quantum-inspired wolf pack algorithm to solve the 0-1...

Research ArticleQuantum-Inspired Wolf Pack Algorithm toSolve the 0-1 Knapsack Problem

Yangjun Gao Fengming Zhang Yu Zhao and Chao Li

Air Force Engineering University Xirsquoan Shanxi China

Correspondence should be addressed to Yangjun Gao greisy2008gmailcom

Received 3 November 2017 Accepted 8 May 2018 Published 20 June 2018

Academic Editor Emilio Insfran Pelozo

Copyright copy 2018 Yangjun Gao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a Quantum-Inspired wolf pack algorithm (QWPA) based on quantum encoding to enhance the performanceof the wolf pack algorithm (WPA) to solve the 0-1 knapsack problems There are two important operations in QWPA quantumrotation and quantum collapseThe first step enables the population tomove to the global optima and the second step helps to avoidthe trapping of individuals into local optima Ten classical and four high-dimensional knapsack problems are employed to test theproposed algorithm and the results are compared with other typical algorithmsThe statistical results demonstrate the effectivenessand global search capability for knapsack problems especially for high-level cases

1 Introduction

The 0-1 knapsack problem (KP01) is a typical combinatorialoptimization problem It offers various practical applicationssuch as task scheduling resource allocation investmentdecisions and others [1 2] In a given set of items each ofthemwith a value119901j and a volume119908119895 there is a knapsackwitha limited capacity C The question is to select a subset fromthe given set to pack the knapsack so that the items in thisknapsack have a maximal value of overall possible solutionsThe model of the KP01 can be formulated as follows

max 119891 (119909) = 119898sum119894=1

119901119894119909119894st 119898sum

119894=1

119908119894119909119894 le 119862 119909119894 = 0 1(1)

The variable 119909119894 takes values either 0 or 1 which representsrejection or selection of the ith item

There are mainly two classes of approaches to solvingthe KP01 the conventional one is an exact solution basedonmathematical programming andoperational research andthe other one is a stochastic solution based on heuristic algo-rithm [3] It is possible to solve a small-scale KP01 problemby branching definition method and dynamic programming

However a high-dimensional situation is NP-hard and it isunrealistic to obtain optimal solutions using an exactmethodAs a result the use of heuristic algorithms has attracted theextensive attention of scholars in this field

In recent years most classical heuristic algorithms suchas the genetic algorithm particle swarm algorithm antcolony algorithm and modifications of these algorithmshave been applied to KP01 problems for excellent results [4ndash6] Some emerging novel algorithms have also been widelyused like the artificial bee colony algorithm [7] and thecuckoo algorithm [8]Thewolf pack algorithm [9] is tomimicthe hunting behavior of wolves to obtain optimal solutionsand it has been shown to be global convergent and robust andthe binary wolf algorithm has shown effective performancefor the KP01 problem [10] However the wolf pack algorithmis proposed in continuous space and the binary encodingrequired to map continuous space to discrete space maylead to confusion [11] The conventional method used forupdating operations in binary encoding is directly roundingorbit updating These operations are able to reflect the ideasof algorithms to a limited extent However it is difficult todeterminewhether upward or downward rounding should beused in the updating operation and whether the number ofbits is able to represent the position of individuals

To avoid the above difficulties we proposed a quantumwolf pack algorithm (QWPA) based on a new updating

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5327056 10 pageshttpsdoiorg10115520185327056

2 Mathematical Problems in Engineering

mechanism The probability position updating operation isemployed to make the population move to the global optimaAt the same time the collapse operationmaps the probabilityposition to a certain position to prevent the diversity of thepopulation The experimental results show the competitiveperformance of the proposed algorithm

The remainder of the paper is organized as follows InSection 2 the basic wolf pack algorithm and the binarywolf pack algorithm are introduced Section 3 proposesthe quantum wolf pack algorithm based on some relatedconcepts The steps of the proposed algorithm are listed inthis section Section 4 presents the experimental testing of thealgorithm and Section 5 concludes the paper

2 Related Work

21 Wolf Pack Algorithm (WPA) There are three roles inthe wolf pack algorithm [12] leader wolf scouting wolfand fierce wolf The leader wolf represents global optima inpopulation and guides other wolvesThe scouting wolf acts toimprove the randomicity of population by renovating aroundnoninferior solutions The fierce wolf which constitutes themain part of the populationmoves toward the global optimaThe leader wolf will be replaced by other wolves if thescouting or fierce wolf finds a better position In other wordsthe global optimal solution will be replaced if there is a bettersolution in the population

22 Binary Wolf Pack Algorithm The binary wolf packalgorithm (BWPA) can be used to solve knapsack prob-lems The position jth wolf can be described as Xj =[xj1 xj2 xji xjm] (j = 1 2 N i = 1 2 m) Wherem is the number of dimensions N is the number of thepopulation and xji isin 0 1

In BWAP the number of updated dimensions r isemployed to represent the distance between every twowolvesThere is an updated set Mrsquo which is the r-element-containingsubset of the feasible setM updating the position of eachwolfby reversing operation

A simple example is as followsXj = [1 0 0 1 0 0] M = 2 5 r = 1

(2)

The Mrsquo can be obtained from M Mrsquo=2 or Mrsquo=5 thenthe new position is

Xj = [1 1 0 1 0 0]or Xj = [1 0 0 1 1 0] (3)

