New binary bat algorithm for solving 0–1 knapsack problem · portfolio optimization, investment decision-making, project selection, resource distribution, and so on. Unfortunately,

Complex Intell. Syst. (2018) 4:31–53https://doi.org/10.1007/s40747-017-0050-z

ORIGINAL ARTICLE

New binary bat algorithm for solving 0–1 knapsack problem

Rizk M. Rizk-Allah1,3 · Aboul Ella Hassanien2,3

Received: 3 April 2017 / Accepted: 11 July 2017 / Published online: 5 August 2017© The Author(s) 2017. This article is an open access publication

Abstract This paper presents a novel binary bat algorithm(NBBA) to solve 0–1knapsackproblems.Theproposed algo-rithm combines two important phases: binary bat algorithm(BBA) and local search scheme (LSS). The bat algorithmenables the bats to enhance the exploration capability whileLSS aims to boost the exploitation tendencies and, therefore,it can prevent the BBA–LSS from the entrapment in the localoptima. Moreover, the LSS starts its search from BBA foundso far. By this methodology, the BBA–LSS enhances thediversity of bats and improves the convergence performance.The proposed algorithm is tested on different size instancesfrom the literature. Computational experiments show that theBBA–LSS can be promise alternative for solving large-scale0–1 knapsack problems.

Keywords Bat algorithm · Local search scheme · Knapsackproblem

Introduction

Knapsack problem (KP) is one of the most important prob-lems in the combinatorial optimization. It appears in abroadvariety of applications, including scheduling problems,

B Aboul Ella [email protected]://www.egyptscience.net

1 Department of Basic Engineering Science, Faculty ofEngineering, Menoufia University, Shebin El-Kom, Egypt

2 Faculty of Computers and Information, Cairo University,Giza, Egypt

3 Scientific Research Group in Egypt, Cairo, Egypt

portfolio optimization, investment decision-making, projectselection, resource distribution, and so on. Unfortunately,KP is non-polynomial (NP) hard, the complete problem [1].Thus, solving this problem using the gradient methods isinappropriate because this problem may fall in local optimafor large-scale problems. Also, these methods are time-consuming, and they achieve one of the closest local optimato initial random solution. Meanwhile, the metaheuristicalgorithms have the ability to overcome these drawbacks andproved to be a robust alternative to solve complex optimiza-tion problems.

Recently, metaheuristic algorithms are one of the signif-icant stochastic research topics in optimization that imitatenatural phenomena. The features of the metaheuristic algo-rithms are the avoidance of local optima; generate multiplesolutions for each run which assist to produce good-qualitysolutions quickly and no dependence on derivative informa-tion [2].

In recent decades, there have been extensive works basedon metaheuristic algorithms to solve 0–1 KP. Liu and Liu[3] introduced an evolutionary algorithm based on schema-guiding to solve 0–1 KP. Martello et al. [4] proposed asurvey of different approaches to solving 0–1 KP. Shi [5]proposed a modified version of the ant colony optimiza-tion (ACO) to solve 0–1 KP. Lin [6] solved the KP inthe fuzzy environment through imprecise weight using agenetic algorithm (GA). Li and Li [7] presented a binaryparticle swarm optimization using a multi-mutation mech-anism to solve KP. Zhang et al. [8] introduced amoeboidorganism algorithm to solve 0–1 KP. Bhattacharjee andSarmah [9] proposed a shuffled frog-leaping algorithm tosolve 0–1 KP. Kulkarni and Shabir [10] proposed Cohortintelligence algorithm for solving 0–1 KP. In addition, manyalgorithms have been flourished for solving the 0–1 KPsuch as genetic algorithm (GA), particle swarm optimiza-

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32 Complex Intell. Syst. (2018) 4:31–53

tion (PSO), artificial fish-swarmalgorithm (AFSA), harmonysearch algorithm (HS), chemical reaction optimization basedon greedy strategy (CROG), genetic mutation bat algorithm(GMBA), monarch butterfly optimization and hybrid cuckoosearch based on harmony search [11–19]. Owing to theimportance of the knapsack problem in the academic areaand practical applications, developing new algorithms withmore promising performance to solve large-scale types of theknapsack problem applications undoubtedly becomes a truechallenge.

Bat algorithm (BA) is one of the recent metaheuristicalgorithms that are inspired by the echolocation behaviorof micro-bats [20]. During flying, bats emit short and ultra-sonic pulses to the environment and record their echoes. Therecorded information from the echoes helps the bats to buildan airtight image of their surroundings and locate preciselythe distance, shapes and prey’s position. The ability of suchecholocation of micro-bats is charming, as these bats canfind their prey and distinguish different types of insects evenin complete darkness [20]. The earlier applications showedthat BA could solve different optimization problems andproved that its efficiency and robustness compared to dif-ferent algorithms such as GA and PSO [20–22]. A new trendin bat algorithms is focusing on hybridizing BA with differ-ent strategies [23–31]. Fister et al. [23] developed a hybridBA based on various evolution strategies for solving opti-mization tasks, while Baziar et al. [24] proposed a modifiedBA based on adaptive self-strategy. A hybrid BA based onharmony search for solving optimization problems was pro-posed by Wang and Guo [25]. Yilmaz and Kucuksille [26]developed an improved BA using some modifications, whileWang et al. [27] presented a modified BA through adjustingthe flight speed and the flight direction adaptively. Fister et al.[28] introduced a new version of BA based on self-adaptationof control parameters. Further, binary versions of BA weredeveloped in [29–31]. Mirjalili et al. [29] introduced a binaryversion of BA by employing a V-shaped transfer functionto overcome the drawback of the sigmoid transfer functionwhich keeps the positions unchanged during the iterationsof the algorithm. In [30], authors developed an integratingversion of the binary BA based on Naïve Bayes classifier forfeature selection problem. In [31], a binary vision of BA isestablished based on the sigmoid transfer function for solvingdifferent optimization problems. Due to continuous natureof BA, it is still in its infancy for solving combinatorial opti-mization problems, so this is also the motivation behind thisstudy.

This paper is motivated by several features. First, incor-porating the rough set with bat algorithm to solve large-scale0–1 KP has not been yet studied. Second, many optimiza-tion algorithms suffer from entanglement in local optimawhen solving large-scale problems. Last, solving large-scaleknapsack problems have not received adequate attention yet.

Hence solving large-scale knapsack problems to optimalityundoubtedly becomes a true challenge.

In this paper, we propose a novel binary bat algorithm(NBBA) to solve 0–1 knapsack problems. In contrast to thebinary version of BA in [29], the multi-V-shaped transferfunction for generating the solutions, the inclusion of therough set scheme (RSS) as a local search strategy (LSS)and updating the solution through one-to-one strategy areintroduced. The proposed algorithm combines two impor-tant phases: binary bat algorithm (BBA) and local searchscheme (LSS). The bat algorithm enables the bats to enhanceexploration capability while LSS aims to boost the exploita-tion tendency and, therefore, it can prevent the BBA–LSSfrom the entrapment in the local optima. Moreover, the LSSstarts its search from BBA found so far. By this method-ology, the BBA–LSS enhances the diversity of bats andimproves the convergence performance. The proposed algo-rithm is tested on different size instances from the literature.Computational experiments show that the BBA–LSS canpromise alternative for solving large-scale 0–1 knapsackproblems.

The main contributions of this approach are to (1) intro-duce a novel binary bat algorithm (NBBA) for solvinglarge-scale 0–1 knapsack problems, (2) integrate intelligentlythe merits of two phases, namely binary bat algorithm (BBA)and rough set scheme (RSS) as a local search scheme, so itcan avoid the sucking in the local optima, (3) improve theexploration capabilities of the BBA phase to seek the overallsearch space while incorporating RSS phase as a counter-part to enhance the exploitation tendencies, (4) implementthe injective (one-to-one) strategy for updating mechanismbetween the two phases such that the fit ones among twophases replace the worst ones based on feasibility rule and(5) to integrate BBA and RSS to improve the quality ofsolutions and speed up the convergence to the global solu-tion.

On the other hand, the proposed algorithm is effectivelyapplied for small- , medium- and large-size problems. Theexperimental results demonstrated the superiority of the pro-posed algorithm in achieving a high quality of solutions. Thesimulation results affirm that the application of RSS may bean effective scheme to improve the performances of opti-mization algorithms.

The novelty of the proposed approach is cleared regardingproposing the multi-V-shaped transfer function for generat-ing the solutions in the BBA phase, and then this can providemore explorations in the search space. Further, adopting theRSS as a local search scheme and introducing the injective(One-to-One) strategy can pick the fit solutions quickly andavoid the running of the algorithm without any improvementin the solutions.

The rest of this paper is organized as follows: In Sect. 2,we describe the preliminaries of the 0–1 knapsack problems.

