Research Article A Hybrid Soft Computing Approach for ...downloads.hindawi.com/journals/mpe/2013/716069.pdf · Introduction Set covering problem (SCP) and set partitioning problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 716069 12 pageshttpdxdoiorg1011552013716069

Research ArticleA Hybrid Soft Computing Approach for Subset Problems

Broderick Crawford12 Ricardo Soto13 Eric Monfroy4 Carlos Castro5

Wenceslao Palma1 and Fernando Paredes6

1 Pontificia Universidad Catolica de Valparaıso Valparaıso 2362807 Chile2 Universidad Finis Terrae Santiago 7500000 Chile3 Universidad Autonoma de Chile Santiago 7500000 Chile4 CNRS LINA Universite de Nantes Nantes 44322 France5 Universidad Tecnica Federico Santa Marıa Valparaıso 2390123 Chile6 Escuela de Ingenierıa Industrial Universidad Diego Portales Santiago 8370179 Chile

Correspondence should be addressed to Broderick Crawford broderickcrawforducvcl

Received 7 April 2013 Revised 21 June 2013 Accepted 22 June 2013

Academic Editor Ker-Wei Yu

Copyright copy 2013 Broderick Crawford et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Subset problems (set partitioning packing and covering) are formal models for many practical optimization problems A setpartitioning problem determines how the items in one set (S) can be partitioned into smaller subsets All items in S must becontained in one and only one partition Related problems are set packing (all itemsmust be contained in zero or one partitions) andset covering (all itemsmust be contained in at least one partition) Here we present a hybrid solver based on ant colony optimization(ACO) combined with arc consistency for solving this kind of problems ACO is a swarm intelligence metaheuristic inspired onants behavior when they search for food It allows to solve complex combinatorial problems for which traditional mathematicaltechniques may fail By other side in constraint programming the solving process of Constraint Satisfaction Problems candramatically reduce the search space bymeans of arc consistency enforcing constraint consistencies either prior to or during searchOur hybrid approach was tested with set covering and set partitioning dataset benchmarks It was observed that the performanceof ACO had been improved embedding this filtering technique in its constructive phase

1 Introduction

Set covering problem (SCP) and set partitioning problem(SPP) have many applications including those involvingrouting scheduling stock cutting electoral redistricting andother important real-life situations [1 2] Although the bestknown application of the SPP is airline crew scheduling [3 4]several other applications exist including vehicle routingproblems (VRP) [5 6] and query processing [7] The maindisadvantage of SPP-based models is the need to explicitlygenerate a large set of possibilities to obtain good solutionsAdditionally in many cases a prohibitive time is needed tofind the exact solution

Furthermore Set Partitioning Problems occur as sub-problems in various combinatorial optimization problems[8] In airline scheduling a subtask called crew scheduling

takes as input data a set of crew pairings where the selectionof crew pairings which cause minimal costs and ensure thateach flight is covered exactly once can be modeled as a setpartitioning problem [1 9] In [10 11] solving a particularcase of VRP the dial-a-ride problem (DARP) also uses anSPP decomposition approach

Because the SPP formulation has demonstrated to beuseful modeling important industrial problems (or theirphases) it is our interest to solve it with novel techniques Inthis work we solve some test instances of SPP and SCP(SCP is considered a relaxation of SPP) with ant colony opti-mization (ACO) algorithms and some hybridizations of ACOwith a constraint programming (CP) technique constraintpropagation [12]

ACO is a swarm intelligence metaheuristic which isinspired from the foraging behavior of real ant colonies The

2 Mathematical Problems in Engineering

ants deposit pheromone on the ground marking the path foridentification by other members of the colony of the routesfrom the nest to food [13] From the early nineties ACOattracted the attention of researchers and many successfulapplications solving optimization problems are done [14]

There exist already some good approaches applying ACOto subset problems [15] In [16] ACO is applied on the setpacking problem using two solution construction strategiesbased on exploration and exploitation In general the sameoccurs in relation with SCP applying ACO only as a con-struction algorithm and testing the approach only on somesmall SCP instances More recent works apply ant computingto the SCP and related problems using techniques to removeredundant columns and local search to improve solutions[17ndash22]

The best performing metaheuristics for SPP are geneticalgorithms [23 24] Taking into account these results it seemsthat the incomplete approach of ant computing could beconsidered as a good alternative to solve these problemswhencomplete techniques are not able to get the optimal solutionin a reasonable time

ACO is of limited effectiveness solving very stronglyconstrained problems They are problems for which neigh-borhoods contain few solutions or none at all and localsearch is of very limited use Probably the most significantof such problems is the SPP A direct implementation ofthe basic ACO framework is incapable of obtaining feasiblesolutions for many standard tested instances of SPP [25ndash27]

Trying to solve larger instances of SPP with the originalant system (AS) [28] or ant colony system (ACS) [29] imple-mentation derives in a lot of unfeasible labeling of variablesand the ants cannot obtain complete solutions using theclassic transition rule when theymove in their neighborhoodThe root of the problem is that simply following the randomproportional transition rule that is learningreforcing goodpaths is no longer enough as this does not check forconstraint consistency

In order to improve this aspect of ACO we are workingin the addition of a constraint programming mechanism inthe construction phase of ACO thus only feasible partialsolutions are generated The CP mechanism allows theincorporation of information about the instantiation of vari-ables after the current decision In general ACO algorithmsare competitive with other optimization techniques whenapplied to problems that are not overly constrained Howeverwhen solving highly constrained problems the performanceof ACO algorithms degrades When a problem is highlyconstrained the difficulty is in finding feasible solutionsThisis where CP comes into play because these problems arethe target problems for CP solvers CP is a programmingparadigm in which a combinatorial optimization problemis modeled as a discrete optimization problem specifyingthe constraints that a feasible solution must meet The CPapproach to search for a feasible solution often works bythe iteration of constraint propagation and the additionof additional constraints and it transforms the problemwithout changing its solutions Constraint propagation is themechanism that reduces the domains of the decision variableswith respect to the given set of constraints [30]

Although the idea of obtaining sinergy from hybridiza-tion of ACO with CP is not novel [31ndash36] our proposal is abit different We explore the addition to the ACO algorithmof a mechanism to check constraint consistency usuallyused in complete techniques arc consistency Other kinds ofcooperation between ACO and CP are shown in [37] wherecombinatorial optimization problems are solved in a genericway by a two-phase algorithm The first phase aims to createa hot start for the second it samples the solution space andapplies reinforcement learning techniques as implemented inACO to create pheromone trails During the second phase aCP optimizer performs a complete tree search guided by thepheromone trails previously accumulated

Here we propose the addition of a lookaheadmechanismin the construction phase of ACO in order that only feasiblesolutions are generated The lookahead mechanism allowsthe incorporation of information about the instantiation ofvariables after the current decisionThe idea differs from thatproposed by [31 32] and these authors proposed a lookaheadfunction evaluating the pheromone in the shortest commonsupersequence problem and estimating the quality of a partialsolution of an industrial scheduling problem respectively

This paper is organized as follows In Section 2 we explainthe problem In Section 3 we describe the ACO frame-work In Section 4 we present the definitions considered inconstraint propagation Our hybrid proposal is described inSection 5 In Section 6 we present the experimental resultsobtained Finally in Section 7 we conclude the paper and givesome perspectives for future research

2 Problem Description

SPP is the problem of partitioning a given set into manu-ally independent subsets while minimizing a cost functiondefined as the sum of the costs associated with each of theeligible subsets

In the SPP matrix formulation we are given a 119898 times 119899

matrix 119860 = (119886119894119895) in which all the matrix elements are either

zero or one Additionally each column is given a nonnegativecost 119888119895

Let 119868 = 1 119898 and 119869 = 1 119899 be the row set andcolumn set respectively

We say that a column 119895 covers a row 119894 if 119886119894119895= 1 Let 119909

119895

be a binary variable which is one if column 119895 is chosen andzero otherwise The SPP can be defined formally asminimize(1) subject to (2) These constraints enforce that each row iscovered by exactly one columnThe SCP is an SPP relaxationThe goal in the SCP is to choose a subset of the columnsof minimal weight formally using constraints to enforce thateach row is covered by at least one column as (3)

