit from exact location data and are not constrained by the swath heuristic; (2) because the solution space is restricted by the routing zones, the computational time
required is much lower than the unconstrained discrete model; and (3) because the points lie within the designated vehicle
routing zones, the solutions are as operationally complex as the policy solutions.
We implement hybrid solutions by applying a restricted tabu search heuristic. Because there is only one route constructed in each routing zone, the only permitted types of moves are string relocation and 2-Opt.
4.3.3. Evaluating solutions
In this section, we evaluate the continuous approximation solutions relative to the discrete solutions. Using notation from
the formulation for the discrete model, once the communities are sorted into the order in which they are visited, the values
for the routing binary variables xij for each pair of nodes may be determined; further the arrival times ai of each node may be
P
calculated. The objective value of the solution is the sum of all the arrival times: Z Policy=Hybrid ¼ Ni¼1 ai . Additionally the realized total linehaul and average local distance can be calculated. The total linehaul is found by summing the distance from the
P
depot to the first node visited in each routing zone: Ni¼1 t0i x0i . The average local distance can be found by averaging the realPN PN
1
ized distance between consecutively visited community nodes: NK
i¼1
j¼1 t ij xij .
5. Computational tests
In this section we present computational tests to assess (1) the effectiveness of the cost approximations from the continuous model in terms of predicting performance of policy solutions and (2) the effect of using simple policies over more
sophisticated techniques in terms of objective value and implementation complexity. Section 5.1 describes the test instances. Section 5.2 presents evaluation of the approximation and Section 5.3 compares the policy and hybrid solutions
against the solutions from the tabu search.
29
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
5.1. Test instances
We consider a range of parameter settings in our tests instances. Three different service regions with equal area are examined: a wide and shallow 200 50 rectangle, a 100 100 square and a narrow and long 50 200 rectangle. The node density of the service region is varied by changing the number of nodes: low-density instances use 480 nodes and high-density
instances use 960 nodes. The effect of the number of vehicles is examined by comparing solutions with 10 and 20 vehicles.
Lastly, two different locations for the depot are considered. We consider instances where the depot is located at the edge of
the service region (p = 0) and instances where the depot is located farther away (p = 100). In total, we consider 2400
instances.
Coordinates for the community node locations are generated by obtaining two random uniform (0, 1) variables and scaling them to the dimensions of the service region. In order to properly compare the effect of different parameter values, the 24
types of instances use the same streams of random numbers to generate community locations for 100 instances. As a result
of using identical streams, with all other parameters constant, the community locations are identical regardless of depot
location; and the relative location of the nodes are the same for the three different shaped service regions.
5.2. Evaluation of approximation
We evaluate the accuracy of the approximations described in Section 4.2 by comparing the approximated values against
the true costs when the policy is implemented for a given realization of community locations.
The VST policy and the HDT policy are applied to the 2400 instances. The key performance metric for comparison is the
objective value of total arrival time of all nodes. In addition, we evaluate the approximations of the total linehaul distance,
the average local distance, and the average transverse and longitudinal components of the local distance. For each instance
and each statistic, the percentage error between the approximated value and the realized value is calculated; e.g., in the case
of the objective value DðzÞ ¼ jZ
CA
Z Policy j
Z Policy
is calculated.
Tables 1 and 2 present summaries of the percentage error for the different types of instances for the VST and HDT policies,
respectively. In both tables, Columns 1–4 describe the type of instance characterized by the node density, number of vehicles, depot location and shape of the service region, respectively. Columns 5–9 report the average (and standard deviation of
the) percentage error over the 100 instances for the different recorded statistics for the instance type specified by the row.
Note that the local distance results reported in Columns 7–9 are the same in the near and far cases because the location of
the depot has no impact on the distance between successive nodes. We make the following observations based on the results
in Tables 1 and 2 regarding the accuracy of the continuous approximation and the sources of approximation errors.
Observation 1. The objective values of VST and HDT policies can be accurately approximated with very small percentage
errors.
Table 1
Accuracy of objective function and distance approximations for the VST policy: average error and standard deviation by instance type.
Parameters
Average error (std. deviation)
Density
Vehicle
Depot
Shape
Objective
Linehaul
Local dist.
Trans. dist.
Long. dist.
Low
10
Near
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
2.3%
2.6%
1.8%
1.5%
1.5%
1.0%
1.9%
1.3%
2.8%
0.9%
0.7%
1.9%
(1.3%)
(1.8%)
(1.5%)
(0.8%)
(1.1%)
(0.9%)
(1.2%)
(1.1%)
(2.1%)
(0.7%)
(0.6%)
(1.2%)
1.4%
2.7%
5.0%
0.7%
0.5%
0.9%
1.0%
1.6%
6.5%
0.3%
0.6%
1.2%
(0.9%)
(2.2%)
(4.2%)
(0.5%)
(0.4%)
(0.8%)
(0.7%)
(1.2%)
(4.6%)
(0.3%)
(0.5%)
(1.0%)
2.1%
2.2%
1.0%
2.1%
2.2%
1.0%
2.1%
1.8%
3.7%
2.1%
1.8%
3.7%
(1.3%)
(1.5%)
(0.8%)
(1.3%)
(1.5%)
(0.8%)
(1.6%)
(1.3%)
(1.4%)
(1.6%)
(1.3%)
(1.4%)
8.1%
3.0%
3.0%
8.1%
3.0%
3.0%
3.0%
3.0%
3.0%
3.0%
3.0%
3.0%
(2.9%)
(2.1%)
(2.1%)
(2.9%)
(2.1%)
(2.1%)
(2.3%)
(2.3%)
(2.3%)
(2.3%)
(2.3%)
(2.3%)
1.3%
0.7%
0.7%
1.3%
0.7%
0.7%
1.1%
1.1%
1.1%
1.1%
1.1%
1.1%
(1.1%)
(0.6%)
(0.6%)
(1.1%)
(0.6%)
(0.6%)
(0.9%)
(0.9%)
(0.9%)
(0.9%)
(0.9%)
(0.9%)
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
2.5%
2.0%
2.1%
1.8%
1.3%
1.3%
1.8%
1.6%
1.3%
1.1%
0.8%
0.8%
(1.2%)
(1.2%)
(1.1%)
(0.8%)
(0.8%)
(0.7%)
(1.0%)
(1.2%)
(1.0%)
(0.6%)
(0.6%)
(0.6%)
0.9%
1.9%
3.2%
0.5%
0.5%
0.5%
0.5%
1.2%
3.7%
0.3%
0.3%
0.7%
(0.8%)
(1.5%)
(2.3%)
(0.4%)
(0.3%)
(0.3%)
(0.4%)
(0.9%)
(3.1%)
(0.3%)
(0.3%)
(0.5%)
2.6%
1.9%
1.7%
2.6%
1.9%
1.7%
1.8%
1.3%
0.8%
1.8%
1.3%
0.8%
(1.1%)
(1.0%)
(0.9%)
(1.1%)
(1.0%)
(0.9%)
(1.0%)
(0.8%)
(0.6%)
(1.0%)
(0.8%)
(0.6%)
6.3%
2.8%
1.8%
6.3%
2.8%
1.8%
4.5%
2.0%
2.0%
4.5%
2.0%
2.0%
(2.2%)
(1.7%)
(1.3%)
(2.2%)
(1.7%)
(1.3%)
(2.3%)
(1.4%)
(1.4%)
(2.3%)
(1.4%)
(1.4%)
0.6%
0.5%
0.4%
0.6%
0.5%
0.4%
0.7%
0.6%
0.6%
0.7%
0.6%
0.6%
(0.5%)
(0.3%)
(0.2%)
(0.5%)
(0.3%)
(0.2%)
(0.5%)
(0.4%)
(0.4%)
(0.5%)
(0.4%)
(0.4%)
Far
20
Near
Far
High
10
Near
Far
20
Near
Far
30
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
Table 2
Accuracy of objective function and distance approximations for the HDT policy: average error and standard deviation by instance type.
