Immune clonal selection algorithm for capacitated arc ... · an immune clonal selection algorithm...

Soft ComputDOI 10.1007/s00500-015-1634-4

METHODOLOGIES AND APPLICATION

Immune clonal selection algorithm for capacitated arc routingproblem

Ronghua Shang · Hongna Ma · Jia Wang ·Licheng Jiao · Rustam Stolkin

© Springer-Verlag Berlin Heidelberg 2015

Abstract In existing metaheuristics for solving the capaci-tated arc routing problem, traversal local search operators areoften used to explore neighbors of the current solutions. Thismechanism is beneficial for finding high-quality solutions;however, it entails a large number of function evaluations,causing high computational complexity. Hence, there is aneed to further enhance the efficiency of such algorithms.This paper proposes a high-efficiency immune clonal selec-tion algorithm for capacitated arc routing instances within alimited number of function evaluations. First, an improvedconstructive heuristic is used to initialize the antibody popu-lation. The initial antibodies generated by this heuristic helpaccelerate the algorithm’s convergence. Second, we showhow an immune clonal selection algorithm can select infavor of these high-quality antibodies. By adopting a vari-ety of different strategies for different clones of the sameantibody, it not only promotes cooperation and informationexchanging among antibodies, but also increases diversity

Communicated by V. Loia.

R. Shang (B) · H. Ma · J. Wang · L. JiaoKey Laboratory of Intelligent Perception and ImageUnderstanding of Ministry of Education of China,Xidian University, Xi’an 710071, Chinae-mail: [email protected]

L. Jiaoe-mail: [email protected]

J. WangXi’an Institute of Huawei Technology Co. Ltd.,Xi’an 710075, Chinae-mail: [email protected]

R. StolkinSchool of Computer Science, University of Birmingham,Edgbaston, Birmingham B15 2TT, UKe-mail: [email protected]

and speeds up convergence. Third, two different antibodyrepair operations are proposed for repairing various kinds ofinfeasible solutions. These operations cause infeasible solu-tions to move towards global optima. Experimental studiesdemonstrate improved performance over state-of-art algo-rithms, especially on medium-scale instances.

Keywords Immune clonal selection · Capacitated arcrouting problem · Antibody repair · Metaheuristic

1 Introduction

The well-known arc routing problem (ARP) belongs to theclassical combinatorial optimization problems (Dror 2001),which can be described as follows: given a graph G, someedges of G are required to be served by a fleet of vehicleswhich are located at a depot. ARP aims to find the least-cost path, subject to various constraints (Euler 1736; Kimet al. 2008). This paper considers an important constraint,known as a capacity constraint, and a particular version ofthe ARP model, known as the capacitated arc routing problem(CARP). In CARP, each edge (or arc) requiring service hasa particular demand, d, while each vehicle has an identicalcapacity Q. The capacity constraint refers to the total demandof the tasks served by each vehicle, which does not exceedits capacity Q (Mei et al. 2011). Here, an edge is undirected,and an arc is directed.

The CARP can be described as follows: a mixed graph,representing a road network, is defined as G = (V ,E , A),with the vertex set V , the arc set A and the edge set E . mvehicles, with limited capacity Q, are based at a predefinedvertex, known as a depot. Each edge (or arc) is associated witha non-negative traveling cost and a non-negative demand.The edges (or arcs) requiring service have positive demands,

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whereas other edges (arcs) can be traversed any number oftimes without incurring any service cost and, therefore, havezero demand. Solving CARP is defined as finding an optimalrouting plan for the vehicles under the following conditions(Mei et al. 2011):

• the route formed by each vehicle must start and end atthe depot.

• All edges (or arcs) requiring service must be visited byexactly one vehicle.

• The total demand served by any vehicle cannot exceedthe vehicle’s capacity Q.

In the past 20 years, there has been a growing interest inCARP. Since CARP is an NP-hard problem, heuristics andmetaheuristics are often considered by researchers. Well-known heuristics include augment-merge and path scanning(Golden et al. 1983; Niu et al. 2014), tour splitting algorithm(Ulusoy 1985) and cycle assignment (Benavent et al. 1990).Of these popular heuristics, the most commonly used is“path-scanning”, which refers to the concept of scanning theremaining unserved tasks to determine which one should beserved next on the current vehicle route. Recently, Santos etal. introduced an improved path-scanning heuristic for solv-ing CARP problems, which includes a novel “ellipse rule”(Santos et al. 2009). When the load of a vehicle is close toits capacity, the path-scanning will focus on only those taskswhich lie inside an ellipse. Ellipse-rule heuristic can obtainbetter results than conventional path-scanning methods, withcomparable CPU time. However, when applying it to large-scale instances, the results may be sub-optimal. Since practi-cal applications of CARP tend to be large scale, researchersshifted their attention toward metaheuristics, which generatemuch better solutions, but at high computational cost. Withthe emergence of intelligent optimization methods such asevolutionary algorithms (Dror 2001), tabu search algorithm(Hertz et al. 2000) and simulated annealing (Rutenbar 1989),metaheuristics began to play an increasingly important rolein solving CARP problems. In 2000, Hertz et al. (2000) pro-posed a tabu search for CARP, called CARPET. In CARPET,a solution is represented as a set of routes which are codedby vertices. Because of the framework of tabu search andsome new local search operators, CARPET generates bettersolutions than previous heuristics. However, CARPET’s useof vertex encoding increased computational complexity andresulted in very low search efficiency. Subsequently, in 2001,Hertz and Mittaz applied a new algorithm to solve CARP thatis the Variable Neighborhood Search algorithm (Hertz andMittaz 2001). The biggest contribution of the algorithm is thatit replaces the framework of the tabu search algorithm withthe framework of the variable neighborhood search algorithmand it has achieved better solutions. In 2003, Beullens et al.proposed a guided local search (GLS) method (Beullens et al.

2003), which uses 2-opt (Lacomme et al. 2004) as the localsearch strategy. In the same year (2003), Greistorfer cameup with the Tabu Distributed Search algorithm based on thecombination of distributed search and tabu search (Lacommeet al. 2006). Then, in 2004, Lacomme et al. (2004) proposedthe Lacomme Memetic Algorithm (LMA) for CARP, whichcombines a genetic algorithm (GA) with a variety of localsearch methods and a task encoding scheme in which a solu-tion is represented as an ordered list of tasks. In Greistorfer(2003), Lacomme extended the LMA to solve bi-objectiveCARP (Greistorfer 2003), resulting in high-quality solutions.Handa et al. proposed a robust routing optimization algo-rithm to optimize the path planning of salt vehicles in 2006(Belenguer and Benavent 2003). In 2008, Eglese proposed adeterministic tabu search algorithm to solve CARP (Handaet al. 2006). Recently, in 2009, Tang et al. proposed a memeticalgorithm with extended neighborhood search (MAENS) forCARP (Brandão and Eglese 2008). This work proposed anovel local search operator, which is capable of searchingusing large step sizes and is less likely to become trapped inlocally optimal solutions. Later, in 2011, Mei et al. combinedMAENS with other strategies to deal with various extendedmodels of CARP, such as a memetic algorithm for periodicCARP (Usberti et al. 2011), a decomposition-based memeticalgorithm for multi-objective CARP (Mei et al. 2011) andcooperative co-evolution with route distance grouping forlarge-scale CARP (Longo et al. 2006). Although MAENSand its variants have shown excellent performance in large-scale instances, it requires large numbers of function evalua-tions. Function evaluation is an important criterion in manyapplications of CARP, especially in those where the edgedemand is uncertain or variable (Santos et al. 2009).

Other metaheuristic methods include variable neighbor-hood descent method (Hertz and Mittaz 2001), tabu scattersearch (Lacomme et al. 2006), cutting plane algorithm (Tanget al. 2009), evolutionary algorithm approaches (Handa et al.2006), tabu search algorithm (TSA) (Handa et al. 2006),GRASP with evolutionary path-relinking (Usberti et al.2011) and solving CARP using a transformation to the capac-itated vehicle routing problem (CVRP) (Longo et al. 2006).These metaheuristics are usually superior to heuristics interms of quality of the solutions in large-scale instances,although their computational cost tends to be higher thannon-metahueristic methods.

The local search phase in MAENS can be divided intothree stages: (1) conventional move operators, (2) merge-split operator and (3) further conventional move operators(Brandão and Eglese 2008). During move operators (such assingle insertion, double insertion and swap), all the neighborsthat can be reached from the current solution are examined tofind the best possible improvement, according to a traversalstyle search. For example, given n edge tasks, a single inser-tion operator will require 2n2 function evaluations. As to the

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Immune clonal selection algorithm for capacitated arc routing problem

merge-split operator, p routes are selected from m possibleroutes for reconstruction, which also incurs high computa-tional cost (Brandão and Eglese 2008).

We need to provide timely vehicle arrangements accordingto the changing demand of edge task.

In order to obtain an acceptable solution within a lim-ited number of function evaluations, this paper proposesan immune clonal selection algorithm for CARP (ICSA-CARP). In ICSA-CARP, first, we use a recently proposed,state-of-the-art heuristic (Hertz et al. 2000) to generate initialsolutions which are most likely to accelerate convergence.Second, by cloning antibodies with high affinities, ICSA-CARP can focus on the most valuable parts of the solutionspace. By adopting different strategies for different clonesof the same antibody, it not only promotes cooperation andinformation exchanging among antibodies, but also increasesdiversity and speeds up convergence. Finally, two differentantibody repair operations (Montes and Coello 2005; Meiet al. 2009) are utilized, which cause infeasible solutions tomove in the direction of the global optimum, further accel-erating convergence.

The remainder of this paper is organized as follows: Sect. 2describes fundamental background material, including math-ematical treatments of CARP and artificial immune systems(AIS). Section 3 describes the ICSA-CARP procedure: anti-body initialization, immune clonal operation, immune geneoperation, antibody repair operation and the clonal selectionoperation. In Sect. 4, three publicly available ground-truthdatasets (gdb DeArmon 1981, val Benavent et al. 1992 andegl Eglese 1994) are used to evaluate the performance ofICSA-CARP in comparison to MAENS (Brandão and Eglese2008) and its two important variants (D-MAENS Mei et al.2011 and RDG-MAENS Longo et al. 2006). In Sect. 5, con-cluding remarks are presented.

2 Related concepts

2.1 Mathematical representation of the CARP problem

For the performance of an algorithm, the appropriate expres-sion of a solution is vital for conducting an effective searchand thus has a significant impact on the quality of solutions.In Hertz’s CARPET (2000), a CARP solution is expressed asa set of routes, each of which is based on the vertex encod-ing scheme. Each route is associated with an indicated vec-tor, composed of a series of binary variables at the end ofthe route. For each such binary variable, 0 means that thetask in the route only to be traveled and not to be servedwhereas 1 indicates that the task not only to be traveled butalso to be served. LMA represents solutions as ordered listsof tasks (Hertz and Mittaz 2001). ICSA-CARP uses a similarconcise task encoding scheme. The sequence of those tasks

Fig. 1 An example of the task ID encoding

can be transformed into a sequence of vertices, by connect-ing every two subsequent tasks by the shortest path betweenthem. CARP is defined on a mixed graph, G = (V, E, A),with required edges, ER ⊆ E, and required arcs, AR ⊆ A.First, we set all tasks (required edges and required arcs) in thegraph as a collection, T,where T = ER ∪ AR . Each requiredarc of the graph is assigned a unique ID. Each required edgeis regarded as a pair of arcs, one for each direction. Thus,each edge task is assigned two IDs (Mei et al. 2011). Forconvenience, all IDs are set to non-negative integers. EachID, e, has six properties, namely h(e), t (e), tc(e), sc(e), d(e)and i(e), which represent the head vertex, tail vertex, trav-eling cost, serving cost, demand and the inversion of taske, respectively (Mei et al. 2011). An illustration of task IDencoding is given in Fig. 1.

In Fig. 1, there are five edge tasks and one arc task inall. The relationships between ID 4 and 10 are h(4) =t (10), t (4) = h(10), d(4) = d(10), tc(4) = tc(10), sc(4) =sc(10) and i(4) = 10. Since all the IDs are positive,i(e) = −1 indicates that the inverse ID of edoes notexist, for example i(3) = −1. In addition, ID 0 is definedas the ID of the depot loop with the following defini-tions: h(0) = t (0) = depot, d(0) = tc(0) = sc(0) =0, i(0) = 0 (Mei et al. 2011). Using the above notations,a CARP solution can be represented as a set of routesx = (R1, R2, . . ., Rm), where the kth-route is showed asRk . In order to ensure that each route starts and ends at thedepot, the first task and the last task are all ID 0 in Rk . InFig. 1, assuming the number of vehicles is 2, a potentialsolution x = (0, 4, 3, 2, 0, 1, 12, 11, 0) is obtained, whereR1 = (0, 4, 3, 2, 0) and R2 = (0, 1, 12, 11, 0). For eachroute, Rk , its total cost TC (Rk) and total demand D(Rk) canbe calculated as follows:

TC(Rk) =length(Rk )−1∑

i=1

{sc(Rki ) + m

c[t (Rki )][

h(

Rk(i+1)

)]}

(1)

D(Rk) =length(Rk)∑

i=1

d(Rki ), (2)

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where the two-dimensional array, mc[i][ j], is the shortest trav-eling cost from vertex i to vertex j , and length(Rk) representsthe length of the sequence of tasks in Rk and Rki denotes thei th task in route Rk .