119891 (119909) =

119899

sum

119895=1

119888119895119909119895 (1)

119899

sum

119895=1

119886119894119895119909119895= 1 forall119894 = 1 119898 (2)

Mathematical Problems in Engineering 3

119899

sum

119895=1

119886119894119895119909119895ge 1 forall119894 = 1 119898 (3)

119909119895isin 0 1 forall119895 = 1 119899 (4)

The notations in (5) are often used to complete the de-scription of the problem

119869119894= 119895 isin 119869 | 119886

119894119895= 1 subset of columns covering row 119894

119868119895= 119894 isin 119868 | 119886

119894119895= 1 subset of rows covered by column 119895

119889 =

sum119898

119894=1sum119899

119895=1119886119894119895

119898119899

density that is

the ratio of non-zero entries in 119860

(5)

3 Ant Colony Optimization for SetPartitioning Problems

In this section we briefly present ACO algorithms and give adescription of their use to solve SPP More details about ACOalgorithms can be found in [13]

The basic idea of ACO algorithms comes from thecapability of real ants to find the shortest paths between thenest and food source From a combinatorial optimizationpoint of view the ants are looking for good solutions Real antscooperate in their search for food by depositing pheromoneon the ground An artificial ant colony simulates this behaviorimplementing artificial ants as parallel processes whose roleis to build solutions using a randomized constructive searchdriven by pheromone trails and heuristic information of theproblem

An important topic in ACO is the adaptation of thepheromone trails during algorithm execution to take intoaccount the cumulated search experience reinforcing thepheromone associated with good solutions and consideringthe evaporation of the pheromone on the components overtime in order to avoid premature convergence ACO can beapplied in a very straightforward way to SPPThe columns arechosen as the solution components and have associated a costand a pheromone trail [13] Each column can be visited by anant only once and then a final solution has to cover all rowsA walk of an ant over the graph representation correspondsto the iterative addition of columns to the partial solutionobtained so far Each ant starts with an empty solution andadds columns until a cover is completed A pheromone trail120591119895and a heuristic information 120578

119895are associated with each

eligible column 119895 A column to be added is chosen with aprobability that depends on pheromone trail and the heuristicinformation The most common form of the ACO decisionpolicy (Transition Rule Probability) when ants work withcomponents is

119901119896

119895(119905) =

120591120572

119895120578120573

119895

sum119897notin119878119896 120591119897

[120578119897]120573

if 119895 notin 119878119896 (6)

where 119878119896 is the partial solution of the ant 119896 120572 and 120573 aretwo parameters which determine the relative influence ofthe pheromone trail and the heuristic information in theprobabilistic decision [13 18]

31 Pheromone Trail 120591119895 One of the most crucial design

decisions to be made in ACO algorithms is the modeling ofthe set of pheromones In the original ACO implementationfor TSP the choice was to put a pheromone value on everylink between a pair of cities but for other combinatorialproblems pheromone values can be often assigned to thedecision variables (first-order pheromone values) [13] In thiswork the pheromone trail is put on the problemrsquos component(each eligible column 119895) instead of the problems connectionsAnd setting a good pheromone quantity is not a trivial taskeither The quantity of pheromone trail laid on columnsis based on the idea that the more pheromone trail on aparticular item the more profitable the item is [15] Then thepheromone deposited in each component will be in relationto its frequency in the ants solutions (In this work we dividedthis frequency by number of ants)

32 Heuristic Information 120578119895 In this paper we use a dynamic

heuristic information that depends on the partial solutionof an ant It can be defined as 120578

119895= 119890119895119888119895 where 119890

119895is the

so-called cover value that is the number of additional rowscoveredwhen adding column 119895 to the current partial solutionand 119888119895is the cost of column 119895 In other words the heuristic

informationmeasures the unit cost of covering one additionalrow An ant ends the solution construction when all rows arecovered

33 AS and ACS In this work we hybridize with CP twoinstances of ACO ant system (AS) [28] and ant colony system(ACS) [29] algorithms the original and the most famousalgorithms in the ACO family

ACS differs from AS in the following aspects First itexploits the search experience accumulated by the ants morestrongly than AS does through the use of a more aggressiveaction choice rule Second pheromone evaporation andpheromone deposit take place only on the columns belongingto the best so far solution Third each time an ant chooses acolumn 119895 it removes some pheromone from the componentincreasing the exploration ACS has demonstrated betterperformance than AS in a wide range of problems

ACS exploits a pseudorandom transition rule in thesolution construction ant 119896 chooses the next column 119895 withcriteria

Argmax119897notin119878119896

120591119897[120578119897]120573 if 119902 le 119902

119900 (7)

following the Transition Rule Probability (6) or otherwisewhere 119902 is a random number uniformly distributed in[0 1] and 119902

0is a parameter that controls how strongly the

ants exploit deterministically the pheromone trail and theheuristic information It should be mentioned that ACS usesa candidate list to restrict the number of available choices tobe considered at each construction step The candidate list


(1)Begin(2) InitParameters()

(3) While remain iterations do(4) Foreach ant do(5) While solution is not completed and (119879119886119887119906119871119894119904119905 ltgt 119895) do(6) Choose next column 119895 with Transition Rule Probability(7) AddColumnToTabuList(119895)

(8) EndWhile(9) End Foreach(10) UpdateOptimum()

(11) UpdatePheromone()

(12) EndWhile(13) Return119861119890119904119905119878119900119897119906119905119894119900119899(14)End

Algorithm 1 ACO for SCP and SPP

contains a number of the best rated columns according to theheuristic criterion

Trying to solve larger instances of SPP with the originalAS or ACS implementation derives in a lot of unfeasiblelabeling of variables and the ants cannot obtain completesolutions In this paper we explore the addition of anarc consistency mechanism in the construction phase ofACO thus only feasible solutions are generated A directimplementation of the basic ACO framework is incapable ofobtaining feasible solution for many SPP instances

34 The ACO Framework Each ant starts with an emptysolution and adds column until a cover is completed But todetermine if a column actually belongs or not to the partialsolution (119895 = 119878

119896) is not good enoughThe traditional ACO decision policy (4) does not work

for SPP because the ants in this traditional selection processof the next columns ignore the information of the problemconstraint when a variable is instantiated And in the worstcase in the iterative steps it is possible to assign values tosome variables thatwillmake it impossible to obtain completesolution (see Algorithm 1)

4 Constraint Propagation

Constraint propagation is crucial in CP and it appears underdifferent names constraint relaxation filtering algorithmsnarrowing algorithms constraint inference simplificationalgorithms label inference local consistency enforcing rulesiteration and chaotic iteration [38]

Constraint propagation embeds any reasoning whichconsists in explicitly forbidding values or combinations ofvalues for some variables of a problem because a givensubset of its constraints cannot be satisfied otherwise Arcconsistency is the most well-known way of propagatingconstraints

41 Arc Consistency Arc-consistency is one of the most usedfiltering techniques in constraint satisfaction for reducing thecombinatorial space of problems Arc-consistency is formally

defined as a local consistency within the constraint program-ming field [39] A local consistency defines properties that theconstraint problemmust satisfy after constraint propagationConstraint propagation is simply the process when the givenlocal consistency is enforced to the problem In the followingsome necessary definitions are stated [38]

Definition 1 (constraint) A constraint 119888 is a relation definedon a sequence of variables 119883(119888) = (119909

1198941 119909

119894|119883(119888)|) called

the scheme of 119888 119888 is the subset of Z|119883(119888)| that contains thecombinations of values (or tuples) 120591 isin Z|119883(119888)| that satisfy119888 |119883(119888)| is called the arity of 119888 A constraint 119888 with scheme119883(119888) = (119909

1 119909

119896) is also noted as 119888(119909

1 119909

119896)

Definition 2 (constraint network) A constraint network alsoknown as constraint satisfaction problem (CSP) is defined bya triple119873 = ⟨119883119863 119862⟩ where

(i) 119883 is a finite sequence of integer variables 119883 = (1199091

119909119899)

(ii) 119863 is the corresponding set of domains for 119883 that is119863 = 119863(119909