Parameters
Average error (std. deviation)
Density
Vehicle
Depot
Shape
Objective
Linehaul
Local dist.
Trans. dist.
Long. dist.
High
10
Near
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
2.2%
2.0%
3.5%
1.2%
1.2%
2.3%
1.4%
2.2%
2.0%
0.7%
1.1%
1.1%
(1.6%)
(1.5%)
(2.1%)
(0.9%)
(0.9%)
(1.4%)
(1.3%)
(1.7%)
(1.5%)
(0.6%)
(0.9%)
(0.9%)
3.2%
1.3%
0.7%
0.6%
0.3%
0.3%
1.8%
1.1%
0.4%
0.2%
0.4%
0.2%
(2.4%)
(1.0%)
(0.5%)
(0.5%)
(0.2%)
(0.2%)
(1.4%)
(0.8%)
(0.3%)
(0.2%)
(0.3%)
(0.1%)
1.8%
2.0%
5.5%
1.8%
2.0%
5.5%
1.8%
1.7%
1.6%
1.8%
1.7%
1.6%
(1.3%)
(1.4%)
(1.8%)
(1.3%)
(1.4%)
(1.8%)
(1.0%)
(1.3%)
(1.4%)
(1.0%)
(1.3%)
(1.4%)
2.7% (2.2%)
8.1% (3.2%)
14.4% (2.7%)
2.7% (2.2%)
8.1% (3.2%)
14.4% (2.7%)
2.6% (2.0%)
2.6% (2.0%)
14.5% (2.9%)
2.6% (2.0%)
2.6% (2.0%)
14.5% (2.9%)
0.7%
1.2%
1.8%
0.7%
1.2%
1.8%
1.0%
1.0%
2.5%
1.0%
1.0%
2.5%
(0.6%)
(1.0%)
(1.5%)
(0.6%)
(1.0%)
(1.5%)
(0.9%)
(0.9%)
(1.7%)
(0.9%)
(0.9%)
(1.7%)
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
2.1%
2.2%
3.5%
1.3%
1.4%
2.4%
1.6%
1.5%
1.7%
0.8%
0.8%
1.1%
(1.4%)
(1.3%)
(1.5%)
(0.9%)
(0.8%)
(1.0%)
(1.1%)
(1.1%)
(1.2%)
(0.5%)
(0.6%)
(0.7%)
1.8%
1.0%
0.6%
0.3%
0.3%
0.2%
1.3%
0.6%
0.3%
0.2%
0.2%
0.1%
(1.6%)
(0.7%)
(0.4%)
(0.2%)
(0.2%)
(0.1%)
(1.0%)
(0.4%)
(0.2%)
(0.2%)
(0.1%)
(0.1%)
1.8%
2.5%
5.0%
1.8%
2.5%
5.0%
1.1%
1.7%
2.7%
1.1%
1.7%
2.7%
(1.0%)
(1.1%)
(1.3%)
(1.0%)
(1.1%)
(1.3%)
(0.8%)
(0.9%)
(1.2%)
(0.8%)
(0.9%)
(1.2%)
2.5% (1.9%)
6.0% (2.4%)
10.9% (2.0%)
2.5% (1.9%)
6.0% (2.4%)
10.9% (2.0%)
2.0% (1.4%)
4.3% (2.2%)
11.0% (1.9%)
2.0% (1.4%)
4.3% (2.2%)
11.0% (1.9%)
0.4%
0.6%
0.8%
0.4%
0.6%
0.8%
0.5%
0.7%
1.1%
0.5%
0.7%
1.1%
(0.3%)
(0.4%)
(0.7%)
(0.3%)
(0.4%)
(0.7%)
(0.3%)
(0.5%)
(0.9%)
(0.3%)
(0.5%)
(0.9%)
Far
20
Near
Far
High
10
Near
Far
20
Near
Far
According to the results, the average percentage error is 1.7% across all cases; further, the worst case error is 10%, and the
error is less than 3% in 85% of all the cases. The magnitude of errors is similar in both policies. These small errors compromise
of random error from the random realizations of the instances and two types of systematic errors arising from the sources
discussed below.
Observation 2. In general, policy approximations are more accurate when the swath makes fewer turns.
One component of the accuracy error is due to ignored swath turns in continuous approximation. The continuous approximation tends to underestimate local distances because turns of the swath are not considered in the approximation of the
local distance and its components (Eqs. (2) and (3)). This underestimates the zone’s total transverse distance and to a lesser
degree overestimates the zone’s total longitudinal distance with a net result of underestimating the total distance. The frequency with which the approximations underestimate and overestimate the realized values is reported in Tables 7 and 8 in
Appendix B. As a result of ignoring turns of the swath, errors for the transverse and longitudinal distance are greater in instances where the swath travels the length of the vehicle routing zone multiple times. For the VST policy these instances
correspond to instances with a wide service region, and for the HDT policy these instances correspond to instances with
a narrow service region. For example, under the VST policy, in the low-density 10-vehicle instances the swath only travels
the length once for the square and narrow instances, but travels the length three times in the wide instances. As a result, the
error of the transverse distance is 8.1% in the wide instances, and 3.0% in square and narrow instances; similarly, the error of
the longitudinal distance is 1.3% in the wide instances, and 0.7% in square and narrow instances (Table 1). The average percentage errors for the VST policy for wide, square and narrow instances are 1.7%, 1.5% and 1.6%, respectively; and for the HDT
policy the average percentage errors for wide, square and narrow instances are 1.4%, 1.5% and 2.2%, respectively.