According to this solution representation scheme, theCARP can be represented as follows (Mei et al. 2011):

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min tot_cost(x) =m∑

k=1T C(Rk)

wherem∑

k=1(length(Rk) − 2) = |T |

Rk1i �= Rk2 j , Rk1i �= i(Rk2 j ), ∀ 1 ≤ k1 <k2 ≤ m;Rk1i �= Rk2 j , Rk1i �= i(Rk2 j ), ∀ k1 =k2 =k, i �= j;D(Rk) ≤ Q, ∀ 1 ≤ k ≤ m.

(3)

2.2 Overview of the artificial immune system approach

Immune System (IS) (Shang et al. 2014; Castro and Timmis2003) is a highly evolved, parallel and distributed adaptivesystem, which can be defined in terms of a computationalsystem. Inspired by IS, a new intelligent method simulat-ing natural immune system, called Artificial Immune System(AIS), was proposed and it has enormous potential to sup-ply novel methods to solve complex problems (Shang et al.2006, 2014). Because AIS is an abstraction of the biologicalimmune system (BIS); many of the concepts and operators ofAIS are associated with those found in BIS. The term “anti-gen” generally refers to the problem to be optimized. Anantibody (B cell) corresponds to a solution of the optimiza-tion problem. “Affinity” represents the quality of a feasiblesolution.

Based on the above correspondence between AIS and BIS,the process of simulating a biological immune response for-mulates the artificial immune algorithm for optimization. Thealgorithm contains the following modules (Shang et al. 2006,2012):

• Antigen recognition and the initial antibody production.According to the characteristics of the problem to be opti-mized, first we design an appropriate antibody encod-ing rule. Under this rule, we can use a-priori knowledgeto generate an initial antibody population (Longo et al.2006).

• Antibody evaluation. In the step of antibody quality eval-uation, the evaluation criterion is known as the anti-body’s “affinity”. High-quality antibodies will continueto evolve, and inferior antibodies will be updated.

• Clonal selection. By simulating immune variety of oper-ations observed in biological immune response, we gen-erate various useful operators, such as immune selection,

cloning, mutation and population updating. These opera-tors are used to form a clone selection principle, based onBIS evolution rules, enabling the solution of numericaloptimization problems (Shang et al. 2006).

3 Immune clonal selection for carp (Icsa-Carp)

The first step of ICSA-CARP is antibody initialization. Thisstep generates an initial set of solutions using a recently pub-lished state-of-the-art CARP heuristic, namely the construc-tion phase of the greedy randomized adaptive search pro-cedure (GRASP) (Handa et al. 2006; Usberti et al. 2011).Using this heuristic to generate initial solutions helps accel-erate convergence. After initialization, the iteration processbegins and all subsequent operations are carried out on theantibody population. The subsequent iterated steps are anti-body evaluation and the calculation of the cloning propor-tion. The cloning proportion is determined by the affinitybetween the antibody and the antigen and the similaritybetween the antibody and other antibodies. By selectivelycloning those antibodies which have high affinity and lowsimilarity, ICSA-CARP is able to focus on the most valuableregions of the solution space. Following the immune geneoperation, multiple mutation strategies are used to increasethe antibody population diversity. Finally, two different anti-body repair operations are employed, which help infeasi-ble solutions approach towards the global optimum, provid-ing additional improvements in convergence speed. Figure 2below shows the flowchart of ICSA-CARP and the follow-ing sections provide detailed descriptions of every stage ofICSA-CARP.

3.1 Antibody initialization

The initialization of ICSA-CARP uses an improved construc-tive heuristic, based on the construction phase of GRASP(Usberti et al. 2011). Prototyping is performed using the path-scanning method of (Golden et al. 1983). The heuristic firstcreates an empty route and then all tasks are inserted into theroute consecutively. Those tasks which meet the α criterionwill be selected to build the restricted candidate list (RCL).Next, we evaluate whether the vehicle is close to its full loadby using the β ellipse rule. If the current route can no longeraccommodate any additional tasks, then the end of the cur-rent route is attached to the depot via the shortest availablepath, and a new empty route is created. If the vehicle is notyet close to its full load, an additional task, e, is randomlyselected from the RCL and inserted into the end of the route.

In ICSA-CARP, we use the “five rules” of (Golden et al.1983) to choose the next task e in RCL: (1) minimize sc(e)/d(e); (2) maximize sc(e) /d(e); (3) minimize the cost back todepot; (4) maximize the cost back to depot; (5) use criterion

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Initialize the number of functionevaluations and antibody population

Determine whether the number ofevaluations reaches the set value

Evaluate the antibodies

Calculate the clone proportionand conduct clone operation

Conduct immune genetic operator bymultiple mutation strategies

Conduct antibody repair operationon the infeasibles olutions

Select the best Psize noncloneantibodies for the next generation by

stochasti cranking operator

Output the optimal feasible solution

No

Yes

Fig. 2 Flowchart of ICSA-CARP

3 if the vehicle has used more than half of its capacity; oth-erwise, criterion 4 is used. Each problem instance is solvedfive times, using a different criterion each time, and the bestof the five antibodies is exported. This method can constructa better feasible solution (Golden et al. 1983). The initialsolutions, generated by heuristics, improve the algorithm’sconvergence speed, compared to randomly generated initialsolutions. Details of the improved constructive heuristic forCARP are shown in Algorithm 1.

This improved constructive heuristic has two parameters,α and β, which directly affect the algorithm’s performanceand must be properly adjusted (Usberti et al. 2011). The α

rule is directly used to generate the RCL, which controlsthe convergence and diversity of the initial antibodies. Whenα = 0, the RCL has strong convergence, because the taskwhich has the nearest distance with u is selected to constructthe RCL. When α = 1, the RCL has good diversity becausetasks are randomly selected (Usberti et al. 2011). The ellipserule parameter β is responsible for controlling the ellipseshape. In other words, when a vehicle is near its full capac-ity, this rule forces the vehicle to stay closer to the depotto reduce its returning cost. According to the experimentalresults in Santos et al. (2009), β setting to 1.5 provides an effi-

Algorithm 1: an improved constructive heuristic for CARPInput: a CARP graph G, the vehicle capacity Q, β=1.5, α1=0, α2=0.25, α3=0.5, α4=0.75, α5=1;Output: a feasible CARP solution x; Begin

k=1; // the route indexRk1=0; // the first task 0 in each routefor (i=1 to |T|) do

RCL= ; for ( e T∀ ∈ ) do

if (mc[ t (u) ][ h (e) ] ≤ α * (ζmax - ζmin) + ζmin) // the α criterion is used to build the RCL

RCL=RCL {e};end if

end forif (rvc ≤ β * (td / |T|) ) // the β ellipse rulefor ( RLCe∀ ∈ ) do

if ( mc[ t (u) ][ h (e) ]+sc(e)+ mc[ t (e) ][ dep ] > ( tc / |T| + mc[ t (u) ][ dep ] )

RCL=RCL\{e}; // Remove those tasks which are furthest awayfrom the depot

end ifend for

end ifif (RCL= = ) // start a new route k++, Rk1=0;

elseThe five rules of Golden are used to choose the next task e in RCL;Insert e into the end of the current route, rvc = rvc-d(e);

end ifend forReturn x;

endwhere T represents the collection of all tasks, td is the total demand of all the tasks, tc is the total cost of all the tasks, rvc is the remaining vehicle capacity, RCL is the restricted candidate list, e is the task to be selected in the remaining task set, u is the last task in the current route, ζmin is the distance (traveling cost) between u and the task which has the farthest distance with u, ζmax is the distance between u and the task which has the nearest distance with u.

cient ellipse shape for many problems. In ICSA-CARP, α is,respectively, set to {0, 0.25, 0.5, 0.75, 1}. In order to obtainsix different initial solutions, the heuristic runs q (q ≥ 6)times independently under each α, so the population sizepsize = 5 ∗ 6 = 30.

3.2 Immune clonal operation

In immunology, cloning means asexual reproduction,wherein a parent cell is split into a plurality of progeny cells.Cloning provides multi-strategy conditions for mutation andICSA provides a mechanism for selecting in favour of thehighest quality antibodies. In ICSA-CARP, before the clonaloperation, the first step is antibody evaluation. Affinity (sim-ilar to the concept of fitness in GA methods) represents thebonding strength between the antigen and an antibody. Theantibody with highest affinity is most likely to be selectedfor cloning (Runarsson and Yao 2000). For each antibody xi

in the parent population P = {x1, x2, …, xpsize }, the affinityis calculated as

affinity(xi ) =(

L B

tot_cost(xi )

)3

, i = 1, 2, . . . , psize, (4)

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where L B is the lower bound of the CARP instance andtot_cost (xi ) is the total cost of antibody xi .

The clonal proportion of each antibody also has a relation-ship with the similarity between two different antibodies. Thehigher the similarity, the stronger the inhibition which arisesbetween antibodies. In ICSA-CARP, a two-dimensionalarray D[i][ j] describes the distance between antibody xi andx j , thereby encoding information about diversity.

In D, the maximum is dmax and di represents the mini-mum in the i th row. Then we get the clonal ratio of antibodyxi :

clone(xi ) = ceil

⎛

⎜⎜⎜⎝affinity(xi )

psize∑j=1

affinity(x j )

∗ Psize ∗ λ ∗ edi

dmax

⎞

⎟⎟⎟⎠ ,

i = 1, . . . , psize (5)

The function ceil (a) represents the smallest integer greaterthan a. In ICSA-CARP, λ = 3, where λ is a parameterrelated to the proportion of cloning. 0 ≤ (di / dmax) <1,when certain antibodies have a high similarity with anti-body xi . In such cases di / dmax is close to 0 and theclonal proportion of xi will be suppressed. After the cloningoperation, the population can be described as follows:

P1 = {x ′

1, x ′2, ..., x ′

psize

}, (6)

where x ′i = {x ′

i j } = {xi1, xi2, ..., xiclone(xi )}, x ′i j = xi j =

xi , j = 1, 2, ..., clone(xi ).After the immune clonal operation has been carried out,

each antibody is substituted by a set of identical clones. How-ever, the clonal proportion of each antibody is different. Thecloning proportion is determined by the affinity between theantibody and the antigen, as well as the similarity betweenthe antibody and other antibodies.

Because of the discreteness of CARP, ICSA-CARP cal-culates the similarity between antibodies based on thesequences composed of task ID. In this paper, we think theantibody xi whose two consecutive tasks in any route areserved also as two consecutive tasks in the same order inantibody yi has no distance with yi when calculating the dis-tance between antibody xi and yi about the two tasks. Forsimplicity, ICSA-CARP presents the relationship betweenantibodies by array D. D[i][ j] indicates the distance betweenthe i-th antibody and the j-th antibody and D[i][ j] is equal toD[ j][i]. The greater the distance between two antibodies, thelower the similarity between them. Calculating the similaritybetween antibodies in this way is not only easy to operate butalso conforms to the theory of CARP.

ICSA-CARP enables rapid amplification of those antibod-ies which exhibit both high affinity and also low similarity

Fig. 3 Single insertion

Fig. 4 Double insertion

Fig. 5 Swap

with other antibodies. The details of the calculation of thesimilarity array D are shown in Algorithm 2.

3.3 Immune gene operation

In immunology, hyper-mutation is the main mechanism bywhich the immune system recognizes external patterns in theform of antibody gene mutation and compilation, so as to gainhigher affinity. In the ICSA-CARP method, proposed in thispaper, for the Immune gene operation, five different mutationstrategies are randomly used with a predefined probabilitypm on the clone population (Mei et al. 2011), explained asfollows (Figs. 3, 4, 5, 6):

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Fig. 6 Part inverse

Algorithm 2: the calculation of the similarity array DInput: the antibody xi and xj , i≠j, 1≤i≤psize , 1≤j≤psize ; Output: D[i][j] and D[j][i] ; Begin

distance=0;for ( k=1; k < length(xi); k++) do

if ( the (k+1)-th task in antibody xi ≠the (z+1)-th task in antibody xj

)&&( the inverse ID of the (k+1)-th task in antibody xi ≠the (s-1)-th task in antibody xj)

distance++;end if

end forD[i][j]= D[j[i] =distance; Return D[i][j] and D[j][i];

endwhere distance describes the distance between two different antibodies, t is the k-th task in antibody xi , i(t) is the inverse task ID of task t, we suppose that task t is located at the z-th place in xj, or i(t) is located at the s-th place in xj .