1)times sdot sdot sdottimes119863(119909

119899) where119863(119909

119894) sub Z is the finite

set of values that variable 119909119894can take

(iii) 119862 is a set of constraints 119862 = 1198881 119888

119890 where var-

iables in119883(119888119895) are in119883

Definition 3 (projection) A projection of 119888 on119884 is denoted as120587119884(119888)

which defines the relation with scheme 119884 that containsthe tuples that can be extended to a tuple on 119883(119888) satisfying119888

As previously mentioned arc-consistency is one of themost used ways of propagating constraints Arc-consistencywas initially defined for binary constraint [40 41] thatis constraints involving two variables We here give themore general definition for nonarbitrary constraints namedgeneralized arc-consistency (GAC)

Definition 4 ((generalized) arc consistency) Given a network119873 = ⟨119883119863 119862⟩ a constraint 119888 isin 119862 and a variable 119909

119894isin 119883(119888)


0

1

2

0

1

2

0

1

2

0

1 1

2

1

2

x1 x2 x3 x1 x2 x3

x1 = x2x1 lt x2 x1 = x2x1 lt x2

Figure 1 Enforcing arc-consistency

Input 119909119894 119888

Output 119862119867119860119873119866119864(1) 119862119867119860119873119866119864 larr false(2) Foreach V

119894isin 119863(119909

119894) do

(3) If ∄120591 isin 119888 cap 120587119883(119888)

(119863)with 120591[119909119894] = V119894then

(4) remove V119894from119863(119909

119894)

(5) 119862119867119860119873119866119864 larr true(6) End If(7) End Foreach(8) Return 119862119867119860119873119866119864

Algorithm 2 Revise3

(i) a value V119894isin 119863(119909

119894) is consistent with 119888 isin 119863 if and

only if there exists a valid tuple 120591 satisfying 119888 such thatV119894= 120591[119909

119894] Such a tuple is called a support for (119909

119894 V119894)

on 119888

(ii) the domain 119863 is (generalized) arc-consistent on 119888 for119909119894if and only if all the values in 119863(119909

119894) are consistent

with 119888 in119863 that is119863(119909119894) sube 120587119909119894(119888 cap 120587

119883(119888)(119863))

(iii) the network 119873 is (generalized) arc-consistent if andonly if 119863 is (generalized) arc-consistent for all vari-ables in119883 on all constraints in 119862

As an example let us consider the non-arc-consistentnetwork 119873 depicted on left side of Figure 1 It considersthree variables 119909

1 1199092 and 119909

3 domains 119863(119909

1) = 119863(119909

2) =

119863(1199093) = 0 1 2 and constraints 119888

12 (1199091lt 1199092) and 119888

23

(1199092= 1199093) Enforcing arc-consistency allows one to eliminate

some inconsistent values For instance when constraint 11988812is

verified the value 2 from 119863(1199091) is removed since there is no

value greater than it in119863(1199092) The value 0 from119863(119909

2) is also

removed since no support for it exists in 119863(1199091) Removing 0

from 119863(1199092) leads to the removal of 0 from 119863(119909

3) when 119888

23is

checked The resulting arc-consistent network is depicted onthe right side of Figure 1

Such a filtering process can be carried out by usingAlgorithms 2 and 3

As previously illustrated the main idea of this process isthe revision of arcs that is to eliminate every value in119863(119883

119894)

that is inconsistent with a given constraint 119888 This notion isencapsulated in the function Revise3 This function takeseach value V

119894in 119863(119909

119894) (line 2) and analyses the space 120591 isin 119888 cap

120587119883(119888)

(119863) searching for a support on constraint 119888 (line 3) Ifsupport does not exist the value V

119894is eliminated from119863(119909

119894)

Finally the function informs if 119863(119883119894) has been changed by

returning true or false otherwise (line 8)Algorithm3 is responsible for ensuring that every domain

is consistent with the set of constraintsThis is done by using aloop that verifies arcs until no change happens The functionbegins by filling a list 119876 with pairs (119909

119894 119888) such that 119909

119894isin 119883(119888)

The idea is to keep the pairs for which 119863(119909119894) is not ensured

to be arc-consistent with respect to 119888 This allows to avoiduseless calls to Revise3 as done in more basic algorithmssuch as AC1 and AC2 Then a loop that takes the pairs (119909

119894 119888)

from 119876 (line 2) and Revise3 is called (line 4) If Revise3returns true 119863(119883

119894) is checked whether it is an empty set If

so the algorithm returns false Otherwise normally a valuefor another variable 119909

119895has lost its support on 119888Thus all pairs

(119909119894 119888) such that 119909

119894isin 119883(119888)must be reinserted in the list119876The

algorithm ends once 119876 is empty and it returns true when allarcs have been verified and remaining values of domains arearc-consistency with respect to all constraints

5 Hybridization of Ants andConstraint Programming

Hybrid algorithms provide appropriate compromises be-tween exact (or complete) search methods and approximate(or incomplete) methods some efforts have been done inorder to integrate constraint programming (exact methods)to ants algorithms (stochastic local search methods) [31ndash36]

A hybridization of ACO and CP can be approached fromtwo directions we can either take ACO or CP as the basealgorithm and try to embed the respective other method intoit A form to integrate CP into ACO is to let it reduce thepossible candidates among the not yet instantiated variablesparticipating in the same constraints that current variableA different approach would be to embed ACO within CPThe point at which ACO can interact with CP is during thelabeling phase using ACO to learn a value ordering that ismore likely to produce good solutions

In this work ACOuses CP in the variable selection (whenACO adds a column to partial solution) The CP algorithmused in this paper is the AC3 filtering procedure [42] Itperforms consistency between pairs of a not yet instantiatedvariable and an instantiated variable that is when a valueis assigned to the current variable any value in the domainof a future variable which conflicts with this assignment isremoved from the domain

The AC3 filtering procedure taking into account theconstraints network topology (ie which sets of variables arelinked by a constraint and which are not) guaranties that ateach step of the search all constraints between ready assignedvariables and not yet assigned variables are consistent itmeans that columns are chosen if they do not generate anyconflicts with the next column to be chosen Then a newtransition rule is developed embedding AC3 in the ACOframework (see Algorithm 4 lines 7 to 12)

51 ACO + CP to SCP In Figure 2 we present an explanationof how ACO + CP works solving SCP Here 119909

119894represents the


Input119883119863 119862Output Boolean(1) 119876 larr (119909

119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)

(2) While 119876 = 0 do(3) select and remove (119909

119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then

(6) Return false(7) Else(8) 119876 larr 119876 cup (119909

119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894

(9) End If(10) End If(11) EndWhile(12) Return true

Algorithm 3 AC3GAC3


(3) While remain iterations do(4) Foreach ant do(5) While solution is not completed and (119879119886119887119906119871119894119904119905 ltgt 119895) do(6) Choose next column 119895 with Transition Rule Probability(7) feasible(119894)= AC3(119883 notin 119878

119896 119863119883notin119878119896 119862119868119895

) lowast Filtering 119894 isin 119868lowast119895

(8) If feasible(119894)forall119894 then(9) AddColumnToSolution(119895)

(10) Else(11) Backtracking(119895)lowast Set 119895 uninstantiated lowast(12) End If-Else(13) AddColumnToTabuList(119895)




Algorithm 4 Hybrid ACO + CP for SCP and SPP

1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000

ximinus2 ximinus1 xi xi+1 xi+2

Figure 2 ACO + CP to SCP

current variable 119909119894minus1

and 119909119894minus2

are variables already instanti-ated and 119909

119894+1and 119909

119894+2are not yet instantiated variables (The

sequence sdot sdot sdot 119909119894minus1

119909119894minus2

119909119894 119909119894+1

119909119894+2

sdot sdot sdot is the order of variableinstantiations given by the enumeration strategy in use)

(1) Constraint Propagation Assigning Value 0 in FutureVariablesWhen instantiating a column for each rowthat can cover that column the other columns thatcan also cover it can be put to 0 as long as all rows thatcan cover each of these columns are already covered(ie the search space is reduced and also favoring theoptimization)

(2) Constraint Propagation Assigning Value 1 in FutureVariables If there is a column which is the only onethat can cover a row this column can be put to 1