Observation 3. Small error is introduced in approximating Euclidean distances for local and linehaul distances.
In general, the error introduced through the use of Daganzo’s approximation for calculating the local distance by Eq. (3) is
acceptably small. However, due to the substitutions made in the derivation of Daganzo’s approximation, there are certain
values of the swath width w in which the approximation is less accurate. For example, when w is close to 0, the error increases. This is seen under the VST policy in Table 1 for the low-density 20-vehicle instances. Despite having the same transverse and longitudinal percent error for all the service region shapes (3.0% and 1.1%, respectively), the average percentage
error for the local distance differs (2.1%, 1.8% and 3.7% for wide, square and narrow service regions, respectively).
Error is also introduced by estimating the linehaul distance as the square root of the sum of squares of the expected transverse and longitudinal components; this underestimates the true expected linehaul. The error is greatest when the constant
portion of the longitudinal and transverse components of the linehaul distance is the smallest. In general this occurs when
the depot is located on the border of the service region. From the results, for the VST policy, the average linehaul errors for
instances with near and far depot are 2.5% and 0.6%, respectively; similarly, for the HDT policy, the average percentage errors
for the linehaul distance are 1.2% and 0.3% for instances with near and far depot, respectively.
31
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
Table 3
Using approximations to determine routing policy: approximated and simulated results.
Parameters
Chosen policy
Gap between two policies
Approximated values (%)
Average of realized values (%)
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
HDT
VST
VST
VST
VST
VST
HDT
VST
VST
VST
VST
VST
12.7
12.7
44.5
6.0
16.9
29.5
24.1
24.1
26.0
8.3
22.7
17.3
13.3
12.4
48.6
5.4
16.6
32.2
25.7
25.8
30.2
7.3
23.8
19.9
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
HDT
VST
VST
VST
VST
VST
HDT
VST
VST
VST
VST
VST
12.3
12.3
45.4
3.5
15.1
32.5
23.8
23.8
39.7
5.1
23.7
26.2
12.8
12.5
47.6
3.0
15.2
34.1
24.2
23.8
42.0
4.6
2.8
27.7
Density
Vehicle
Depot
Shape
Low
10
Near
Far
20
Near
Far
High
10
Near
Far
20
Near
Far
Overall, the low percentage errors imply that the policy solutions are not very sensitive to the true realized locations of
the community nodes; that is, one can make reasonably accurate approximations of policy performance based on aggregate
data. The accuracy of the approximations justifies the use of the continuous models to quickly evaluate which family of policies (and subsequently which policy) should be used for a type of instance. For example, by comparing the approximated
performance of the VST and HDT policies, it can be determined which policy should be implemented for a given instance
type. For different instance types, Table 3 reports the policy suggested by the approximation, the percentage gap between
the
approximations
Z Policy ðNot ChosenÞZ Policy ðChosenÞ
,
Z Policy ðChosenÞ
CA
Z CA ðNot ChosenÞZ CA ðChosenÞ
,
Z CA ðChosenÞ
and
the
percentage
gap
between
the
average
realized
costs
where Z(Chosen) is the objective value of the policy solution determined to perform best using
and Z(Not Chosen) is the objective value of the other policy solution.
According to Table 3, the gap between the approximations of the two policies (reported in Column 6) is very similar to the
gap of the average realized costs (reported in Column 7). This is consistent with the low percentage error between the
approximations and the realized values. Moreover, when the realized instances are examined, in all but one of the 2400 instances, the policy chosen by the approximation has a lower realized cost; in that one instance where the chosen policy has
higher cost, the gap of the realized costs is less than 0.1%.
Table 3 allows us to compare the performances of VST and HDT policies for different types of instances. From the table, we
make the following observations.
Z
Observation 4. In general, the VST policy performs better than the HDT policy in instances with narrow service regions.
As seen in Table 3, VST is the chosen policy in all instances except some instances with wide service regions. This is expected as when the service region is wide, the VST policy results in wide vehicle routing zones and the total local distance
increases due to increased zigzagging behavior in each swath. Also, as the service region becomes wider, total linehaul
distance increases. Therefore, VST will perform better for narrow service regions and HDT may become the chosen policy
as the size of the service region increases. The point at which HDT may begin to outperform VST also depends on the values
of other parameters such as the depot distance.
Observation 5. In general, the VST policy performs better than the HDT policy when the distance between the depot and the
service region is far.
As observed from Table 3, the VST outperforms the HDT policy in all instances in which the depot is far. Even in the instances with wide service regions, in which using the HDT policy may be more advantageous (by Observation 4), the VST is
the recommended policy when the depot is far. This is intuitive as an increase in depot distance does not affect the
32
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
transverse component in both policies; however, the increase in the longitudinal distance has a larger effect on the total linehaul distance in the HDT policy compared to the VST policy. Therefore, VST will likely to outperform HDT for far depot
distances.
While service zone width and the depot distance can be important factors in choosing a policy as discussed in Observations 4 and 5, the other policy parameters (i.e., the number of vehicles and the node density) do not appear to affect policy
selection decisions according to the results in Table 3. When we perform some additional tests to check the sensitivity of the
difference between policy performances with respect to these parameters, the results confirm our observations from Table 3.
That is, the same policy is recommended as the values of the node density or the number of vehicles is systematically changed under fixed values for other parameters. However, the changes in these parameters may affect the extent to which one
can do better with the selected policy over the other policy.
5.3. Evaluation of policy derived solutions
As discussed in Section 3, the use of aggregated data facilitates the derivation of simple, robust solutions with limited
computational requirements. However, the quality of the solutions in terms of arrival times may be inferior to those obtained with more sophisticated approaches. By restricting solutions to simplified policies, the solution space itself is restricted, which may lead to suboptimal solutions. We compare the policy and hybrid solutions with unrestricted
solutions which utilize all of the instance data. We analyze under what conditions the gap in objective values is small
and evaluate the implementation complexity of the solutions.
The evaluations are performed on 10 instances for each instance type using both the policy derived approach and a tabu
search. The tabu search procedure (described in Appendix A) is coded in C++ and tested on a 2.4 GHz 64-bit (4 MB L2 cache)
CPU machine with 8 GB of RAM. To improve the quality of the benchmark, the tabu search is run with two different initial
solutions. The best resulting solution is used to evaluate ZTabu. The policy (either VST or HDT) that performs best for the instance using ZCA is used to determine ZPolicy.
5.3.1. Arrival time comparison
Policy
Z Tabu
Table 4 reports the percentage gaps between ZPolicy and ZTabu in terms of Z ZTabu
for each problem setting aggregated over
the 10 instances. Similarly, Table 5 reports the percentage gaps between ZHybrid and ZTabu in terms of Z
Hybrid
Z Tabu
.