Fig. 7 2-opt mutation operation

Algorithm 3: antibody repair method 1 for infeasible antibodiesBegin

Input an infeasible antibody x = (R1, R2, …, Rm), i=1;Determine those routes which violate the capacity constraints. SupposeW ={ R1, R2, …, Rτ } is the set which consists of τ overload routes;

=1

= ( )t

jj

UMS D R∑ ;

The remaining m-τ routes are sorted in ascending order according to their different task demands, expressed as {R'

1, R'2, …, R'

(m-τ)}; while (SUM + D(R'

i) > (i+τ)*Q) doPut R'

i into W; SUM = SUM + D(R'

i);i++;

end whilePut R'

i into W; Use path-scanning heuristic to sort these unordered tasks in W and

reconstruct a feasible solution; Apply Ulusoy’s splitting to optimally split the ordered lists into new;Embed the new routes back into the original antibody x and obtain a new feasible antibody x';

end

Algorithm 4: antibody repair method 2 for infeasible antibodiesBegin

Input an infeasible antibody x=(R1, R2, …, Rm). For all tasks {e1, e2, …, etask_num}, if task ei is traversed (deadheading or serving as task) in j-th route, Ω[j][i]=1, else Ω[j][i]=0, where Ω is a two-dimensional array of mrows and task_num columns; Initialize M={ e1, e2, …, etask_num}, where M contains the remaining tasks

which have not yet been inserted into any route;while (M ≠ ) do

For all the ei in M, determine the route set Φ(ei) which task ei can be inserted into, Φ(ei)={j| Ω[j][i]=1&& d(ei) ≤ rvcj}; The task with the minimum |Φ(ei)| ( the size of Φ(ei) ) will first be selected; If multiple tasks meet the above condition, the one with the largest d(ei) will be selected; Suppose the task is ek; Determine the route with minimum rvc in Φ(ek). If more than one route meets this condition, randomly choose a route and insert ek ;

Remove ek from M and update the current load in each route;end while

endwhere Ri (1≤i≤m) represents the route by i-th vehicle, rvcj is the remaining vehicle capacity of the j-th vehicle, e is the task required to be served, task_numis the total task number.

Algorithm 5: Immune clonal selection algorithm for CARPBegin

Initialize the number of function evaluations it=0 and antibody population P;

while (anti_best ≠ LB && it<function_evaluation_number) doAfter antibody evaluation and calculation of the similarity matrix,

get a clonal antibody group P1; Conduct immune gene operator on P1 and get an antibody group P2; Conduct antibody repair operation on P2 and get an antibody group P3

(if an infeasible antibody’s total cost is less than 1.3*LB, Method 2 is used, otherwise Method 1 is used);Sort P3 using stochastic ranking and assign the former psize

antibodies to P; end whileOutput the best feasible antibody anti_best;

endwhere LB represents the lower bound of the test instance, anti_best is the best feasible antibody, pm is the mutation probability, in this paper, pm=0.6.

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Table 1 Average total_cost of compared algorithms on gdb set under 10,000 function evaluations

Name |V | |T | |E | MAENS D-MAENS RDG-MAENS ICSA-CARP Lower bound

Man SD Mean SD Mean SD Mean SD

gdb1 12 22 22 322.2 4.1 326.8 3.9 326.6 3.8 317.0 2.8 316

gdb2 12 26 26 357.8 4.5 363.5 4.5 362.1 5.0 348.5 3.7 339

gdb3 12 22 22 284.5 4.3 290.8 4.2 292.1 4.5 278.2 3.4 275

gdb4 11 19 19 290.6 4.4 295.0 5.0 295.8 4.8 287.2 2.0 287

gdb5 13 26 26 396.1 5.0 402.1 5.0 404.1 4.1 384.9 3.7 377

gdb6 12 22 22 315.0 3.9 320.5 4.1 322.4 4.4 306.5 4.2 298

gdb7 12 22 22 330.0 4.2 333.3 4.6 335.4 4.3 325.0 0.0 325

gdb8 27 46 46 421.4 5.3 426.0 4.7 425.6 4.9 403.3 6.7 348

gdb9 27 51 51 366.8 4.8 362.7 4.5 365.1 4.9 360.5 5.3 303

gdb10 12 25 25 285.1 4.2 291.4 3.7 294.3 4.5 276.8 3.3 275

gdb11 22 45 45 443.3 4.4 445.7 4.4 451.2 5.0 432.4 5.5 395

gdb12 13 23 23 553.4 7.9 585.9 6.0 578.7 6.5 485.4 5.9 458

gdb13 10 28 28 561.1 4.5 564.0 4.5 566.9 4.7 547.4 2.4 536

gdb14 7 21 21 102.2 2.8 104.7 3.2 110.4 3.5 100.1 1.5 100

gdb15 7 21 21 58.5 1.9 59.8 2.3 60.5 2.2 58.0 0.0 58

gdb16 8 28 28 131.3 2.8 133.7 2.6 133.5 2.7 128.5 2.2 127

gdb17 8 28 28 91.6 2.0 91.9 2.0 92.1 2.5 91.0 0.0 91

gdb18 9 36 36 176.0 3.4 175.5 2.9 178.0 3.0 165.6 2.8 164

gdb19 8 11 11 55.2 1.6 55.1 1.3 56.2 2.9 55.0 0.0 55

gdb20 11 22 22 124.8 2.7 124.3 2.6 125.6 2.8 122.6 1.8 121

gdb21 11 33 33 166.5 3.4 167.5 3.2 165.9 2.9 159.1 2.7 156

gdb22 11 44 44 206.7 2.4 206.3 2.4 206.7 2.3 203.3 2.6 200

gdb23 11 55 55 246.7 2.7 246.5 2.5 246.1 3.2 242.9 3.3 233

Mean _ _ _ 273.3 3.8 277.1 3.7 278.1 3.9 264.3 2.9 253.8

1. Single insertion: randomly select a task and move it infront of another task. If the selected task is an edge task,both its directions will be considered.

2. Double insertion: Randomly select two consecutive tasksand move them in front of another task. If the selectedtasks are edge tasks, both their directions will be consid-ered.

3. Swap: randomly select two different tasks and exchangetheir positions. If the selected tasks are edge tasks, boththeir directions will be considered.

4. Part inverse: randomly select a route Rk in antibody xand Rk is randomly cut into three subroutes Rk = (Rk1,Rk2, Rk3). Reverse the second subroute Rk2 and obtain anew Rk = (Rk1, inverse (Rk2), Rk3).

5. 2-opt operation: there are two types of 2-opt move opera-tors, including both single route and double routes (Beul-lens et al. 2003). In ICSA-CARP, the double routes 2-optoperator is used. We select two routes R1 and R2 from thecurrent antibody x randomly. These two routes are as fol-lows: R1 = (t1, t2, …, ti , …, ts) and R2 = (u1, u2, …, u j ,…, un). Then the two routes are further divided into four

subroutes randomly. R11 = (t1, t2, …, ti ), R12 = (ti+1,…, ts), R21 = (u1, u2, …, u j ), R22 = (u j+1, …, un).Through different connections, two different candidatesolutions are generated by reconnecting the four sub-routes. One candidate solution can be obtained by con-necting R11 with R22, and R21 with R12, R

′1 = (t1, …,

ti , u j+1, …, un), R′2 = (u1, …, u j , ti+1, …, ts). And the

other candidate solution can be obtained by connectingR11 with the inversion of R21, and the inversion of R12

with R22, R′′1 = (t1, …, ti , i(u j ), …, i(u1)), R

′′2 = (i(ts),

…, i(ti+1), u j+1, …, un). Finally, we can choose the routewith the smallest cost. Specific operations are shown inFig. 7.

In ICSA-CARP, hypermutation acts like a local searchstrategy; however, it is random and undirected. Each hyper-mutation only changes a small portion of the current antibody,with the purpose of increasing the diversity of the population.In contrast, most of meta-heuristic algorithms for CARP usetraverse local search operators which are all based on greedysearch approaches (Brandão and Eglese 2008; Belenguer and

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Table 2 Average total_cost of the compared algorithms on val set under 10,000 function evaluations



1A 24 39 39 184.3 3.3 184.3 3.3 183.3 2.9 180.7 3.5 173

1B 24 39 39 193.3 3.2 192.4 3.7 193.1 4.1 191.0 3.9 173

1C 24 39 39 298.6 4.9 300.2 5.0 300.2 5.1 271.6 5.8 245

2A 24 34 34 243.6 4.2 245.3 4.2 246.6 4.3 243.9 3.9 227

2B 24 34 34 282.3 4.6 283.7 4.5 287.1 3.9 275.0 4.5 259

2C 24 34 34 509.3 4.9 510.5 4.9 510.3 5.0 500.7 5.4 457

3A 24 35 35 86.3 2.5 86.4 2.5 86.6 2.6 86.0 2.7 81

3B 24 35 35 97.0 2.8 97.5 2.9 96.4 3.0 93.7 2.8 87

3C 24 35 35 150.3 3.3 150.5 3.3 150.6 3.0 148.0 3.2 138

4A 41 69 69 438.0 4.6 440.2 4.7 442.4 5.1 437.6 5.0 400

4B 41 69 69 468.2 4.4 438.8 4.5 468.4 5.1 470.4 4.8 412

4C 41 69 69 501.7 5.0 499.0 5.0 502.1 5.4 443.8 4.2 428

4D 41 69 69 642.8 5.1 640.6 4.9 643.5 4.8 639.2 5.7 526

5A 34 65 65 464.2 4.9 463.0 4.3 466.2 4.9 464.6 5.0 423

5B 34 65 65 496.1 5.3 496.6 5.1 499.6 4.7 495.1 5.2 446

5C 34 65 65 532.5 4.9 532.7 5.6 533.2 5.7 528.5 5.5 473

5D 34 65 65 696.1 5.1 692.8 5.7 693.9 5.3 691.3 5.2 573

6A 31 50 50 243.9 3.8 245.3 4.0 242.7 3.6 240.3 3.8 223

6B 31 50 50 262.5 4.2 264.4 3.9 262.2 3.3 259.6 4.0 233

6C 31 50 50 379.4 4.4 376.5 5.0 377.3 4.4 368.5 5.4 317

7A 40 66 66 315.0 4.0 313.5 4.2 322.3 4.0 312.3 4.4 279

7B 40 66 66 325.1 4.4 325.6 4.2 327.6 4.5 327.0 4.3 283

7C 40 66 66 402.9 4.8 401.2 4.5 398.6 5.2 398.8 5.1 334

8A 30 63 63 428.4 4.7 428.6 4.7 429.2 4.7 426.8 4.7 386

8B 30 63 63 454.7 4.8 452.2 4.9 453.4 4.2 453.4 4.7 395

8C 30 63 63 603.7 4.9 605.5 4.3 610.5 5.1 601.1 5.0 518

9A 50 92 92 351.2 3.6 351.7 3.5 353.4 3.8 348.9 3.4 323

9B 50 92 92 363.7 4.3 364.4 3.6 366.1 3.9 356.4 3.7 326

9C 50 92 92 380.3 4.1 381.2 4.1 379.8 3.8 377.1 4.0 332

9D 50 92 92 467.9 4.2 467.1 4.0 471.5 4.8 463.0 4.2 385

10A 50 97 97 460.9 3.8 458.8 3.9 461.9 4.5 450.7 3.7 428

10B 50 97 97 477.8 4.2 478.7 4.1 477.1 3.6 472.3 4.0 436

10C 50 97 97 494.9 4.4 497.9 4.3 500.4 5.0 486.6 4.0 446

10D 50 97 97 607.3 4.6 609.7 4.5 609.8 4.8 603.4 4.8 525

Mean _ _ _ 391.3 4.3 390.5 4.3 392.6 4.4 385.5 4.4 343.2

Bold values indicate the best results

Benavent 2003; Li and Yao 2011; Mei et al. 2009). Thesetraverse local search operators continually search the neigh-bors of the current solution leading to high computationalexpense. Meanwhile, the performance of these traverse localsearch operators is strongly dependent on the quality of theinitial solutions. MAENS conducts exhaustive search withinthe neighborhoods defined by the single insertion, doubleinsertion, swap and merge-split operators. The single inser-tion operation first selects one task out of the total n tasksand then inserts this task into n + m − 1 potential positions,

where m is the number of routes. Thus, the single inser-tion operation requires n2 function evaluations (Brandão andEglese 2008). If each single move operator is consecutivelyapplied k times, then the function evaluations will be n2k .Furthermore, the complex merge-split operator is also sig-nificantly time-consuming. In MAENS, all offspring shareequal probability for local search, so that poor offspring leadto numerous unsuccessful searches. In contrast, by cloningthose antibodies which have the highest affinity and the low-est similarity, ICSA-CARP focuses computational effort on

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Table 3 The average total_cost of the compared algorithms on egl set under 10,000 function evaluations



E1-A 77 51 98 4111.9 10.0 4107.0 11.8 4125.9 11.4 4102.2 9.8 3548

E1-B 77 51 98 5105.6 11.2 5109.9 10.4 5117.9 10.2 5078.9 10.7 4498

E1-C 77 51 98 6938.7 11.2 6913.2 11.0 6949.4 10.5 6903.1 10.4 5566

E2-A 77 72 98 5852.2 11.2 5874.7 12.2 5836.1 14.1 5818.8 10.7 5018

E2-B 77 72 98 7583.9 13.6 7594.8 15.4 7636.2 13.0 7571.7 9.0 6305

E2-C 77 72 98 9858.9 14.4 9851.4 14.1 9897.5 13.3 9849.4 11.5 8243

E3-A 77 87 98 6932.0 14.1 6928.6 13.2 6962.7 12.7 6917.7 10.5 5898

E3-B 77 87 98 9126.1 14.2 9178.7 15.2 9181.4 13.9 9124.6 12.6 7704

E3-C 77 87 98 12,096.5 15.3 12138 15.1 12,113.8 16.4 12,086.6 13.8 10,163

E4-A 77 98 98 7627.6 14.2 7625.7 13.2 7605.7 13.9 7603.0 9.7 6408

E4-B 77 98 98 10,427.1 14.9 10,438.5 14.3 10,507.0 12.7 10,422.8 11.4 8884

E4-C 77 98 98 13,798.3 15.3 13,830.9 16.0 13,806.0 15.7 13,786.1 13.4 11,427

S1-A 140 75 190 6098.5 13.0 6102.0 12.1 6110.7 11.7 6087.4 11.0 5018

S1-B 140 75 190 8301.3 9.3 8305.2 9.9 8286.1 9.0 8267.6 10.3 6384

S1-C 140 75 190 10,022.7 13.3 10,031.2 13.3 10,005.3 14.5 9938.0 10.2 8493

S2-A 140 147 190 12,001.2 14.3 12,036.6 13.7 12,011.8 14.9 11,995.4 12.6 9824

S2-B 140 147 190 15,582.9 18.3 15,610.8 18.1 15,613.9 17.7 15,548.7 14.0 12,968

S2-C 140 147 190 19,699.2 15.5 19,677.5 16.0 19,688.0 16.5 19,647.2 12.0 16,353

S3-A 140 159 190 12,696.1 14.3 12,737.7 13.5 12,754.3 14.5 12,651.1 12.2 10,143

S3-B 140 159 190 16631.0 14.4 16,670.0 16.9 16,664.2 15.8 16,585.3 11.8 13,616

S3-C 140 159 190 20,867.8 17.6 20,918.8 16.4 20,889.2 17.3 20,800.1 11.6 17,100

S4-A 140 190 190 15,199.9 16.4 15,235.9 14.2 15,221.5 14.7 15,163.8 10.8 12,143

S4-B 140 190 190 19,718.9 14.9 19,718.9 15.7 19,681.9 16.0 19,696.1 11.4 16,093

S4-C 140 190 190 24,875.0 18.2 24,848.4 17.2 24,890.2 17.9 24,818.6 13.8 20,375

Mean _ _ _ 11,714.7 14.1 11,728.3 14.1 11,733.4 14.1 11,686.3 11.5 9673.8

the more valuable regions of the solution space, acceleratingconvergence.