In fact BWPA essentially adopts a bit updating mech-anism which may lead to confusion [11] in the computingprocess Instead the QWPA is proposed in this paper

3 Quantum Wolf Pack Algorithm (QWPA)

Quantum information and quantum computation processeswere extensively developed in the 1990s The most popular

quantum algorithms are the short large number decompo-sition algorithm [13 14] and Grover search algorithm [15]It is worth mentioning that the Grover algorithm can solvethe search problem of the scale of N in the case of the timecomplexity 119874(radic119873) In this section we propose the QWPAthrough several quantum concepts related to the Groveralgorithm

31 Related Definitions As described in the previous sectionthe position of each wolf is described as a vector Unlikea certain value (0 or 1) of each dimension in BWPA thepositions of wolves are uncertain in quantum theory Takingthe knapsack problem as an example the value of anydimension is not 0 or 1 but the probability superposition of0 and 1 The definitions what we have been used in QWPAare shown in Figure 1

As seen in Figure 1 the updating of quantum encoding issimilar to the hidden Markov process changing the impliedstates by state transitionmatrix (quantum-rotating gate) thenthe observation states are obtained by Confusion Matrix(collapse operation) In the following the related definitionsinvolved in Figure 1 will be elaborated

Definition 1 (probability position) In a finite set the positioncan be defined by the linear combinations of states

119909 = sum119904

120582119904 |119904 ⟩ |119904 ⟩ isin 119880 119904 = 1 2 sdot sdot sdot 119881 (4)

where U represents the finite set and |s⟩ is the element ofU |120582s|2 is the probability of obtaining |s⟩ and |120582s| isin [0 1]

In KP01 each item has only two states selected or not (1or 0)This character will bring it easy to describe the solutionsby 2-dimension quantum superposition state of 1 and 0Whatwe need to do is to control the probability of obtaining 1or 0 to update the superposition The ranges of variables inKP fit well with 2-dimension cases for updating on the onehand a simple linear transformation can be used to increase(or decrease) the probability of one state and decrease (orincrease) another one on the other hand the sum of the twoprobabilities is always kept as 1

For the jth wolf Xj in the knapsack problem the ithdimension xji can be formulated as xji = 1205820|0⟩+1205821|1⟩ (|1205820|2+|1205821|2 = 1)Definition 2 (position probability) The probability of obtain-ing a certain position is defined as the position probabilityFor example |120582s|2 is the position probability of obtaining |s⟩in formulation (4)

Definition 3 (certain position) The certain position refers tothe conventional position in Euclidean space In (4) each |s⟩is a certain position

Definition 4 (collapse) Themapping from the probable posi-tions to certain positions is defined as a collapse operation Aprobability position may map to multiple certain positions

Mathematical Problems in Engineering 3

Collapseoperation

Quantumrotating gate

[0000] [0001] [1111]

[0000] [0001] [1111]

Collapseoperation

Probabilityposition

Certainposition

Certainposition

Positionprobability

Positionprobability

c1 d1 c2 d2 ci di cm dm

c1 d1

c2 d2

ci di

cm dm

0 10 10 10 1

0 10 10 10 1

p1 =c1

c2middot middot middot cm

2 p2 =c 1

c2middot middot middot dm

2

p2 =d 1

d2middot middot middot dm

2

p1 =c 1 c2 middot middot middot cm

2 p2 =c 1 c2 middot middot middot dm

2 p2 =d 1d2 middot middot middot dm

2

x1Q x2

Q xiQ xm

Q

middot middot middot

middot middot middot middot middot middot


x1Q

x2Q

xiQ

xmQ






Figure 1 Related concepts

Definition 5 (quantum-rotating gate) The quantum-rotatinggate is the process of updating the probability positionOrthogonal transformation is usually used to change theprobability position One of the most popular transforma-tions is

119867 = [cos 120579 minus sin 120579sin 120579 cos 120579 ] (5)

where 120579 is the quantum rotation angle The probabilityposition is updated as follows

[119888119894119899119890119908119889119894119899119890119908] = 119867119894 [119888119894119889119894] = [

cos 120579119894 minus sin 120579119894sin 120579119894 cos 120579119894 ][

119888119894119889119894] (6)

Formulation (6) shows the updating process of eachdimension

The 2m possible combinations may appear if any ci =0 1 (i = 1 2m) Supposing that xi = ci|0⟩ + di|1⟩ is the ithdimension then119898prod119894=1

(119888119894 |0 ⟩ + 119889119894 |1 ⟩) = 11988811198882 119888119898 |00 0 ⟩+ 11988811198882 119889119898 |00 1 ⟩ + + 11988911198892 119889119898 |11 1 ⟩

(7)

where |c1c2 cm|2+|c1c2 dm|2+ +|d1d2 dm|2 = 1|c1c2 cm|2 |c1c2 dm|2 |d1d2 dm|2 represent theprobability of certain positions |00 0⟩ |00 1⟩ |11 1⟩ respectively

We can update the probability of 2m certain positions byadjustingm quantumpositions through the quantumparalleloperation

32 Rule Description of QWPA The concepts of probabilityposition and certain position are employed in QWPA Inknapsack problems the certain position of the jth wolf canbe defined as Xj = [xj1 xj2 xji xjm] where xji isin 0 1and the probability position can be formulated as