Z Tabu
The full results
for individual instances are presented in Tables 9 and 10 in the Appendix C. In the tables, Columns 2–4 report the percentage
error for the different-shaped instances where the depot is located at the border of the service region, and Columns 5–7 report the percentage error for the different-shaped instances where the depot is located away from the service region. Note
Table 4
Solution gap between policy and tabu search.
Near
Far
Wide
(HDT)
Square
(VST)
Narrow
(VST)
Wide
(VST)
Square
(VST)
Narrow
(VST)
Low density
10 Vehicles
20 Vehicles
36.0%
19.3%
37.2%
19.6%
7.8%
5.5%
17.4%
10.5%
12.1%
2.3%
2.2%
0.6%
High density
10 Vehicles
20 Vehicles
34.3%
22.8%
37.3%
24.0%
14.6%
6.8%
19.6%
12.9%
15.9%
4.8%
6.7%
1.2%
Wide
(HDT)
Square
(VST)
Narrow
(VST)
Wide
(VST)
Square
(VST)
Narrow
(VST)
Low density
10 Vehicles
20 Vehicles
24.1%
18.5%
26.6%
18.5%
6.9%
5.5%
14.2%
7.2%
6.5%
1.9%
1.7%
0.5%
High density
10 Vehicles
20 Vehicles
28.2%
19.3%
29.8%
19.8%
13.1%
6.6%
12.5%
11.3%
11.9%
3.1%
4.0%
1.1%
Table 5
Solution gap between hybrid and tabu search.
Near
Far
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
33
with the exception of instances reported in Column 2, all reported instances use the VST policy/routing zones; instances reported in Column 2 use the HDT policy/routing zones. We make the following observations from the results.
Observation 6. The solutions obtained with the policies derived from the continuous approximation can yield large gaps in
solution quality, relative to the tabu search solutions, depending on the problem instance characteristics.
According to Table 4, the average gap for policy solutions is 15.5%; the gaps range from 0.4% to 43.3% across different instances. The results also show that the performances of policy solutions depend largely on the type of instance. Specifically,
continuous approximation solutions perform better when: (i) the shape of the service region dictates narrower routing zones
(narrow service region for the VST policy; wide service region for the HDT policy), (ii) more vehicles are available (resulting
in narrower routing zones), (iii) the depot is located far away from the service region, and (iv) the node density is low. Under
such favorable combinations of parameters, the percentage gaps between ZPolicy and ZTabu are reasonably small; specifically,
the average gap is 1.4% for instances with narrow or square routing zones, a large number of vehicles, a far depot, and low
node density. We discuss the factors which makes continuous approximation more competitive below. Accordingly, one can
choose to use continuous approximation versus more sophisticated methods in practice depending on the instance
characteristics.
The shapes of the vehicle routing zones (as dictated by the implemented policy) and the service region and the number of
vehicles have the largest effect on the gap between the tabu solutions and the policy solutions. According to results, gaps are
quite small (ranging from 0.4% to 7.6%) for the instances with narrow service regions and a large number of vehicles. Newell
and Daganzo (1986) report an approximate 20% gap between the swath heuristic and empirical methods for the TSP. The
authors identify that the gap may be reduced by (1) allowing vehicles to cross the swath border (2) allowing ‘‘longitudinal
backtracking’’ which relaxes the requirement that nodes are visited strictly in order of appearance along the swath and (3)
altering the method in which the swath covers the region. Newell and Daganzo (1986) also note that the swath heuristic is
more competitive when solving the VRP because elongation of the routing zones reduces the opportunity for those improvements. Similarly, narrow routing zones for the ARP reduce the opportunity for improvement of policy solutions resulting in a
lower percentage gap. To improve the gap when instance conditions force the routing zones to be wide, the VST and HDT
policies might adopt some of the gap-reducing improvements suggested in Newell and Daganzo (1986) for the TSP swath
heuristic. Although not related to the shape of the routing zones, the reduction of opportunities for improvement also explains the preference for low node densities; that is, fewer nodes yield fewer opportunities for improvement resulting in
a smaller percentage gap. Indeed, according to results, given low densities, instances with narrow but a small number of
vehicles are also quite competitive; the average gap for these instances is 4.9% and the gaps range from 1.4% to 9.5%.
According to results, the percentage gap is lower when the depot is far from the service region. Specifically, across all
problem instances, the average and maximum gaps for policy solutions when the depot is far are 8.8% and 21.1%, respectively, while the average and maximum gaps are 22.1% and 43.3% when the depot is near the service region. Under the
VST and HDT policy solutions, we observe an increase in the linehaul distance as vehicles must travel a moderate distance
to access their routing zones while vehicles in the tabu solutions visit nodes immediately upon reaching the service region.
The longer linehauls in the policy solutions lead to higher objective values contributing to the percentage gap. The linehaul
affects the percentage gap the least when the depot is located far from the service region as the tabu solutions must travel a
similar linehaul distance to access the service region. The effect of the linehaul may be reduced if a policy were to partition
the service region so that vehicle routing zones were shaped as wedges pointed toward the depot instead of rectangles. By
shaping routing zones as wedges, the distance required to access the routing zones would be minimized reducing the difference in linehaul. Wedge shaped service regions may be created by adopting polar co-ordinates (with the depot at the origin) instead of Cartesian co-ordinates. However, wedge shaped zones are complicated in a rectangular service region. The
boundary of the service region results in routing zones with uneven length. With different routing zone lengths, additional
calculations must be made to determine the angles used to partition the service region into routing zones of equal area.
Moreover, the routing zone length changes within a routing zone; thus the swath ‘‘width’’ or angle would not be constant
within a routing zone, changing each traversal of the routing zone.
Observation 7. The hybrid solution approach reduces the solution gap by improving route sequences.
According to the results in Tables 4 and 5, hybrid solutions provide more competitive results than the policy solutions for
each group of instances. The average gap across all instances is 12.2% for hybrid solutions. The results also show that the
performance of hybrid solutions is similarly affected by parameter combinations. Under favorable instance characteristics
(discussed in Observation 6), the average percentage gap between ZHybrid and ZTabu is 1.2%. In other instances with undesirable conditions, percentage gap can be as high as 37%.