3.4 Antibody repair operation

There are many common methods to deal with constrainedoptimization problems, such as penalty function method (Meiet al. 2011; Brandão and Eglese 2008), multi-objective strat-egy (Montes and Coello 2005) and repair operators for theinfeasible solutions (Mei et al. 2009). The penalty functionmethod is a commonly used method to deal with capacityconstraints in CARP. Because the degree of antibody con-straint violations is decided by a penalty function, researchersput most of their studies on the designing of the penalty coef-ficient to design a suitable penalty function. In CARP, Thepenalty function is generally expressed as follows (Mei et al.2011):

f (x) = tot_cost(x) + δ∗m∑

k=1

(max{D(Rk) − Q, 0}), (7)

where tot_cost(x) is the total cost of antibody x, δ is thepenalty coefficient and Q is the vehicle’s capacity. In prac-tice, the selection of the penalty coefficient δ is non-trivial.If δ is too small, the algorithm may converge on a solu-tion which is far from the true optimum. So it is difficultto generate feasible solutions in this case. If δ is too large,it may cause excessive computational expense and lead topremature convergence (Mei et al. 2011). Converting con-strained optimization problems into multi-objective opti-mization problems has attracted considerable recent atten-tion. These approaches treat the constraints as one or moreobjective functions (Shang et al. 2012). However, CARP isdifferent from most of the conventional function optimizationproblems. It belongs to the discrete combinatorial optimiza-tion problem, and the solution space is divided into severalseparate feasible regions by constraints. So that such methodsare difficult to apply.

The optimal solution of an integer linear programmingproblem must be located inside the boundary of the feasibleconvex hull. Since this is adjacent to the infeasible region, it

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Table 4 The average total_cost of the compared algorithms on gdb set under 100,000 function evaluations



gdb1 12 22 22 316.2 2.2 316.7 2.7 324.1 3.1 316.0 0.0 316

gdb2 12 26 26 345.2 3.1 348.6 3.3 355.9 4.1 341.1 2.9 339

gdb3 12 22 22 275.4 2.4 277.4 3.2 286.6 4.0 275.0 0.0 275

gdb4 11 19 19 287.0 0.0 287.0 0.0 289.6 3.6 287.0 0.0 287

gdb5 13 26 26 382.2 2.8 386.0 3.8 389.6 4.0 378.8 2.9 377

gdb6 12 22 22 303.3 4.2 305.5 4.3 310.3 4.0 298.0 0.0 298

gdb7 12 22 22 325.0 0.0 325.0 0.0 329.3 3.0 325.0 0.0 325

gdb8 27 46 46 379.3 6.1 409.0 6.1 412.4 4.9 358.5 3.2 348

gdb9 27 51 51 346.3 6.3 360.2 5.4 365.0 4.2 320.0 3.5 303

gdb10 12 25 25 276.1 2.8 277.5 3.6 278.8 4.2 275.0 0.0 275

gdb11 22 45 45 416.8 5.4 434.8 5.3 440.6 4.9 412.9 3.4 395

gdb12 13 23 23 484.7 5.8 492.4 5.9 552.0 8.1 462.2 3.5 458

gdb13 10 28 28 546.3 2.6 547.5 2.8 555.6 3.8 543.9 2.2 536

gdb14 7 21 21 100.0 0.0 100.1 1.5 102.5 3.0 100.0 0.0 100

gdb15 7 21 21 58.0 0.0 58.0 0.0 58.5 2.0 58.0 0.0 58

gdb16 8 28 28 127.8 2.0 128.5 2.1 132.3 3.0 127.0 0.0 127

gdb17 8 28 28 91.0 0.0 91.0 0.0 91.5 2.0 91.0 0.0 91

gdb18 9 36 36 164.8 2.1 165.6 2.5 170.0 3.1 164.0 0.0 164

gdb19 8 11 11 55.0 0.0 55.0 0.0 55.0 0.0 55.0 0.0 55

gdb20 11 22 22 122.5 1.8 122.7 1.9 124.3 2.7 121.2 1.5 121

gdb21 11 33 33 158.3 2.4 160.1 2.6 166.0 3.2 156.1 1.4 156

gdb22 11 44 44 202.2 2.1 203.0 2.3 206.2 2.4 200.2 1.4 200

gdb23 11 55 55 238.6 2.8 244.7 3.5 245.9 2.5 235.2 2.0 233

Mean _ _ _ 261.0 2.5 265.1 2.7 271.3 3.5 256.1 1.2 253.8


makes the exploration of the infeasible solution space partic-ularly important. In ICSA-CARP, we use two repair operators(referred to later as “method 1” and “method 2”) to improvethese infeasible antibodies. First, these methods provide away of transforming infeasible antibodies into feasible anti-bodies. Second, they are equivalent to a local search, whichsignificantly enhances searching of the infeasible/feasibleboundaries of the solution space. Many experiments alsoshow that the performance of repair operators is better thanother methods in handling constrained optimization prob-lems (Montes and Coello 2005). In ICSA-CARP, two differ-ent antibody repair methods are used to deal with differentinfeasible antibodies. Method 1 can handle these infeasibleantibodies with high total cost by reconstructing the routes ofoverloaded vehicles. Method 2 tries to minimize the capacityviolation of these low total cost infeasible antibodies whileretaining their original driving path. Here, we define the hightotal cost as the total cost more than 1.3*L B , and low totalcost as the total cost less than 1.3*L B and L B representsthe lower bound of the test instance. Therefore, Method 1is used for infeasible antibodies with a large degree of con-

straint violation, whereas Method 2 is used for infeasibleantibodies with a small degree of constraint violation.

Method 1 For an infeasible antibody x which contains mroutes, the first step of method 1 is to find out those routeswhich violate the capacity constraints. Suppose the numberof overload routes is τ and these τ routes’ total task demandsare described as SU M . Next, the remaining m − τ routes aresorted in ascending order according to their different taskdemands. The route ranked top is first added to SU M . IfSU M > (τ + 1)*Q, then we continue iterating to add thenext highest ranked route to the SU M . Next, we judge whetherSU M is greater than (τ + 2)*Q. If the condition is not metthen the algorithm jumps out of the cycle and goes to thesecond step. In the second step, we disrupt the order of thosetasks belonging to the routes in SU M . These dis-ordered tasksare then handled as a small-scale CARP instance, and a path-scanning heuristic (Golden et al. 1983) is employed to re-sortthese tasks to reconstruct a feasible solution. Next, Ulusoy’ssplitting procedure (Ulusoy 1985) is used to split the orderedlists into new routes in an optimal way. This method can

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Table 5 The average total_cost of the compared algorithms on val set under 100,000 function evaluations



1A 24 39 39 182.1 3.3 182.4 3.0 185.4 2.8 180.0 0.0 173

1B 24 39 39 184.8 3.4 189.0 4.0 193.5 3.6 180.9 2.7 173

1C 24 39 39 263.3 4.3 273.0 5.5 302.5 4.4 251.6 3.1 245

2A 24 34 34 242.5 4.4 243.2 4.5 246.1 4.0 234.2 3.4 227

2B 24 34 34 269.6 4.0 273.7 4.2 284.1 3.8 260.3 2.0 259

2C 24 34 34 485.6 4.7 494.1 5.3 507.8 4.9 471.8 3.9 457

3A 24 35 35 86.0 2.4 86.8 2.5 86.8 2.5 83.1 1.9 81

3B 24 35 35 91.8 2.6 94.1 3.3 93.9 2.8 88.2 1.7 87

3C 24 35 35 145.5 2.7 146.9 3.1 148.1 3.3 140.3 2.5 138

4A 41 69 69 440.6 4.3 441.7 4.4 441.0 4.7 419.5 4.4 400

4B 41 69 69 465.2 5.1 468.3 4.7 469.2 5.0 435.2 4.4 412

4C 41 69 69 496.5 5.4 498.3 4.5 500.1 5.0 459.8 4.5 428

4D 41 69 69 638.0 5.5 641.3 5.2 641.7 5.0 573.2 4.6 526

5A 34 65 65 460.6 5.0 463.1 4.8 468.3 4.8 438.9 4.1 423

5B 34 65 65 494.6 5.4 496.4 4.8 498.7 4.9 457.6 4.2 446

5C 34 65 65 528.5 5.8 529.8 5.8 535.0 5.0 488.4 3.8 473

5D 34 65 65 682.6 7.0 694.0 5.3 694.8 4.8 619.0 4.4 573

6A 31 50 50 241.0 3.8 243.8 4.2 242.0 3.8 231.0 3.4 223

6B 31 50 50 252.7 3.8 258.9 4.1 261.4 4.0 242.6 2.8 233

6C 31 50 50 349.6 5.8 371.3 5.2 369.7 5.5 333.0 3.0 317

7A 40 66 66 310.2 4.3 311.8 4.3 310.8 4.5 294.6 3.8 279

7B 40 66 66 325.8 4.8 323.8 4.5 323.1 4.4 298.2 4.0 283

7C 40 66 66 396.8 5.7 401.1 4.4 397.8 4.9 353.3 4.1 334

8A 30 63 63 424.7 4.6 426.6 4.8 430.9 4.3 414.4 4.3 386

8B 30 63 63 444.5 5.3 451.3 4.9 455.2 4.7 415.9 4.2 395

8C 30 63 63 593.0 6.2 603.2 5.2 602.5 5.4 568.6 4.5 518

9A 50 92 92 352.9 3.8 353.0 3.5 352.7 3.6 344.4 3.8 323

9B 50 92 92 365.3 3.9 365.3 3.8 364.7 4.0 349.2 4.0 326

9C 50 92 92 379.0 4.5 380.7 3.7 378.6 4.0 355.6 4.4 332

9D 50 92 92 470.0 4.2 469.1 4.4 466.8 4.6 426.1 4.6 385

10A 50 97 97 459.3 3.8 461.3 3.5 459.6 3.7 448.7 4.3 428

10B 50 97 97 476.1 4.2 478.1 4.1 485.5 3.8 457.3 4.6 436

10C 50 97 97 496.6 4.0 497.7 3.9 501.6 4.1 474.3 4.7 446

10D 50 97 97 608.0 4.5 609.7 4.2 607.7 4.4 564.2 4.3 525

Mean _ _ _ 385.4 4.5 388.9 4.3 391.4 4.3 363.3 3.7 343.2

choose the best split point to minimize the additional con-sumption which is caused by splitting. Finally, we may obtaina new antibody x’ by embedding the new routes back into theoriginal antibody x. The procedure for the method 1 antibodyrepair operation is shown in Algorithm 3.

Method 2 Handa et al. proposed a repair operator for infea-sible solutions (Belenguer and Benavent 2003) which ran-domly chooses a task e from whichever route has maximumviolation of the capacity constraint. This task is then moved

to other routes which have an opening for the task e (such thatthe task e must be traversed as a deadheading path). How-ever, this method merely corrects a small part of the solutionand does not consider it as a whole. Mei et al. proposed analternative repair operator, called GRO, which is designedto handle these infeasible solutions with low total cost (Meiet al. 2009). However, GRO is limited to algorithms withvertex encoding.

By making use of some global repair strategies of GRO, wedesign method 2 and apply it to the task encoding algorithm.