119883119895119876 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (8)

where |cji|2 + |dji|2 = 1 cji dji isin [0 1] |cji|2 and |dji|2respectively represent the probability of the ith dimensionobtaining 0 or 1 The value of each dimension is uncertain sothat the collapse operation is necessary to obtain the certainpositions

Suppose that the number of wolves in the population isP pop In knapsack problems a certain position represents asolution whose quality is determined by the value of fitnessfunction as follows

119891 (119909) =

119898sum119894=1

119901119894119909119895119894 119898sum119894=1

119908119894119909119895119894 le 119862119898sum119894=1

119901119894119909119895119894 minus119872 119898sum119894=1

119908119894119909119895119894 gt 119862(9)

where M is a large enough real number and xji is the ithdimension of the jth (j=12 P pop) wolf Here we adopted a


penalty function in (9) to ensure the solutions under volumeconstraint

There are three steps before updating the starts setthe probability positions of the wolf pack obtain certainpositions by applying the collapse operation and select theoptimal certain position as the position of the leader wolfin the population (The position of leader wolf is a certainposition and the other wolves can have either a probabilityposition or a certain position) As reported previously [16]the efficiency calculation is better if the number of scoutwolves and fiercewolves is as large as possible Here the num-ber of the above two kinds of wolves can each be set to N-1

321 Scout Behavior There are p certain positions of thejth wolf that can be obtained from the application of thecollapse operation according to its probability position (p isinH H = 1 2 h) Then the p certain positions of thejth wolf are used to determine whether or not to update thecertain position of the leader The process above is repeateduntil the jth wolf finds a certain position that is better than theleaderrsquos position or the number of scouting behavior exceedsthe limit

The value of h is a random integer between hmin and hmaxas defined in the literature [10]

322 Beleaguer Behavior Theposition of the leader wolf willbe extended to the probability position according to formula(10)

119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [10] 119909119897119890119886119889119890119903119894 = 0

119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [01] 119909119897119890119886119889119890119903119894 = 1

(10)

where i=12 m The Manhattan distance can be calcu-lated by the extended position and the jth wolf as follows

119889 (119883119897119890119886119889119890119903119876 119883119895119876) = 119898sum119894=1

10038161003816100381610038161003816119888119897119890119886119889119890119903119894119876 minus 11988811989511989411987610038161003816100381610038161003816 (11)

If the distance above is more than a threshold distancednear the jth wolf will approach the leader in a large step120579a otherwise it will use a small step 120579b 120582 is a randomnumber between 0 and 1 Extensive experiments were donein this study to determine the values of 120579a and 120579b for a high-dimension knapsack

120579119886 = 0002119898 times 120587120579119887 = (plusmn00001119898 times 120587) times 119896maxminus119896119896max

(12)

wherem is the dimension of the knapsack k is the currentiteration and kmax is the maximum iteration

323 Elimination Mechanism If the certain positions ob-tained by the probability position of a wolf in scout behaviorare always inferior to others in a cycle this probabilityposition will be eliminated Then a new probability positionwill randomly be obtained for this wolf

33 Procedure for QWPA Solving Knapsack Problem

Step 1 Parameter and the probability position of wolf packinitialization

Step 2 Obtain certain positions of wolves and determine theposition of the leader wolf

Step 3 Thepopulation enters the scout stage For each wolf pcertain positions are obtained and compared with that of theleader to determine whether to update the certain positionof the leader The procedure above is repeated Tmax times orwhen stopped by the conditions of termination

Step 4 The population enters the beleaguer stage The prob-ability positions of wolves are updated by (6) accordingto quantum-rotating angles which are determined by theManhattan distances and dnear

Step 5 Eliminate R probability positions and generate Rprobability positions to maintain the population diversity

Step 6 Repeat Step 3simStep 5 until the terminal condition issatisfied

34Theoretical Analysis of theAlgorithm AMarkov chain is aMarkov process with discrete parameters and state space setsThe procedure of QWPA is only related to the current stateand the parameter and state space sets are discrete So we canconclude that the population sequence is a Markov chain

Definition 6 If the limit of the transition probability matrix[9] of Markov chain exists and is unrelated to s the Markovchain is ergodic

lim119911rarrinfin

119901119904119905 (119911) = 119901119905 119904 119905 isin 119864 (13)

where E is the state space and z is the number of transitionsteps

Theorem 7 In a finite Markov chain if there is a positiveinteger v satisfying the condition

119901119904119905 (V) gt 0 119904 119905 = 1 2 (14)

Then the Markov chain is ergodic [17 18]

Proposition 8 QWPA is globally convergentAssuming the probability position of the jth wolf as formula

(8) then each dimension

1199091198951 = [11988811989511198891198951]

1199091198952 = [11988811989521198891198952] 119909119895119898 = [119888119895119898119889119895119898]

(15)

The probability positions are only updated in beleaguerbehavior The xji (i=12 m) will be updated in step 120579a or 120579bby (8) It is specified in definition 5 that the quantum-rotatinggates are orthogonal transformations


Table 1 The parameters of KP01

Number Dimension Constraint OptimaKP1 10 269 295KP2 15 35496 48169KP3 20 871 1024KP4 23 9768 9767KP5 50 1000 3103KP6 50 1000 3119KP7 50 11231 16102KP8 60 2393 8362KP9 80 1170 5183KP10 100 2818 15170

