A Memetic Algorithm for Periodic Capacitated Arc Routing Problem

TL;DR: A new Memetic Algorithm (MA) is proposed that adopts a new solution representation scheme and a novel crossover operator, and a Route-Merging procedure is devised and embedded in the algorithm to tackle the insensitive objective of PCARP.

Abstract: This paper investigates the Periodic Capacitated Arc Routing Problem (PCARP), which is often encountered in the waste collection application. PCARP is an extension of the well-known Capacitated Arc Routing Problem (CARP) from a single period to a multi-period horizon. PCARP is a hierarchical optimization problem which has a primary objective (minimizing the number of vehicles ) and a secondary objective (minimizing the total cost ). An important factor that makes PCARP challenging is that its primary objective is little affected by existing operators and thus difficult to improve. We propose a new Memetic Algorithm (MA) for solving PCARP. The MA adopts a new solution representation scheme and a novel crossover operator. Most importantly, a Route-Merging (RM) procedure is devised and embedded in the algorithm to tackle the insensitive objective . The MA with RM (MARM) has been compared with existing meta-heuristic approaches on two PCARP benchmark sets and a real-world data set. The experimental results show that MARM obtained better solutions than the compared algorithms in much less time, and even updated the best known solutions of all the benchmark instances. Further study reveals that the RM procedure plays a key role in the superior performance of MARM.

Summary

Introduction

• This is derived from the fact that in the waste collection application, the untreated waste after one service will be retained until the next service reaches.
• Besides the constraints, PCARP is different from CARP in the objectives as well.

II. NOTATIONS AND PROBLEM DEFINITION

• In PCARP, a connected graph G(V,E) and a p-period horizon are given, where V and E are the vertex and edge sets.
• Constraints (9) and (10) indicate that each task is served no more than once in each period.
• The reason lies in that CARP only considers minimizing tc while PCARP considers minimizing mnv prior to minimizing tc.
• Compared with tc, mnv is much less sensitive to the search operators, including the crossover and local search operators.
• During the local search, the existing operators generally define a small neighborhood around the current solution by moving only one or two tasks.

IV. MEMETIC ALGORITHM WITH ROUTE-MERGING

• It can be viewed as a class of population-based meta-heuristic approaches that incorporates local search procedures with the traditional genetic algorithms, and has been successfully applied to many realworld problems (e.g., [16], [28], [38]) with better solutions achieved and the ability of exploring the solution spaces more efficiently than traditional genetic algorithms.
• In the field of PCARP, the only two meta-heuristic approaches LMA and SS can both be viewed as adopting the framework of MA by combining global search operators with local search process.

A. Framework of the Algorithm

• At first, the population pop is set empty.
• To keep the diversity of the population, identical solutions, also called clones, are not allowed in the population throughout the search process.
• Once an initial solution has been generated, it is compared with all the solutions in pop.
• The search process stops after Gmax generations.
• Next, the authors will describe the details of MARM, including the solution representation and evaluation, solution initialization, crossover operator and local search process.

B. Solution Representation and Evaluation

• Different representation schemes will build different fitness landscapes in the solution space, and thus lead to different difficulties to search for the global optimum.
• For an edge task, the two corresponding IDs have the same serving costs, deadheading costs, demand vectors, service frequencies and allowed period combination sets, which are exactly those of the edge task itself.
• It should be noted that under the explicit task encoding scheme, the capacity constraints may be violated and infeasible solutions may appear during the search.
• Thus, it is no longer appropriate to use stochastic ranking procedure since it can only maintain a set of relatively good solutions, but cannot tell which solution is the best.
• 2) For infeasible solutions, the one with less constraint violation is better;.

C. Solution Initialization

• In the initialization phase, each initial solution is generated by the following three steps: Step 1) Randomly choose a period combination for each task from its allowed period combination set; Step 2).
• For each period, apply the path scanning heuristic [18] to all the task that should be served in that period to generate a corresponding single-period sub-solution; Step 3) Combine all the single-period sub-solutions to form a complete PCARP solution.
• The path scanning heuristic was proposed by Golden et al. [18] in 1983 for the Vehicle Routing Problem (VRP), which is the node routing counterpart of CARP, and was extended by Lacomme et al. in 2002 to solve CARP [26].
• Therefore, employing the path scanning heuristic in the solution initialization can generate good PCARP initial solutions and accelerate the convergence of MARM.
• Therefore, the feasibility of the final solution can be guaranteed, and the final solution should be no worse than the best initial solution.