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Table 6 The average total_cost of the compared algorithms on egl set under 100,000 function evaluations



E1-A 77 51 98 4054.3 13.5 4126.0 11.6 4134.8 12.8 3727.5 11.2 3548

E1-B 77 51 98 4978.6 13.8 5071.6 11.8 5089.9 11.0 4655.5 9.8 4498

E1-C 77 51 98 6570.7 17.2 6882.5 12.6 6881.1 12.3 5923.9 11.6 5566

E2-A 77 72 98 5810.6 12.0 5828.0 12.4 5855.1 11.74 5368.4 11.8 5018

E2-B 77 72 98 7564.4 14.7 7628.2 12.9 7558.6 13.0 6717.7 11.1 6305

E2-C 77 72 98 9845.7 14.0 9838.1 13.8 9825.5 14.6 8956.6 8.7 8243

E3-A 77 87 98 6936.1 13.5 6899.4 13.2 6978.8 14.0 6432.7 12.5 5898

E3-B 77 87 98 9119.5 14.5 9169.0 13.8 9111.4 14.1 8496.7 11.7 7704

E3-C 77 87 98 12,084.7 15.1 12,046.7 15.4 12,082.0 16.9 11,122.8 11.1 10,163

E4-A 77 98 98 7611.8 13.8 7624.3 14.0 7127.9 14.1 7136.3 14.2 6408

E4-B 77 98 98 10,432.1 13.9 10,410.5 14.7 10,504.7 13.7 9843.3 10.5 8884

E4-C 77 98 98 13,803.0 14.6 13,793.3 15.7 13,833.8 15.3 12,604.3 10.0 11,427

S1-A 140 75 190 6068.8 12.8 6081.0 12.5 6089.9 12.2 5570.2 10.5 5018

S1-B 140 75 190 8171.9 17.0 8259.2 12.8 8248.7 12.9 7018.7 12.4 6384

S1-C 140 75 190 9985.7 14.4 10,004.8 14.4 10,014.3 13.2 9268.8 10.7 8493

S2-A 140 147 190 12,080.7 14.2 12,038.8 15.1 12,090.9 14.6 11,880.7 11.4 9824

S2-B 140 147 190 15,618.6 17.8 15,675.9 16.9 15,567.8 17.7 15,477.3 12.9 12,968

S2-C 140 147 190 19,718.7 16.1 19,613.2 16.1 19,697.2 15.8 19,430.1 14.7 16,353

S3-A 140 159 190 12,678.6 15.3 12,681.2 15.3 12,728.3 13.8 12,540.8 10.7 10,143

S3-B 140 159 190 16,733.8 17.0 16,656.3 16.4 16,624.7 16.3 16,458.7 16.2 13,616

S3-C 140 159 190 20,893.4 16.3 21,011.2 17.3 20,911.2 17.3 20,807.2 17.2 17,100

S4-A 140 190 190 15,190.2 16.3 15,264.3 13.7 15,180.2 14.3 15,195.7 14.4 12,143

S4-B 140 190 190 19,679.2 15.4 19,683.5 17.4 19,690.6 15.8 19,581.4 12.7 16,093

S4-C 140 190 190 24,844.7 17.2 24,865.8 18.1 24,877.3 16.4 24,835.1 15.5 20,375

Mean _ _ _ 11,686.5 15.0 11,714.7 14.5 11,714.0 14.2 11,210.5 12.2 9673.8


The main ideas of Method 2 are described as follows: foran infeasible antibody x which contains m routes, the totalnumber of tasks is denoted as task_num. For all the tasks{e1, e2, …, etask_num}, if task ei is traversed (deadheading orserving as task) in the j-th route, �[ j][i] = 1, else �[ j][i] = 0,where � is a two-dimensional array of m rows and task_numcolumns. By preserving the original vehicle traversing pathand regrouping all tasks according to certain rules, we canget a better task allocation scheme without any additionalcost and violation of the capacity constraints. The specificprocess of Method 2 is shown in Algorithm 4.

3.5 Clonal selection operation

ICSA-CARP employs stochastic ranking operator, a differentclonal selection operator from conventional immune clonaloperators, to select the best antibodies from the offspringgenerated by clonal proliferation. Proposed by Runarssonand Yao (2000), the stochastic ranking method has shown

good performance on many constrained optimization prob-lems. The stochastic ranking criterion is as follows: if thetwo compared antibodies are both feasible, then comparisonwill be made only with respect to affinity. Otherwise, the twoantibodies will be compared according to their affinity witha probability p f or according to their constraint violationswith a probability (1− p f ). p f is set to 0.45 by the user. Thebest psize nonclone antibodies will be selected and evolvedinto the next generation.

3.6 Summary of proposed immune clonal selectionalgorithm

Algorithm 5 summarizes the new immune clonal selectionalgorithm for CARP (ICSA-CARP). The algorithm makesuse of the various steps described in previous sections ofthe paper: immune clone operation, immune gene operation,antibody repair operation and clonal selection operation.

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Table 7 The average total_cost of the compared algorithms on gdb set under 1,000,000 function evaluations



gdb1 12 22 22 316.0 0.0 316.0 0.0 316.0 0.0 316.0 0.0 316

gdb2 12 26 26 340.9 2.9 343.0 3.0 342.1 3.0 340.6 2.8 339

gdb3 12 22 22 275.0 0.0 275.0 0.0 275.0 0.0 275.0 0.0 275

gdb4 11 19 19 287.0 0.0 287.0 0.0 287.0 0.0 287.0 0.0 287

gdb5 13 26 26 377.4 2.3 380.8 3.0 380.6 3.0 378.5 2.9 377

gdb6 12 22 22 298.0 0.0 298.1 1.7 298.3 2.5 298.0 0.0 298

gdb7 12 22 22 325.0 0.0 325.0 0.0 325.0 0.0 325.0 0.0 325

gdb8 27 46 46 359.4 2.9 361.1 3.5 361.6 3.4 358.8 3.1 348

gdb9 27 51 51 320.7 3.7 324.9 4.1 324.1 4.1 322.2 3.7 303

gdb10 12 25 25 275.0 0.0 275.0 0.0 275.0 0.0 275.0 0.0 275

gdb11 22 45 45 402.1 3.6 405.2 4.1 405.2 3.9 402.8 3.7 395

gdb12 13 23 23 461.9 3.7 465.7 3.7 465.3 3.8 462.1 3.6 458

gdb13 10 28 28 543.6 2.4 544.0 0.0 544.8 2.4 544.0 0.0 536

gdb14 7 21 21 100.0 0.0 100.0 0.0 100.0 0.0 100.0 0.0 100

gdb15 7 21 21 58.0 0.0 58.0 0.0 58.0 0.0 58.0 0.0 58

gdb16 8 28 28 127.0 0.0 127.0 0.0 127.0 0.0 127.0 0.0 127

gdb17 8 28 28 91.0 0.0 91.0 0.0 91.0 0.0 91.0 0.0 91

gdb18 9 36 36 164.0 0.0 164.0 0.0 164.0 0.0 164.0 0.0 164

gdb19 8 11 11 55.0 0.0 55.0 0.0 55.0 0.0 55.0 0.0 55

gdb20 11 22 22 121.3 1.7 121.9 1.9 121.8 1.9 121.1 1.7 121

gdb21 11 33 33 156.2 1.5 156.7 1.9 156.4 1.8 156.1 1.5 156

gdb22 11 44 44 200.1 1.1 200.5 1.7 200.7 1.8 200.0 1.0 200

gdb23 11 55 55 235.1 1.6 235.4 2.2 235.9 2.0 235.2 2.0 233

Mean _ _ _ 256.1 1.2 257.0 1.3 256.9 1.5 256.2 1.1 253.8


4 Experimental analysis

4.1 Experimental setup

In this section, ICSA-CARP is evaluated on three publicbenchmark data sets of CARP instances, known as the gdbset (DeArmon 1981), the val set (Benavent et al. 1992), andthe egl set (Eglese 1994). 81 test instances are included inthese test sets which are all based on undirected graphs. Inthis paper, we select three algorithms (MAENS, D-MAENSand RDG-MAENS) to compare with ICSA-CARP. AlthoughD-MAENS is designed for solving multi-objective CARP,it has shown a certain competitive advantage when solv-ing single-objective CARP. On the other hand, to determinewhether ICSA-CARP can be as effective as MAENS andall the other variants of MAENS, it is appropriate to selectthe three algorithms. RDG-MAENS uses novel clusteringmethods to decompose the whole problem into multiple sub-components and then it adopts MAENS to solve each sub-component. Therefore, function evaluation is not suitable forRDG-MAENS. To make a fair comparison, RDG-MAENS

is set in such a way that the computational time is comparableto ICSA-CARP. Furthermore, in RDG-MAENS, the numberof subcomponents is 2 and the fuzziness control parameteris 10. All three groups of experiments are performed for 30independent runs, and we report the mean results and stan-dard deviations over the 30 runs. For lower bound values,we use those published in Handa et al. (2006), Tang et al.(2009), Longo et al. (2006), Beullens et al. (2003), Chenget al. (2012), Baldacci and Maniezzo (2006), Shang et al.(2014a, b).

4.2 Experimental results and parametric statistical analyses

The function evaluations of the four algorithms are sepa-rately set to 10,000, 100,000 and 1,000,000 throughout theexperiments.

Table 1 shows a comparison of the four algorithms on asmall-scale test under a limit of 10,000 function evaluations.The columns headed |V |, |T | and |E | indicate the number ofvertices, the number of tasks and the number of edges, respec-

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Table 8 The average total_cost of the compared algorithms on val set under 1,000,000 function evaluations



1A 24 39 39 179.1 3.0 181.9 3.3 182.4 3.1 179.8 3.0 173

1B 24 39 39 180.3 3.1 182.4 2.7 182.8 2.8 180.7 3.0 173

1C 24 39 39 252.5 3.1 254.5 3.2 255.1 3.6 251.3 3.1 245

2A 24 34 34 233.5 3.6 234.8 3.8 236.0 3.8 233.5 3.6 227

2B 24 34 34 261.1 2.9 261.8 3.1 262.6 3.1 260.2 2.0 259

2C 24 34 34 470.7 4.1 472.1 4.2 477.0 4.6 471.3 3.9 457

3A 24 35 35 82.8 2.2 83.0 2.0 83.4 2.1 82.6 2.1 81

3B 24 35 35 88.4 2.0 88.7 2.1 88.6 2.2 88.6 2.0 87

3C 24 35 35 140.5 2.6 141.1 2.7 141.9 2.9 140.9 2.7 138

4A 41 69 69 417.7 4.1 425.5 4.7 425.9 4.5 420.2 4.7 400

4B 41 69 69 435.4 4.5 439.3 4.5 441.1 4.9 434.2 4.4 412

4C 41 69 69 457.4 4.5 468.8 5.0 467.7 5.1 459.6 4.1 428

4D 41 69 69 573.4 4.5 585.0 6.2 576.5 5.8 567.9 4.8 526

5A 34 65 65 439.8 4.5 446.8 4.5 448.1 4.9 442.1 4.2 423

5B 34 65 65 438.6 4.1 464.8 4.8 462.8 4.6 458.4 4.1 446

5C 34 65 65 488.9 3.9 495.0 4.8 497.4 5.2 486.6 4.0 473

5D 34 65 65 619.5 4.3 629.2 4.9 628.0 5.0 623.8 4.3 573

6A 31 50 50 231.8 3.5 233.7 3.7 235.1 3.2 230.8 3.1 223

6B 31 50 50 241.9 3.1 244.2 3.3 243.7 3.2 242.5 3.0 233

6C 31 50 50 331.0 3.2 334.2 3.7 333.8 3.5 333.0 3.3 317

7A 40 66 66 295.7 4.1 298.2 4.4 298.6 4.6 293.7 3.9 279

7B 40 66 66 294.9 4.0 301.3 4.7 303.1 4.8 296.4 3.7 283

7C 40 66 66 353.1 4.0 356.9 4.7 355.6 5.0 352.0 4.0 334

8A 30 63 63 399.0 4.3 405.5 4.8 407.5 4.9 400.2 4.5 386

8B 30 63 63 414.1 4.2 418.5 4.9 418.9 5.0 411.1 4.3 395

8C 30 63 63 562.5 4.4 565.4 4.7 567.2 4.8 562.3 4.4 518

9A 50 92 92 342.6 4.0 346.6 4.1 346.7 4.2 342.8 4.1 323

9B 50 92 92 348.3 4.0 355.2 4.5 354.5 5.1 347.7 4.7 326

9C 50 92 92 356.6 4.7 367.2 5.2 365.3 5.2 354.8 4.6 332

9D 50 92 92 430.9 5.3 436.2 5.7 435.2 5.8 425.7 5.2 385

10A 50 97 97 445.7 4.3 454.5 4.9 453.6 4.3 448.1 4.5 428

10B 50 97 97 457.3 4.4 464.0 5.1 466.5 5.5 457.2 4.5 436

10C 50 97 97 472.5 4.7 484.2 5.7 482.2 5.4 467.9 4.4 446

10D 50 97 97 565.9 4.4 577.2 5.9 573.7 6.1 564.4 4.3 525

Mean _ _ _ 361.9 3.9 367.6 4.3 367.6 4.4 362.1 3.8 343.2


tively. Moreover, at the bottom of each table, we added therow “mean” to represent the average value of the algorithmon all instances. For each experiment, the results highlightedin bold indicate the algorithm obtaining the minimum meanvalue of total_cost.

The table shows that none of the algorithms can convergeto the lower bound of test instances within 10,000 functionevaluations. But it can be concluded that ICSA-CARP con-verges faster than MAENS, D-MAENS and RDG-MAENS,

because ICSA-CARP obtained the minimal mean total_coston all gdb test instances.