The beleaguer operation is Ha or Hb Then the updatingprocedure can be formulated as follows in a cycle

[119888119895119894119899119890119908119889119895119894119899119890119908] =

119867119886 [[119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) gt 119889119899119890119886119903119867119887 [[

119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) le 119889119899119890119886119903(16)

If 120579a 120579b = 0 the matrix H = Ha or H = Hb in whichwithout 0 element will continue to update To say this inanother way a matrix without 0 element exists when v=1 andsatisfies the condition of Theorem 7 Therefore the QWPA isergodic

Because the feasible solution of a knapsack is finite theQWPA could obtain the global optimal solution in infiniteiterations

The position of the leader wolf is saved in the nextiteration and when the global optimal solution is found theleader will not update

When the number of iteration kgeK (K is a large enoughpositive number) the leader will be constant The probabilitypositions of other wolves approach the leader by a quantum-rotating gate

lim119896=119896maxrarrinfin

11986711198672 119867119896 [119888119895119894119889119895119894] = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] (17)

where k = kmax gt KFormula (17) shows that the probability of the jth

(j=12 P pop) will eventually converge to the extendedposition of the leaderThe certain position of the jth can onlybe the position of leader

4 Experiments and Analyses

Two groups of data sets were employed to evaluate theperformance of QWPA to solve KP01 The first is ten classicsets of data described in the literature [19] and used to test theperformance in a simple situation There are 100 250 500

and 1000 dimension sets of data by formula (18) to test theperformance of QWPA in a high-dimension situation

119908119894 = rand int [1 10] 119901119894 = 119908119894 + 5119862 = (12)

119898sum119894=1

119908119894(18)

where wi is the volume and p119894 is the value of the ith itemC represents the constraint of volume and m represents thenumber of dimensions All experiments were conducted withMatlab2012 Core(TM)i7-479 CPU 360GHz processorand Windows 7 ultimate edition

Experiment 1 (study of ten classic KP01 problems) Theten classic knapsack problems are employed to test theperformance ofQWPAOther outcomes of typical algorithmssuch as the binary wolf pack algorithm (BWPA) geneticalgorithm (GA) harmony search algorithm (HS) and greedyalgorithm were compared The numbers of the dimensionof the ten problems range from 10 to 100 In QWPA andBWPA the number of iteration is 100 and the population sizeis 40 In order to obtain reliable results we ran the two abovealgorithms 20 timesThe parameters in QWPA are as follows120579a=02 120579b isin[0010015] dnear = 1 R=8 The parameters inBWPA are given in the literature [10] The related parametersof the ten classic knapsack problems are given in Table 1

Table 2 shows the test results of QWPA and BWAP Theresults of the best solution worst solution and average arelisted in rows 3-5 The standard deviation and the timesof getting the optimal results are showed in rows 6 and 7respectively The last row lists the results of other algorithmsas reported in the literature [10]

The QWPA shows very competitive results to those ofBWPAand other algorithms for the ten classic KP01 problemsas shown in Table 2 For KP1 to KP4 BWAP and QWPAobtained the optimal solutions with 100 probability It isimportant to note that BWPA and other algorithms wereunable to obtain the global optimal solutions for KP5 andKP6 butQWPAobtained the optimal solutions several timesHowever QWPA also became trapped in local optimum in


Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105

Differentevaluationbasedon

hybrid

encoding

3112GreedyGeneticalgorithm

3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020

14865Geneticalgorithm

15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15


Table 3 Parameters of BWPA and QWPA

Parameters of steps Common parametersQWPA 119889119899119890119886119903 = 2119879max = 10ℎ = fix(rand(1) times (10 minus 20) + 20)

119877 = fix(rand (1) times ( 1198992119887 minus 119899119887) + 119887119899)119887 = 3

120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1

the two sets of data especially in KP6 where QWPA onlyobtained the global optimum 7 times out of 20 attempts Incomparison QWPA showed better performance in stabilityand statistically For KP7-KP9 QWPA and BWPA behavedalmost the same and were obviously superior to other algo-rithms For KP10 although QWPA could obtain the optimalsolutions it was trapped in local optima several timesOverall from the analysis above wemay conclude that BWPAand QWPA perform obviously better than other algorithmsdescribed previously [10] and behave relatively the same

Experiment 2 (study of four high-dimension KP01) InExperiment 1 the results of KP10 may suggest that QWPAis not adapted to high-dimension situations To test thisExperiment 2 was designed Four high-dimension knapsackswith 100 250 500 and 1000 dimensions were generated byformula (18) We compared the performance of the quantumgenetic algorithm (QGA) the artificial fish algorithm (AF)BWPA and QWPA to solve the above high-dimensionknapsack problems The parameters of the four algorithmsare given as follows The population size is 40 and thenumber of iteration is 100 for all algorithms In QGA thelength of binary encoding is 20 and the size of quantum-rotating angles is 005120587 0025120587 001120587 and 0005120587 In AF theperception distance is 05 the crowding factor is 0618 andthe random selection time is 50 The parameters of BWPAand QWPA are shown in Table 3

Figures 2(a)ndash2(d) show the separation curve of thefour above algorithms in different dimensions The abscissarepresents the computation time and the ordinate representsthe calculated results All problems were calculated 20 timesfor each algorithm