1 1

1 1

1 1

1

(2)

(3)

0 0 0

xiximinus1 xi+1 xi+2

(3)

(1)

Figure 3 ACO + CP to SPP

52 ACO + CP to SPP In Figure 3 we present an explanationof how ACO + CP works solving SPP Here 119909



is a variable already instantiated and119909119894+1

and 119909119894+2

are not yet instantiated variables

(1) Backtracking in Current Variable If any of the rowsthat can cover a column is already been coveredthe current column cannot be instantiated because itviolates a constraint and then it should be done byusing backtracking

(2) Constraint Propagation Assigning Value 0 in FutureVariables When instantiating a column for eachuncovered row that can cover that column the othercolumns that can also cover (ie by constraint prop-agation it reduces the search space should be put to0 and in practice for the uninstantiated variable thevalue 1 of its domain was eliminated)

(3) Backtracking in Current Variable and Constraint Prop-agation Assigning Value 1 in Future Variables If anyof these columns (which were assigned the value 0)is the only column that can cover another row thecurrent column cannot be instantiated because it doesnot lead to a solution and then it should be done abacktracking

6 Experimental Evaluation

Wehave implementedAS andACS and the proposedAS+CPand ACS + CP algorithms The effectiveness of the proposedalgorithms was evaluated experimentally using SCP and SPPtest instances from Beasleyrsquos OR-library [43] Each instancewas solved (else it is indicated with 119899119891) 12 times and thealgorithms have been run with 100 ants and a maximumnumber of 200 iterations Table 1 shows the value consideredfor each standard ACO parameter 120572 is the relative influenceof the pheromone trail 120573 is the relative influence of the theheuristic information 120588 is the pheromone evaporation rate1199020is used in ACS pseudorandom proportional action choice

rule 120598 is used in ACS local pheromone trail update and

Table 1 Parameter settings for ACO algorithms

ACO algorithm 120572 120573 120588 1199020

120598 lsAS 1 05 04 mdash mdash mdashACS 1 05 099 05 01 300

the ACS list size 119897119904 was 300 For each instance the initialpheromone 120591

0was calculated as follows

1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)

Algorithms were implemented using ANSI C GCC 336under a 20GHz Intel Core2 Duo T5870 with 1Gb RAM run-ning Microsoft Windows XP Professional

Tables 2(a) and 2(b) describe problem instances and theyshow the problem code the number of rows119898 (constraints)the number of columns 119899 (decision variables) the 119863119890119899119904119894119905119910(ie the percentage of nonzero entries in the problemmatrix) and the best known cost value for each instance Opt(IP optimal) of the SCP and SPP instances used in the exper-imental evaluation

Computational results (best cost obtained) are shown inTables 3(a) 3b) 4(a) 4(b) 5(a) 5(b) 6(a) and 6(b) andin Figures 4 and 5 The quality of a solution is evaluatedusing the relative percentage deviation (RPD) and the relativepercentage improvement (RPI)measures [19]TheRPD valuequantifies the deviation of the objective value 119885 from 119885optwhich in our case is the best known cost value for eachinstance (see the third column) and the RPI value quantifiesthe improvement of 119885 from an initial solution 119885

119868(see the

fourth column)These measures are computed as follows

RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)

For all the implemented algorithms the solution qualityand computational effort (Secs) are related using themarginalrelative improvement (MIC) measure (see the fifth column)Thismeasure quantifies the improvement achieved per unit ofCPU time (RPISecs) The solution time is measured in CPUseconds and it is the time that each algorithm takes to firstreach the final best solution

The results expressed in terms of the averageRPD averageRPI and average MIC show the effectiveness of AS + CP andACS + CP over AS and ACS to solve SCP (see Tables 3(a)3(b) 4(a) and 4(b)) Our hybrid solver provides high qualitynear optimal solutions and it has the ability to generate themfor a variety of instances

From Tables 5(a) 5(b) 6(a) and 6(b) it can be observedthat AS + CP and ACS + CP solving SPP obtained betterresults than AS and ACS (viewing average RPD and average


Table 2 SCP and SPP instances

(a) Details of the SCP test instances

Problem 119898 119899 Density Optscp410 200 1000 2 514scpa1 300 3000 2 253scpa2 300 3000 2 252scpa3 300 3000 2 232scpa4 300 3000 2 234scpa5 300 3000 2 236scpb1 300 3000 5 69scpb2 300 3000 5 76scpb3 300 3000 5 80scpc1 400 4000 2 227scpc2 400 4000 2 219scpc3 400 4000 2 243scpc4 400 4000 2 219scpcyc07 672 448 2 144scpcyc08 1792 1024 2 344scpcyc09 4608 2304 2 780scpcyc10 11520 5120 2 1792scpd1 400 4000 5 60scpd2 400 4000 5 66scpd3 400 4000 5 72

(b) Details of the SPP test instances

Problem 119898 119899 Density Optsppaa01 823 8904 1 56137sppaa02 531 5198 13 30494sppaa03 825 8627 1 49649sppaa05 801 8308 099 55839sppaa06 646 7292 11 27040sppnw06 50 6774 182 7810sppnw08 24 434 224 35894sppnw09 40 3103 162 67760sppnw10 24 853 212 68271sppnw12 27 626 20 14118sppnw15 31 467 195 67743sppnw18 124 10757 68 340160sppnw19 40 2879 219 10898sppnw23 19 711 248 12534sppnw26 23 771 238 6796sppnw32 19 294 243 14877sppnw34 20 899 281 10488sppnw39 25 677 266 10080sppnw41 17 197 221 11307

RPI) but they are a bit worse in the average MIC It indicatesthat the computational effort is slightly higher using thehybridization but our approach can obtain optimal solutionsin some instances where AS or ACS failed Our hybrid

Table 3 Experimental results of SCP benchmarks using AS and AS+ CP

(a) AS experimental results

Problem AS RPD RPI MIC Secsscp410 539 486 999 97 103scpa1 592 13399 966 118 82scpa2 531 11071 972 074 1305scpa3 473 10388 973 078 125scpa4 375 6026 985 087 1133scpa5 349 4788 988 941 105scpb1 243 18406 953 329 29scpb2 207 21974 944 533 177scpb3 442 15875 959 05 1905scpc1 484 9471 976 139 70scpc2 551 121 969 473 205scpc3 523 12675 968 074 1305scpc4 272 13881 964 16 602scpcyc07 512 8889 977 138 707scpcyc08 1297 4884 988 079 1246scpcyc09 3123 6628 983 07 1405scpcyc10 1969 7427 981 065 1503scpd1 184 20667 947 928 102scpd2 209 21667 944 104 907scpd3 221 20694 947 05 1906avg 1207 969 273

(b) AS + CP experimental results

Problem AS + CP RPD RPI MIC Secsscp410 556 817 9979 1029 97scpa1 288 1383 9965 125 795scpa2 285 131 9966 083 120scpa3 270 1638 9958 199 501scpa4 278 188 9952 429 232scpa5 272 1525 9961 655 152scpb1 75 87 9978 832 12scpb2 87 1447 9963 967 103scpb3 89 1125 9971 076 1304scpc1 261 1498 9962 143 698scpc2 260 1872 9952 655 152scpc3 268 1029 9974 068 1473scpc4 259 1826 9953 194 513scpcyc07 148 278 9993 199 503scpcyc08 364 581 9985 1 1001scpcyc09 816 462 9988 1 1003scpcyc10 1969 988 9975 11 907scpd1 72 20 9949 1015 98scpd2 74 1212 9969 142 702scpd3 83 1528 9961 053 1879avg 1264 9968 359

approach shows an excellent tradeoff between the quality ofthe solutions obtained and the computational effort required