Table 2 shows a comparison of four algorithms on amiddle-scale test under a limited number of function eval-uations of 10,000. Table 2 again shows that none of the algo-rithms can converge to the lower bound of test instancesunder 10,000 function evaluations. However, Table 2 alsoshows that ICSA-CARP obtains a better result on 28 out of34 val instances, compared to the other algorithms, while

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Table 9 The average total_cost of the compared algorithms on egl set under 1,000,000 function evaluations



E1-A 77 51 98 3703.5 12.2 3789.2 12.4 3784.1 11.8 3693.3 12.4 3548

E1-B 77 51 98 4658.4 10.1 4690.3 11.2 4663.4 10.2 4637.8 8.9 4498

E1-C 77 51 98 5910.7 11.7 5999.8 14.9 5965.8 12.2 5920.1 12.4 5566

E2-A 77 72 98 5367.8 12.7 5490.4 14.4 5334.7 12.3 5338.1 11.0 5018

E2-B 77 72 98 6745.3 15.1 6858.3 17.0 6663.4 13.2 6728.9 10.8 6305

E2-C 77 72 98 8981.4 14.7 9107.0 18.3 8889.5 14.8 8936.4 9.4 8243

E3-A 77 87 98 6457.9 15.5 6672.3 19.3 6487.7 17.5 6405.9 11.5 5898

E3-B 77 87 98 8484.0 15.8 8760.1 18.6 8356.2 15.7 8388.6 10.5 7704

E3-C 77 87 98 11,037.0 16.9 11,357.1 18.3 11,138.3 17.4 11,069.4 10.6 10,163

E4-A 77 98 98 7161.7 19.2 7358.4 18.2 7098.3 18.8 7094.6 14.4 6408

E4-B 77 98 98 9943.0 17.0 10,241.2 19.2 9855.5 19.2 9818.1 11.6 8884

E4-C 77 98 98 12639.1 16.5 13,175.0 21.0 12,488.7 17.5 12,538.6 11.4 11,427

S1-A 140 75 190 5553.9 13.3 5633.1 15.0 5519.8 13.5 5530.7 11.4 5018

S1-B 140 75 190 7004.8 14.8 7157.6 16.2 6997.2 15.0 7017.2 10.1 6384

S1-C 140 75 190 9137.0 15.6 9508.6 16.3 9191.4 14.5 9236.2 11.7 8493

S2-A 140 147 190 11,841.0 19.3 11,999.2 15.5 11,925.0 15.6 11,863.4 12.9 9824

S2-B 140 147 190 15,235.0 19.9 15,569.6 18.3 15,418.9 20.6 15,424.8 13.9 12,968

S2-C 140 147 190 19,254.1 20.1 19,645.5 16.9 19,105.6 19.7 19,351.1 13.4 16,353

S3-A 140 159 190 12,447.1 18.1 12,657.5 15.9 12,253.8 19.5 12,500.0 15.1 10,143

S3-B 140 159 190 16,253.2 20.6 16,630.3 15.6 16,235.1 21.4 16,297.8 17.2 13,616

S3-C 140 159 190 20,413.7 21.2 20,886.6 18.3 20,429.2 22.4 20,574.4 15.8 17,100

S4-A 140 190 190 15,077.5 18.0 15,131.6 16.6 14,958.5 19.6 15,099.4 14.8 12,143

S4-B 140 190 190 19,427.3 18.0 19,612.7 17.0 19,460.4 20.1 19,511.0 15.3 16,093

S4-C 140 190 190 24,707.2 20.1 24,855.3 18.2 24,737.0 19.8 24,776.6 15.1 20,375

Mean _ _ _ 11,151.1 16.5 11,366.1 16.8 11,112.6 16.8 11,156.4 12.6 9673.8


D-MAENS, MAENS and RDG-MAENS perform compara-tively poorly.

Table 3 shows a comparison of the four algorithms on alarge-scale test under a limited number of function evalua-tions of 10,000. Through the table we again find that none ofthe algorithms converge to the lower bound of test instancesunder 10,000 function evaluations. On the egl set, ICSA-CARP produces the best result on all 24 instances, while thethree comparison algorithms fail to perform best on any ofthe egl instances.

The above results in Tables 1, 2 and 3 deserve further dis-cussion. In this group of experiments, all the algorithms wereset to 10,000 function evaluations. Under this condition, thecomputing resources are far from being sufficient to opti-mally solve the problem. Even on the smallest-scale gdb set,none of the four algorithms can stably converge to the lowerbound of the test instances. However, in this limited numberof function evaluations, we find that ICSA-CARP demon-strates the fastest convergence rate. As mentioned previously,both MAENS and its variants adopt a complex local search

operator. Moreover, the offspring are randomly selected forlocal search, and the search performance is strongly depen-dent on the quality of the initial solution. If the initial solutionis located in an unpromising region of the solution space, itwill lead to wasting of computing resources. In contrast, theinitialization of ICSA-CARP uses a competitive construc-tive heuristic, which has been proved to be effective (Longoet al. 2006). Additionally, ICSA-CARP adopts the frame-work of immune clonal selection algorithm, which enablesit to focus computational resources on the highest qualityantibodies. The high-frequency mutation operator increasesthe diversity of the population and two different antibodyrepair operations are introduced, which help infeasible solu-tions move towards the global optimum. All these featuresincrease the convergence rate of ICSA-CARP.

Table 4 shows a comparison of the four algorithms on gdbtest under a limited number of 100,000 function evaluations.

Table 4 shows that ICSA-CARP can stably converge tothe lower bound during all 30 independent runs on all 12 gdbinstances. MAENS can stably converge to the lower bound

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Table 10 The anova test on gdb under 10,000, 100,000 and 1,000,000function evaluations

Name F/104 F/105 F/106

gdb1 2.36 3.68 0

gdb2 3.16 9.59 0.47

gdb3 3.83 8.93 0

gdb4 6.98 2.54 0

gdb5 9.13 9.14 0.71

gdb6 6.68 4.27 0.19

gdb7 1.87 1.83 0

gdb8 4.36 41.16 0.62

gdb9 1.93 39.41 0.38

gdb10 3.36 3.19 0

gdb11 2.98 3.95 0.35

gdb12 43.18 33.48 0.73

gdb13 3.85 4.82 0.12

gdb14 2.59 2.54 0

gdb15 2.36 2.13 0

gdb16 1.34 3.28 0

gdb17 2.29 2.25 0

gdb18 2.66 4.29 0

gdb19 1.13 0 0

gdb20 2.07 3.48 0.24

gdb21 2.68 2.68 0.27

gdb22 1.85 2.92 0.19

gdb23 1.67 4.18 0.14


during all the 30 dependent runs on only 6 gdb instances.The numbers of instances on which D-MAENS and RDG-MAENS achieve the same indicator are 5 and 1, respectively.

Table 5 shows a comparison of the four algorithms on theval data set under 100,000 function evaluations.

It can be seen from Table 5 that none of the algorithmsconverge to the lower bound of test instances under 100,000function evaluations. However, ICSA-CARP demonstratesthe fastest convergence rate. ICSA-CARP obtains the small-est mean total_cost result on all 34 val instances.

Table 6 shows a comparison of the four algorithms onthe egl data set using 100,000 function evaluations. Table6 again shows that none of the algorithms converges to thelower bound of test instances under 100,000 function evalu-ations. However, ICSA-CARP obtains a better result on 22out of 24 egl instances, compared to MAENS, D-MAENSand RDG-MAENS. ICSA-CARP shows poorer performancethan RDG-MAENS on only 2 out of 24 instances. NeitherD-MAENS nor MAENS shows the best performance on anyegl instances.

Table 7 shows a comparison of the four algorithms ongdb test using 1,000,000 function evaluations. On most gdb

Table 11 The Anova test on val under 10,000, 100,000 and 1,000,000function evaluations

Name F/104 F/105 F/106

val1A 4.17 4.94 1.45

val1B 2.47 6.13 1.53

val1C 15.76 32.28 1.74

val2A 1.86 4.31 1.47

val2B 6.98 8.37 1.25

val2C 1.31 10.55 1.39

val3A 0.83 4.75 0.83

val3B 2.19 5.18 0.28

val3C 1.94 4.29 0.41

val4A 1.26 7.62 1.29

val4B 17.74 13.82 1.68

val4C 31.28 38.24 1.37

val4D 1.47 37.27 0.92

val5A 0.71 16.76 1.42

val5B 1.42 36.72 1.91

val5C 0.92 38.93 2.36

val5D 1.27 41.18 1.37

val6A 2.31 5.27 1.52

val6B 2.47 23.52 0.63

val6C 3.08 32.36 0.71

val7A 2.63 18.42 0.52

val7B 0.84 32.19 1.16

val7C 0.81 42.48 0.78

val8A 1.27 9.62 1.62

val8B 0.95 36.92 0.84

val8C 1.21 41.17 0.49

val9A 2.64 4.18 1.47

val9B 3.51 8.35 1.64

val9C 1.94 24.02 1.97

val9D 2.37 15.28 1.83

val10A 1.86 7.41 1.36

val10B 1.42 21.69 1.79

val10C 2.79 19.93 2.63

val10D 1.16 38.46 1.21


test instances, all four algorithms can converge to the lowerbound using the larger computational resources of 1,000,000function evaluations. The advantage of ICSA-CARP is lessobvious in such cases. ICSA-CARP obtained smaller meantotal_cost than the other algorithms on 6 out of the total23 gdb test instances, while there are 6 instances on whichMAENS significantly outperforms the other algorithms. Nei-ther D-MAENS nor RDG-MAENS performs best on any ofthe gdb instances.

Table 8 shows a comparison of the four algorithms on theval data set using 1,000,000 function evaluations. On most of

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Table 12 The Anova test on egl under 10,000, 100,000 and 1,000,000function evaluations

Name F/104 F/105 F/106

egl-e1-A 1.75 41.36 0.94

egl-e1-B 1.37 37.92 0.31

egl-e1-C 0.96 45.62 0.43

egl-e2-A 1.89 39.08 1.35

egl-e2-B 0.53 42.59 1.99

egl-e2-C 0.88 41.84 1.51

egl-e3-A 0.42 36.29 2.23

egl-e3-B 1.14 27.16 2.47

egl-e3-C 0.84 35.22 1.76

egl-e4-A 0.81 13.27 1.62

egl-e4-B 0.74 23.49 2.49

egl-e4-C 0.92 38.92 1.45

egl-s1-A 0.76 38.14 0.37

egl-s1-B 1.08 47.82 0.62

egl-s1-C 0.94 24.73 0.94

egl-s2-A 0.42 3.88 1.67

egl-s2-B 1.32 1.97 1.46

egl-s2-C 0.75 3.26 1.38

egl-s3-A 0.37 2.57 1.74

egl-s3-B 1.43 2.84 2.11

egl-s3-C 1.27 1.79 1.53

egl-s4-A 1.48 0.63 0.49

egl-s4-B 0.32 1.42 0.37

egl-s4-C 1.24 0.47 1.32


the val test instances, none of the four algorithms converge tothe lower bound within 1,000,000 function evaluations. Onthis experiment, the performance of MAENS is no worse thanthat of ICSA-CARP. They both achieve very similar resultson most instances, each achieving slightly better scores onroughly half of the 34 val instances. The other two algorithmsdo not perform best on any of the val instances.

Table 9 compares the four algorithms on the egl dataset using 1,000,000 function evaluations. On most egl testinstances, none of the four algorithms is able to convergeto the lower bound within 1,000,000 function evaluations.Over the 24 egl instances, of the number on which RDG-MAENS performs best has increased to 13. This shows thatRDG-MAENS has strong ability to tackle large-scale prob-lem. MAENS performs significantly better than the othermethods on 8 egl instances. ICSA-CARP is significantly bet-ter on 4 egl instances and D-MAENS fail to outperform othermethods on any egl instances.

In summary, when the number of function evaluationsis set to 1,000,000, the performance of MAENS is noworse than that of ICSA-CARP and D-MAENS performs

worse than ICSA-CARP only on gdb2 and gdb12. The1,000,000 function evaluations provide enough computa-tional resources to ensure the ability of MAENS to search theentire solution space of the gdb small-scale data set. Hence,the difference between ICSA-CARP and MAENS is mini-mal. On the egl test data, the performance of ICSA-CARPis not as good as that of MAENS and RDG-MAENS. Withthe increasing number of tasks, the solution space becomeslarge and contains many local optima. In this case, the localsearch method of MAENS becomes more suitable, as it canconduct the search effectively using the large computingresources. However, ICSA-CARP cannot easily escape fromlocal optima and cannot easily shift its search from one feasi-ble region to another without the traversal style local searchstrategy. As to RDG-MAENS, a novel divide-and-conquerapproach is used which has been proved to be effective forsolving such large-scale CARP data.

In all experiments, we find that MAENS produces betterresults than D-MAENS for the same number of function eval-uations. This is because D-MAENS is a multi-objective opti-mization algorithm, and it needs to optimize multiple objec-tives simultaneously. So D-MAENS must uniformly allocateits computational resources to search the whole Pareto front.