D. Crossover Operator

• Since a new solution representation scheme is employed in MARM, a corresponding crossover operator should be designed.
• To this end, the authors extend the Route-Based Crossover (RBX) operator [35] from the case of CARP to PCARP.
• For each period of Sx, remove the tasks whose period combinations have been changed and should no longer be served in that period; Step 7).
• Note that the insertion of a task may increase the tc and tvd.
• If more than one such position exists, one of them is chosen arbitrarily; Step 8) Return Sx. 2.

E. Local Search

• Sx undergoes the local search process with a predefined probability Pls.
• The neighborhood N (S) is generated by the single-insertion, double-insertion and swap operators.
• The above three operators are described as follows: Single-insertion: move a task service from its original position to another; Double-insertion: move two adjacent task services from their original positions to another; Swap: exchange the positions of two task services.
• In all the three operators, both directions of the involved tasks are considered.
• Note that the period combinations of the involved tasks may change due to the movements.

V. EXPERIMENTAL STUDIES

• Two sets of experiments have been carried out to evaluate the performance of MARM.
• The first experiment is carried out on two relatively simple test sets, i.e., the pgdb and pval test sets [10] generated from the corresponding gdb and val CARP benchmark sets.
• MARM is applied to them and the results are compared with that obtained by LMA (provided in [26]) and SS (provided in [10]).
• Besides, to observe the influence of the RM procedure on the performance of MARM, the authors remove it from the framework of MARM and compare the resultant algorithm with MARM on the pgdb and pval test sets.
• The real-world data set was first generated by Brandão and Eglese [7] and consists of 10 large CARP instances defined on a road network in Lancashire, U.K.

A. Experimental Setup

• The pgdb and pval test sets used in the first experiment were generated by Chu et al. in [10] by extending two well-known CARP benchmark sets, i.e., the gdb and val sets from singleperiod case to multi-period case.
• If the service frequency of (u, v) is 2 or 3, then consecutive services over the horizon (e.g., (Monday, Tuesday) and (Monday, Tuesday, Wednesday)) are forbidden.
• Unfortunately, this set is unavailable online, nor could the authors implement it due to the ambiguous description provided in the original literature.
• The real-world data set used in the second experiment, called the pG set, is extended from the G set generated by Brandão and Eglese in [7], which consists of 10 large CARP instances based on a road network in Lancashire, U.K.

B. Experimental Results on the Benchmark Sets

• In the first experiment, MARM is compared with LMA and SS on the pgdb and pval test sets.
• From the column headed “Best”, it can be observed that the best performance of MARM was better than LMA, SS, and even the best known results.
• Table V presents the average runtime (in seconds) of each compared algorithm on the two test sets.
• In summary, the authors can conclude that embedding RM can enhance the ability of the algorithm to find better solutions, especially those with smaller mnv’s.

D. Results on the Real-World Data Set

• For the pG real-world data set, LMA, SS, and MARM were implemented for 30 independent runs, and their best and average performance are shown in Tables VIII and IX.
• For each compared algorithm, Table VII gives the mnv and tc of the best solution obtained in the 30 independent runs, while Table IX presents the mean and standard deviation of the mnv and tc of the 30 final solutions obtained by the 30 runs (in the same form as in Tables III and IV).
• Besides, the properties of each instance, including the number of vertices, edges and services over the horizon and LBV are provided.
• From Table VIII, it is seen that MARM was able to achieve solutions with smaller mnv than that of LMA and SS on all the 6 pG instances.
• This verifies the superiority of MARM in optimizing the primary objective mnv.

E. Summary and Further Discussion

• The experimental studies showed that MARM outperformed the other two compared algorithms significantly, especially in terms of the primary objective mnv.
• Besides, it was found that the average tc obtained by MARM was also smaller than those of the compared algorithms.
• Therefore, smaller tc’s can be obtained through the more efficient local search process.
• This observation indirectly implies that MARM is not an ideal approach to CARP.
• Thus, the authors suggest practitioners employ MARM only in the case of PCARP.

Figures

TABLE I NOTATIONS USED IN THE PCARP DEFINITION

Fig. 1. Example of the explicit task encoding scheme. (a) The given graph: all the edges are tasks, and v0 is the depot; (b) a PCARP solution: the solid lines indicate the services while the dashed lines indicate the deadheading paths.

TABLE V RUNTIME (IN CPU SECONDS) OF THE COMPARED ALGORITHMS ON THE TEST SETS

TABLE IV EXPERIMENTAL RESULTS OF THE COMPARED ALGORITHMS ON THE pgdb TEST SET. THE OPTIMAL RESULTS ARE WITH “∗,” AND THE NEW BEST RESULTS FOUND BY MARM ARE IN BOLD

TABLE III EXPERIMENTAL RESULTS OF THE COMPARED ALGORITHMS ON THE pgdb TEST SET. THE OPTIMAL RESULTS ARE WITH “∗,” AND THE NEW BEST RESULTS FOUND BY MARM ARE IN BOLD

Fig. 2. Example of the PRBX operator.