Table 4 Experimental results of SCP benchmarks using ACS andACS + CP

(a) ACS experimental results

Problem ACS RPD RPI MIC Secsscp410 669 3016 9923 1272 78scpa1 348 3755 9904 121 816scpa2 378 50 9872 083 1189scpa3 319 375 9904 207 479scpa4 333 4231 9892 518 191scpa5 353 4958 9873 866 114scpb1 101 4638 9881 978 101scpb2 117 5395 9862 1389 71scpb3 112 40 9897 064 1538scpc1 305 3436 9912 149 664scpc2 309 411 9895 687 144scpc3 367 5103 9869 065 1513scpc4 324 4795 9877 193 511scpcyc07 321 12292 9685 248 39scpcyc08 769 12355 9683 175 554scpcyc09 1723 1209 969 134 723scpcyc10 4097 12863 967 086 112scpd1 92 5333 9863 2013 49scpd2 96 4545 9883 164 602scpd3 111 5417 9861 056 1775avg 6054 9845 473

(b) ACS + CP experimental results

Problem ACS + CP RPD RPI MIC Secsscp410 664 2918 9925 993 10scpa1 331 3083 9921 179 553scpa2 376 4921 9874 164 602scpa3 295 2716 993 298 333scpa4 301 2863 9927 537 185scpa5 335 4195 9892 989 10scpb1 115 6667 9829 806 122scpb2 110 4474 9885 1498 66scpb3 117 4625 9881 114 87scpc1 317 3965 9898 223 444scpc2 311 4201 9892 1902 52scpc3 328 3498 991 097 1017scpc4 303 3836 9902 356 27scpcyc07 321 12292 9685 32 303scpcyc08 769 12355 9683 203 477scpcyc09 1445 8526 9781 176 555scpcyc10 3506 9565 9755 196 497scpd1 105 75 9808 1886 52scpd2 113 7121 9817 218 45scpd3 119 6528 9833 109 903avg 5792 985 563

7 Conclusion

Highly constrained combinatorial optimization problemshave proved to be a challenge for constructive metaheuristic

Table 5 Experimental results of SPP benchmarks using AS and AS+ CP


Problem AS RPD RPI MIC Secssppaa01 96256 7147 9871 009 11003sppaa02 39883 3079 9921 033 3007sppaa03 63734 2837 9927 011 9006sppaa05 61703 105 9973 013 7783sppaa06 42015 5538 9858 025 4005sppnw06 9200 178 9954 11 908sppnw08 nf nf nf nf nfsppnw09 70462 399 999 704 142sppnw10 nf nf nf nf nfsppnw12 15406 912 9977 1296 77sppnw15 67755 002 100 1471 68sppnw18 385596 1336 9966 05 2002sppnw19 11678 716 9982 143 697sppnw23 14304 1412 9964 1465 68sppnw26 6976 265 9993 1332 75sppnw32 14877 0 100 1031 97sppnw34 13341 272 993 1324 75sppnw39 11670 1577 996 1606 62sppnw41 11307 0 100 1493 67avg 181 9954 713


Problem AS + CP RPD RPI MIC Secssppaa01 60246 732 9981 01 10007sppaa02 37452 2282 9941 04 2504sppaa03 55082 1094 9972 012 8003sppaa05 58158 415 9989 015 680sppaa06 33524 2398 9939 032 3135sppnw06 8160 448 9989 118 847sppnw08 35894 0 100 014 7003sppnw09 70222 363 9991 799 125sppnw10 nf nf nf nf nfsppnw12 14466 246 9994 952 105sppnw15 67743 0 100 1818 55sppnw18 345762 165 9996 09 1108sppnw19 11060 149 9996 127 789sppnw23 13932 1115 9971 169 59sppnw26 6880 124 9997 1204 83sppnw32 14877 0 100 935 107sppnw34 10713 215 9994 884 113sppnw39 11322 1232 9968 2121 47sppnw41 11307 0 100 1538 65avg 61 998 689

In this paper ACO framework has been modified so thatit may be applied to any constraint satisfaction problem ingeneral and hard constrained instances in particular


Table 6 Experimental results of SPP benchmarks using ACS andACS + CP


Problem ACS RPD RPI MIC Secssppaa01 94720 6793 9826 016 6008sppaa02 57632 8899 9772 036 2704sppaa03 93304 8793 9775 013 7805sppaa05 91134 6321 9838 014 7123sppaa06 54964 10327 9735 038 2549sppnw06 9788 2533 9935 126 787sppnw08 nf nf nf nf nfsppnw09 nf nf nf nf nfsppnw10 nf nf nf nf nfsppnw12 16060 1376 9965 81 123sppnw15 67746 0 100 1754 57sppnw18 365398 742 9981 083 1207sppnw19 12350 1332 9966 135 737sppnw23 14604 1652 9958 1747 57sppnw26 6956 235 9994 1281 78sppnw32 14886 006 100 806 124sppnw34 11289 764 998 1468 68sppnw39 10758 673 9983 192 52sppnw41 11307 0 100 1667 6avg 3153 992 745


Problem ACS + CP RPD RPI MIC Secssppaa01 88435 5041 9871 014 7071sppaa02 52211 7122 9817 032 3094sppaa03 81177 635 9837 016 6005sppaa05 84362 5108 9869 017 5907sppaa06 48703 8011 9795 033 3009sppnw06 8038 292 9993 33 303sppnw08 36682 22 9994 032 3097sppnw09 69332 232 9994 245 408sppnw10 nf nf nf nf nfsppnw12 14252 095 9998 934 107sppnw15 67743 0 100 1613 62sppnw18 345130 146 9996 147 678sppnw19 11858 881 9977 142 701sppnw23 12880 276 9993 1666 6sppnw26 6880 124 9997 1587 63sppnw32 14877 0 100 98 102sppnw34 10797 295 9992 1753 57sppnw39 10545 461 9988 1585 63sppnw41 11307 0 100 1923 52avg 1925 995 725

A direct implementation of the basic ACO framework isincapable of obtaining feasible solutions for many stronglyconstrained problems In order to improve this aspect ofACO we integrated a constraint programming mechanism

Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3

Figure 4 Experimental results for SCP

Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104

Figure 5 Experimental results for SPP

in the construction phase of ACO thus only feasible partialsolutions are generated

The effectiveness of the proposed rule was tested onbenchmark problems and we solved SCP and SPP with ACOusing constraint propagation in its transition rule and resultswere compared with pure ACO algorithms About efficiencythe computational effort required is almost the same

An interesting extension of this work would be relatedto hybridization of AC3 with other metaheuristics [44] Theuse of autonomous search (AS) [45] in conjunction withconstraint programming would be also a promising directionto follow AS represents a new research field and it providespractitioners with systems that are able to autonomously self-tune their performance while effectively solving problems Itsmajor strength and originality consist in the fact that problemsolvers can now perform self-improvement operations basedon analysis of the performances of the solving processIn [46 47] the order in which the variables are selectedfor instantiation is determined by a choice function thatdynamically selects from a set of variable ordering heuristics


the one that best matches the current problem state and thiscombination can accelerate the resolution process especiallyin harder instances In [48] the results show that somephases of reactive propagation are beneficial to the mainhybrid algorithm and the hybridization strategies are thuscrucial in order to decide when to perform or not constraintpropagation

Furthermore we are considering to use different prepro-cessing steps from the OR literature which allow to reducethe problem size [49 50]

Acknowledgment

The author Fernando Paredes is supported by FONDECYT-Chile Grant 1130455

References

[1] E Balas and M Padberg ldquoSet partitioning a surveyrdquo Manage-ment Sciences Research Report Defense Technical InformationCenter Fort Belvoir Va USA 1976

[2] T A Feo and M G C Resende ldquoA probabilistic heuristic fora computationally difficult set covering problemrdquo OperationsResearch Letters vol 8 no 2 pp 67ndash71 1989

[3] A Mingozzi M A Boschetti S Ricciardelli and L BiancoldquoA set partitioning approach to the crew scheduling problemrdquoOperations Research vol 47 no 6 pp 873ndash888 1999

[4] M Mesquita and A Paias ldquoSet partitioningcovering-basedapproaches for the integrated vehicle and crew schedulingproblemrdquoComputers and Operations Research vol 35 no 5 pp1562ndash1575 2008 special issue algorithms and computationalmethods in feasibility and infeasibility

[5] J P Kelly and J Xu ldquoA set-partitioning-based heuristic for thevehicle routing problemrdquo INFORMS Journal on Computing vol11 no 2 pp 161ndash172 1999