5.3.2. Implementation complexity comparison
While one can obtain high quality solutions with tabu search within reasonable times with a fast computer, the continuous approximation method is just a simple calculation and does not require a high quality computer. Moreover, even if that
the time required for tabu search may not be an obstacle for some agencies, the proposed methods may be preferable in
post-disaster situations due to their solutions with low operational complexity. More specifically, the policy and hybrid solutions are easier to implement since they require each driver to only visit communities in its vehicle routing zone and hence
34
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
drivers of the vehicles do not need to be familiar with the entire service region. In contrast, the tabu solutions are not restricted to routing zones and routes may visit a large portion of the service region; further, in certain areas, the zones for
tabu solution routes may overlap requiring more than one driver to be familiar with the area. Moreover, we find that the
routes generated by policy solutions may fit well to the sampling schemes followed by survey teams doing needs assessments in the field (see e.g., Daley et al., 2001; Johnson and Wilfert, 2008). For instance, in the clustering sampling method
explained in Johnson and Wilfert (2008), the disaster region is divided into mutually exclusive blocks with well-defined
boundaries, where street grids or geographical features such as rivers are considered to be good boundaries. Also, the sampling scheme described in Daley et al. (2001) leads to routes one can obtain by using the swath heuristic. Therefore, vehicle
routing zones created by using continuous approximation would be easy to apply in real-world applications. Depending on
the agencies’ access to computational resources, the policy or hybrid approach can be used.
Although the increased complexity of the tabu solutions is apparent when the solutions are visualized, the implementation complexity of a route is difficult to quantify. As discussed in Section 2, the existing studies address implementation complexity of routes in a multi-period setting, and hence may not directly apply to ARP which involves a single period. We use
three different metrics for quantifying and evaluating the implementation complexity of the ARP solutions. Because the
routes of the hybrid solutions differ from the routes of policy solutions only in the order of the visited nodes for each vehicle
(and not by which nodes are visited), for the remainder of this section we focus only on the comparison between the policy
solutions and the tabu solutions.
Driver coverage. To measure operational complexity of ARP solutions, a method is adapted from Francis et al. (2007),
which introduce a driver coverage metric to quantify operational complexity for the PVRP. The driver coverage metric measures the portion of the service region visited by a vehicle. Drivers who routinely visit nodes in the same geographic region
throughout the multi-period planning horizon become more familiar with that region, and hence, performance improves. To
compute the metric, Francis et al. (2007) partition the service region into cells. A cell is considered covered by a driver if the
driver visits one node within a cell at least once over the planning horizon; solutions are considered less complex when the
number of cells (NC) covered by each individual driver is minimized. We adapt the driver coverage metric to the ARP by
counting the number of cells visited in a day by each driver and report the average.
Bounding box. The bounding box approach measures, for each vehicle, the area of the smallest rectangle oriented along
the axes which envelopes all the community nodes visited by the vehicle. That is, the area of coverage is measured directly
without partitioning into cells.
Convex hull. Similar to the bounding box approach, the convex hull approach focuses on the area covered by each vehicle.
Specifically, we measure the area of the convex hull of the visited nodes.
Table 6 compares the policy solutions and the tabu solutions for implementation complexity in terms of each metric defined above. Columns 1–4 describe the instance type. Columns 5–7 report the percent difference in number of cells visited
Tabu Policy NC
NC
for cells of dimensions 25 25, 10 10 and 5 5 for the driver coverage metric. Columns 8–9 report the
NC Policy
Table 6
Comparison of implementation complexity.
Parameters
Cell
Area
Density
Vehicle
Depot
Shape
25 25 (%)
10 10 (%)
5 5 (%)
Bounding box (%)
Convex hull (%)
Low
10
Near
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
63.4
17.5
4.3
46.5
15.4
5.4
84.1
13.1
10.2
39.9
12.6
2.6
62.2
60.4
15.2
48.1
60.7
16.0
108.1
15.7
1.8
77.1
18.0
9.8
7.2
9.1
18.6
4.7
8.3
18.9
17.9
19.3
5.9
10.7
19.6
10.3
206.2
226.9
256.7
157.7
182.4
241.0
393.5
403.7
399.3
250.8
99.8
163.5
93.4
103.1
144.1
75.1
88.0
115.2
91.1
110.9
154.6
70.0
39.9
64.2
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
22.3
27.1
8.6
15.6
24.6
10.6
77.3
10.9
5.8
43.7
6.7
0.4
89.1
95.9
24.7
84.2
91.3
22.3
49.4
4.0
1.9
26.8
1.2
2.4
17.6
19.1
29.7
15.8
17.4
28.6
13.1
0.3
0.2
4.4
0.1
3.3
241.9
261.0
288.3
217.7
195.5
212.0
422.7
454.9
449.3
278.9
146.6
149.0
123.6
121.2
169.2
119.4
105.1
113.4
146.5
166.9
208.1
111.4
72.7
73.2
Far
20
Near
Far
High
10
Near
Far
20
Near
Far
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
percent difference of the coverage area
AreaTabu AreaPolicy
AreaPolicy
35
for the bounding box and the convex hull metrics, respectively. We
make the following observation from based on the results.
Observation 8. In general, tabu solutions are more complex to implement than policy solutions.
According to the results in Table 6, we observe that the tabu solutions are more complex to implement as each metric has
a positive percent difference for the majority of its results. This strongly suggests that the tabu solutions require drivers to be
familiar with larger areas. Hence, although the policy solutions yield solutions with a higher objective value than the tabu
solutions, the policy solutions are operationally simpler and easier to implement.
Although results clearly suggest that policy solutions are easier to implement than tabu solution in practice, it is difficult
to assess the exact degree of implementation complexity from these results since each metric may underestimate/overestimate the complexity of the solutions in different instances. The shortcomings of each approach to accurately quantify implementation complexity are discussed next.
Observation 9. The dimensions of the cells in the driver coverage metric may cause overestimation or underestimation of
the actual area covered by vehicles.
When adapting the driver coverage metric to the ARP, results appear to be influenced by the size of the cells. Specifically,
when the cells are large, the metric may overstate the area covered by the vehicles and a large area can be attributed to a
vehicle for visiting a single node in the cell. Fig. 7a shows three routes each of which travels the length of two cells along a
swath 1/3 of the width of a cell. Despite visiting only a third of each cell, under the driver coverage metric, each route
counts both cells in its entirety. This overstatement of area may significantly impact the VST policy solution in instances
with narrow service region. As shown in the figure, the routes in the solution visit and count all the cells along the service
region’s length, despite visiting only a thin fraction of each cell. With the exception of the instances with narrow service
regions, the overstatement of area visited has a larger effect in the measurement of tabu solutions than the policy solutions.
From Table 6, the results obtained by partitioning the service region into 25 25 cells and 10 10 cells suggest that the
cells are too large; we observe large gaps when the service region is wide, which reduces dramatically for narrow service
regions.
When the cells are smaller, area may be understated especially in the tabu solutions. As shown in Table 6, when the service region is partitioned into 5 5 cells, the gaps are smaller in general. When counting the number of cells, the measure
fails to consider where the cells are located. Fig. 7b shows a route from a tabu solution and the cells which are covered. The
visited cells highlighted in gray do not form a contiguous block; hence, the number of cells do not accurately represent the
area of the service region with which a driver would be familiar. Results may also be influenced by where the cell borders lay.