The results of Figure 2 show that QWPA has the competi-tive performance to that of the other three algorithmsQWPAfound the global optima 618 for 100-dimension problemsall 20 times but the other algorithms did not do as wellAs the dimensions increased the difference in the qualityof results from the four algorithms increased continuouslyWhen the number of dimensions reached 1000 the best value(obtained by QWPA) was approximately 600 better than theworst value (obtained by BWPA) It is obvious that QWPAis well adapted for the solving of high-dimension knapsackproblems Another finding is that algorithms based on aquantum mechanism performed better in high-dimensionproblems It is shown in (a)sim(d) of Figure 2 that QGA and

QWPA are better for solving these problems This researchproblem will be investigated in further studies

In order to more completely analyze the performance ofQWPA Figure 3 shows the convergence curves of the fouralgorithms for the high-dimensional problems The data inFigure 3 are mean values of one hundred repeats of eachalgorithm

It is evident from Figure 3 that QWPA and QGA evolvemore quickly than AF and BWPA in high dimensions In500 and 1000 dimensions QWPA and QGA still have thepotential to evolve even after the 100th iteration We canconclude that algorithms based on a quantum mechanismwill preserve the diversity of population to avoid local optima

The diversity of the population is discussed in QWPAfor solving a 100-dimension knapsack problem Assumethe position of leader is Xleader=[0100110 ] which can beextended as

119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)

The other wolves approach the extended position by aquantum-rotating gate After multiple iterations the positionof the jth wolf is

119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)

where |cj1|2 asymp 099 |dj2|2 asymp 099 and so on This meansthat the probability of obtaining 0 in the first dimensionis approximately 099 the probability to obtain 1 in thesecond dimension is approximately 099 and so onThen theprobability of the jth wolf being the same as the leader is099100 asymp 0366 In other words although each dimensionof the probability position of the jth is very close to that ofthe leader the certain position of the jth is not the sameas the leader Thus the diversity of the population is bettermaintained in QWPA

5 Conclusion

A quantum wolf pack algorithm is proposed to solve KP01problems in this paper New concepts of probability positionand certain position are included in the proposed algorithmThe updating of the probability position plays a guidingrole and the collapse operation from probability to a certain


610

620Va

lues

600

590

580

570

560

550151050

Times of computing20

AFKPBWPA

QWPAQGA

(a) Separation curve of 100-dimension problem

Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA

(b) Separation curve of 250-dimension problem

Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA

(c) Separation curve of 500-dimension problem

6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA

(d) Separation curve of 1000-dimension problem

Figure 2 Separation curves of high-dimension problems

position is a random process in QWPA Ten classic and fourhigh-dimensional knapsack problems were employed to testthe performance of the proposed algorithmThe results showthe competitive performance of the proposed algorithm forKP01 problems especially for high-dimension cases

We are going to study the influence of parameters onthe algorithm in the future In addition we found that themethod of 2-dimension quantum encoding is well adapted

to knapsack problems and quantum mechanism can beapplied to other algorithms to solve different knapsack prob-lems

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper


Valu

es

590

600

610

620

580

570

560

550

540

Iteration400 100806020

QGAAF WPA

QWPA

(a) Convergence curve of 100-dimension problem

1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

(b) Convergence curve of 250-dimension problem

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600

(c) Convergence curve of 500-dimension problem

Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA

(d) Convergence curve of 1000-dimension problem

Figure 3 Convergence curves of high-dimension problems

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant 71601183

References

[1] H Kellerer U Pferschy and D Pisinger Knapsack ProblemsSpringer 2004

[2] X ZWang andY CHe ldquoEvolutionary algorithms for knapsackproblemsrdquo Journal of Software Ruanjian Xuebao vol 28 no 1pp 1ndash16 2017

[3] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing vol 11 no 2 pp 1556ndash1564 2011

[4] G L Chen X F Wang Z Q Zhuang and D S Wang Gene-tic Algorithm and Its Applications Beijing The posts and Tele-communications Press 2003


[5] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tionrdquo Swarm Intelligence vol 1 no 1 pp 33ndash57 2007

[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCombridge MA USA 2004

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization Artificial beecolony(ABC) algorithmrdquo Journal of Global Optimization vol39 no 3 pp 459ndash471 2007

[8] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of World Congress nature ampamp Biologically InspiredComputing India pp 210ndash214 USA IEEE Publications 2009

[9] H-S Wu F-M Zhang and L-S Wu ldquoNew swarm intelligencealgorithm-wolf pack algorithmrdquo Systems Engineering and Elec-tronics vol 35 no 11 pp 2430ndash2438 2013

[10] W Husheng Z Fengming and Z Renju ldquoA binary wolf packalgorithm for solving 0-1 knapsack problemrdquo Systems Engi-neering and Electronics vol 8 pp 1660ndash1667 2014

[11] M G Gong Q Cai X W Chen and L J Ma ldquoComplexnetwork clustering by multiobjective discrete particle swarmoptimization based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 18 no 1 pp 82ndash97 2014

[12] H-S Wu and F-M Zhang ldquoWolf pack algorithm for uncon-strained global optimizationrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 465082 17 pages 2014

[13] H-W Chen K Li and S-M Zhao ldquoQuantumwalk search algo-rithm based on phase matching and circuit cmplementationrdquoWuli XuebaoActa Physica Sinica vol 64 no 24 2015