[6] M L ] Balinski and R E Quandt ldquoOn an integer program for adelivery problemrdquo Operations Research vol 12 no 2 pp 300ndash304 1964

[7] R D Gopal and R Ramesh ldquoQuery clustering problem a setpartitioning approachrdquo IEEE Transactions on Knowledge andData Engineering vol 7 no 6 pp 885ndash899 1995

[8] G B Alvarenga and G R Mateus ldquoA two-phase genetic and setpartitioning approach for the vehicle routing problemwith timewindowsrdquo in Proceedings of the 4th International Conference onHybrid Intelligent Systems (HIS rsquo04) M Ishikawa S HashimotoM Paprzycki et al Eds pp 428ndash433 IEEE Computer Society2004

[9] F Barahona and R Anbil ldquoOn some difficult linear programscoming from set partitioningrdquo Discrete Applied Mathematicsvol 118 no 1-2 pp 3ndash11 2002 special Issue devoted to theALIO-EURO Workshop on Applied Combinatorial Optimiza-tion

[10] R Borndorfer M Grotschel F Klostermeier and C KuttnerldquoTelebus berlin vehicle scheduling in a dial-a-ride systemrdquoTech Rep SC 9723 Konrad-Zuse-Zentrum fur Information-stechnik Berlin Germany 1997

[11] B Crawford C Castro and E Monfroy ldquoSolving dial-a-rideproblems with a low-level hybridization of ants and constraintprogrammingrdquo in Proceedings of the 2nd International Work-Conference on the Interplay between Natural and Artificial

Computation (IWINAC rsquo07) J Mira and J R Alvarez Eds vol4528 of Lecture Notes in Computer Science pp 317ndash327 Springer2007

[12] K Apt Principles of Constraint Programming Cambridge Uni-versity Press New York NY USA 2003

[13] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[14] B Chandra Mohan and R Baskaran ldquoA survey ant colonyoptimization based recent research and implementation onseveral engineering domainrdquo Expert Systems with Applicationsvol 39 no 4 pp 4618ndash4627 2012

[15] G Leguizamon and Z Michalewicz ldquoA new version of antsystem for subset problemsrdquo in Proceedings of Congress on Evo-lutionary Computation (CEC rsquo99) P Angeline Z MichalewiczM Schoenauer X Yao and A Zalzala Eds IEEE Press July1999

[16] XGandibleux XDelorme andV TrsquoKindt ldquoAn ant colony opti-misation algorithm for the set packing problemrdquo in Proceedingsof the 4th International Workshop on Ant Colony Optimizationand Swarm Intelligence (ANTS rsquo04) pp 49ndash60 2004

[17] M Rahoual R Hadji and V Bachelet ldquoParallel ant system forthe set covering problemrdquo inAnt Algorithms pp 262ndash267 2002

[18] L Lessing I Dumitrescu and T Stutzle ldquoA comparisonbetween aco algorithms for the set covering problemrdquo inProceedings of the 4th International Workshop on Ant ColonyOptimization and Swarm Intelligence (ANTS rsquo04) pp 1ndash12 2004

[19] Z-G Ren Z-R Feng L-J Ke and Z-J Zhang ldquoNew ideas forapplying ant colony optimization to the set covering problemrdquoComputers and Industrial Engineering vol 58 no 4 pp 774ndash784 2010

[20] R M D A Silva and G L Ramalho ldquoAnt system for theset covering problemrdquo in Proceedings of IEEE InternationalConference on Systems Man and Cybernetics vol 5 pp 3129ndash3133 October 2001

[21] R HadjiM Rahoual E Talbi andV Bachelet ldquoAnt colonies forthe set covering problemrdquo in Proceedings of 2nd InternationalWorkshop on Ant Algorithms (ANTS rsquo00) M Dorigo Ed pp63ndash66 Brussels Belgium 2000

[22] M H Mulati and A A Constantino ldquoAnt-line a lineorientedaco algorithm for the set covering problemrdquo inProceedings of the30th International Conference of the Chilean Computer ScienceSociety (SCCC rsquo11) pp 265ndash274 Curico Chile November 2011

[23] P C Chu and J E Beasley ldquoConstraint handling in geneticalgorithms the set partitioning problemrdquo Journal of Heuristicsvol 4 no 4 pp 323ndash357 1998

[24] D Levine ldquoA parallel genetic algorithm for the set partitioningproblemrdquo Tech Rep Illinois Institute of Technology ChicagoIll USA 1994

[25] V Maniezzo and M Milandri ldquoAn ant-based framework forvery strongly constrained problemsrdquo in Ant Algorithms pp222ndash227 2002

[26] V Maniezzo and M Roffilli ldquoVery strongly constrained prob-lems an ant colony optimization approachrdquo Cybernetics andSystems vol 39 no 4 pp 395ndash424 2008

[27] M Randall and A Lewis ldquoModifications and additions toant colony optimisation to solve the set partitioning problemrdquoin Proceedings of the 6th IEEE International Conference on e-Science Workshops (e-ScienceW rsquo10) pp 110ndash116 Los AlamitosCalif USA December 2010

[28] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions on


Systems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[29] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[30] C Blum ldquoAnt colony optimization introduction and recenttrendsrdquo Physics of Life Reviews vol 2 no 4 pp 353ndash373 2005

[31] R Michel and M Middendorf ldquoAn island model based ant sys-temwith lookahead for the shortest supersequence problemrdquo inParallel Problem Solving from Nature vol 1498 of Lecture Notesin Computer Science pp 692ndash701 Springer Berlin Germany1998

[32] C Gagne M Gravel and W Price ldquoA look-ahead addition tothe ant colony optimization metaheuristic and its applicationto an industrial scheduling problemrdquo in Proceedings of the 4thMetaheuristics International Conference (MIC rsquo01) J Sousa Edpp 79ndash84 Porto Portugal July 2001

[33] B Meyer and A T Ernst ldquoIntegrating aco and constraintpropagationrdquo in Proceedings of the 4th International Workshopon Ant Colony Optimization and Swarm Intelligence (ANTS rsquo04)pp 166ndash177 2004

[34] B Crawford and C Castro ldquoAco with lookahead proceduresfor solving set partitioning and covering problemsrdquo in Pro-ceedings of Workshop on Combination of Metaheuristic andLocal Search with Constraint Programming Techniques NantesFrance November 2005

[35] B Crawford and C Castro ldquoIntegrating lookahead and postprocessing procedures with aco for solving set partitioningand covering problemsrdquo in Proceedings of the 8th InternationalConference onArtificial Intelligence and SoftComputing (ICAISCrsquo06) L Rutkowski R Tadeusiewicz L A Zadeh and J MZurada Eds vol 4029 of Lecture Notes in Computer Sciencepp 1082ndash1090 Springer 2006

[36] D RThiruvady C Blum B Meyer and A T Ernst ldquoHybridiz-ing beam-aco with constraint programming for single machinejob schedulingrdquo in Hybrid Metaheuristics M J Blesa C BlumL D Gaspero A Roli M Sampels and A Schaerf Eds vol5818 of Lecture Notes in Computer Science pp 30ndash44 SpringerBerlin Germany 2009

[37] M Khichane P Albert and C Solnon ldquoStrong combinationof ant colony optimization with constraint programming opti-mizationrdquo in Proceedings of the 7th International Conferenceon Integration of Artificial Intelligence and Operations Research(CPAIOR rsquo10) A Lodi MMilano and P Toth Eds vol 6140 ofLecture Notes in Computer Science pp 232ndash245 Springer 2010

[38] C Bessiere ldquoConstraint propagationrdquo in Handbook of Con-straint Programming F Rossi P van Beek and T Walsh Edspp 29ndash84 Elsevier Rio de Janeiro Brazil 2006

[39] F Rossi P van Beek and T Walsh Handbook of ConstraintProgramming Elsevier Rio de Janeiro Brazil 2006

[40] A K Mackworth ldquoConsistency in networks of relationsrdquoArtificial Intelligence vol 8 no 1 pp 99ndash118 1977

[41] A Mackworth ldquoOn reading sketch mapsrdquo in Proceedings of theInternational Joint Conference on Artificial Intelligence (IJCAIrsquo77) pp 598ndash606 1977