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Complex Intell. Syst. (2018) 4:31–53 33

In Sect. 3, the basics of both BA and rough set theory (RST)are reviewed. The proposed algorithm is explained in detailin Sect. 4. The numerical experiments are given in Sect. 5to show the superiority of the proposed algorithm. Section 6gives the conclusions and the further work.

Preliminaries

Problem description

There are N items and the knapsack capacity is C · w j is theweight of the j th item, p j is the profit of the j th item. Thensolve which items are let into the knapsack to make the totalweight of the items no more than capacity of the knapsackand get the maximum total of the profit.

Mathematical description

The mathematical description of the 0–1 knapsack problemcan be formulated as follows:0–1 KP:

Max f (x) =N∑

j=1

p j x j ,

s.t. :⎧⎨

⎩

∑Nj=1 w j x j ≤ C,

x j = {0, 1}, j = 1, 2, . . . , Np j > 0, w j ≥ 0, C > 0

⎫⎬

⎭ (1)

The binary decision variables x j are used to determinewhether the item j is put in the knapsack or not.

In large-scale instances, the total weights of the items thatcan be packed in the knapsack may violate the constraint,and this violation is unacceptable and must be handled. Theprominent way to handle the constraint is the penalty func-tion method. It imposes the penalty on unfeasible solutionsand, therefore, it can evolve the unfeasible solutions untilthey move to candidate feasible regions. By use of penaltyfunction, the 0–1 KP can be reformulated as follows:0–1 KP:

Max f (x) =N∑

j=1

p j x j − λ

⎛

⎜⎝max

⎧⎨

⎩0,N∑

j=1

w j x j − C

⎫⎬

⎭

2⎞

⎟⎠

(2)

s.t. :{x j = {0, 1}, j = 1, 2, . . . , Np j > 0, w j ≥ 0,C > 0,

}

where λ defines the penalty coefficient where it is set to 1010

for all test instances.

Overview of bat algorithm (BA) and rough settheory (RST)

This section is devoted to describing the basics of bat algo-rithm (BA) and rough set theory (RST).

Real behavior algorithm

Bat algorithmwas established based on echolocation processof bats. In the echolocation process, pulses will be created bybats which are alive for 8–10 ms at a constant frequency andcorresponding wavelength as given in Fig. 1. The features ofbatswhich are exhibited for the development of bat algorithmare as follows: (i) Even without visibility, bats can sense andestimate the distance between food and the obstacles behindthem, (ii) the bats are associated with velocity, position, fixedfrequency, varying loudness and wavelength when they startflying to find their food and (iii)many strategies are attributedto change in values of loudness from a small constant valueto a maximum positive value.

Bat algorithm (BA)

Velocity and position

BA starts with the random initial population of bats in an-dimensional search space where the position of the bati denoted by xti and its velocity denoted by vti at time t .Therefore, the new positions xt+1

i and new velocities vt+1i at

time step t + 1 can be determined by

αi = αmin + (αmax − αmin)β (3)

vt+1i = vti + (xti − xbest)α (4)

xt+1i = xti + vt+1

i , (5)

Fig. 1 Real behavior of bats

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where β is a random number in [0, 1] and xbest representsthe current global optimal solution. αi represents the pulsefrequency emitted by bat iat the current moment, and αmin

and αmax represent the minimum and maximum values ofpulse frequency, respectively. Initially, the pulse frequencyis assigned randomly for each bat which is elected uniformlyfrom [αmax, αmin].

In this scenario, a bat is chosen randomly from the batpopulation, and then the corresponding position of this batis updated according to Eq. (6). This random walk can becomprehended as a process of local search that generates anew solution by the selected solution.

xnew = xold + εAt (6)

where xold represents a random solution chosen from thecurrent best solutions, At is the loudness and ε is a randomvector that is drawn from [−1, 1].

Loudness and pulse emission

It is worth noting that loudness (A(i)) and pulse rate (r(i))are responsible for balancing the combination between thelocal and global moves, where the loudness is strong, andpulse emission is small at the beginning of the search pro-cess. Once the bat has got its prey, the loudness decreaseswhile pulse emission gradually increases. A(i) And r(i) areupdated according to Eqs. (7) and (8):

r t+1(i) = r0(i) × [1 − e−γ t ] (7)

At+1(i) = δAt (i), (8)

where both δ and γ are constants. A(i) = 0 means that thebat has just found its prey and temporarily stopped emittingany sound. For any 0 < δ < 1 and γ > 0, we have

At (i) → 0, r t (i) → r0(i), as t → ∞ (9)

The implementation steps of bat algorithm

Step 1: Set the basic parameters: population size (PS), attenu-ation coefficient of loudness δ, increasing coefficient of pulseemission γ , the maximum loudness A0 and maximum pulseemission r0 and the maximum number of iterations T .

Step 2: Define objective function f (xi ), i = 1, 2, . . . ,PS.

Step 3: Initialize pulse frequency αi ∈ [αmin, αmax];Step 4: Initialize the bat population x and v.

Step 5: Start the main loop. If rand < ri , generate newsolutions by updating process for both velocity and currentposition by using Eqs. (4) and (5). Otherwise, generate newposition of bat by making a random disturbance, and go tostep 5.

Step 6: If rand < Ai and f (xi ) < f (xbest), accept the newsolutions and fly to the new position.

Step 7: If f (xi ) < fmin, replace the best bat and adjustA(i)and r(i)according to Eqs. (7) and (8).

Step 8: Evaluate the bat population, and return the best batand its position.

Step 9: If the termination condition is met (i.e., satisfy thesearch accuracy condition or reach a maximum number ofiterations), go to step 10; else, go to step 5, and perform thenext search.

Step 10: Get the output (i.e., global solution and the bestfitness).where, rand is a uniform distribution in [0, 1].

Rough set theory (RST)

The fundamental concept of the RST is the indiscernibilityrelation, which is produced by the information of interestedobjects [32].Because of discerningknowledge is lacking, onecannot identify some objects based on the available informa-tion. The indiscernibility relation relies on the granules ofindiscernible objects as a fundamental basis. Some relevantconcepts of the RST are as follows [32,33]:

Definition 1 (Information system) An information system(IS) is denoted as a triplet T = (U, A, f ), whereU is a non-empty finite set of objects and A is a non-empty finite set ofattributes. An information function f maps an object to itsattribute, i.e., fa : U → Va for every a ∈ A, where Va isthe value set for attribute a. A posteriori knowledge (denotedby d) is denoted by one distinguished attributed. A decisionsystem is an IS with the form DT = (U, A ∪ {d}, f ), whered /∈ A is used as supervised learning. The elements of A arecalled conditional attributes.

Definition 2 (Indiscernibility) For an attribute set B ⊆A, the equivalence relation induced by B is called aB-indiscernibility relation, i.e., INDT(B) = {(x, y) ∈U 2|∀a ∈ B, fa(x) = fa(y)} The equivalence classes ofthe B-indiscernibility relation are denoted as IB(x).

Definition 3 (Set approximation) Let X ⊆ U and B ⊆A in an IS, the B-lower approximation of X is the setof objects that belongs to X with certainty, i.e., BX ={x ∈ U |IB(x) ⊆ X}. The B-upper approximation is theset of objects that possibly belongs to X , where B̄ X ={x ∈ U |IB(x) ∩ X �= φ}.Definition 4 (Reducts) If X1

DT, X2DT, . . . , Xr

DT are the deci-sion classes of DT, the set POSB(d) = BX1 ∪ BX2 ∪ · · · ∪BXr is the B-positive region of DT. A subset B ⊆ A is a setof relative reducts of DT if and only if POSB(d) = POSC (d)

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NEGB(X)

BNB(X)

POSB(X)

Fig. 2 Definitions regarding rough set approximations

and POSB−{b}(d) �= POSC (d),∀b ∈ B. In the same wayPOSB(X), BNB(X) and NEGB(X) are defined below (referto Fig. 2).

• POSB(X) = BX ⇒ certainly member of X• NEGB(X) = U − B̄ X ⇒ certainly nonmember of X• BNB(X) = B̄ X − BX ⇒ possibly the member of X.

The proposed algorithm (IBBA-RSS)

In this section, we present the injective binary bat algorithmbased rough set scheme (IBBA-RSS) to solve the KP. Differ-ent from the conventional BA, first, a discrete binary stringis adopted to represent a solution; second the updating pro-cess of position using Eq. (4) cannot be used to handle thebinary space directly; therefore, a new transfer function isintroduced to map velocity values to probability values forupdating process of the position; third the RSS is adoptedto exploit the neighborhood in search process; fourth, afterthe binary BA procedures, the updating mechanism is imple-mented based on the injective (one-to-one) strategy, wherethe fit one replaces the worst one based on feasibility rule. Bythis methodology, the IBBA-RSS enhances the diversity ofbats and improves the convergence performance. The detailsof the proposed algorithm are given below.