TABLE IX AVERAGE PERFORMANCE OF LMA, SS, AND MARM ON THE pG REAL-WORLD DATA SET. THE mnv’s AND tc’s OBTAINED BY MARM THAT ARE SIGNIFICANTLY BETTER THAN THAT OF LMA AND SS (WITH CONFIDENCE PROBABILITY OF 95%) ARE IN BOLD

TABLE VIII BEST PERFORMANCE OF LMA, SS, AND MARM ON THE pG REAL-WORLD DATA SET. THE OPTIMAL mnv’s ARE WITH “∗”

TABLE VI COMPARISON BETWEEN THE MA WITH AND WITHOUT RM ON THE TEST SETS

TABLE VII STATISTICAL COMPARISON BETWEEN THE RESULTS OBTAINED BY 30 INDEPENDENT RUNS OF MAs WITH AND WITHOUT RM

TABLE II PARAMETER SETTINGS OF MARM

Full text

1654 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 6, DECEMBER 2011
A Memetic Algorithm for Periodic
Capacitated Arc Routing Problem
Yi Mei, Student Member, IEEE, Ke Tang, Member, IEEE,andXinYao,Fellow, IEEE
Abstract—This paper investigates the Periodic Capacitated Arc
Routing Problem (PCARP), which is often encountered in the
waste collection application. PCARP is an extension of the well-
known Capacitated Arc Routing Problem (CARP) from a single
period to a multi-period horizon. PCARP is a hierarchical opti-
mization problem which has a primary objective (minimizing the
number of vehicles mnv) and a secondary objective (minimiz-
ing the total cost tc). An important factor that makes PCARP
challenging is that its primary objective mnv is little affected
by existing operators and thus difficult to improve. We propose
a new Memetic Algorithm (MA) for solving PCARP. The MA
adopts a new solution representation scheme and a novel crossover
operator. Most importantly, a Route-Merging (RM) procedure is
devised and embedded in the algorithm to tackle the insensitive ob-
jective mnv. The MA with RM (MARM) has been compared with
existing meta-heuristic approaches on two PCARP benchmark
sets and a real-world data set. The experimental results show that
MARM obtained better solutions than the compared algorithms
in much less time, and even updated the best known solutions of
all the benchmark instances. Further study reveals that the RM
procedure plays a key role in the superior performance of MARM.
Index Terms—Capacitated arc routing problems, combinatorial
optimization, evolutionary algorithms, memetic algorithms.
I. INTRODUCTION
A
S A CLASSICAL combinatorial optimization problem,
Capacitated Arc Routing Problem (CARP) [13] has at-
tracted much research interest from various scientific and in-
dustrial fields. It has many real-world applications in the fields
of logistics and transportation management, such as waste col-
lection, post delivery, winter gritting and snow removal. CARP
can be described as follows: Given a connected graph, some
edges (called the tasks) of the graph are required to be served
by a vehicle fleet located at the depot vertex. CARP aims to
Manuscript received March 6, 2011; accepted May 11, 2011. Date of pub-
lication July 14, 2011; date of current version November 18, 2011. This work
was partially supported by an EPSRC Grant (No. EP/E058884/1) on “Evolu-
tionary Algorithms for Dynamic Optimisation Problems: Design, Analysis and
Applications,” the National Natural Science Foundation of China under Grants
U0835002 and 61028009, and the grant for Distinguished Young Scholars of
Anhui Province. This paper was recommended by Associate Editor H. Takagi.
Y. Mei and K. Tang are with the Nature Inspired Computation and Appli-
cations Laboratory, School of Computer Science and Technology, University
of Science and Technology of China, Hefei 230027, China (e-mail: meiyi@
mail.ustc.edu.cn; ketang@ustc.edu.cn).
X. Yao is with the Nature Inspired Computation and Applications Lab-
oratory, School of Computer Science and Technology, University of Sci-
ence and Technology of China, Hefei 230027, China. He is also with
CERCIA, the School of Computer Science, University of Birmingham, B15
2TT Birmingham, U.K. (e-mail: x.yao@cs.bham.ac.uk).
Digital Object Identifier 10.1109/TSMCB.2011.2158307
determine a least-cost plan subject to the following constraints:
1) Each vehicle must start and end at the depot vertex;
2) Each task is served exactly once by one vehicle;
3) The total demand of the tasks served by each route cannot
exceed its capacity.
CARP has been proven to be NP-hard in [19]. Thus, exact
methods are only available for small and medium instances.
When solving real-world instances which often have large
problem size, heuristics and meta-heuristics are promising ap-
proaches for finding near-optimal solutions in the given time
budget. So far, numerous constructive heuristics (e.g., [18],
[19], [32], [33], [39]) and meta-heuristics (e.g., the guided local
search [6], the tabu search [7], [20], [22], [30], the variable
neighborhood search [23], [34] and the memetic algorithm
[21], [26], [37]) have been proposed for CARP, and typically
been evaluated by comparing the obtained solutions to the tight
lower bounds calculated with the relaxation methods proposed
in [2], [4].
Despite the intensive investigations conducted on CARP,
there is still a wide gap between this simple model and real-
world situations. Hence, for many applications in reality, the
CARP approaches can hardly be directly applied. To model the
real-world applications more closely, an extended CARP, called
the Periodic CARP (PCARP), is studied in this paper. PCARP is
often encountered in the waste collection application, in which
the department in charge aims to design a plan to collect the
daily waste produced by each street in the city. In practice,
there may be a large difference in the amount of daily waste
production and the tolerance of the waste duration between the
streets, depending on the region, population size, the type of
buildings around, etc. To be more specific, the apartment blocks
and heavy industrial regions produce a large amount of daily
waste that cannot be stored for a long time. It is necessary to
collect waste for such streets everyday. On the other hand, the
rural areas could keep the produced waste for a relatively long
time. These streets can be treated less frequently (e.g., every
two days or twice a week). Therefore, it is more proper to make
a schedule for a multi-period horizon (e.g., a week or month)
rather than a single day. PCARP is thus introduced to model
such a situation.
As first introduced by Lacomme et al. [25], PCARP can be
described as follows: Given a connected graph and a multi-
period horizon, the tasks need to be served for a certain num-
ber of times (say their service frequencies) over the horizon.
PCARP is to make an optimal service plan so that the following
1083-4419/$26.00 © 2011 IEEE