Fig. 7c shows how the same route can cover considerably fewer cells when the cell borders are shifted. In some instances, the
borders of the vehicle routing zone line up with the cell borders causing the policy solution to visit far fewer cells than the
tabu solution.
Observation 10. Bounding box and convex hull approaches may highly overestimate the coverage areas of vehicles.
As shown in Fig. 8a, the bounding box approach can dramatically overstate the area. The gray areas at the corners of the
box are counted in the calculation of the area but are not actually visited by the route. Further, the metric can be inconsistent
in its measurement for a route. Fig. 8b shows how the rotation of a route by a small angle can more than double the area of
Fig. 7. Problems with the driver coverage metric for the ARP: (a) when cells are large, area is overstated: each route only uses 1/3 of the area of each cell but
the area of the entire cell is counted; (b) when cells are small, the metric does not distinguish which cells are counted: the cells covered by a driver in a tabu
solution may be discontiguous; and (c) the metric is dependent on cell borders: shifting the boundaries of the cells can reduce metric value.
36
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
Fig. 8. Problems with directly measuring the area. In bounding box approach: (a) the area is overstated if route travels at an angle, (b) the same route
undergoing a small rotation can double the area; In convex hull approach: (c) the area is preserved under rotation, and (d) the area that is not served can
still be counted.
the bounding box. As a result, this metric is favorable to policy solutions whose vehicle routing zones form natural bounding
boxes. The tabu solutions which are more likely to travel at an angle, typically have bounding boxes with 2–5 times larger
area than the policy solutions (Table 6).
As Fig. 8c illustrates, the area of the convex hull is unaffected by the rotation of a route; thus, it provides a more accurate
comparison than the bounding box method. However, as Fig. 8d demonstrates, the convex hull can still overstate the covered
area by including an area which is not actually visited by a route.
In summary, the results show that while each metric has shortcomings, when different metrics are used concurrently, the
benefit of the vehicle routing zones from the continuous-approximation-based policy becomes apparent. The shortcomings
of the metrics prevent us from accurately comparing the implementation complexity of policy and tabu solution. Therefore,
there is a need for developing further methods for measuring implementation complexity in single-period routing settings
and evaluating the true value of the gap between policy and tabu solutions in ARP.
6. Conclusion
In this paper, we introduce the ARP in disaster relief and the use of continuous approximation in solving the problem.
Specifically, we show how a continuous model may be used to generate easy-to-solve policies for the ARP, approximations
to evaluate these policies, and hybrid solutions which can be generated with modest computing resources. While a simplified setting is used to illustrate the analysis, the ideas presented can be adapted for service regions of different dimensions,
different locations of the depot and non-constant demand densities.
Our numerical tests show that the approximations for the policy can approximate the true cost of implementing the policy quickly and accurately. We also compare the policy and hybrid solutions to solutions from more sophisticated techniques
and show that solutions obtained using a continuous model can be very competitive in many settings while yielding solutions which are easier to implement. Hence, decisions may be determined quickly addressing the time-sensitive nature of
assessment routing. The successful use of aggregate data also suggest that the policy solutions are insensitive to parameter
error, which is important in the humanitarian context given the limited information.
Avenues of future work include determining the effect of relaxing simplifications used in our setting such as allowing
for irregularly shaped service regions and non-constant demand densities. Most of the continuous approximation models
in the literature assume either constant or slowly-varying demand over the service region. There are few studies that
consider general demand density functions in continuous location and transportation models (e.g., Murat et al., 2010).
Therefore, studying ARP with non-constant node density would contribute both to disaster relief and continuous
approximation literature. In ARP with variable node density, the vehicle routing zones will no longer be of the same
shape. In certain settings, one may be able to decompose the problem to allow zone of different shape, which may also
37
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
lead to the development of new routing policies. Future work can also focus on developing new metrics for accurately
evaluating the implementation complexity of routes in the single-period routing setting of ARP. Moreover, the degree of
robustness of the policy solutions may be further investigated. Policy solutions can be compared against stochastic
two-stage models to properly evaluate how well the continuous model addresses parameter error. Lastly, one may also
consider utilizing continuous approximation on other disaster relief routing problems such as routing for the delivery of
aid.
Appendix A. Description of the tabu search heuristic
Tabu search is a popular metaheuristic (see Glover, 1990a,b) that attempts many local changes to a current solution in
sequence. A tabu list is created that prohibits certain recent moves in an effort to avoid cycling. However, if a move in
the tabu list yields a solution better than the current best solution, the move is made ignoring the tabu list. When a new
best solution is found, the tabu list is cleared and the number of iterations is reset. The heuristic terminates if no better solution is found after a given number of (in our case 50) iterations.
Tabu search is a deterministic procedure that depends on the initial solution. In our algorithm, we use two different initial
solutions, one of which is obtained from a randomized insertion heuristic and the other is a policy solution from continuous
approximation. We consider a number of different types of moves including string relocation, string swap, 2-Opt and string
cross, which are illustrated in Fig. 9 and described below. The tabu list keeps track of two types of moves: node relocation to
prevent nodes returning to their original routes, and string crosses to prevent the same string cross being executed twice. We
keep 35 moves in the tabu list.
String relocation takes a string of n (610) consecutive nodes on a route and moves them either elsewhere in the same
route or onto a different route (i.e., node relocation). A string relocation which moves nodes on the same route is only allowed if it improves the current solution. The move does not require checking the tabu list and is not recorded onto the tabu
list. A relocation of nodes to a different route is treated differently. If any of the nodes in the string are forbidden by the tabu
list to move to the destination route, the entire relocation is not allowed. The relocation is recorded onto the tabu list as n
separate node relocations.
A string swap move swaps the location of two equal n length strings on separate routes. When checking if a swap is on the
tabu list, only the first nodes of each string is allowed to move onto their respective destination routes, and hence it is possible for other nodes in the string to return to a route through a swap move. Swaps are recorded onto the tabu list as 2n
separate node relocations.
Different routes:
Initial solution
String relocation
String swap
String cross
Same route
Initial solution
String relocation
2-Opt
Fig. 9. Illustration of moves investigated in tabu search.
38
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
2-Opt swaps the endpoint of one edge with the origin of another edge on the same route. The portion of the route between
the two edges is reversed. Similar to string relocation, the move is only considered if it improves the current solution and the
move is not recorded onto the tabu list.
String cross swaps the endpoints of two edges on different routes. Because this move can relocate many nodes leading to
very large changes in the current solution, string crosses are entered into the tabu list as their own type of move. Additionally, we require that the difference between the number of crossed edges is less than or equal to the difference in route
Table 7
Number of instances where approximated value underestimates/overestimates the realized value for the VST policy.