Binary position scheme

In this step, each bat of the population is a solution to theKP, where each bat is represented by the n-bit binary string,where n is the number of decision variables (items) in theKP. For example, considering that xi represents the bat bits,then its j th bit xi j = (xi1, xi2, . . . , xin) is a binary variable,0 or 1.

Binary velocity scheme

In bat algorithm, the velocity of the bat is responsible forupdating the position. To update the position, the transferfunction is introduced to force bat to fly in a binary space.

-8 -6 -4 -2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity range

Sig.

tran

sfer

func

tion

Fig. 3 Sigmoid transfer function [34]

Therefore, the transfer function is responsible for the switch-ing between “0” and “1” values. The traditional transferfunction that has been used for binary particle swarm opti-mization (PSO) is defined as Eq. (10) and Fig. 3 [34].

Sig(vki (t)) = 1

1 + e−vki (t), (10)

where Sig denotes the Sigmoid transfer function and vki (t)denotes the velocity of the bat i in kth dimension at itera-tion t . After calculating the transfer function values, the newposition updating equation is necessary to update particles’position as follows [34]:

xki (t + 1) ={0 if rand < S(vki (t + 1))1 if rand ≥ S(vki (t + 1)),

(11)

where xki (t) indicates the position and vki (t) indicates thevelocity of i th the bat at iteration t in kth dimension. But thedrawback of the sigmoid transfer function is that the parti-cles’ positions remain unchanged when their velocity valuesincrease. To overcome this drawback, amulti V-shaped trans-fer function (see Fig. 4) is introduced to oblige the batswith high velocity to diverse their positions. A multi V-shaped transfer function and the new position are stated asin Eqs. (12) and (13), respectively.

V (vki (t)) =∣∣∣∣2

πarctan

(π

2vki (t)

)∣∣∣∣Q

(12)

xki (t + 1) ={

(xki (t))′ if rand < V (vki (t + 1))

xki (t) if rand ≥ V (vki (t + 1)),(13)

where Q = 0.1 ∼ 3, xki (t) is the position and vki (t) is thevelocity of the i th bat at iteration t in kth dimension and(xki (t))

′ is the complement of xki (t).

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-8 -6 -4 -2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity range

V- tr

ansf

er fu

nctio

n

Q=0.1Q=0.5Q=1Q=2Q=3

Fig. 4 Proposed multi V-shaped transfer function

Evaluation

The distance estimation of the bat is related to objectivefunction (fitness function). In this step, the fitness functionis evaluated based on finding the maximum profit as in theEq. (2). Therefore, the best estimation sourcewith the highestfitness xbest is determined as follows:

xbest = arg(Max{ f (xi )}PSi=1) (14)

Rough set scheme (RSS)

In this step, the RSS is introduced to reduce the redundantbits. In this regard, the obtained population is assumed as aninformation system consisting of bats’ solutions where eachbat is represented by a set of condition attributes and onedecision attribute. For the bat i, xi j the condition attributeillustrates the selected item j , and the decision attributedemonstrates the feasibility of this bat. The term feasibilitymeans that the candidate bat satisfies the knapsack capac-ity. When the candidate bat is feasible, the decision attributetakes one value; otherwise it takes 0 value.After that, all solu-tions are formulating as augmented matrix consisting of thecondition and decision attributes [xi1, xi2, . . . , xin |{D}]PSi=1,where D denotes decision attribute that takes 1 or 0 value.Therefore, D splits the population into two classes: mem-bers that picked value of one in D and members thatpicked value of zero in D. Let U be the set of objects(solutions) and X ⊆ U that contains the one valuesof D and B = {x1, x2, . . . , xn} is the set of conditionattribute in an IS. Then according to Definition 4, the redun-dant items are eliminated where BX, B̄ X BNB(X) andNEGB(X) of X are obtained based on the process of attributereduction.

Afterward, the population is updated by putting one valuefor items that belong to the BX , picking zero value for

Fig. 5 An illustration of mutation process

items that included in NEGB(X) and picking random valuewhether 0 or 1 for items that included in BNB(X). If wehave an empty BNB(X) or zero values for D, then muta-tion strategy can be implemented by generated N neighborsaround each solution as follows: for each string generatedN -solutions randomly, K bits are selected randomly from xiof bat i ; then the selected K bits are inverted. Figure 5 illus-trates an example for this step with N = 3 and B = 2, andthe colored bits denote the selected ones.

Injective updating based on feasibility rule

Now the feasibility rule must be carried out to obtain a newpopulation. In this step, the population is updated basedon injective scheme (one-to-one), where the solution afterimplementing the rough set scheme is compared with cor-responding one obtained by the bat procedures. The winnersolution is selected for updated process based on the feasi-bility rule that was introduced by Deb [35]. These rules aredefined as follows:

1. Through comparing two feasible solutions, the chosen isthe one that has a better objective.

2. Through comparing a feasible and an infeasible solution,the chosen is the feasible one.

3. Through comparing two infeasible solutions, the chosenis the one with the lower sum of constraint violation.

The sum of constraint violation for a solution given by

CV(x) = max

⎛

⎝N∑

j=1

w j x j − C, 0

⎞

⎠ (15)

The flowchart of the proposed IBBA-RSS algorithm is shownin Fig. 6, where the changes are highlighted.

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No

Mutation strategy

Replace the items of BX with 1, ( )BNEG X with 0, ( )BBN X with (0 or 1), values

No

Yes

Yes No

Yes

Initialization: (0,1)ix rand= ,

0iv i= ∀ and initialize α

Select a solution among the best solutions and change some dimensions of solution randomly

End

Start

No

Initialize pulse rates ir and

loudness iA

> irand r

Generate a new solution by flying randomly

Evaluate the fitness of all bats

Accept the new solutions and then increase ir and

reduce iA

irand A< &( ) ( )best

if x f x<

Rank the bats and update global best solution

Population of bats (POP)

Is the iteration satisfied?

Calculate ( )if x and the

average ( avef ) for all solutions

Update v and adjust α

Calculate transfer function value using equation (12)

Update positions using equation (13)

If ( )i avef x f< then D=1 else D=2 (for feasible solutions) and

D=0 for unfeasible solutions

Formulate the information system matrix ([POP. D])

Select some knowledge randomly from D as X D⊆

Find , ,BX BX( ), ( )B BBN X NEG X

( )BBN X = φ

Population after rough set phase

Injective strategy based on feasibility rule

Survival Population

Return the global best solution

Yes

Fig. 6 Flowchart of the proposed IBBA-RSS approach

Experimental results and analysis

In this section, the performance of the IBBA-RSS algorithmis extensively investigated by a large number of experimentalstudies. Ten low-dimensional, ten medium size and twelvelarge-scale instances are considered to validate the robust-ness of the proposed IBBA-RSS algorithm. The algorithm iscoded in MATLAB 7, running on a computer with an Intel

Core I 5 (1.8 GHz) processor and 4 GB RAM memory andWindows XP operating system.

Low-dimensional 0–1 knapsack problems

In this section, the performance of proposed algorithm isinvestigated to solve ten low-dimensional 0–1 knapsackproblems, where these instances are taken from [36,37]. The

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Table 1 The parameters,dimension and optimum of tentest instances

Problem Parameter Dimension Optimum

KP1 w = (95, 4, 60, 32, 23, 72, 80, 62, 65, 46), C = 269,p = (55, 10, 47, 5, 4, 50, 8, 61, 85, 87)

10 295

KP2 w = (92, 4, 43, 83, 84, 68, 92, 82, 6, 44, 32, 18, 56, 83, 25, 96,70, 48, 14, 58), C = 878, p = (44, 46, 90, 72, 91, 40, 75, 35,8, 54, 78, 40, 77, 15, 61, 17, 75, 29, 75, 63)

20 1024

KP3 w = (6, 5, 9, 7), C = 20, p = (9, 11, 13, 15) 4 35

KP4 w = (2, 4, 6, 7), C = 11, p = (6, 10, 12, 13) 4 23

KP5 w = (56.358531, 80.874050, 47.987304, 89.596240,74.660482, 85.894345, 51.353496, 1.498459, 36.445204,16.589862, 44.569231, 0.466933, 37.788018, 57.118442,60.716575), C = 375, p = (0.125126, 19.330424,58.500931, 35.029145, 82.284005, 17.410810, 71.050142,30.399487, 9.140294, 14.731285, 98.852504, 11.908322,0.891140, 53.166295, 60.176397)

15 481.07

KP6 w = (30, 25, 20, 18, 17, 11, 5, 2, 1, 1), C = 60,p = (20, 18, 17, 15, 15, 10, 5, 3, 1, 1)

10 52

KP7 w = (31, 10, 20, 19, 4, 3, 6), C = 50,p = (70, 20, 39, 37, 7, 5, 10)

7 107

KP8 w = (983, 982, 981, 980, 979, 978, 488, 976, 972, 486, 486,972, 972, 485, 485, 969, 966, 483, 964, 963, 961, 958, 959),C = 10,000, p = (981, 980, 979, 978, 977, 976, 487, 974,970, 485, 485, 970, 970, 484, 484, 976, 974, 482, 962, 961,959, 958, 857)

23 9767

KP9 w = (15, 20, 17, 8, 31), C = 80, p = (33, 24, 36, 37, 12) 5 130

KP10 w = (84, 83, 43, 4, 44, 6, 82, 92, 25, 83, 56, 18, 58, 14, 48, 70,96, 32, 68, 92), C = 879, p = (91, 72, 90, 46, 55, 8, 35, 75,61, 15, 77, 40, 63, 75, 29, 75, 17, 78, 40, 44)

20 1025

required information about test instances such as dimensionand parameters is listed in Table 1. The maximum numberof iterations is set to 400 iterations for each instance with30 bats for the population size, where each instance is testedwith 30 independent algorithm runs. To completely evalu-ate the IBBA-RSS performance statistical measures such assuccess rate (SR) among all runs in reaching the appointedOptima, “Best”, “Median”, “Worst”, “Mean” and standarddivision (Std.) are calculated.