MEI et al.: MEMETIC ALGORITHM FOR PCARP 1655
constraints are satisfied:
1) In each period, each vehicle must start and end at the
depot vertex;
2) In each period, each task is served no more than once;
3) The number of services of each task over the horizon
equals its service frequency;
4) The period combination of each task must be in the
allowed period combination set;
5) In each period, the total accumulated demand of the tasks
served by each vehicle cannot exceed its capacity.
In Constraint 4) of PCARP, the period combination of a task
means the periods in which the task is served. The allowed
period combination set is predefined by the problem nature.
For instance, consecutive services may be forbidden for the
tasks with low service frequencies and thus not in the allowed
period combination set. In Constraint 5), the demand of tasks
are replaced by the accumulated demand. This is derived from
the fact that in the waste collection application, the untreated
waste after one service will be retained until the next service
reaches. Therefore, a service for a task in PCARP is to serve
the accumulated demand since its previous service.
When there is only one period and all the tasks are to be
served in this period, PCARP is reduced to CARP. In this sense,
PCARP can be seen as a generalization of CARP from single-
period to multi-period, and CARP can be seen as a single-
period special case of PCARP.
Besides the constraints, PCARP is different from CARP in
the objectives as well. As mentioned in [25], waste management
departments usually consider minimizing the investment cost
(depending on the fleet size) prior to minimizing the operating
costs (i.e., the total cost). Hence, PCARP has a primary objec-
tive of minimizing the number of vehicles used over the horizon
(denoted as mnv) and a secondary objective of minimizing the
total cost (denoted as tc).
Being a relatively new topic, research on PCARP is still in
its infancy and there have been only a few literatures available
so far. It was first defined by Lacomme et al. [25]. Then,
Chu et al. built a linear programming model and proposed
several constructive heuristics [8], and proposed lower bounds
for it [9]. Lacomme et al. proposed a Memetic Algorithm (MA)
[27] (referred to as LMA hereafter) and Chu et al. proposed
a Scatter Search (SS) [10]. LMA and SS extend a solution
representation scheme for CARP and the corresponding search
operators to the multi-period case, and include an elite solution
generated by a greedy constructive heuristic in the initial pop-
ulation for performance enhancement. Recently, Kansou and
Yassine [24] proposed an efficient constructive heuristic and an
ant colony system.
Compared with PCARP, the Periodic Vehicle Routing Prob-
lem (PVRP), which can be viewed as the vertex routing coun-
terpart of PCARP, arose earlier and received more research
interests (e.g., [1], [3], [11], [14], [15], [17]). Nevertheless,
previous work on PVRP ignored the primary objective mnv.
Besides, CARP has more complicated problem characteristics
than VRP, e.g., a CARP with n tasks can be transformed to a
VRP with 2n +1vertices [29]. Thus, solving PCARP will be
more complicated than solving PVRP.
In this paper, it is found that the primary objective mnv can
hardly be improved by existing search operators. To address this