[14] PW Shor ldquoPolynomial-time algorithms for prime factorizationand discrete logarithms on a quantum computerrdquo SIAM Journalon Computing vol 26 no 5 pp 1484ndash1509 1997

[15] K L Grover Proceedings of 28th ACM Symposium onTheory ofComputation Philadelphia USA 1996

[16] L Guoliang Research and application ofWolf Colony AlgorithmEast China University of Technology 2016

[17] R Wall An Introduction to Mathematical Statistics and ItsApplications Prentice-Hall 1986

[18] SM Ross Introduction to ProbabilityModels 10th edition 2011[19] L Juan F Ping and Z Ming ldquoA hybrid genetic algorithm for

knapsack problemrdquo Journal of Nanchang Institute of Aeronauti-cal Technology vol 3 pp 35ndash39 1998

Hindawiwwwhindawicom Volume 2018

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Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom


mechanism The probability position updating operation isemployed to make the population move to the global optimaAt the same time the collapse operationmaps the probabilityposition to a certain position to prevent the diversity of thepopulation The experimental results show the competitiveperformance of the proposed algorithm

The remainder of the paper is organized as follows InSection 2 the basic wolf pack algorithm and the binarywolf pack algorithm are introduced Section 3 proposesthe quantum wolf pack algorithm based on some relatedconcepts The steps of the proposed algorithm are listed inthis section Section 4 presents the experimental testing of thealgorithm and Section 5 concludes the paper

2 Related Work

21 Wolf Pack Algorithm (WPA) There are three roles inthe wolf pack algorithm [12] leader wolf scouting wolfand fierce wolf The leader wolf represents global optima inpopulation and guides other wolvesThe scouting wolf acts toimprove the randomicity of population by renovating aroundnoninferior solutions The fierce wolf which constitutes themain part of the populationmoves toward the global optimaThe leader wolf will be replaced by other wolves if thescouting or fierce wolf finds a better position In other wordsthe global optimal solution will be replaced if there is a bettersolution in the population

22 Binary Wolf Pack Algorithm The binary wolf packalgorithm (BWPA) can be used to solve knapsack prob-lems The position jth wolf can be described as Xj =[xj1 xj2 xji xjm] (j = 1 2 N i = 1 2 m) Wherem is the number of dimensions N is the number of thepopulation and xji isin 0 1

In BWAP the number of updated dimensions r isemployed to represent the distance between every twowolvesThere is an updated set Mrsquo which is the r-element-containingsubset of the feasible setM updating the position of eachwolfby reversing operation

A simple example is as followsXj = [1 0 0 1 0 0] M = 2 5 r = 1

(2)

The Mrsquo can be obtained from M Mrsquo=2 or Mrsquo=5 thenthe new position is

Xj = [1 1 0 1 0 0]or Xj = [1 0 0 1 1 0] (3)

In fact BWPA essentially adopts a bit updating mech-anism which may lead to confusion [11] in the computingprocess Instead the QWPA is proposed in this paper

3 Quantum Wolf Pack Algorithm (QWPA)

Quantum information and quantum computation processeswere extensively developed in the 1990s The most popular

quantum algorithms are the short large number decompo-sition algorithm [13 14] and Grover search algorithm [15]It is worth mentioning that the Grover algorithm can solvethe search problem of the scale of N in the case of the timecomplexity 119874(radic119873) In this section we propose the QWPAthrough several quantum concepts related to the Groveralgorithm

31 Related Definitions As described in the previous sectionthe position of each wolf is described as a vector Unlikea certain value (0 or 1) of each dimension in BWPA thepositions of wolves are uncertain in quantum theory Takingthe knapsack problem as an example the value of anydimension is not 0 or 1 but the probability superposition of0 and 1 The definitions what we have been used in QWPAare shown in Figure 1

As seen in Figure 1 the updating of quantum encoding issimilar to the hidden Markov process changing the impliedstates by state transitionmatrix (quantum-rotating gate) thenthe observation states are obtained by Confusion Matrix(collapse operation) In the following the related definitionsinvolved in Figure 1 will be elaborated

Definition 1 (probability position) In a finite set the positioncan be defined by the linear combinations of states

119909 = sum119904

120582119904 |119904 ⟩ |119904 ⟩ isin 119880 119904 = 1 2 sdot sdot sdot 119881 (4)

where U represents the finite set and |s⟩ is the element ofU |120582s|2 is the probability of obtaining |s⟩ and |120582s| isin [0 1]

In KP01 each item has only two states selected or not (1or 0)This character will bring it easy to describe the solutionsby 2-dimension quantum superposition state of 1 and 0Whatwe need to do is to control the probability of obtaining 1or 0 to update the superposition The ranges of variables inKP fit well with 2-dimension cases for updating on the onehand a simple linear transformation can be used to increase(or decrease) the probability of one state and decrease (orincrease) another one on the other hand the sum of the twoprobabilities is always kept as 1

For the jth wolf Xj in the knapsack problem the ithdimension xji can be formulated as xji = 1205820|0⟩+1205821|1⟩ (|1205820|2+|1205821|2 = 1)Definition 2 (position probability) The probability of obtain-ing a certain position is defined as the position probabilityFor example |120582s|2 is the position probability of obtaining |s⟩in formulation (4)