Parameters
() Underestimation (+) overestimation of approximation
Density
Vehicle
Depot
Shape
Objective
Linehaul
Local distance
Trans. dist.
Long. dist.
Low
10
Near
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
()88
()76
()66
()90
()77
()61
()73
()45
()15
()78
()36
()12
(+)12
(+)24
(+)34
(+)10
(+)23
(+)39
(+)27
(+)55
(+)85
(+)22
(+)64
(+)88
()65
()56
()67
()52
()55
()46
()64
()75
()67
()56
()49
()47
(+)35
(+)44
(+)33
(+)48
(+)45
(+)54
(+)36
(+)25
(+)33
(+)44
(+)51
(+)53
()90 (+)10
()69 (+)31
()65 (+)35
()90 (+)10
()69 (+)31
()65 (+)35
()46 (+)54
()15 (+)85
()0 (+)100
()46 (+)54
()15 (+)85
()0 (+)100
()100 (+)0
()56 (+)44
()56 (+)44
()100 (+)0
()56 (+)44
()56 (+)44
()50 (+)50
()50 (+)50
()50 (+)50
()50 (+)50
()50 (+)50
()50 (+)50
()38
()49
()49
()38
()49
()49
()40
()40
()40
()40
()40
()40
(+)62
(+)51
(+)51
(+)62
(+)51
(+)51
(+)60
(+)60
(+)60
(+)60
(+)60
(+)60
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
()93
()95
()94
()97
()95
()91
()92
()86
()49
()94
()86
()41
(+)7
(+)5
(+)6
(+)3
(+)5
(+)9
(+)8
(+)14
(+)51
(+)6
(+)14
(+)59
()54
()64
()65
()51
()52
()46
()55
()63
()74
()53
()51
()48
(+)46
(+)36
(+)35
(+)49
(+)48
(+)54
(+)45
(+)37
(+)26
(+)47
(+)49
(+)52
()99
()96
()95
()99
()96
()95
()89
()80
()21
()89
()80
()21
()100 (+)0
()86 (+)14
()64 (+)36
()100 (+)0
()86 (+)14
()64 (+)36
()97 (+)3
()58 (+)42
()58 (+)42
()97 (+)3
()58 (+)42
()58 (+)42
()38
()53
()54
()38
()53
()54
()33
()48
()48
()33
()48
()48
(+)62
(+)47
(+)46
(+)62
(+)47
(+)46
(+)67
(+)52
(+)52
(+)67
(+)52
(+)52
Far
20
Near
Far
High
10
Near
Far
20
Near
Far
(+)1
(+)4
(+)5
(+)1
(+)4
(+)5
(+)11
(+)20
(+)79
(+)11
(+)20
(+)79
Table 8
Number of instances where approximated value underestimates/overestimates the realized value for the HDT policy.
Parameters
() Underestimation (+) overestimation of approximation
Density
Vehicle
Depot
Shape
Objective
Linehaul
Local distance
Trans. dist.
Long. dist.
Low
10
Near
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
()76
()79
()95
()74
()76
()95
()45
()74
()68
()38
()73
()67
(+)24
(+)21
(+)5
(+)26
(+)24
(+)5
(+)55
(+)26
(+)32
(+)62
(+)27
(+)33
()52
()58
()60
()49
()53
()54
()77
()56
()70
()59
()52
()62
(+)48
(+)42
(+)40
(+)51
(+)47
(+)46
(+)23
(+)44
(+)30
(+)41
(+)48
(+)38
()62 (+)38
()81 (+)19
()100 (+)0
()62 (+)38
()81 (+)19
()100 (+)0
()11 (+)89
()52 (+)48
()71 (+)29
()11 (+)89
()52 (+)48
()71 (+)29
()46 (+)54
()100 (+)0
()100 (+)0
()46 (+)54
()100 (+)0
()100 (+)0
()55 (+)45
()55 (+)45
()100 (+)0
()55 (+)45
()55 (+)45
()100 (+)0
()52 (+)48
()40 (+)60
()22 (+)78
()52 (+)48
()40 (+)60
()22 (+)78
()40 (+)60
()40 (+)60
()9 (+)91
()40 (+)60
()40 (+)60
()9 (+)91
wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
Wide
Square
Narrow
()92 (+)8
()96 (+)4
()100 (+)0
()91 (+)9
()96 (+)4
()100 (+)0
()84 (+)16
()86 (+)14
()85 (+)15
()82 (+)18
()86 (+)14
()84 (+)16
()56
()56
()62
()55
()52
()55
()69
()58
()55
()61
()51
()54
(+)44
(+)44
(+)38
(+)45
(+)48
(+)45
(+)31
(+)42
(+)45
(+)39
(+)49
(+)46
()97 (+)3
()95 (+)5
()100 (+)0
()97 (+)3
()95 (+)5
()100 (+)0
()77 (+)23
()89 (+)11
()99 (+)1
()77 (+)23
()89 (+)11
()99 (+)1
()81 (+)19
()98 (+)2
()100 (+)0
()81 (+)19
()98 (+)2
()100 (+)0
()51 (+)49
()95 (+)5
()100 (+)0
()51 (+)49
()95 (+)5
()100 (+)0
()45
()40
()26
()45
()40
()26
()49
()38
()22
()49
()38
()22
Far
20
Near
Far
High
10
near
Far
20
Near
Far
(+)55
(+)60
(+)74
(+)55
(+)60
(+)74
(+)51
(+)62
(+)78
(+)51
(+)62
(+)78
39
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
length +30 to prevent a large disparity in the resulting route lengths. We also ensure that the swapped edges are not the first
edges in their respective routes to avoid obtaining the same solution.
Appendix B. Frequency of underestimation/overestimation
Tables 7 and 8 display for the VST and HDT policy respectively, the number of realizations for each type of instance where
the approximated values underestimate/overestimate the realized values.
The tables show that the continuous approximation for the objective tends to underestimate the true cost of implementing the policy. Between the two policies, the policy solution underestimates the true cost 76.5% of the instances. As discussed
in Section 5.2, the local distance is underestimated for instances with wide vehicle routing zones due to not factoring turns of
the swath. The linehaul is also underestimated but to a lesser degree.
Appendix C. Solution gaps for individual instances
Table 9
Solution gap between policy and tabu search.