issue, a specific Route-Merging (RM) procedure is developed.
Then, the RM procedure is embedded in the MA framework and
the MA with RM (MARM) is thereby proposed. In MARM,
the RM procedure is located before the local search process.
Thus, a solution is first improved in mnv by the RM procedure,
and then improved in tc by the subsequent local search. The
experimental results on two benchmark sets and a real-world
data set demonstrate the superior performance of MARM.
The rest of the paper is organized as follows: First, Section II
gives the notations and problem definition. Afterwards, the
newly introduced primary objective mnv is investigated and
the RM procedure is introduced in Section III. Then, Section IV
describes MARM in detail. After that, experimental studies are
carried out in Section V. Finally, Section VI concludes this
paper.
II. N
OTATIONS AND PROBLEM DEFINITION
In PCARP, a connected graph G(V, E) and a p-period hori-
zon are given, where V and E are the vertex and edge sets.
Each edge (u, v) ∈ E is associated with a serving cost sc(u, v),
a deadheading cost dc(u, v), a demand vector d(u, v)=
(d
1
(u, v),...,d
p
(u, v)) and a service frequency freq(u, v).
The serving cost indicates the cost induced by serving (u, v),
while the deadheading cost is the cost induced by traversing
(u, v) without service. d
i
(u, v) of d(u, v) is the demand of
(u, v) increased in period i. The task set T consists of the
edges with positive demand in at least one period, and the
number of tasks is n = |T|. Note that the serving cost is only
induced by serving a task, we have sc(u, v) > 0, ∀(u, v) ∈ T
and sc(u, v)=0, ∀(u, v) ∈ T . Besides, each task (u, v) ∈ T is
associated with an allowed period combination set AP C(u, v),
which consists of a number of period combinations being rep-
resented as a p-dimensional 0–1 vector. The ith component of
the vector takes 1 if (u, v) is served in period i, and 0 otherwise.
For example, (1, 0, 0, 1, 0, 0, 0) indicates that (u, v) is served on
Monday and Thursday of the week. Without loss of generality,
the depot vertex is denoted as v
0
∈ V . These services are to be
served by m vehicles with capacity of Q.
Under the above notations, the route X
ij
traversed by
vehicle j in period i can be denoted as a sequence of
vertices (with length of l
ij
), i.e., X
ij
=(x
ij1
,...,x
ijl
ij
),
where x
ijk
∈ V , k =1,...,l
ij
. Besides, a 0–1 vector Y
ij
=
(y
ij1
,...,y
ij(l
ij
−1)
) is used to indicate the positions of the task
services in X
ij
. Concretely, y
ijk
=1 if (x
ijk
,x
ij(k+1)
) is a
service, and y
ijk
=0 otherwise. Then, a PCARP solution S
can be denoted as the combination of the X
ij
’s and Y
ij
’s, i.e.,
S = {(X
ij
,Y
ij
)|i =1,...,p; j =1,...,m}.
Given a solution S, the period combination PC(u, v, S)
of each task (u, v) is represented as PC(u, v, S)=
(PC
1
(u, v, S),...,PC
p
(u, v, S)), where PC
i
(u, v, S) takes 1
if (u, v) is served in period i of S, and 0 otherwise. It
can be obtained as follows: For period i, if there exists
j ∈{1,...,m} and k ∈{1,...,l
ij
} so that y
ijk
=1 and
(x
ijk
,x
ij(k+1)
)=(u, v) or (x
ijk
,x
ij(k+1)
)=(v, u), then
PC
i
(u, v, S)=1. Otherwise, PC
i
(u, v, S)=0.
Based on the period combination, the accumulated demand
ad(u, v, i, S) of a task (u, v) in period i can be further obtained