[42] R Dechter and D Frost ldquoBackjump-based backtracking forconstraint satisfaction problemsrdquoArtificial Intelligence vol 136no 2 pp 147ndash188 2002

[43] J E Beasley ldquoOr-library distributing test problems by elec-tronic mailrdquo Journal of the Operational Research Society vol 41no 11 pp 1069ndash1072 1990

[44] R Soto B Crawford C Galleguillos E Monfroy and F Pare-des ldquoA hybrid ac3-tabu search algorithm for solving sudokupuzzlesrdquo Expert Systems with Applications vol 40 no 15 pp5817ndash5821 2013

[45] Y Hamadi E Monfroy and F Saubion ldquoAn introductionto autonomous searchrdquo in Autonomous Search Y HamadiE Monfroy and F Saubion Eds pp 1ndash11 Springer BerlinGermany 2011

[46] B Crawford C Castro E Monfroy R Soto W Palma andF Paredes ldquoDynamic selection of enumeration strategies forsolving constraint satisfaction problemsrdquo Romanian Journal ofInformation Science and Technology vol 15 no 2 pp 106ndash1282012

[47] B Crawford R Soto E Monfroy W Palma C Castro and FParedes ldquoParameter tuning of a choice-function based hyper-heuristic using particle swarm optimizationrdquo Expert Systemswith Applications vol 40 no 5 pp 1690ndash1695 2013

[48] E Monfroy C Castro B Crawford R Soto F Paredes and CFigueroa ldquoA reactive and hybrid constraint solverrdquo Journal ofExperimental and Theoretical Artificial Intelligence vol 25 no1 pp 1ndash22 2013

[49] T Muller ldquoSolving set partitioning problems with constraintprogrammingrdquo in Proceedings of the 6th International Confer-ence on the Practical Application of Prolog and the 4th Inter-national Confernce on the Practical Application of ConstraintTechnology (PAPPACT rsquo98) pp 313ndash332 London UK March1998

[50] M Krieken H Fleuren and M Peeters ldquoProblem reduction inset partitioning problemsrdquo Discussion Paper 2003-80 TilburgUniversity Center for EconomicResearch TilburgTheNether-lands 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


ants deposit pheromone on the ground marking the path foridentification by other members of the colony of the routesfrom the nest to food [13] From the early nineties ACOattracted the attention of researchers and many successfulapplications solving optimization problems are done [14]

There exist already some good approaches applying ACOto subset problems [15] In [16] ACO is applied on the setpacking problem using two solution construction strategiesbased on exploration and exploitation In general the sameoccurs in relation with SCP applying ACO only as a con-struction algorithm and testing the approach only on somesmall SCP instances More recent works apply ant computingto the SCP and related problems using techniques to removeredundant columns and local search to improve solutions[17ndash22]

The best performing metaheuristics for SPP are geneticalgorithms [23 24] Taking into account these results it seemsthat the incomplete approach of ant computing could beconsidered as a good alternative to solve these problemswhencomplete techniques are not able to get the optimal solutionin a reasonable time

ACO is of limited effectiveness solving very stronglyconstrained problems They are problems for which neigh-borhoods contain few solutions or none at all and localsearch is of very limited use Probably the most significantof such problems is the SPP A direct implementation ofthe basic ACO framework is incapable of obtaining feasiblesolutions for many standard tested instances of SPP [25ndash27]

Trying to solve larger instances of SPP with the originalant system (AS) [28] or ant colony system (ACS) [29] imple-mentation derives in a lot of unfeasible labeling of variablesand the ants cannot obtain complete solutions using theclassic transition rule when theymove in their neighborhoodThe root of the problem is that simply following the randomproportional transition rule that is learningreforcing goodpaths is no longer enough as this does not check forconstraint consistency

In order to improve this aspect of ACO we are workingin the addition of a constraint programming mechanism inthe construction phase of ACO thus only feasible partialsolutions are generated The CP mechanism allows theincorporation of information about the instantiation of vari-ables after the current decision In general ACO algorithmsare competitive with other optimization techniques whenapplied to problems that are not overly constrained Howeverwhen solving highly constrained problems the performanceof ACO algorithms degrades When a problem is highlyconstrained the difficulty is in finding feasible solutionsThisis where CP comes into play because these problems arethe target problems for CP solvers CP is a programmingparadigm in which a combinatorial optimization problemis modeled as a discrete optimization problem specifyingthe constraints that a feasible solution must meet The CPapproach to search for a feasible solution often works bythe iteration of constraint propagation and the additionof additional constraints and it transforms the problemwithout changing its solutions Constraint propagation is themechanism that reduces the domains of the decision variableswith respect to the given set of constraints [30]

Although the idea of obtaining sinergy from hybridiza-tion of ACO with CP is not novel [31ndash36] our proposal is abit different We explore the addition to the ACO algorithmof a mechanism to check constraint consistency usuallyused in complete techniques arc consistency Other kinds ofcooperation between ACO and CP are shown in [37] wherecombinatorial optimization problems are solved in a genericway by a two-phase algorithm The first phase aims to createa hot start for the second it samples the solution space andapplies reinforcement learning techniques as implemented inACO to create pheromone trails During the second phase aCP optimizer performs a complete tree search guided by thepheromone trails previously accumulated

Here we propose the addition of a lookaheadmechanismin the construction phase of ACO in order that only feasiblesolutions are generated The lookahead mechanism allowsthe incorporation of information about the instantiation ofvariables after the current decisionThe idea differs from thatproposed by [31 32] and these authors proposed a lookaheadfunction evaluating the pheromone in the shortest commonsupersequence problem and estimating the quality of a partialsolution of an industrial scheduling problem respectively

This paper is organized as follows In Section 2 we explainthe problem In Section 3 we describe the ACO frame-work In Section 4 we present the definitions considered inconstraint propagation Our hybrid proposal is described inSection 5 In Section 6 we present the experimental resultsobtained Finally in Section 7 we conclude the paper and givesome perspectives for future research

2 Problem Description

SPP is the problem of partitioning a given set into manu-ally independent subsets while minimizing a cost functiondefined as the sum of the costs associated with each of theeligible subsets

In the SPP matrix formulation we are given a 119898 times 119899

matrix 119860 = (119886119894119895) in which all the matrix elements are either

zero or one Additionally each column is given a nonnegativecost 119888119895

Let 119868 = 1 119898 and 119869 = 1 119899 be the row set andcolumn set respectively

We say that a column 119895 covers a row 119894 if 119886119894119895= 1 Let 119909

119895

be a binary variable which is one if column 119895 is chosen andzero otherwise The SPP can be defined formally asminimize(1) subject to (2) These constraints enforce that each row iscovered by exactly one columnThe SCP is an SPP relaxationThe goal in the SCP is to choose a subset of the columnsof minimal weight formally using constraints to enforce thateach row is covered by at least one column as (3)

119891 (119909) =

119899

sum

119895=1

119888119895119909119895 (1)

119899

sum

119895=1

119886119894119895119909119895= 1 forall119894 = 1 119898 (2)


119899

sum

119895=1

119886119894119895119909119895ge 1 forall119894 = 1 119898 (3)

119909119895isin 0 1 forall119895 = 1 119899 (4)


119869119894= 119895 isin 119869 | 119886


119868119895= 119894 isin 119868 | 119886


119889 =

sum119898

119894=1sum119899

119895=1119886119894119895

119898119899

density that is


(5)







119901119896

119895(119905) =

120591120572

119895120578120573

119895

sum119897notin119878119896 120591119897

[120578119897]120573

if 119895 notin 119878119896 (6)






119895= 119890119895119888119895 where 119890

119895is the






Argmax119897notin119878119896

120591119897[120578119897]120573 if 119902 le 119902

119900 (7)






















1198941 119909

119894|119883(119888)|) called


1 119909

119896) is also noted as 119888(119909

1 119909

119896)



119909119899)



119899) where119863(119909




119890 where var-

iables in119883(119888119895) are in119883





119894isin 119883(119888)