On the other hand, the performance of the proposed IBBA-RSS algorithm is compared with six different algorithmsthat are reported in [37]: NGHS1 [36], SBHS [37], BHS[38], DBHS [39], ABHS [40] and ABHS1 [41]. Table 2shows the comparisons between the proposed algorithm andsix algorithms, where best results are highlighted in bold.The obtained results showed that the proposed algorithmcould achieve the optima for the low-dimensional knap-sack problems, where the proposed IBBA-RSS algorithm iscompetitive with SBHS, ABHS and DBHS and outperformsBHS, NGHS1 and ABHS1.

Medium size 0–1 knapsack problems

This section is devoted to investigate the performance of theproposed algorithm to solvemedium size 0–1 knapsack prob-

lems. Ten instances are taken from [42], where sizes of theseinstances include 30, 35, 40, 45, 50, 55, 60, 65, 70 and 75items. The information about these instances such as dimen-sion, parameters and optimum solution are listed in Table 3.Extensive experimental tests were carried out to adjust themaximum numbers of iterations. Based upon these tests, themaximum numbers of iterations are accordingly set to 400iterations for KP10–KP15 and 500 iterations for KP16–KP20.The proposed IBBA-RSS is run 30 times for each instancewith 30 bats for the population size.

To demonstrate the effectiveness and robustness of theproposed IBBA-RSS, it is implemented and compared withthe BBA phase. The statistical measures for the each instanceis obtainedusingBBAand IBBA-RSSand reported inTable 4where best results are highlighted in bold. The statisticalmeasures such as the best, median, worst, mean values andstandard deviations are determined.

The proposed IBBA-RSS is compared with BBA, CohortIntelligence (CI) and Branch and Bound method (B&B) asin Table 4. From these Tables, we can see that the proposedIBBA-RSS is statistically superior to other algorithms for themost KP instances and similar for the some KP instances.It can be perceived from Table 4 that the proposed IBBA-RSS is competent to obtain very competitive solutions withother algorithms. For KP15, KP16, KP18, KP19 and KP20

123


Table 2 Comparisons of the small sizes KP

KP1 KP2 KP3 KP4 KP5 KP6 KP7 KP8 KP9 KP10 TSR

BHS

SR 0.78 0.92 0.98 1 0.96 0.9 0.56 0.82 0.98 0.94 1

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 293 1018 28 23 437.94 50 93 9762 118 1019

Mean 294.58 1023.52 34.86 23 479.55 51.84 104.34 9766.34 129.76 1024.64

Std 0.81 1.64 0.99 0 7.59 0.51 4.5 1.52 1.7 1.44

DBHS

SR 1 1 1 1 1 1 1 1 1 1 10

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 295 1024 35 23 481.07 52 107 9767 130 1025

Mean 295 1024 35 23 481.07 52 107 9767 130 1025

Std 0 0 0 0 0 0 0 0 0 0

NGHS1

SR 1 1 1 1 1 0.96 1 0.94 1 1 8

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 295 1024 35 23 481.07 51 107 9765 130 1025

Mean 295 1024 35 23 481.07 51.96 107 9766.88 130 1025

Std 0 0 0 0 0 0.2 0 0.48 0 0

ABHS

SR 1 1 1 1 1 1 1 1 1 1 10

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 295 1024 35 23 481.07 52 107 9767 130 1025

Mean 295 1024 35 23 481.07 52 107 9767 130 1025

Std 0 0 0 0 0 0 0 0 0 0

ABHS1

SR 0.86 0.96 1 0.98 0.98 0.84 0.48 0.82 1 1 3

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 105 9767 130 1025

Worst 293 1018 35 22 475.48 49 96 9762 130 1025

Mean 294.72 1023.76 35 22.98 480.96 51.68 105.18 9766.44 130 1025

Std 0.7 1.19 0 0.14 0.8 0.82 2.95 1.33 0 0

SBHS

SR 1 1 1 1 1 1 1 1 1 1 10

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 295 1024 35 23 481.07 52 107 9767 130 1025

Mean 295 1024 35 23 481.07 52 107 9767 130 1025

Std 0 0 0 0 0 0 0 0 0 0

123


Table 2 continued

KP1 KP2 KP3 KP4 KP5 KP6 KP7 KP8 KP9 KP10 TSR

IBBA-RSS

SR 1 1 1 1 1 1 1 1 1 1 10

Best 295 1024 35 23 481.07 52 107 9767 130 1025

Median 295 1024 35 23 481.07 52 107 9767 130 1025

Worst 295 1024 35 23 481.07 52 107 9767 130 1025

Mean 295 1024 35 23 481.07 52 107 9767 130 1025

Std 0 0 0 0 0 0 0 0 0 0

Table 3 The parameters, dimension and optimum of ten test problems


KP11 w = [46, 17, 35, 1, 26, 17, 17, 48, 38, 17, 32, 21, 29, 48, 31, 8, 42, 37, 6, 9, 15, 22, 27, 14,42, 40, 14, 31, 6, 34], p = [57, 64, 50, 6, 52, 6, 85, 60, 70, 65, 63, 96, 18, 48, 85, 50, 77,18, 70, 92, 17, 43, 5, 23, 67, 88, 35, 3, 91, 48], C = 577

30 1437

KP12 w = [7, 4, 36, 47, 6, 33, 8, 35, 32, 3, 40, 50, 22, 18, 3, 12, 30, 31,13, 33, 4, 48, 5, 17, 33, 26,27, 19, 39, 15, 33, 47, 17, 41, 40], p = [35, 67, 30, 69, 40, 40, 21, 73, 82, 93, 52, 20, 61,20, 42, 86, 43, 93, 38, 70, 59, 11, 42, 93, 6, 39, 25, 23, 36, 93, 51, 81, 36, 46, 96], C = 655

35 1689

KP13 w = [28, 23, 35, 38, 20, 29, 11, 48, 26, 14, 12, 48, 35, 36, 33, 39, 30, 26, 44, 20, 13, 15, 46,36, 43, 19, 32, 2, 47, 24, 26, 39, 17, 32, 17, 16, 33, 22, 6, 12], p = [13, 16, 42, 69, 66, 68,1, 13, 77, 85, 75, 95, 92, 23, 51, 79, 53, 62, 56, 74, 7, 50, 23, 34, 56, 75, 42, 51, 13, 22, 30,45, 25, 27, 90, 59, 94, 62, 26, 11], C = 819

40 1816

KP14 w = [18, 12, 38, 12, 23, 13, 18, 46, 1, 7, 20, 43, 11, 47, 49, 19, 50, 7, 39, 29, 32, 25, 12, 8,32, 41, 34, 24, 48, 30, 12, 35, 17, 38, 50, 14, 47, 35, 5, 13, 47, 24, 45, 39, 1], p = [98, 70,66, 33, 2, 58, 4, 27, 20, 45, 77, 63, 32, 30, 8, 18, 73, 9, 92, 43, 8, 58, 84, 35, 78, 71, 60, 38,40, 43, 43, 22, 50, 4, 57, 5, 88, 87, 34, 98, 96, 99, 16, 1, 25], C = 907

45 2020

KP15 w = [15, 40, 22, 28, 50, 35, 49, 5, 45, 3, 7, 32, 19, 16, 40, 16, 31, 24, 15, 42, 29, 4, 14, 9,29, 11, 25, 37, 48, 39, 5, 47, 49, 31, 48, 17, 46, 1, 25, 8, 16, 9, 30, 33, 18, 3, 3, 3, 4,1],p = [78, 69, 87, 59, 63, 12, 22, 4, 45, 33, 29, 50, 19, 94, 95, 60, 1, 91, 69, 8, 100, 84, 100,32, 81, 47, 59, 48, 56, 18, 59, 16, 45, 54, 47, 98, 75, 20, 4, 19, 58, 63, 37, 64, 90, 26, 29,13, 53, 83], C = 882