1656 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 6, DECEMBER 2011
TAB LE I
N
OTATIONS USED IN THE PCARP DEFINITION
by the following steps:
i) Set l = i, ad(u, v, i, S)=d
l
(u, v);
ii) l ← l − 1.Ifl =0,setl = p;
iii) If PC
l
(u, v, S)=1, return ad(u, v, i, S);
iv) ad(u, v, i, S) ← ad(u, v, i, S)+d
l
(u, v). go back to ii).
Then, the total cost and total accumulated demand of the
tasks served by each route can be obtained as:
tc(X
ij
,Y
ij
)=
l
ij
−1

k=1
sc

x
ijk
,x
ij(k+1)

·y
ijk
+
l
ij
−1

k=1
dc

x
ijk
,x
ij(k+1)

·(1−y
ijk
) (1)
d(X
ij
,Y
ij
)=
l
ij
−1

k=1
ad

x
ijk
,x
ij(k+1)
,i,S

·[y
ijk
=1] (2)
where the indicator function [A] of an event A is defined as:
[A]=

1,Ais true;
0, otherwise.
(3)
Obviously, if vehicle j is not used in period i, then both X
ij
and Y
ij
are empty vectors and l
ij
=0. Therefore, the number
of vehicles nv(i, S) in period i can be obtained as:
nv(i, S)=
m

j=1
[l
ij
> 0]. (4)
Finally, the primary objective mnv(S) and the secondary
objective tc(S) of solution S can be computed as:
mnv(S) = max
i∈{1,...,p}
nv(i, S) (5)
tc(S)=
p

i=1
m

j=1
tc(X
ij
,Y
ij
). (6)
For clarity, all the notations used in the problem definition
are presented in Table I.
Then, PCARP can be defined as follows:
min f(S)=α · mnv(S)+tc(S) (7)
s.t. : x
ij1
=x
ijl
ij
=v
0
, ∀i=1,...,p, j=1,...,m; (8)

x
ijk
1
,x
ij(k
1
+1)

=

x
ijk
2
,x
ij(k
2
+1)

,
∀y
ijk
1
=1,y
ijk
2
=1,i=1,...,p, j=1,...,m; (9)

x
ijk
1
,x
ij(k
1
+1)

=

x
ij(k
2
+1)
,x
ijk
2

,
∀y
ijk
1
=1,y
ijk
2
=1,i=1,...,p, j=1,...,m; (10)
PC(u, v, S) ∈ APC(u, v), ∀(u, v) ∈ T (11)
d(X
ij
,Y
ij
)  Q, ∀i =1,...,p,j=1,...,m. (12)
In the objective function (7), the primary objective mnv(S)
and the secondary objective tc(S) are combined into a single
objective function f(S) by the weighted sum approach. α is
a sufficiently large constant that can guarantee the priority
of mnv (e.g., α = 100 000 in the subsequent experiments).
Constraint (8) means that each route starts and ends at the
depot. Constraints (9) and (10) indicate that each task is served
no more than once in each period. Constraint (11) restricts
the period combinations of the tasks in their allowed period
combination sets. Constraint (12) indicates the capacity con-
straint. Note that the equality between the number of services
of each task and its service frequency is implicitly ensured by
Constraints (8)–(11).
III. I
MPROVING THE mnv WITH ROUTE-MERGING
PCARP is different from CARP in various problem charac-
teristics. First, PCARP contains multiple CARPs, each for a pe-
riod. Second, the restriction imposed on the period combination
of the tasks makes the feasible regions in the solution space
smaller and more separated. Finally, and most importantly,
PCARP introduces a primary objective mnv whereas the tc,
which is the only objective of CARP, becomes a secondary
objective. This makes PCARP a hierarchical optimization prob-
lem rather than a simple single-objective optimization problem.
Since PCARP is an extension of CARP, it is reasonable to
consider extending CARP approaches to solve PCARP. This
can be accomplished by modifying the solution representation
scheme and corresponding search operators. Both LMA and SS
adopted this idea, and extended the solution representation and
search operators of a competitive MA for CARP [26] to solve
PCARP. However, merely extending the search operators can
by no means lead to high-quality PCARP solutions. The reason
lies in that CARP only considers minimizing tc while PCARP
considers minimizing mnv prior to minimizing tc. The search
operators extended from those for minimizing tc will also keep
their eyes on tc and thus will be unable to attain high-quality
solutions in PCARP, which must have small mnv’s as well as
small tc’s.
LMA and SS have already made some effort to deal with
mnv. LMA maintains a small-mnv solution in the population
throughout the search process so that mnv of the final solution
can be guaranteed. SS adopts elite solution scheme as well, and
modifies the evaluation phase to focus more on mnv.Never-
theless, generating good solutions and evaluating them are both