Definition 3 (certain position) The certain position refers tothe conventional position in Euclidean space In (4) each |s⟩is a certain position

Definition 4 (collapse) Themapping from the probable posi-tions to certain positions is defined as a collapse operation Aprobability position may map to multiple certain positions


Collapseoperation


[0000] [0001] [1111]

[0000] [0001] [1111]

Collapseoperation

Probabilityposition

Certainposition

Certainposition

Positionprobability

Positionprobability


c1 d1

c2 d2

ci di

cm dm

0 10 10 10 1

0 10 10 10 1

p1 =c1


2 p2 =c 1


2

p2 =d 1


2




2

x1Q x2

Q xiQ xm

Q




x1Q

x2Q

xiQ

xmQ










[119888119894119899119890119908119889119894119899119890119908] = 119867119894 [119888119894119889119894] = [


119888119894119889119894] (6)



(119888119894 |0 ⟩ + 119889119894 |1 ⟩) = 11988811198882 119888119898 |00 0 ⟩+ 11988811198882 119889119898 |00 1 ⟩ + + 11988911198892 119889119898 |11 1 ⟩

(7)




119883119895119876 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (8)



119891 (119909) =

119898sum119894=1

119901119894119909119895119894 119898sum119894=1

119908119894119909119895119894 le 119862119898sum119894=1

119901119894119909119895119894 minus119872 119898sum119894=1

119908119894119909119895119894 gt 119862(9)








119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [10] 119909119897119890119886119889119890119903119894 = 0

119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [01] 119909119897119890119886119889119890119903119894 = 1

(10)


119889 (119883119897119890119886119889119890119903119876 119883119895119876) = 119898sum119894=1

10038161003816100381610038161003816119888119897119890119886119889119890119903119894119876 minus 11988811989511989411987610038161003816100381610038161003816 (11)



(12)












lim119911rarrinfin

119901119904119905 (119911) = 119901119905 119904 119905 isin 119864 (13)



119901119904119905 (V) gt 0 119904 119905 = 1 2 (14)




1199091198951 = [11988811989511198891198951]

1199091198952 = [11988811989521198891198952] 119909119895119898 = [119888119895119898119889119895119898]

(15)






[119888119895119894119899119890119908119889119895119894119899119890119908] =

119867119886 [[119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) gt 119889119899119890119886119903119867119887 [[

119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) le 119889119899119890119886119903(16)






11986711198672 119867119896 [119888119895119894119889119895119894] = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] (17)






119908119894 = rand int [1 10] 119901119894 = 119908119894 + 5119862 = (12)

119898sum119894=1

119908119894(18)






Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105


hybrid

encoding


3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020


15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15





120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









Collapseoperation


[0000] [0001] [1111]

[0000] [0001] [1111]

Collapseoperation

Probabilityposition

Certainposition

Certainposition

Positionprobability

Positionprobability


c1 d1

c2 d2

ci di

cm dm

0 10 10 10 1

0 10 10 10 1

p1 =c1


2 p2 =c 1


2

p2 =d 1


2




2

x1Q x2

Q xiQ xm

Q




x1Q

x2Q

xiQ

xmQ










[119888119894119899119890119908119889119894119899119890119908] = 119867119894 [119888119894119889119894] = [


119888119894119889119894] (6)



(119888119894 |0 ⟩ + 119889119894 |1 ⟩) = 11988811198882 119888119898 |00 0 ⟩+ 11988811198882 119889119898 |00 1 ⟩ + + 11988911198892 119889119898 |11 1 ⟩

(7)




119883119895119876 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (8)



119891 (119909) =

119898sum119894=1

119901119894119909119895119894 119898sum119894=1

119908119894119909119895119894 le 119862119898sum119894=1

119901119894119909119895119894 minus119872 119898sum119894=1

119908119894119909119895119894 gt 119862(9)








119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [10] 119909119897119890119886119889119890119903119894 = 0

119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [01] 119909119897119890119886119889119890119903119894 = 1

(10)


119889 (119883119897119890119886119889119890119903119876 119883119895119876) = 119898sum119894=1

10038161003816100381610038161003816119888119897119890119886119889119890119903119894119876 minus 11988811989511989411987610038161003816100381610038161003816 (11)



(12)












lim119911rarrinfin

119901119904119905 (119911) = 119901119905 119904 119905 isin 119864 (13)



119901119904119905 (V) gt 0 119904 119905 = 1 2 (14)




1199091198951 = [11988811989511198891198951]

1199091198952 = [11988811989521198891198952] 119909119895119898 = [119888119895119898119889119895119898]

(15)






[119888119895119894119899119890119908119889119895119894119899119890119908] =

119867119886 [[119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) gt 119889119899119890119886119903119867119887 [[

119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) le 119889119899119890119886119903(16)






11986711198672 119867119896 [119888119895119894119889119895119894] = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] (17)






119908119894 = rand int [1 10] 119901119894 = 119908119894 + 5119862 = (12)

119898sum119894=1

119908119894(18)






Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105


hybrid

encoding


3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020


15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15





120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018














119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [10] 119909119897119890119886119889119890119903119894 = 0

119909119897119890119886119889119890119903119894119876 = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] = [01] 119909119897119890119886119889119890119903119894 = 1

(10)


119889 (119883119897119890119886119889119890119903119876 119883119895119876) = 119898sum119894=1