50 2440

KP16 w = [27, 15, 46, 5, 40, 9, 36, 12, 11, 11, 49, 20, 32, 3, 12, 44, 24, 1, 24, 42, 44, 16, 12, 42,22, 26, 10, 8, 46, 50, 20, 42, 48, 45, 43, 35, 9, 12, 22, 2, 14, 50, 16, 29, 31, 46, 20, 35, 11,4, 32, 35, 15, 29, 16], p = [98, 74, 76, 4, 12, 27, 90, 98, 100, 35, 30, 19, 75, 72, 19, 44, 5,66, 79, 87, 79, 44, 35, 6, 82, 11, 1, 28, 95, 68, 39, 86, 68, 61, 44, 97, 83, 2, 15, 49, 59, 30,44, 40, 14, 96, 37, 84, 5, 43, 8, 32, 95, 86, 18], C = 1050

55 2643

KP17 w = [7, 13, 47, 33, 38, 41, 3, 21, 37, 7, 32, 13, 42, 42, 23, 20, 49, 1, 20, 25, 31, 4, 8, 33, 11,6, 3, 9, 26, 44, 39, 7, 4, 34, 25, 25, 16, 17, 46, 23, 38, 10, 5, 11, 28, 34, 47, 3, 9, 22, 17, 5,41, 20, 33, 29, 1, 33, 16, 14], p = [81, 37, 70, 64, 97, 21, 60, 9, 55, 85, 5, 33, 71, 87, 51,100, 43, 27, 48, 17, 16,27, 76, 61, 97, 78, 58, 46, 29, 76, 10, 11, 74, 36, 59, 30, 72, 37, 72,100, 9, 47, 10, 73, 92, 9, 52, 56, 69, 30, 61, 20, 66, 70, 46, 16, 43, 60, 33, 84], C = 1006

60 2917

KP18 w = [47, 27, 24, 27, 17, 17, 50, 24, 38, 34, 40, 14, 15, 36, 10, 42, 9, 48, 37, 7, 43, 47, 29,20, 23, 36, 14, 2, 48, 50, 39, 50, 25, 7, 24, 38, 34, 44, 38, 31, 14, 17, 42, 20, 5, 44, 22, 9, 1,33, 19, 19, 23, 26, 16, 24, 1, 9, 16, 38, 30, 36, 41, 43, 6], p = [47, 63, 81, 57, 3, 80, 28,83, 69, 61, 39, 7, 100, 67, 23, 10, 25, 91, 22, 48, 91, 20, 45, 62, 60, 67, 27, 43, 80, 94, 47,31, 44, 31, 28, 14, 17, 50, 9, 93, 15, 17, 72, 68, 36, 10, 1, 38, 79, 45, 10, 81, 66, 46, 54, 53,63, 65, 20, 81, 20, 42, 24, 28, 1], C = 1319

65 2814

the solutions of the proposed IBBA-RSS demonstrate that itis capable of outperforming the BBA phase. The solutionsof the proposed IBBA-RSS are also superior to the resultsof the other evaluated techniques in the most of the testcases.

Further, the convergence behavior for each instance isdepicted in Fig. 7, where the KP11 is depicted in Fig. 7a,the KP12 is depicted in Fig. 7b and so on. As shown intheses graphs the proposed IBBA-RSS gives better resultsthan BBA, and consequently, the profit for each instance isimproved.

123


Table 3 continued


KP19 w = [4, 16, 16, 2, 9, 44, 33, 43, 14, 45, 11, 49, 21, 12, 41, 19, 26, 38, 42, 20, 5, 14, 40, 47,29, 47, 30, 50, 39, 10, 26, 33, 44, 31, 50, 7, 15, 24, 7, 12, 10, 34, 17, 40, 28, 12, 35, 3, 29,50, 19, 28, 47, 13, 42, 9, 44, 14, 43, 41, 10, 49, 13, 39, 41, 25, 46, 6, 7, 43], p = [66, 76,71, 61, 4, 20, 34, 65, 22, 8, 99, 21, 99, 62, 25, 52, 72, 26, 12, 55, 22, 32, 98, 31, 95, 42, 2,32, 16, 100, 46, 55, 27, 89, 11, 83, 43, 93, 53, 88, 36, 41, 60, 92, 14, 5, 41, 60, 92, 30, 55,79, 33, 10, 45, 3, 68, 12, 20, 54, 63, 38, 61, 85, 71, 40, 58, 25, 73, 35], C = 1426

70 3221

KP20 w = [24, 45, 15, 40, 9, 37, 13, 5, 43, 35, 48, 50, 27, 46, 24, 45, 2, 7, 38, 20, 20, 31, 2, 20, 3,35, 27, 4, 21, 22, 33, 11, 5, 24, 37, 31, 46, 13, 12, 12, 41, 36, 44, 36, 34, 22, 29, 50, 48, 17,8, 21, 28, 2, 44, 45, 25, 11, 37, 35, 24, 9, 40, 45, 8, 47, 1, 22, 1, 12, 36, 35, 14, 17, 5],p = [2, 73, 82, 12, 49, 35, 78, 29, 83, 18, 87, 93, 20, 6, 55, 1, 83, 91, 71, 25, 59, 94, 90,61, 80, 84, 57, 1, 26, 44, 44, 88, 7, 34, 18, 25, 73, 29, 24, 14, 23, 82, 38, 67, 94, 43, 61, 97,37, 67, 32, 89, 30, 30, 91, 50, 21, 3, 18, 31, 97, 79, 68, 85, 43, 71, 49, 83, 44, 86, 1, 100,28, 4,16], C = 1433

75 3614

Large-scale 0–1 knapsack problems

To further prove the proficiency of the proposed IBBA-RSSalgorithm, twelve large-scale 0–1 knapsack instances wereutilized. The sizes of these instances include 100, 200, 300,500, 700, 1000, 1200, 1500, 1800, 2000, 2600 and 3000items. Each large-scale KP (KP11–KP22) is generated as fol-lows: the volume of each item is randomly chosen from 0.5and 2 and its corresponding profit is randomly set between0.5 and 1. The maximal volume capacity of the knapsack islimited to 0.75 times of the sum volumes of the items gen-erated following the above procedure. It is worth noting thatthese instances are created only once using a random gen-erator and kept constant for all the experiments. Extensiveexperimental tests were carried out to adjust the maximumnumbers of iterations. Based upon these tests, the maximumnumbers of iterations are accordingly set to 300, 600, 600,1000, 1800, 1800, 2500, 10,000, 10,000, 16,000, 18,000 and18,000 respectively. The proposed IBBA-RSS is run 30 timesfor each instance with 30 bats for the population size.

The proposed IBBA-RSS and BBA phase are comparedwith the V-shaped binary bat algorithm (V-BBA) which wasdeveloped in [29]. The statistical measures for each instanceusing the proposed IBBA-RSS and other comparative algo-rithms are presented in Tables 5 and 6 while best resultsare highlighted in bold. The statistical measures such asthe best, median, worst, mean values and standard devia-tions are determined where the success rate (SR) results arenot reported because the optimal profits of KP21–KP32 areunknown.

The proposed IBBA-RSS is compared with 16 differentalgorithms as in Tables 5 and 6. From these tables, we cansee that the proposed IBBA-RSS outperforms the other algo-rithms for all KP instances (KP21–KP32). Also, the proposedalgorithm saves the commotional time, where it is consumeda small number of iterations compared with the other algo-rithms [37].

Further, the convergence behavior for each instance isdepicted in Fig. 8, where the KP21 is depicted in Fig. 8a,the KP22 is depicted in Fig. 8b and so on. As shown fromthese graphs, the proposed IBBA-RSS achieves better simu-lation results than the BBA phase andV-BBA. Consequently,the profit for each instance has improved significantly. Theimproved ratio for each instance is equivalent to 1.4313,1.0708, 1.2748, 1.1861, 0.4242, 0.6764, 0.3515, 0.3498,0.3656, 0.3742, 0.4306 and 1.9425%, respectively, whencomparing IBBA-RSS with BBA phase, while the improvedratio obtained by comparing IBBA-RSS with V-BBA isequivalent to 5.4714, 6.6404, 2.4526, 2.6791, 1.7067, 1.7673,2.2394, 0.9915, 0.5675, 1.6160, 1.4548 and3.2288%, respec-tively. Further, comparing BBA with V-BBA achieves thefollowing improved ratio as follows: 4.0988, 5.6298, 1.1929,1.5108, 1.2879, 1.0983, 1.8945, 0.6439, 0.2027, 1.2467,1.0287 and 1.3117%, respectively. Although these ratiosseem small for some instances, it is very significant fromthe practical point of view for large-scale problems. Basedon the above-improved ratios, it can be concluded that pro-posed IBBA-RSS algorithm has better ratios. Therefore, theproposed IBBA-RSS is robust approach and has powerfulsearching quality.