MEI et al.: MEMETIC ALGORITHM FOR PCARP 1657
crucial to the success of an algorithm. Although SS makes the
latter more effective, nothing has been done to the former.
Compared with tc, mnv is much less sensitive to the search
operators, including the crossover and local search operators.
When applying a crossover operator, the offspring will inherit
the characteristics of its parents. Thus, there is no trend that
crossover operators can generate solutions with smaller mnv’s
than their parents. During the local search, the existing opera-
tors generally define a small neighborhood around the current
solution by moving only one or two tasks. However, reducing
mnv by one requires eliminating a route in all the periods that
have the most routes over the horizon, which means selecting
one route in each such period and move all the involved tasks
to other routes. This requires many steps of the local search.
Thus, it is difficult for the existing local search operators to
improve mnv.
Based on the above discussions, it is necessary to design a
specific operator to deal with mnv directly. To this end, the
RM procedure is developed. It is described in Algorithm 1.
Algorithm 1: The Route-Merging (RM) procedure
Input: A solution S, minmnv;
Output: A solution S

;
1: Set S

= S;
2: for i =1→ p do
3: if nv(i, S

) >minmnvthen
4: for j =1→ nv(i, S

) − minmnv do
5: Call (R
1
,R
2
)=MinAccDemands(S

,i);
6: Call Merge(R
1
,R
2
);
7: end for
8: end if
9: end for
10: return S

The function (R
1
,R
2
)=MinAccDemands(S

,i) returns the
two routes R
1
and R
2
whose total accumulated demands are
minimized among all the route couples in S

i
. The function
Merge(R
1
,R
2
) merges the two routes by appending the latter
route to the end of the former.
It is obvious that the performance of the RM procedure
depends on the user-defined parameter minmnv. To keep the
balance between mnv and other objectives such as tc and the
violation to the capacity constraints, minmnv should be set to
an appropriate value. Here, we set minmnv to the lower bound
of mnv, which can be obtained as follows:
minmnv =


(u,v)∈T

p
i=1
d
i
(u, v)
p
w
Q

(13)
where p
w
indicates the number of working periods (the periods
that can be used to allocate services) over the horizon. Since
the above value is the theoretical minimal mnv value that can
be achieved in the solution space, it should be a proper setting
for minmnv.
The computational complexity of the RM procedure is

p
i=1
max{nv(i, S) − minmnv, 0} = O(pm).
The RM procedure can keep the mnv no larger than
minmnv. However, it may increase the violation to the capac-
ity constraints since the the tasks that are originally served in
two routes will be served in the single merged route. Therefore,
the RM procedure can be seen as reducing mnv at the cost
of increasing the violation to the capacity constraints, and
efforts are still to be made before successfully combining the
RM procedure and existing algorithms. In our study, the RM
procedure is carefully embedded in the framework of MA and
the resultant MARM is thus proposed. Next, we will introduce
MARM in detail.
IV. M
EMETIC ALGORITHM WITH ROUTE-MERGING
As a recently growing area in evolutionary computation, MA
was firstly introduced by Moscato in [31]. It can be viewed
as a class of population-based meta-heuristic approaches that
incorporates local search procedures with the traditional genetic
algorithms, and has been successfully applied to many real-
world problems (e.g., [16], [28], [38]) with better solutions
achieved and the ability of exploring the solution spaces more
efficiently than traditional genetic algorithms. In the field of
PCARP, the only two meta-heuristic approaches LMA and SS
can both be viewed as adopting the framework of MA by
combining global search operators with local search process.
Without loss of generality, the framework of MA can be pre-
sented in Algorithm 2.
Algorithm 2: The framework of MA
1: Initialization: Generate an initial population;
2: while stopping criteria are not met do
3: Evaluate all individuals in the population;
4: Evolve a new population using evolutionary operators;
5: for each individual in the new population do
6: Perform local search with probability P ;
7: end for
8: end while
The framework of MARM is derived from Algorithm 2, but
with certain problem-specific modifications and extensions so
as to solve PCARP more effectively.
A. Framework of the Algorithm
Algorithm 3 gives the framework of MARM. At first, the
population pop is set empty. Then, initial solutions are gener-
ated and inserted into pop one by one. To keep the diversity of
the population, identical solutions, also called clones, are not
allowed in the population throughout the search process. Once
an initial solution has been generated, it is compared with all the
solutions in pop. If it is not a clone of any solution in pop, then
it is accepted and inserted into pop. Otherwise, it is abandoned
and a new trial starts. The initialization phase terminates when
pop is full (its size equals popsize) or no eligible solution has
been generated after Mtrial consecutive trials.