4.3 Non-parametric statistical tests

ANOVA (Hogg and Ledolter 1987) is used as the first non-parametric statistical test to analyze whether there are obvi-ous differences between the means of ICSA-CARP and thecompared algorithms. Because different algorithms get dif-ferent numbers of solutions with the same number of functionevaluations, we select ten representative solutions, respec-tively, from solution sets of the four algorithms to compare.The following Tables 10, 11 and 12 show the values of theinterval under the confidence equal 95 %. After calculating,we get the degree of freedom f1 is 3 and f2 is 36. The standardcritical value of F is about 2.87 after checking the criticalvalue table of F-ratio distribution. In the following tables,we mark the results which show obvious differences by boldfonts. “F/104” means the values of F under 104 functionevaluations. “F/105” and “F/106” have the similar meaningwith “F/104”. Table 10 shows the ANOVA test on gdb under10,000, 100,000 and 1,000,000 function evaluations.

We can see from Table 10 that the four algorithms are sig-nificantly different on gdb under 105 function evaluations.And they have no obvious difference with 106 function evalu-ations. Under the condition of 104 function evaluations, thereare about half of the instances in gdb on which the four algo-rithms perform differently. Table 11 shows the ANOVA teston val under 10,000, 100,000 and 1,000,000 function evalu-ations.

In a similar way, the four algorithms have obvious differ-ences when tested the set egl under 105 function evaluations.

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Fig. 8 Boxplots of the compared algorithms on some representative instances under 10,000 function evaluations


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Table 13 The Mann–WhitneyU test followed by Bonferronicorrection on gdb under 10,000,100,000 and 1,000,000 functionevaluations

Bold values indicate the bestresults

Name U/104 U/105 U/106

MS-IC DMS-IC RD-IC MS-IC DMS-IC RD-IC MS-IC DMS-IC RD-IC

gdb1 31.5 22 21.5 31 23 19.5 50 50 50

gdb2 22.5 18.5 13 21 15.5 13 45 40 42.5

gdb3 21 16.5 12 22.5 19.5 18.5 50 50 50

gdb4 24.5 17.5 12.5 50 50 34.5 50 50 50

gdb5 17 15.5 9 19.5 16.5 14 45 34.5 42

gdb6 21.5 19 15.5 28.5 21.5 19.5 50 49.5 49

gdb7 33 28 26.5 50 50 31.5 50 50 50

gdb8 23 18.5 19 17 2.5 0.5 43.5 42.5 35.5

gdb9 27.5 32 29 1.5 1.5 1 42 32 42.5

gdb10 31 23 18.5 21.5 19 17.5 50 50 50

gdb11 26.5 23 21.5 23 20.5 21.5 43 42 42

gdb12 5.5 0.5 2 20.5 14.5 0.5 49 34.5 35

gdb13 24 21.5 18 25.5 22 19 48 50 45

gdb14 28 27 25.5 50 49 24 50 50 50

gdb15 29 27.5 26 50 50 33.5 50 50 50

gdb16 37 31 35 29 21 19.5 50 50 50

gdb17 35 33 21 50 50 40 50 50 50

gdb18 27 31 24.5 31.5 21 15.5 50 50 50

gdb19 41.5 42 36 50 50 50 50 50 50

gdb20 37.5 39 35 22.5 17.5 11.5 43 35 37

gdb21 36.5 36.5 35 32 22.5 18 48 43 43

gdb22 38 39.5 38 32.5 28.5 22 49 43 42.5

gdb23 34 34.5 35.5 26.5 20.5 19.5 48 45 42.5

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Table 14 The Mann–WhitneyU test followed by Bonferronicorrection on val under 10,000,100,000 and 1000,000 functionevaluations


Name U/104 U/105 U/106


val1A 19 19 20.5 29 16.5 10.5 43 33.5 30.5

val1B 33 37 35.5 23.5 13.5 2 44.4 35 31.5

val1C 19.5 18 18 13.5 0.5 0 40 37.5 32.5

val2A 46.5 37 35 12.5 15.5 12.5 50 40.5 35

val2B 21 18 17.5 14 9.5 1 43 40 48

val2C 28 26.5 28.5 16 13 1 47.5 43 30.5

val3A 43 45.5 47 14 12.5 12 45 41.5 38

val3B 29.5 28 32 12 1.5 1.5 45.5 48.5 50

val3C 32.5 30 31.5 15 13 12 44 45.5 38.5

val4A 43 37.5 36 13.5 12.5 13 39 30 30

val4B 27 2.5 14.5 1.5 1 1 44 30.5 28.5

val4C 3.5 6 1.5 1 0.5 0.5 42 32.5 29.5

val4D 28 31.5 26 0.5 0.5 0.5 35 30.5 29.5

val5A 46.5 42 38 4 3.5 1 45.5 35 33.5

val5B 45 43.5 37.5 1.5 1.5 1 25 30 35

val5C 34.5 39 35 1 1 0.5 43 29.5 29

val5D 29 36 33 1 1 0.5 33 33.5 46.5

val6A 31 28 39 13 12 11.5 41 31.5 29.5

val6B 33 29.5 34 14 0.5 12 45.5 40.5 36.5

val6C 19 25 22.5 13 0.5 0.5 40 42.5 46

val7A 32 47 22.5 12.5 11.5 12 35 30 29.5

val7B 38 43 47 0.5 0.5 0.5 42.5 29.5 27.5

val7C 36 39 48 0.5 0.5 0.5 45 30.5 29.5

val8A 46 44 41 25 17.5 12.5 45 30 28

val8B 43.5 43 50 5 1 1 35 29 29

val8C 30.5 28 20.5 0 1 1 47.5 41 31

val9A 42 38.5 32 23.5 24 24 49 33 32.5

val9B 23 22 17.5 16 15 13.5 45 32 29.5

val9C 34 31 38 4.5 5.5 5.5 28.5 28 27.5

val9D 31.5 37 29 13 13 13.5 30 29 27.5

val10A 25 27 21.5 22 21.5 21.5 40 30.5 30

val10B 29 24.5 32 14 14 13 50 33 28.5

val10C 29.5 26 23 15 15 12.5 30 25.5 26

val10D 32 27 26.5 1 0.5 0.5 45.5 27.5 28

Table 12 shows the ANOVA test on egl under 10,000, 100,000and 1,000,000 function evaluations.

From Table 12 we can see that there are obvious differ-ences on means between the four algorithms under 104 func-tion evaluations. However, there is no remarkable differenceon the means of the four algorithms under 105 function eval-uations and 106 function evaluations.

The following figures show the boxplots of MAENS, D-MAENS, RDG-MAENS and ICSA-CARP on some repre-sentative instances under different numbers of function eval-uations. The representative instances include gdb8, gdb17,

gdb20, gdb22, val2C, val4D, val7A, val9B, egl-e2-B, egl-e3-A, egl-s2-B and egl-s3-A.

Figure 8 shows the boxplots of the four algorithms onsome representative instances under 10,000 function evalua-tions. As we can see from Fig. 8 the ICSA-CARP can get thelowest average total cost obviously. Therefore, ICSA-CARPis a competitive and effective algorithm under 10,000 func-tion evaluations.

Figure 9 shows the boxplots of the compared algorithmson some representative instances under 100,000 functionevaluations. We can see that the four algorithms work equally

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Table 15 The Mann–WhitneyU test followed by Bonferronicorrection on egl under 10,000,100,000 and 1,000,000 functionevaluations


Name U/104 U/105 U/106


egl-e1-A 43 46 41.5 2.5 1.5 1 33 37 24

egl-e1-B 47 43 42 3.5 2 1.5 41 33.5 39

egl-e1-C 39 43 35 1 0.5 0.5 43 31 37.5

egl-e2-A 42.5 39 32 6.5 5 4 42 21 47

egl-e2-B 46 46 35 0.5 0.5 1 49 26 39

egl-e2-C 43 48 25 0.5 0.5 0.5 45 29 43

egl-e3-A 43 46.5 31 3.5 5 2.5 41 21.5 35

egl-e3-B 47.5 27 25.5 7 3.5 8 29 19.5 41

egl-e3-C 45 28 31 0.5 0.5 0.5 43 24 38

egl-e4-A 43 37 46 6.5 3.5 43 41 26 48

egl-e4-B 47 45 35 13 15.5 9 33 25 47

egl-e4-C 46 36.5 43 0.5 0.5 0.5 44 19.5 49

egl-s1-A 47 45.5 41 3 2 1.5 45 31 47

egl-s1-B 42 39 46.5 0 0.5 0.5 48 35 45

egl-s1-C 37 33 49 2.5 1.5 1 43.5 29.5 47

egl-s2-A 45 36 39 20.5 25.5 19 47 35 43

egl-s2-B 42 37 35.5 29 23.5 37 32 37 46.5

egl-s2-C 34 39 37 19 21.5 27 43.5 29 26

egl-s3-A 42 38 36.5 25 24 21 44 40 32

egl-s3-B 41 37 39.5 22 26.5 29 42.5 27 39.5

egl-s3-C 34 39 32 38 28 35 34 28 36

egl-s4-A 37 29.5 35 43 39 36.5 41 38 31

egl-s4-B 41 41 46 41 38 37 36.5 33 37

egl-s4-C 39 37 43 45 41 39.5 45 44.5 47

well on gdb17 except RDG-MAENS. That is because gdb17is a very small-scale instance, and the decomposition oper-ation in RDG-MAENS leads to much waste. On the otherinstances, ICSA-CARP performs very well under 100,000function evaluations.

Figure 10 shows the boxplots of the compared algo-rithms on some representative instances under 1,000,000function evaluations. ICSA-CARP performs well on mostof the instances. All the compared algorithms can reach thelower bound of gdb17. However, ICSA-CARP has poor per-formance on val2C and egl-s3-A, because the four algorithmscan give full play to their performance when solving CARPunder so large numbers of function evaluations.

Next, we use Mann–Whitney U test (Hettmansperger andMcKean 1998) followed by Bonferroni correction (Dunn1961; Dunnett 1955) as the second nonparametric statisticaltest to pairwise compare ICSA-CARP with the other algo-rithms. The confidence interval is set to be 95 % and then fol-lowed by Bonferroni correction. We use solutions the same asthe ANOVA test. As shown in Tables 13, 14, 15, we test all theinstances, respectively, under 10,000, 100,000 and 1,000,000function evaluations. In tables, “MS-IC” means the com-

parison between MAENS and ICSA-CARP and “DMS-IC”represents the comparison between D-MAENS and ICSA-CARP. “RD-IC” compares RDG-MAENS and ICSA-CARP.“U/104” means the values of U under 104 function evalua-tions. “U/105” and “U/106” have the similar meaning with“U/104”. We find the critical value of U is 23. The bold fontsin the following tables mean an obvious difference betweentwo algorithms. Table 13 shows the Mann-whitney U test fol-lowed by Bonferroni correction on gdb under 104, 105 and106 function evaluations.

We can see from Table 13 that ICSA-CARP has certaindifferences with the three compared algorithms on someinstances in gdb under 104 function evaluations. And ICSA-CARP has significant differences with them under 105 func-tion evaluations. There is no obvious difference between thefour algorithms under 106 function evaluations. Table 14shows the Mann–whitney U test followed by Bonferroni cor-rection on val under 104, 105 and 106 function evaluations.

Analogously, ICSA-CARP performs differently on all theinstances in the set val under 105 function evaluations. Also,there are few instances on which ICSA-CARP has certaindifferences with the compared algorithms under 104 func-

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Table 16 The average computational time of the compared algorithmson gdb under 10,000 function evaluations

Name MAENS D-MAENS ICSA-CARP

gdb1 2.4 2.5 1.9

gdb2 2.3 3.0 2.1

gdb3 1.6 2.8 2.2

gdb4 1.6 2.7 2.1

gdb5 1.8 3.0 2.3

gdb6 2.5 3.1 1.9

gdb7 2.4 3.2 1.8

gdb8 3.1 9.3 3.4

gdb9 3.7 6.3 3.7

gdb10 1.9 7.8 2.5

gdb11 3.1 9.4 3.1

gdb12 1.6 3.1 1.5

gdb13 1.8 2.9 2.5

gdb14 1.2 2.2 1.8

gdb15 1.5 1.9 1.5

gdb16 2.2 3.0 1.5

gdb17 1.2 3.1 1.2

gdb18 3.1 4.7 2.8

gdb19 0.9 1.6 1.2

gdb20 1.6 3.2 2.2

gdb21 2.2 3.1 2.2

gdb22 2.1 7.8 2.1

gdb23 2.3 4.3 2.1


tion evaluations. Table 15 shows the Mann–whitney U testfollowed by Bonferroni correction on egl under 104, 105 and106 function evaluations.

In Table 15, we can see that ICSA-CARP has obviousdifferences with MAENS, D-MAENS and RDG-MAENSon most of the egl instances under 105 function evaluations.However, there is no obvious difference between the fouralgorithms when solving CARP under 104 and 106 functionevaluations. The test results show that ICSA works betterunder 105 function evaluations.