Regarding overall results on Tables 2, 4, 5 and 6, amongstthe evaluated optimizers, IBBA-RSS could achieve the bestperformance. The main reason for the superior performanceof the proposed IBBA-RSS lies behind the multi-V-shapedtransfer function. Themulti-V-shaped transfer function helpsthe proposed algorithm to preserve the diversity of the solu-tions and thus refine the convergence rate of the proposedalgorithm. Also incorporating of RSS-based approximationscan efficiently redistribute the search bats to enhance theirdiversity and to the emphasis on more explorative steps incase of convergence to the local optimum. The new trans-fer function and RSS strategies have improved the searchingcapacities and quality of the solutions of the proposed algo-rithm. Therefore, these strategies can assist the proposed

123


Table 4 Comparison results for the medium sizes KP (KP11–KP20)

Fun Algorithm Obtained solution Best Mean Worst Std

KP11 IBBA-RSS 111110111111001110110101111011 1437 1437 1437 0

BBA 111110111111001110110101111011 1437 1437 1437 0

CI NA 1437 1418 1398 11.79

B&B NA 1437 NA NA NA

KP12 IBBA-RSS 11011111111010111111101101110111111 1689 1689 1689 0

BBA 11011111111010111111101101110111111 1689 1689 1689 0

CI NA 1689 1686.5 1679 3.8188


KP13 IBBA-RSS 0011110011111011111101011111001111111111 1821 1821 1821 0

BBA 0011110011111011111101011111001111111111 1821 1821 1821 0

CI NA 1816 1807.5 1791 9.604


KP14 IBBA-RSS 11110100111111011111011111111111101011111 1001 2033 2033 2033 0

BBA 11110100111111011111011111111111101011111 1001 2033 2030.3333 2016 6.0988

CI NA 2020 2017 2007 4.749


KP15 IBBA-RSS 11111001111111110110111111111010111111011101111111 2448 2448 2448 0

BBA 11111001111111110110111111111010011111011111111111 2440 2439.633333 2435 1.1591

CI NA 2440 2436.166 2421 6.841


KP16 IBBA-RSS 1110011111011111011111101001111111111001101101110101111 2643 2642.6000 2632 2.0103

BBA 1111011111011111011111101001111111111001101101111101110 2642 2640.4000 2614 5.5930

CI NA 2643 2605 2581 22.018


KP17 IBBA-RSS 1111101011011111011001111111110111111111011 11011111111101111 2917 2917 2917 0

BBA 111110101101111101100111111111011111111101111011111111101111 2917 2915 2893 6.1923

CI NA 2917 2915 2905 4.472


KP18 IBBA-RSS 11110101111011101111101111111111111001011111100111011111111101010

2818 2817.6333 2814 1.0661

BBA 11111101111011101101101111111111111001011111100111111111111101010

2809 2808.3333 2802 1.881549

CI NA 2814 2773.66 2716, 18.273


KP19 IBBA-RSS 1111101110101101110111111101011101011111111100111111111011011111111111

3223 3222.6000 3219 1.1017

BBA 1111101110101101100111111101011111011111111111111011111011011111111111

3213 3212.9000 3209 1.4936

CI NA 3221 3216 3211 4.3589


KP20 IBBA-RSS 011011111011001011111111111011111100111101111111011111111001111111111101101

3614 3613.2333 3605 2.4166

BBA 011011111011001011111111111011111100111101111111111111110100111111111101101

3602 3600.3793 3588 4.1611

CI NA 3614 3603.8 3591 8.035


123


Table 5 Comparisons of the large-size KP

IBBA-RSS BBA SBHS IHS GHS SAHS EHS NGHS NDHS

KP21Best 63.2149 62.3101 62.08 61.99 61.81 62.02 61.78 61.82 61.61

Median 63.2149 62.3101 62.04 61.81 61.3 61.86 61.25 61.5 61.02

Worst 62.0222 61.1074 61.97 61.23 60.94 61.65 60.63 61.11 59.59

Mean 63.1545 62.2322 62.04 61.77 61.29 61.85 61.22 61.5 60.86

Std 0.2418 0.2964 0.03 0.15 0.19 0.11 0.3 0.2 0.45

Best 131.1273 129.7232 129.44 128.89 127.09 127.99 128.43 128.34 127.82

Median 131.1273 129.7232 129.38 128.42 125.7 127.21 127.88 127.7 127

KP22Worst 129.2422 128.3646 129.27 127.61 124.47 126.39 127.08 126.87 125.72

Mean 130.9917 129.6492 129.37 128.4 125.69 127.16 127.81 127.66 126.86

Std 0.4352 0.2890 0.04 0.31 0.61 0.41 0.36 0.42 0.54

Best 195.0331 192.5467 192.02 189.94 187.28 188.15 190.96 190.18 189.97

Median 195.0331 192.5467 192.02 189.35 185.77 187.36 190.43 189.31 189.04

KP23Worst 193.2210 192.4450 191.85 188.27 184.16 186.05 189.27 187.9 187.85

Mean 194.9348 192.5431 192.01 189.14 185.77 187.27 190.28 189.23 188.97

Std 0.3587 0.0186 0.03 0.51 0.72 0.53 0.43 0.58 0.61

Best 316.3039 312.5521 314.23 306.89 301.03 302.92 312.04 310.16 309.49

Median 316.1211 312.5521 314.2 305.11 299.78 300.72 311.32 308.28 308.28

KP24Worst 315.8936 312.2119 314.1 303.55 297.25 299.14 310.29 305.67 305.94

Mean 316.1044 312.5294 314.19 305.1 299.6 300.79 311.25 308.33 308.07

Std 0.0789 0.0863 0.03 0.92 0.91 1.03 0.49 1.06 0.93

Best 448.8721 446.9679 448.65 434.04 429.02 431.63 444.91 442.32 442.85

Median 448.8721 446.9679 448.63 431.74 425.75 428.99 443.64 441.13 439.39

KP25Worst 447.2503 446.2406 448.46 429.63 423.35 427.08 442.13 436.45 436.01

Mean 448.7179 446.9049 448.6 431.73 425.68 428.93 443.53 440.83 439.43

Std 0.4232 0.1947 0.05 1.13 1.28 1.23 0.64 1.23 1.37

Best 639.4001 635.0750 638.14 605.88 602.29 606.5 629.29 626.77 621.15

Median 639.4001 635.0750 638.08 603.42 599.07 601.31 626.62 623.9 618.41

KP26Worst 639.0579 632.6213 638 599.53 594.34 597.84 624.99 619.15 614.86

Mean 639.3884 634.8336 638.09 603.26 598.83 601.78 626.76 623.87 618.09

Std 0.0624 0.7367 0.04 1.56 1.9 2.34 1.13 1.37 1.48

Best 767.0228 764.3262 763.81 722.52 721.23 724.4 751.73 750.67 744.72

Median 767.0228 764.3262 763.72 718.39 716.92 721.53 749.16 747.88 739.88

KP27Worst 766.9989 764.1197 763.39 714.39 713.17 716.46 746.38 745.05 735.02

Mean 767.0219 764.3177 763.71 718.29 716.69 721.38 749.15 747.66 739.76

Std 0.0043 0.0383 0.08 1.98 1.84 1.95 1.33 1.41 2.14

Best 966.0450 962.6650 964.91 902.36 903.31 908.1 944.09 945.2 932.32

Median 966.0450 962.6650 964.86 897.78 901.26 904.01 940.76 942.09 926.48

123


Table 5 continued

IBBA-RSS BBA SBHS IHS GHS SAHS EHS NGHS NDHS

KP28Worst 965.5550 962.6020 964.7 891.26 895.58 899.04 937.07 938.31 923.45

Mean 966.0164 962.661 964.85 897.62 900.63 903.83 940.72 941.97 926.62

Std 0.1099 0.0152 0.06 2.68 1.77 2.54 1.68 1.7 2

Best 1157.2337 1153.0032 1155.65 1073.93 1080.1 1086.57 1128.25 1133.44 1110.98

Median 1157.2337 1153.0032 1155.58 1066.02 1076.49 1080.71 1122.29 1128.77 1106.13

KP29Worst 1155.6659 1152.8484 1155.35 1058.6 1072.65 1074.16 1119.25 1125.69 1099.5