0

1

2

0

1

2

0

1

2

0

1 1

2

1

2

x1 x2 x3 x1 x2 x3

x1 = x2x1 lt x2 x1 = x2x1 lt x2


Input 119909119894 119888


119894isin 119863(119909

119894) do

(3) If ∄120591 isin 119888 cap 120587119883(119888)

(119863)with 120591[119909119894] = V119894then

(4) remove V119894from119863(119909

119894)


Algorithm 2 Revise3

(i) a value V119894isin 119863(119909




119894 V119894)

on 119888




119883(119888)(119863))



1 1199092 and 119909

3 domains 119863(119909

1) = 119863(119909

2) =

119863(1199093) = 0 1 2 and constraints 119888

12 (1199091lt 1199092) and 119888

23





2) is also



3) when 119888

23is




119894)


119894in 119863(119909


120587119883(119888)



119894)




119894 119888) such that 119909

119894isin 119883(119888)



119894 119888)





(119909119894 119888) such that 119909












119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)


119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then


119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894


Algorithm 3 AC3GAC3



119896 119863119883notin119878119896 119862119868119895








1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000




and 119909119894minus2


119894+1and 119909



119909119894minus2

119909119894 119909119894+1

119909119894+2





1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119899

sum

119895=1

119886119894119895119909119895ge 1 forall119894 = 1 119898 (3)

119909119895isin 0 1 forall119895 = 1 119899 (4)


119869119894= 119895 isin 119869 | 119886


119868119895= 119894 isin 119868 | 119886


119889 =

sum119898

119894=1sum119899

119895=1119886119894119895

119898119899

density that is


(5)







119901119896

119895(119905) =

120591120572

119895120578120573

119895

sum119897notin119878119896 120591119897

[120578119897]120573

if 119895 notin 119878119896 (6)






119895= 119890119895119888119895 where 119890

119895is the






Argmax119897notin119878119896

120591119897[120578119897]120573 if 119902 le 119902

119900 (7)






















1198941 119909

119894|119883(119888)|) called


1 119909

119896) is also noted as 119888(119909

1 119909

119896)



119909119899)



119899) where119863(119909




119890 where var-

iables in119883(119888119895) are in119883





119894isin 119883(119888)


0

1

2

0

1

2

0

1

2

0

1 1

2

1

2

x1 x2 x3 x1 x2 x3

x1 = x2x1 lt x2 x1 = x2x1 lt x2


Input 119909119894 119888


119894isin 119863(119909

119894) do

(3) If ∄120591 isin 119888 cap 120587119883(119888)

(119863)with 120591[119909119894] = V119894then

(4) remove V119894from119863(119909

119894)


Algorithm 2 Revise3

(i) a value V119894isin 119863(119909




119894 V119894)

on 119888




119883(119888)(119863))



1 1199092 and 119909

3 domains 119863(119909

1) = 119863(119909

2) =

119863(1199093) = 0 1 2 and constraints 119888

12 (1199091lt 1199092) and 119888

23





2) is also



3) when 119888

23is




119894)


119894in 119863(119909


120587119883(119888)



119894)




119894 119888) such that 119909

119894isin 119883(119888)



119894 119888)





(119909119894 119888) such that 119909












119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)


119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then


119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894


Algorithm 3 AC3GAC3



119896 119863119883notin119878119896 119862119868119895








1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000




and 119909119894minus2


119894+1and 119909



119909119894minus2

119909119894 119909119894+1

119909119894+2





1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























1198941 119909

119894|119883(119888)|) called


1 119909

119896) is also noted as 119888(119909

1 119909

119896)



119909119899)



119899) where119863(119909




119890 where var-

iables in119883(119888119895) are in119883





119894isin 119883(119888)


0

1

2

0

1

2

0

1

2

0

1 1

2

1

2

x1 x2 x3 x1 x2 x3

x1 = x2x1 lt x2 x1 = x2x1 lt x2


Input 119909119894 119888


119894isin 119863(119909

119894) do

(3) If ∄120591 isin 119888 cap 120587119883(119888)

(119863)with 120591[119909119894] = V119894then

(4) remove V119894from119863(119909

119894)


Algorithm 2 Revise3

(i) a value V119894isin 119863(119909




119894 V119894)

on 119888




119883(119888)(119863))



1 1199092 and 119909

3 domains 119863(119909

1) = 119863(119909

2) =

119863(1199093) = 0 1 2 and constraints 119888

12 (1199091lt 1199092) and 119888

23





2) is also



3) when 119888

23is




119894)


119894in 119863(119909


120587119883(119888)



119894)




119894 119888) such that 119909

119894isin 119883(119888)



119894 119888)





(119909119894 119888) such that 119909












119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)


119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then


119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894


Algorithm 3 AC3GAC3



119896 119863119883notin119878119896 119862119868119895








1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000




and 119909119894minus2


119894+1and 119909



119909119894minus2

119909119894 119909119894+1

119909119894+2





1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

0

1

2

0

1

2

0

1 1

2

1

2

x1 x2 x3 x1 x2 x3

x1 = x2x1 lt x2 x1 = x2x1 lt x2


Input 119909119894 119888


119894isin 119863(119909

119894) do

(3) If ∄120591 isin 119888 cap 120587119883(119888)

(119863)with 120591[119909119894] = V119894then

(4) remove V119894from119863(119909

119894)


Algorithm 2 Revise3

(i) a value V119894isin 119863(119909




119894 V119894)

on 119888




119883(119888)(119863))



1 1199092 and 119909

3 domains 119863(119909

1) = 119863(119909

2) =

119863(1199093) = 0 1 2 and constraints 119888

12 (1199091lt 1199092) and 119888

23





2) is also



3) when 119888

23is




119894)


119894in 119863(119909


120587119883(119888)



119894)




119894 119888) such that 119909

119894isin 119883(119888)



119894 119888)





(119909119894 119888) such that 119909












119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)


119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then


119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894


Algorithm 3 AC3GAC3



119896 119863119883notin119878119896 119862119868119895








1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000




and 119909119894minus2


119894+1and 119909



119909119894minus2

119909119894 119909119894+1

119909119894+2





1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 119888) | 119888 isin 119862 119909

119894isin 119883 (119888)


119894 119888) from 119876

(4) If Revise3(119909119894 119888) then

(5) If 119863(119909119894) = 0 then


119895 1198881015840) | 1198881015840isin 119862 and 119888

1015840= 119888 and 119909

119894 119909119895isin 119883 (119888

1015840) and 119895 = 119894


Algorithm 3 AC3GAC3



119896 119863119883notin119878119896 119862119868119895








1

1

1

1

1 1

1

1

1

(1)

(1)

(1)

(1)

(1)

(2)(2)

(2)

0000




and 119909119894minus2


119894+1and 119909



119909119894minus2

119909119894 119909119894+1

119909119894+2





1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 1

1 1

1 1

1

(2)

(3)

0 0 0


(3)

(1)






and 119909119894+2









ACO algorithm 120572 120573 120588 1199020




1205910=

2119898

(Opt + sum119899119895=1

119888119895)

(8)




119868(see the


RPD =

(119885 minus 119885opt)

119885opttimes 100

RPI =(119885119868minus 119885)

(119885119868minus 119885opt)

times 100

(9)























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















7 Conclusion















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Cos

t

45004000350030002500200015001000

5000

ASACS

AS + CPACS + CP

Benchmark instance

scp4

10sc

pa1

scpa

2sc

pa3

scpa

4sc

pa5

scpb

1sc

pb2

scpb

3sc

pc1

scpc

2sc

pc3

scpc

4sc

pcyc

07sc

pcyc

08sc

pcyc

09sc

pcyc

10sc

pd1

scpd

2sc

pd3


Cos

t

ASACS

AS + CPACS + CP

Benchmark instance

12

10

8

6

4

2

0

sppa

a01

sppa

a02

sppa

a03

sppa

a05

sppa

a06

sppn

w06

sppn

w08

sppn

w09

sppn

w12

sppn

w15

sppn

w19

sppn

w23

sppn

w26

sppn

w32

sppn

w34

sppn

w39

sppn

w41

times104








Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Hybrid Soft Computing Approach for ...downloads.hindawi.com/journals/mpe/2013/716069.pdf · Introduction Set covering problem (SCP) and set partitioning problem