4.4 Comparison between ICSA-CARP and the comparedalgorithms on computational time

We make comparison on the average computational time for30 independent runs between ICSA-CARP and the comparedalgorithms. For the sake of simplicity, we test all the instancesunder 10,000 function evaluations. However, the principle ofRDG-MAENS is to decompose the whole CARP into manysub-problems and then solve every sub-problem by MAENS.So, it may be not suitable for RDG-MAENS to compare thecomputational time with MAENS and ICSA-CARP. Thus,

Table 17 The average computational time of the compared algorithmson val under 10,000 function evaluations


val1A 3.1 3.4 2.8

val1B 3.1 3.2 2.8

val1C 2.8 3.1 3.1

val2A 3.1 3.2 2.9

val2B 2.8 3.1 2.5

val2C 3.1 4.7 2.2

val3A 2.1 3.2 2.5

val3B 2.5 3.1 2.8

val3C 3.7 4.6 2.2

val4A 6.5 6.4 6.2

val4B 6.2 14.3 5.9

val4C 6.2 17.9 5.6

val4D 5.6 9.8 5.6

val5A 6.2 13.6 6.2

val5B 5.6 6.4 5.9

val5C 5.6 7.2 5.3

val5D 4.9 6.4 5.0

val6A 4.3 7.1 4.7

val6B 4.7 8.9 4.4

val6C 3.7 6.3 4.3

val7A 7.2 12.5 6.6

val7B 7.5 11.4 5.9

val7C 4.6 6.2 5.0

val8A 5.9 12.7 5.6

val8B 5.9 11.8 5.3

val8C 5.0 17.2 5.0

val9A 13.1 31.4 13.1

val9B 11.8 28.8 12.1

val9C 10.9 32.1 9.8

val9D 10.3 30.9 10.3

val10A 10.9 29.8 10.3

val10B 16.8 28.7 11.8

val10C 11.5 32.9 11.4

val10D 10.0 24.3 10.6


we compare the average computational time for 30 indepen-dent runs between ICSA-CARP and RDG-MAENS. In thefollowing tables, better results are shown by bold fonts andthe values of time are measured in seconds. Table 16 showsthe average computational time of MAENS, D-MAENS andICSA-CARP on gdb with 10,000 function evaluations.

We can see from Table 16 that ICSA-CARP uses the short-est time of the three algorithms on 8 of the 23 instances. Onthe other hand, MAENS uses the shortest time on 9 instances.At the same time, there are 6 instances on which MAENS andICSA-CARP uses the same time. D-MAENS fails to use less

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Table 18 The average computational time of the compared algorithmson egl under 10,000 function evaluations


egl-e1-A 4.1 10.9 3.4egl-e1-B 3.1 9.5 3.1egl-e1-C 3.4 11.5 3.4egl-e2-A 5.6 16.3 5.0egl-e2-B 5.9 19.5 5.6egl-e2-C 4.9 23.4 5.0egl-e3-A 6.2 28.1 6.2egl-e3-B 6.8 31.2 6.4egl-e3-C 7.1 29.6 6.6egl-e4-A 8.4 41.5 8.1egl-e4-B 8.1 37.8 7.8egl-e4-C 7.8 33.7 7.1egl-s1-A 6.2 32.4 5.6egl-s1-B 5.9 29.8 4.9egl-s1-C 5.0 27.8 5.1egl-s2-A 15.3 112.6 14.6egl-s2-B 16.2 123.9 14.3egl-s2-C 14.4 119.5 14.6egl-s3-A 17.4 62.2 16.5egl-s3-B 16.8 148.2 16.2egl-s3-C 17.8 61.9 16.1egl-s4-A 24.9 32.7 23.1egl-s4-B 23.1 28.2 22.5egl-s4-C 24.0 27.9 22.1


time under 10,000 function evaluations for its multi-objectivespecialty. The following Table 17 shows the average compu-tational time of MAENS, D-MAENS and ICSA-CARP onval with 10,000 function evaluations.

We can see from Table 17 that ICSA-CARP and MAENSuse less time than D-MAENS and ICSA-CARP performsbetter on 18 of the 34 instances than MAENS. Meanwhile,MAENS performs better on 10 instances than ICSA-CARP.Overall, the performance of ICSA-MAENS on the runningtime is slightly better than MAENS and much better thanD-MAENS.

From Table 18, we can see that MAENS uses the shortesttime on 6 instances of 24 test instances in egl and ICSA-CARP performs better on the most of the instances. Also, theperformance of D-MAENS on the running time is relativelypoor.

In short, the advantages of ICSA-CARP are not very obvi-ous on gdb and val. But for egl, ICSA-CARP has shownits obvious advantages on the running time. This is becausegdb and val are small-scale test sets and clone operationswill waste much time when solving these simple problems.However, for the large-scale CARP, clone operations pro-duce many good solutions quickly, and it is beneficial to getbetter solutions in a short time. D-MAENS is designed formulti-objective optimization problems, so it fails to use lesstime with limited function evaluations.

Table 19 The average computational time of the two algorithms for 30independent runs on gdb under 200 iterations

Name RDG-MAENS ICSA-CARP

gdb1 154.13 150.87

gdb2 339.30 159.39

gdb3 316.32 240.08

gdb4 240.05 238.12

gdb5 446.09 150.22

gdb6 273.13 214.65

gdb7 330.53 219.01

gdb8 206.20 214.80

gdb9 943.25 111.9

gdb10 306.76 123.25

gdb11 365.50 97.35

gdb12 360.06 88.95

gdb13 364.91 168.32

gdb14 198.46 275.70

gdb15 113.01 229.78

gdb16 110.68 208.05

gdb17 161.57 188.55

gdb18 161.55 211.13

gdb19 150.27 79.03

gdb20 333.26 205.05

gdb21 473.52 277.65

gdb22 348.56 210.05

gdb23 142.89 85.38


Tables 19, 20 and 21 show the average computational timeof RDG-MAENS and ICSA-CARP for 30 independent runsunder the same iteration of 200. In the following tables, wealso highlight the shortest computational time by bold fontand the values of time are measured in seconds.

We can see from Table 19 that ICSA-CARP uses much lesstime than IRDG-MAENS on most of the instances under thesame iteration, which proves the faster convergence of ICSA-CARP. The following Table 20 means the average computa-tional time of the two algorithms for 30 independent runs onval.

We can see from Table 20 that ICSA-CARP uses almosthalf of the time of IRDG-MAENS on all the instances inval. The reason for this is that gdb and val are very small-scale sets, and the working principle of IRDG-MAENS is todecompose a problem into many sub-problems. In fact, it isunnecessary and time consuming for the simple CARP. So,the time of ICSA-CARP is much less than IRDG-MAENSon most of the small-scale instances.

We can see from Tble 21 that ICSA-CARP uses less timethan RDG-MAENS on most of the instances in egl. For thelast few instances, ICSA-CARP shows the obvious advan-

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Table 20 The average computational time of the two algorithms for 30independent runs on val under 200 iterations


val1A 304.14 114.23

val1B 352.35 133.95

val1C 529.16 170.70

val2A 148.66 134.72

val2B 228.65 153.02

val2C 392.84 137.10

val3A 299.64 151.65

val3B 390.85 99.60

val3C 469.20 157.5

val4A 210.53 204.75

val4B 378.76 81.75

val4C 731.12 136.95

val4D 404.53 90.45

val5A 615.39 145.95

val5B 451.19 83.55

val5C 711.83 150.32

val5D 563.32 143.62

val6A 335.38 83.85

val6B 492.37 162.63

val6C 467.79 140.85

val7A 413.55 131.74

val7B 361.65 82.95

val7C 611.98 322.16

val8A 442.54 113.10

val8B 260.26 97.05

val8C 575.58 220.8

val9A 475.98 109.95

val9B 631.08 197.52

val9C 543.31 87.75

val9D 537.97 175.87

val10A 441.61 58.95

val10B 576.68 139.65

val10C 498.46 83.25

val10D 533.24 128.72


tages on the speed. From Tables 19, 20 and 21, we can con-clude that compared with RDG-MAENS, ICSA-CARP usesmuch less time obviously under the same iteration.

4.5 Comparative results between different versions ofICSA-CARP

In this paper, we also conduct experiments to analyze whichindependent operator makes more contribution when solvingCARP. Clonal selection operator and antibody repair operatorare selected to analyze. To keep things simple, we study the

Table 21 The average computational time of the two algorithms for 30independent runs on egl under 200 iterations


egl-e1-A 109.72 105.89

egl-e1-B 116.03 120.23

egl-e1-C 124.57 137.91

egl-e2-A 185.34 179.86

egl-e2-B 202.29 232.28

egl-e2-C 215.91 207.47

egl-e3-A 252.87 223.11

egl-e3-B 275.76 286.51

egl-e3-C 293.41 132.04

egl-e4-A 305.87 141.65

egl-e4-B 348.01 154.09

egl-e4-C 363.83 171.01

egl-s1-A 175.65 169.54

egl-s1-B 180.22 226.69

egl-s1-C 281.56 183.75

egl-s2-A 744.09 177.58

egl-s2-B 709.89 159.05

egl-s2-C 702.93 154.94

egl-s3-A 740.62 174.60

egl-s3-B 835.32 183.22

egl-s3-C 841.54 181.34

egl-s4-A 1121.37 159.91

egl-s4-B 1071.14 167.53

egl-s4-C 1098.67 163.04


instances in Tables 12 and 13 under the same function eval-uation 100,000. The following Table 14 shows the results ofdifferent versions of ICSA-CARP. The bold fonts indicatethe better results between “1-CARP” and “2-CARP”.

Table 18 shows the mean total_cost of three versions ofICSA-CARP. “1-CARP” means ICSA-CARP without clonalselection operator. “2-CARP” represents ICSA-CARP with-out antibody repair operator. “3-CARP” is ICSA-CARP. Wecan conclude from the experimental results that both “1-CARP” and “2-CARP” contribute to ICSA-CARP a lot. “1-CARP” works importantly in ICSA-CARP; however, “2-CARP” plays a more important role obviously when solvingmedium-scale and large-scale CARP.

4.6 Convergence curves of the compared algorithmson representative instances

In order to compare the convergence rates of the four algo-rithms, we make a more detailed evaluation on several repre-sentative instances. Figure 11 plots mean total_cost (over30 independent trials) of the best-so-far solutions against

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Fig. 11 The convergence curves of the compared algorithms on gdb11, val10D, egl-e1-A and egl-s1-A

Table 22 The average total_cost of three versions of Icsa-Carp com-pared under 100,000 function evaluations

Name 1-CARP 2-CARP 3-CARP

gdb2 345.1 362.3 341.1

gdb8 372.7 422.2 358.5

gdb13 544.3 566.7 543.9

gdb17 91.5 91.5 91.0

gdb23 240.9 239.9 235.2

val2C 483.7 506.5 471.8

val4A 449.1 451.6 419.5

val5D 680.4 679.5 619.0

val7C 405.8 394.1 353.3

val9B 366.2 372.4 349.2

egl-e1-A 4156.7 4155.8 3727.5

egl-e2-B 7552.8 7579.3 6717.7

egl-e3-A 6828.3 6840.9 6432.7

egl-e4-C 13,775.4 13,869.2 12,604.3

egl-s1-B 8287.6 8335.4 7018.7

egl-s2-C 19,931.9 19,809.3 19,430.1

egl-s3-A 12,878.3 12,700.1 12,540.8

egl-s4-B 19,778.4 19,706.7 19,581.4


the number of function evaluations. ICSA-CARP performsbest when the function evaluations vary from 10,000 to1,000,000. Once the number of function evaluations exceed

1,000,000, the performance of ICSA-CARP becomes worsethan MAENS and its variants, especially on large-scaleinstances.

From Tables 1, 2, 3, 4, 5, 6, 7, 8 and 9, we can see thatICSA-CARP can get better results with limited function eval-uations. Tables 10, 11, 12, 13, 14 and 15 show that ICSA-CARP has obvious differences with the compared algorithmsunder limited function evaluations. Tables 16, 17, 18, 19, 20,21 and 22 show the efficiency of ICSA-CARP and Fig. 11represents that it can converge quickly with limited functionevaluations. High in speed and good in results, ICSA-CARPis an efficient algorithm within a limited number of functionevaluations.

5 Conclusion

This paper investigates the CARP problem within the frame-work of ICSA and makes several contributions. First, wehave proposed a novel ICSA-CARP algorithm. Previousmeta-heuristic approaches predominantly work by randomlyselecting offspring and applying a traversal local search overall neighbors of the current offspring. In contrast, ICSA-CARP focuses computational resources on the highest qual-ity offspring. By taking a variety of mutation strategies ondifferent clones of the same antibody, it promotes coopera-

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tion and information exchanging among antibodies and alsoincreases the diversity and speeds up the convergence. Hence,the ICSA-CARP is capable of efficiency searching within alimited computational budget.

Second, the construction phase of GRASP is used toinitialize the antibody population for ICSA-CARP. Conse-quently, the efficiency and effectiveness of the improvedheuristic will have an important impact on the efficiency andeffectiveness of the ICSA-CARP.

Third, two different antibody repair operators are intro-duced, which cause infeasible solutions to move towardsthe global optimum. Three experimental studies show theconvergence characteristics of ICSA-CARP under differ-ent numbers of function evaluations (10,000, 100,000 and1,000,000). Compared with state-of-art algorithms from theliterature, the experimental studies demonstrate the high effi-cacy of ICSA-CARP, especially on medium-scale CARPinstances and especially under conditions of limited com-putational resource.

Acknowledgments We would like to express our sincere appreciationto Professor Xin Yao and the anonymous reviewers for their valuablecomments, which have greatly helped us in improving the quality of thepaper. This work was partially supported by the National Basic ResearchProgram (973 Program) of China under Grant 2013CB329402, theNational Natural Science Foundation of China, under Grants 61371201,61203303 and 61272279, the Program for Cheung Kong Scholars andInnovative Research Team in University under Grant IRT1170, and theEU FP7 project (Grant no. 247619) on “NICaiA: Nature Inspired Com-putation and its Applications”.

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