Mean 1157.1784 1152.9942 1155.57 1066.1 1076.58 1080.58 1122.61 1129.02 1105.73

Std 0.2861 0.0344 0.08 3.29 2.02 2.83 2.33 1.94 2.86

Best 1289.5521 1284.7260 1283.92 1182.55 1198.69 1202.7 1247.95 1257.45 1229.87

Median 1289.5521 1284.7260 1283.81 1177.52 1192.03 1196.75 1243.8 1252.9 1223.25

KP30Worst 1285.6171 1283.6650 1283.26 1172.02 1188.27 1190.05 1238.26 1249.74 1218.14

Mean 1289.4157 1284.6381 1283.79 1177.59 1192.71 1196.71 1243.07 1252.86 1223.5

Std 0.7177 0.2760 0.12 2.34 2.66 3.34 2.55 1.83 2.94

Best 1668.4021 1661.2185 1653.72 1500.31 1534.74 1536.25 1592.68 1615.64 1570.24

Median 1668.4021 1661.2185 1653.66 1492.52 1526.73 1528.71 1587.53 1611.05 1561.41

KP31Worst 1661.4592 1651.3906 1653.43 1481.67 1521.56 1521.65 1582.16 1604.28 1553.61

Mean 1668.0197 1660.5869 1653.64 1492.57 1527.06 1528.66 1587.06 1610.5 1561.24

Std 1.3841 2.3466 0.06 4.25 3.2 3.33 2.93 2.71 3.68

Best 1927.8000 1890.3517 1917.49 1731.78 1777.72 1785.64 1843.7 1877.6 1818.63

Median 1921.7966 1890.3517 1917.44 1724.57 1771.48 1779.68 1838.22 1872.5 1809.24

KP32Worst 1917.3188 1884.0266 1917.23 1714.03 1767.32 1769.75 1830.47 1868.31 1800.95

Mean 1921.3983 1890.07 1917.42 1724.16 1771.88 1779.06 1838.15 1872.43 1809.34

Std 1.1391 1.323 0.06 3.81 2.78 4 3.09 2.26 4.06

algorithm to switch between exploration and exploitationbehaviors more effectively.

Performance assessment

Regarding the assessment, the performance of the proposedalgorithm is investigated through using the Wilcoxon signedranks (WSRs) test for a better comparison [43]. WSRs testis a nonparametric test that utilized in a hypothesis testingsituation involving a design with two samples [42]. It is apair-wise test that aims to find out significant differencesbetween the behaviors of two algorithms.WSRs test is work-ing as follows: First, the difference between the scores ofthe two algorithms on i th of n problems and the differencesare ranked according to their absolute values. Second R+and R− are determined, where R+ is the sum of positiveranks, while R− is the sum of negative ranks; then the min-imum of R+ and R− is obtained. If the result of the test isreturned in p < 0.05 (i.e. p-value is the probability of the null

hypothesis being true) indicates a rejection of the null hypoth-esis, while p > 0.05 indicates a failure to reject the nullhypothesis.

Therefore, we apply the WSRs test for the proposedIBBA-RSS algorithm against the different algorithms thatappear in Table 2 and the obtained results for WSRs test isreported in Table 7. Also, the WSRs test is employed forthe results of the medium size KP instances that depictedin Table 4, where obtained results for WSRs test forthe medium size instances are reported in Table 8. Alsothe WSRs test is employed for the results of the large-scale KP instances that depicted in Tables 5 and 6, whereobtained results for WSRs test for the large-scale instancesare reported in Table 9. From Tables 7, 8 and 9, it canbe concluded that the proposed IBBA-RSS has superiorcharacteristics both in the high quality of the solutionand robustness of the results. Also, it can keep a signifi-cant balance between the global exploration and the localexploitation.

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Convergence behavior for LKP11 Convergence behavior for LKP12



Fig. 7 The Convergence behavior for medium KP (KP11–KP20)

Most of the p values reported in Tables 7, 8 and 9 are lessthan 0.05 (5% significance level) which is a robust evidenceagainst the null hypothesis, concluding that the obtainedresults by the proposed approach are statistically better andthey have not happened by chance.

Convergence analysis

To analyze the convergence analysis of the proposed algo-rithm, statistical measures, Wilcoxon signed ranks (WSRs)test and improvement ratio were developed. Tables 2, 4, 5

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Fig. 7 continued

and 6 demonstrated the superiority of the proposed approachregarding optimality. Further, the nonparametric WSRs isemployed to offer the winner algorithm, where Tables 7, 8and 9 show that the proposed algorithmoutperforms the othercomparative algorithms regarding the obtained p value. Also,the improvement ratio for the large-scale test instances isrecorded as 1.4313, 1.0708, 1.2748, 1.1861, 0.4242, 0.6764,0.3515, 0.3498, 0.3656, 0.3742, 0.4306 and1.9425%, respec-tively, when comparing IBBA-RSS with BBA phase, whilethe improved ratio obtained by comparing IBBA-RSS withV-BBA is equivalent to 5.4714, 6.6404, 2.4526, 2.6791,1.7067, 1.7673, 2.2394, 0.9915, 0.5675, 1.6160, 1.4548and 3.2288%, respectively. Further, comparing BBA phasewith V-BBA achieves the following improved ratio: 4.0988,5.6298, 1.1929, 1.5108, 1.2879, 1.0983, 1.8945, 0.6439,0.2027, 1.2467, 1.0287 and 1.3117%, respectively. Based onthe above-improved ratios, it can be concluded that proposedIBBA-RSS algorithm has better ratios. From the practi-

cal point of view for large-scale problems, these ratios arevery significant. Despite the high dimensionality of the KPproblems, it is noteworthy that the proposed algorithm iscompetent to provide very significant results in a small num-ber of iterations compared to the other algorithms. Regardingpresented analyses, it can be concluded that the inherentcharacteristic of this improvement is contained in incorpo-rating the RSS as a local search strategy that acceleratesthe convergence behavior and avoids the systematic run-ning of the algorithm without any improvements in theoutcomes. It can be concluded that the proposed IBBA-RSS has a significant performance and that the immatureconvergence inaccuracies of BBA phase are mitigated, effi-ciently.

In this subsection, a comparative study has been carriedout to evaluate the performance of the proposed IBBA-RSSalgorithmconcerning the binarization strategy andhybridiza-tion. In this respect, a new strategy is introduced based on

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Table 6 Comparisons for the large sizes KP (continued)