10038161003816100381610038161003816119888119897119890119886119889119890119903119894119876 minus 11988811989511989411987610038161003816100381610038161003816 (11)



(12)












lim119911rarrinfin

119901119904119905 (119911) = 119901119905 119904 119905 isin 119864 (13)



119901119904119905 (V) gt 0 119904 119905 = 1 2 (14)




1199091198951 = [11988811989511198891198951]

1199091198952 = [11988811989521198891198952] 119909119895119898 = [119888119895119898119889119895119898]

(15)






[119888119895119894119899119890119908119889119895119894119899119890119908] =

119867119886 [[119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) gt 119889119899119890119886119903119867119887 [[

119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) le 119889119899119890119886119903(16)






11986711198672 119867119896 [119888119895119894119889119895119894] = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] (17)






119908119894 = rand int [1 10] 119901119894 = 119908119894 + 5119862 = (12)

119898sum119894=1

119908119894(18)






Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105


hybrid

encoding


3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020


15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15





120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018












[119888119895119894119899119890119908119889119895119894119899119890119908] =

119867119886 [[119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) gt 119889119899119890119886119903119867119887 [[

119888119895119894119889119895119894]]

119889 (119883119897119890119886119889119890119903119876 119883119876) le 119889119899119890119886119903(16)






11986711198672 119867119896 [119888119895119894119889119895119894] = [119888119897119890119886119889119890119903119894119889119897119890119886119889119890119903119894] (17)






119908119894 = rand int [1 10] 119901119894 = 119908119894 + 5119862 = (12)

119898sum119894=1

119908119894(18)






Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105


hybrid

encoding


3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020


15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15





120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









Table2Th

eresultsof

theten

KP01

knapsack

prob

lems

Num

ber

Algorith

ms

Best

Worst

AVG

STD

Obtained

times

Results

from

theliterature

[10]

KP1

BWPA

295

295

295

020

295Geneticalgorithm

209

GreedyAlgorith

m

295Fu

zzyparticlesw

arm

optim

alalgorithm

QWPA

295

295

295

020

KP2

BWPA

4816

94816

94816

90

204816

9Ad

aptiv

eharmon

yalgorithm

QWPA

4816

94816

94816

90

20

KP3

BWPA

1024

1024

1024

020

1018

GreedyAlgorith

m1024Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP4

BWPA

9767

9767

9767

020

9757

Dminsio

nalityredu

ctionalgorithm

9767

Quantum

harm

onyalgorithm

QWPA

1024

1024

1024

020

KP5

BWPA

3096

3066

30805

817

03082

Simulated

annealingalgorithm

3090

Geneticalgorithm

basedon

simulated

annealing

QWPA

3103

3095

3101

323

16

KP6

BWPA

3104

3080

30928

727

03105


hybrid

encoding


3114

Learnedharm

onysearch

algorithm

QWPA

3119

3110

31163

303

7

KP7

BWPA

16102

10102

16102

020


15565

Binary

particlesw

arm

optim

alalgorithm

15955

Simulated

annealingalgorithm

QWPA

16102

16102

16102

020

KP8

BWPA

8362

8356

83614

185

177775

Geneticalgorithm

basedon

greedy

strategy

8362

Ant

colony

optim

izationalgorithm

with

scou

tsub

grou

pQWPA

8362

8362

8362

020

KP9

BWPA

5183

5183

5183

020

5107

Hybrid

particlesw

arm

algorithm

5101

Disc

reteparticlesw

arm

optim

izationalgorithm

QWPA

5183

5183

5183

020

KP10

BWPA

15170

15170

15170

020

15080

Hybrid

discreteparticlesw

arm

optim

izationalgorithm

15089-Disc

reteparticlesw

arm

optim

izationalgorithm

basedon

penalty

functio

nQWPA

15170

15144

151642

878

15





120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018












120579119886 = 02120587120579119887 = 0015120587BWPA119904119905119890119901119886 = 2119904119905119890119901119887 = 2119904119905119890119901119888 = 1









119883119897119890119886119889119890119903 = [1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] (19)


119883119895 = [1198881198951 1198881198952 119888119895119894 1198881198951198981198891198951 1198891198952 119889119895119894 119889119895119898] (20)


5 Conclusion



610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









610

620Va

lues

600

590

580

570

560

550151050


AFKPBWPA

QWPAQGA


Valu

es

151050


1520

1500

1480

1460

1440

1400

1420

1380

1360

1340

AFKPBWPA

QWPAQGA


Valu

es

151050


3000

2950

2900

2850

2800

2750

2700

2650

2600

AFKPBWPA

QWPAQGA


6000

5900

5800

5700

5600

5500

5400

5300

Valu

es

151050


AFKPBWPA

QWPAQGA









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









Valu

es

590

600

610

620

580

570

560

550

540


QGAAF WPA

QWPA


1550

1500

1450

1400

1350

1300

1250

Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA


Iteration

Valu

es

400 100806020

QGAAF WPA

QWPA

3050

3000

2950

2900

2850

2800

2750

2700

2650

2600


Valu

es

5900

6000

5800

5700

5600

5500

5400

5300


QGAAF WPA

QWPA



Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018































Journal of












Journal of







Volume 2018






Volume 2018








quantum-inspired wolf pack algorithm to solve the 0-1...

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