Customer realization
Near
Far
Wide
(HDT)
Square
(VST)
Narrow
(VST)
Wide
(VST)
Square
(VST)
Narrow
(VST)
(a) Low density – 10 vehicles
1
2
3
4
5
6
7
8
9
10
38.2%
33.8%
36.1%
36.7%
36.2%
33.4%
39.3%
35.5%
33.2%
37.7%
42.4%
37.5%
43.3%
34.0%
34.5%
38.9%
34.5%
33.1%
37.5%
36.3%
8.8%
8.2%
8.4%
6.1%
7.1%
6.1%
7.4%
7.4%
9.5%
9.2%
19.9%
17.2%
16.7%
13.9%
18.4%
17.1%
17.7%
16.9%
15.4%
21.1%
13.0%
12.3%
14.0%
10.9%
10.9%
12.0%
11.0%
11.7%
14.2%
11.1%
2.4%
2.4%
2.5%
1.5%
1.4%
2.2%
1.9%
2.4%
3.1%
1.9%
(b) Low density – 20 vehicles
1
2
3
4
5
6
7
8
9
10
19.6%
19.2%
20.4%
18.3%
18.9%
18.3%
18.6%
19.6%
20.2%
20.0%
20.4%
18.4%
22.4%
19.6%
18.1%
18.1%
20.5%
16.7%
19.5%
22.0%
4.7%
5.0%
5.7%
5.8%
6.0%
5.4%
5.6%
4.7%
5.2%
7.0%
11.4%
9.6%
12.4%
10.6%
10.6%
8.2%
11.3%
8.0%
10.7%
11.7%
2.4%
2.1%
2.9%
2.3%
2.3%
1.8%
2.9%
1.6%
2.4%
2.2%
0.6%
0.5%
0.7%
0.6%
0.6%
0.5%
0.7%
0.4%
0.6%
0.5%
(c) High density – 10 vehicles
1
2
3
4
5
6
7
8
9
10
33.6%
34.3%
33.2%
35.4%
32.5%
39.1%
34.1%
33.3%
32.5%
35.2%
38.3%
35.8%
37.6%
37.9%
36.6%
34.1%
39.0%
34.7%
40.8%
38.4%
14.8%
12.7%
13.8%
13.3%
15.9%
15.2%
14.9%
14.9%
14.4%
16.1%
21.0%
18.6%
19.7%
20.4%
20.5%
18.9%
17.8%
19.9%
18.4%
21.1%
15.9%
15.3%
16.6%
18.2%
15.3%
15.0%
15.5%
14.4%
16.1%
16.3%
6.3%
5.8%
6.2%
6.3%
7.2%
7.1%
6.6%
6.9%
6.7%
7.8%
(d) High density – 20 vehicles
1
2
3
4
5
6
7
8
9
10
23.7%
20.7%
21.1%
23.8%
23.2%
24.7%
23.5%
22.8%
22.4%
21.6%
22.7%
23.9%
23.1%
24.1%
25.5%
23.1%
26.3%
20.5%
27.4%
23.3%
6.2%
6.7%
6.8%
6.5%
7.3%
6.4%
7.6%
6.0%
7.4%
6.6%
14.4%
12.7%
13.5%
13.9%
13.7%
13.1%
11.3%
12.1%
11.9%
12.2%
4.2%
5.3%
5.0%
5.3%
5.1%
4.7%
4.9%
4.0%
5.5%
4.1%
1.1%
1.1%
1.2%
1.4%
1.3%
1.0%
1.3%
1.0%
1.4%
1.0%
40
M. Huang et al. / Transportation Research Part B 50 (2013) 20–41
Table 10
Solution gap between hybrid and tabu search.
Customer realization
Near
Far
Wide
(HDT)
Square
(VST)
Narrow
(VST)
Wide
(VST)
Square
(VST)
Narrow
(VST)
(a) Low density – 10 vehicles
1
2
3
4
5
6
7
8
9
10
33.6%
18.8%
21.0%
20.9%
24.6%
23.7%
21.4%
21.6%
19.9%
35.6%
32.7%
36.5%
24.5%
24.7%
20.0%
27.0%
21.7%
21.2%
21.3%
36.1%
7.5%
7.6%
7.4%
5.5%
6.4%
5.2%
6.4%
6.0%
8.3%
8.8%
13.1%
12.0%
12.3%
14.3%
15.6%
15.2%
14.3%
12.4%
15.3%
17.0%
5.2%
7.9%
6.2%
7.4%
5.0%
5.8%
4.9%
7.4%
7.9%
7.3%
1.7%
2.0%
1.9%
1.1%
1.1%
1.7%
1.4%
1.6%
2.4%
1.6%
(b) Low density – 20 vehicles
1
2
3
4
5
6
7
8
9
10
19.1%
18.7%
19.6%
17.6%
17.9%
17.5%
17.5%
19.0%
18.9%
19.2%
18.6%
17.7%
20.7%
18.7%
17.3%
17.0%
19.2%
16.6%
18.5%
21.1%
4.7%
5.0%
5.7%
5.8%
6.1%
5.3%
5.6%
4.7%
5.2%
7.0%
7.5%
8.0%
7.4%
6.7%
7.5%
6.3%
8.5%
6.2%
5.9%
7.9%
2.0%
1.9%
2.3%
1.9%
2.0%
1.4%
2.4%
1.6%
1.9%
2.0%
0.6%
0.5%
0.6%
0.5%
0.6%
0.4%
0.7%
0.4%
0.6%
0.5%
(c) High density – 10 vehicles
1
2
3
4
5
6
7
8
9
10
24.0%
21.3%
37.0%
29.1%
32.0%
25.1%
24.0%
31.1%
28.1%
30.2%
33.6%
24.2%
31.0%
25.9%
30.3%
28.9%
27.4%
34.9%
33.1%
28.2%
27.6%
8.9%
8.5%
8.6%
10.7%
10.9%
24.7%
10.3%
9.5%
10.9%
12.8%
9.1%
12.5%
17.3%
10.6%
10.7%
9.5%
14.7%
13.6%
14.3%
13.5%
9.5%
7.1%
12.3%
10.7%
14.8%
9.2%
15.9%
12.4%
13.3%
3.9%
3.7%
3.2%
3.5%
4.1%
4.5%
4.0%
4.2%
3.8%
4.8%
(d) High density – 20 vehicles
1
2
3
4
5
6
7
8
9
10
19.2%
18.0%
18.0%
19.2%
19.5%
24.7%
18.8%
18.0%
18.7%
18.4%
19.5%
20.1%
18.7%
19.4%
21.5%
19.3%
21.4%
16.8%
22.1%
19.4%
6.2%
6.5%
6.7%
6.3%
7.2%
6.3%
7.3%
5.9%
7.0%
6.4%
12.7%
11.8%
15.1%
10.7%
10.2%
8.5%
9.3%
11.1%
13.3%
10.0%
2.9%
3.5%
3.2%
3.4%
3.4%
3.1%
3.3%
2.5%
3.4%
2.5%
1.1%
1.0%
1.1%
1.3%
1.2%
1.0%
1.2%
0.9%
1.2%
1.0%
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