1658 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 6, DECEMBER 2011
The population initialization is followed by the search
process, which consists of a number of generations. At each
generation, two solutions S
1
and S
2
are randomly selected from
pop. Then, the crossover operator is applied to S
1
and S
2
to
generate an offspring S
x
. After that, the RM procedure and
local search are applied to S
x
in turn. Finally, the offspring
is compared with the solutions in pop. If it is not a clone,
it is inserted into pop and then the worst solution in pop
is removed to keep the size of pop. For the removal, the
solutions in pop is sorted by the stochastic ranking and the
last solution is removed. The search process stops after G
max
generations.
Algorithm 3: The framework of MARM
Input: A PCARP instance, popsize, M trial, P
ls
,G
max
;
Output: A PCARP solution S
bf
;
// Initialization:
1: Set pop = ∅;
2: while |pop| < popsize do
3: Set ntrial =0;
4: repeat
5: Generate a solution S
0
;
6: ntrial ← ntrial +1;
7: until (S
0
is not a clone) or (ntrial = Mtrial)
8: if S
0
is a clone then
9: break;
10: end if
11: pop ← pop ∪{S
0
};
12: end while
// Main Loop:
13: Set ngen =0;
14: while ngen < G
max
do
15: ngen ← ngen +1;
16: Call (S
1
,S
2
)=RandSelect(pop);
17: Call S
x
= Crossover(S
1
,S
2
);
18: Call S
x
= RM(S
x
,minmnv);
19: Sample r ∈ U(0, 1);
// U (0, 1) is an uniform distribution between 0
and 1
20: if r<P
ls
then
21: Call S
ls
= LocalSearch(S
x
);
22: if S
ls
is not a clone then
23: pop ← pop ∪{S
ls
};
24: else if S
x
is not a clone then
25: pop ← pop ∪{S
x
};
26: end if
27: else if S
x
is not a clone then
28: pop ← pop ∪{S
x
};
29: end if
30: if S
ls
or S
x
has been inserted in pop then
31: Call pop = StochasticRanking(pop);
32: pop = pop(1 : |pop|−1);
33: end if
34: end while
35: return the best feasible solution S
bf
in pop;
In PCARP, the complicated solution structure makes an
exact clone examination between PCARP solutions quite
time-consuming. In this situation, the following approximated
scheme is used instead: Solution S
1
is considered as a clone
of solution S
2
if they have the same values of the mnv, tc and
violation to the capacity constraints.
Next, we will describe the details of MARM, including the
solution representation and evaluation, solution initialization,
crossover operator and local search process.
B. Solution Representation and Evaluation
How to represent a solution is a fundamental issue in a
stochastic search algorithm such as MA. Different represen-
tation schemes will build different fitness landscapes in the
solution space, and thus lead to different difficulties to search
for the global optimum. LMA and SS employ the same solution
representation scheme, which can be called the implicit task en-
coding scheme. In the implicit task encoding scheme, a PCARP
solution is represented as a number of task sequences, each
for a period. During the evaluation phase, each task sequence
undergoes a two-phase decoding procedure. First, it is split into
a set of feasible routes with respect to the capacity constraints so
that the additional cutting cost is minimized. Then, the adjacent
tasks of the routes are connected with the shortest path between
them. After the above decoding procedure, the two objectives
mnv and tc can be directly calculated. The implicit task encod-
ing scheme simplifies the design of crossover operator. How-
ever, it makes solution evaluation relatively time-consuming
due to the computational complexity of the decoding procedure.
Besides, a slight modification of the encoded solution may
result in a severe change after decoding. This characteristic
makes the encoded solution space more rugged and hinders
local search. To overcome these drawbacks, an explicit task
encoding scheme is developed and adopted by MARM.
In the explicit task encoding scheme, each edge task is first
associated with two IDs, each for a direction. The IDs are
different from each other, and chosen from the positive integer
set N
+
. For an edge task, the two corresponding IDs have the
same serving costs, deadheading costs, demand vectors, service
frequencies and allowed period combination sets, which are ex-
actly those of the edge task itself. The only differences between
them are their head and tail vertices. To be specific, assuming
atask(u, v) is associated with IDs t
1
and t
2
, then tv(t
1
)=
hv(t
2
)=v and hv(t
1
)=tv(t
2
)=u, where tv(t) and hv(t)
indicate the tail and head vertices of ID t. Besides, a depot loop
0 is defined to act as a delimiter between different routes. Its
head and tail vertices are both v
0
, and its serving cost, dead-
heading cost, demand vector and service frequency are 0 (0).
Using the above IDs, a solution S can be represented as
a set of task sequences S = {S
ij
|i =1,...,p; j =1,...,m},
where S
ij
=(0,t
ij1
,...,t
ijl
ij
, 0) presents the route j in period
i. The depot loop 0 is inserted at the beginning and end of
each task sequence to ensure that it starts and ends at the
depot. l
ij
is the number of tasks served in S
ij
, and l
ij
=
0 (i.e., S
ij
=(0, 0)) indicates that vehicle j is not used in
period i. Fig. 1 gives a simple illustration of the explicit task
encoding scheme.