PSFHS BHS DBHS NGHS1 ABHS ABHS1 ITHS V-BBA

KP21Best 56.3 62.05 59.99 61.76 62.01 62.08 62.06 59.7561

Median 53.26 61.87 58.58 61.46 61.92 61.98 61.95 59.7561

Worst 48.48 61.68 58.04 61.12 61.71 61.76 61.76 56.0839

Mean 53.13 61.87 58.63 61.44 61.9 61.95 61.93 59.51

Std 1.82 0.1 0.43 0.17 0.09 0.1 0.07 0.84

Best 106.52 129.27 118.24 128.41 129.31 129.29 129.24 122.4199

Median 99.89 129.06 115.95 127.72 129 128.95 128.88 121.1979

KP22Worst 94.25 128.76 113.32 125.66 128.51 128.56 128.45 120.6412

Mean 100.15 129.06 115.88 127.59 128.94 128.95 128.87 122.16

Std 2.92 0.13 1.12 0.6 0.21 0.18 0.19 0.60

Best 147.08 191.54 166.55 190.83 191.49 191.46 191.41 190.2497

Median 141.64 190.97 164.1 189.17 191.05 190.71 190.78 188.5120

KP23Worst 136.66 190.5 162.24 187.7 190.32 189.78 190.06 179.5679

Mean 141.2 190.94 164.4 189.14 191.04 190.67 190.74 187.37

Std 2.71 0.25 1.27 0.64 0.24 0.34 0.36 2.54

Best 234.23 311.85 257.61 310.1 312.51 310.28 310.94 307.8299

Median 224.64 310.6 252.58 308.39 311.92 309.32 309.85 307.7595

KP24Worst 218.81 309.56 249.16 306.91 310.67 307.82 308.44 292.0437

Mean 225.45 310.52 252.87 308.38 311.79 309.23 309.82 306.52

Std 4.26 0.6 1.86 0.83 0.48 0.61 0.61 4.13

Best 323.93 443.43 355.45 442.2 446.3 441.51 442.35 441.2110

Median 311.68 441.82 348.81 440.68 445.45 439.71 441.18 440.2339

KP25Worst 301.51 439.93 344.56 437.99 444.42 437.15 438.8 425.8292

Mean 311.5 441.66 349.09 440.52 445.43 439.45 440.93 438.91

Std 4.29 0.75 2.8 1.02 0.51 0.93 0.83 3.84

Best 453.2 626.04 482.59 626.27 632.38 620.31 624.04 628.0995

Median 431.71 623.09 475.73 623.07 630.33 618.12 621.68 628.0995

KP26Worst 420.42 621.53 470.08 619.09 628.65 615.83 618.81 610.5837

Mean 431.97 623.18 475.33 623.17 630.34 617.96 621.7 626.62

Std 6.68 1.25 3.15 1.64 1.02 1.28 1.18 4.30

Best 526.59 746.55 570.95 750.32 756.08 741.27 745.77 749.8460

Median 512.59 744.38 560.84 746.73 754.26 738.82 743.32 745.7390

KP27Worst 497.65 741.5 556.7 744.14 752.1 734.96 738.73 725.7928

Mean 511.65 744.4 561.83 746.95 754.26 738.47 743.03 744.51

Std 7 1.17 3.18 1.51 1.11 1.36 1.59 4.72

Best 659.05 938.36 700.81 944.36 950.7 927.6 937.62 956.4658

Median 631.94 935.21 693.91 941.18 949.42 924.15 933.82 956.4548

KP28Worst 615.25 932.94 687.84 937.3 947.36 920.73 930.6 946.4512

Mean 633.31 935.18 694.53 941.14 949.17 924.25 933.7 955.73

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Table 6 continued

PSFHS BHS DBHS NGHS1 ABHS ABHS1 ITHS V-BBA

Std 8.37 1.29 3.87 2 1 1.64 1.8 2.15

Best 780.89 1118.83 833.43 1129.81 1140.69 1106.12 1121.58 1150.6657

Median 755.93 1115.78 819.96 1127.63 1136.71 1102.06 1115.3 1150.5952

KP29Worst 742.55 1112.07 813.31 1123.39 1133.22 1098.7 1111.32 1150.2488

Mean 755.48 1115.23 821.07 1127.15 1136.57 1102.07 1115.39 1150.57

Std 9.97 1.68 4.59 1.8 1.6 1.93 2.57 0.07

Best 867.81 1238.16 916.07 1254.53 1263.67 1223.38 1240.66 1268.7089

Median 835.85 1234.81 906.09 1252.63 1260.42 1220.21 1234.72 1266.2494

KP30Worst 818.62 1231.58 896.47 1247.62 1257.85 1214.94 1231.26 1263.4345

Mean 835.23 1234.53 905.17 1252.08 1260.46 1219.88 1234.95 1266.23

Std 9.72 1.85 5.44 1.71 1.54 2.07 2.26 1.05

Best 1092.87 1579.8 1148.13 1613.95 1623.3 1559.19 1585.94 1644.1290

Median 1061.59 1577.19 1140.14 1609.36 1618.89 1553.33 1579.92 1644.1290

KP31Worst 1044.6 1573.51 1129.41 1605.75 1613.54 1545.65 1572.33 1562.5883

Mean 1062.7 1577.17 1139.77 1609.53 1618.77 1553.04 1579.7 1640.25

Std 10.58 1.74 4.17 2.17 2.09 2.71 3.43 17.00

Best 1269.54 1830.65 1332.06 1875.71 1879.12 1803.16 1839.01 1865.5547

Median 1222.61 1826.22 1314.44 1871.05 1874.11 1797.45 1830.55 1865.5547

KP32Worst 1205.88 1821.17 1304.5 1866.99 1868.61 1792.59 1824.55 1851.2552

Mean 1224.09 1825.98 1314.9 1870.91 1874.04 1797.55 1831.37 1864.60

Std 12.96 2.38 7 2.28 2.65 2.75 3.88 3.0132

the multi-V-shaped transfer function that has effective explo-ration than the sigmoid transfer function. On the one hand,hybridization mechanism is implemented through incorpo-rating the RSS as a local search step can avoid the trappingon undesirable values. Also, this hybridization can acceleratethe convergence and also preserve its searching efficiency andeffectiveness. On the other hand, the proposed algorithm ishighly competitivewhen comparing itwith the othermethodsregarding calculating the statistical measures and Wilcoxonsigned ranks test. So the use of the hybrid approach has agreat potential for solving large-scale knapsack problems.Moreover, it can be recognized from the obtained results on32 test instances that the proposed IBBA-RSS is capable ofattaining satisfactory solutions with an appropriate exploita-tion potential. The reason is that the proposedRSS-embeddedmechanisms can stimulate the exploitation tendency of theBBA phase, effectively.

Careful observation will reveal the following achieve-ments of the proposed IBBA-RSS algorithm:

(a) IBBA-RSS performs better on all large-scale KP prob-lems while the other algorithms often miss the betterresults.

(b) IBBA-RSS can obtain a relatively stable and better resulton the whole regarding the statistical measures.

(c) IBBA-RSS combines the merits of the BBA and RSS toobtain an elevated performance.

(d) IBBA-RSS gives a promising improvement in the prob-lem profit and can avoid the trapping in local optima.

(e) The proposedmethodology opens up numerous researchdirections for solving the different variants of KPproblems such as multi-dimensional KP and quadraticKP.

Conclusions and future work

This paper presented a novel injective binary bat algorithm-based rough set scheme (IBBA-RSS) for solving 0/1 knap-sack problems. To overcome the BBA’s problem of beingconverged to local optima and to improve its exploration andexploitation tendencies, it is hybridized with the RSS phase.Furthermore, the survival process of bats is achieved based onthe injective (one-to-one) strategy, where the fit one replacesthe worst one based on feasibility rule. The performance of

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Convergence behavior for LKP21Convergence behavior for LKP22



Fig. 8 The Convergence behavior for large scale KP (KP21–KP32)

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Fig. 8 continued

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Table 7 Wilcoxon test forcomparison results in Table 2

Compared methods Solution evaluations

The proposed Compared algorithms R− R+ p Value Winner

IBBA-RSS BHS 45 0 0.007686 IBBA-RSS

IBBA-RSS DBHS 0 0 – –

IBBA-RSS NGHS1 3 0 0.179712 IBBA-RSS

IBBA-RSS ABHS 0 0 – –

IBBA-RSS ABHS1 28 0 0.017960 IBBA-RSS

IBBA-RSS SBHS 0 0 – –

Table 8 Wilcoxon test forcomparison results in Table 4


The proposed Compared algorithms R− R+ p Value Winner algorithm

IBBA-RSS BBA 15 0 0.043114 IBBA-RSS

IBBA-RSS CI 12.5 8.5 0.674987 IBBA-RSS

IBBA-RSS B&B 6 4 0.715001 IBBA-RSS

Table 9 Wilcoxon test forcomparison results in Tables 5and 6


The proposed Compared algorithms R− R+ p Value Winner algorithm

IBBA-RSS BBA 78 0 0.002218 IBBA-RSS

IBBA-RSS SBHS 78 0 0.002218 IBBA-RSS

IBBA-RSS IHS 78 0 0.002218 IBBA-RSS

IBBA-RSS GHS 78 0 0.002218 IBBA-RSS

IBBA-RSS SAHS 78 0 0.002218 IBBA-RSS

IBBA-RSS EHS 78 0 0.002218 IBBA-RSS

IBBA-RSS NGHS 78 0 0.002218 IBBA-RSS

IBBA-RSS NDHS 78 0 0.002218 IBBA-RSS

IBBA-RSS PSFHS 78 0 0.002218 IBBA-RSS

IBBA-RSS BHS 78 0 0.002218 IBBA-RSS

IBBA-RSS DBHS 78 0 0.002218 IBBA-RSS

IBBA-RSS NGHS1 78 0 0.002218 IBBA-RSS

IBBA-RSS ABHS 78 0 0.002218 IBBA-RSS

IBBA-RSS ABHS1 78 0 0.002218 IBBA-RSS

IBBA-RSS ITHS 78 0 0.002218 IBBA-RSS

IBBA-RSS V-BBA 78 0 0.002218 IBBA-RSS

the proposed algorithm has been extensively investigatedthrough using small-scale, medium- scale and large-scaleinstances of 0–1 KP. The proposed algorithm is comparedwith several algorithms from the literature. Based on statis-tical measures, the proposed algorithm can explore/exploitbetter-quality solutions, and it outperforms as the bestamongst the other compared algorithms. Convergence trendfor average best results for IBBA-RSS is preferable to equiv-alent curves for BBA. Also, non-parametric statistical testsalso affirm that the optimality of solutions is enriched, sig-nificantly. The results reveal that IBBA-RSS is competentto provide very competitive results compared to BBA andother investigated algorithms. Regarding presented analyses,

it can be concluded that IBBA-RSS has a desirable perfor-mance and the immature convergence inaccuracies of theBBA phase is mitigated, efficiently.

For future works, it is possible to investigate the pro-posed IBBA-RSS algorithm to solve different forms ofKP problems like multi-dimensional KP and quadratic KP.Further research on using other metaheuristic algorithmssuch as krill herd, monarch butterfly optimization (MBO),earthworm optimization algorithm (EWA), elephant herdingoptimization (EHO) and moth search (MS) algorithm needto be developed to solve different forms of KP problems.Finally, I hope to design a new version of KP that is bi-levelKP.

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http://dx.doi.org/10.1007/s00521-017-2903-1

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