Complex Intell. Syst. DOI 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 © The Author(s) 2017. This article is an open access publication
Abstract This paper presents a novel binary bat algorithm (NBBA) to solve 0–1 knapsack problems. The proposed algorithm combines two important phases: binary bat algorithm (BBA) and local search scheme (LSS). The bat algorithm enables the bats to enhance the exploration capability while LSS aims to boost the exploitation tendencies and, therefore, it can prevent the BBA–LSS from the entrapment in the local optima. Moreover, the LSS starts its search from BBA found so far. By this methodology, the BBA–LSS enhances the diversity of bats and improves the convergence performance. The proposed algorithm is tested on different size instances from the literature. Computational experiments show that the BBA–LSS can be promise alternative for solving large-scale 0–1 knapsack problems. Keywords Bat algorithm · Local search scheme · Knapsack problem
Introduction Knapsack problem (KP) is one of the most important problems in the combinatorial optimization. It appears in a broad variety of applications, including scheduling problems,
B
Aboul Ella Hassanien
[email protected] http://www.egyptscience.net
1
Department of Basic Engineering Science, Faculty of Engineering, 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, project selection, resource distribution, and so on. Unfortunately, KP is non-polynomial (NP) hard, the complete problem [1]. Thus, solving this problem using the gradient methods is inappropriate because this problem may fall in local optima for large-scale problems. Also, these methods are timeconsuming, and they achieve one of the closest local optima to initial random solution. Meanwhile, the metaheuristic algorithms have the ability to overcome these drawbacks and proved to be a robust alternative to solve complex optimization problems. Recently, metaheuristic algorithms are one of the significant stochastic research topics in optimization that imitate natural phenomena. The features of the metaheuristic algorithms are the avoidance of local optima; generate multiple solutions for each run which assist to produce good-quality solutions quickly and no dependence on derivative information [2]. In recent decades, there have been extensive works based on metaheuristic algorithms to solve 0–1 KP. Liu and Liu [3] introduced an evolutionary algorithm based on schemaguiding to solve 0–1 KP. Martello et al. [4] proposed a survey of different approaches to solving 0–1 KP. Shi [5] proposed a modified version of the ant colony optimization (ACO) to solve 0–1 KP. Lin [6] solved the KP in the fuzzy environment through imprecise weight using a genetic algorithm (GA). Li and Li [7] presented a binary particle swarm optimization using a multi-mutation mechanism to solve KP. Zhang et al. [8] introduced amoeboid organism algorithm to solve 0–1 KP. Bhattacharjee and Sarmah [9] proposed a shuffled frog-leaping algorithm to solve 0–1 KP. Kulkarni and Shabir [10] proposed Cohort intelligence algorithm for solving 0–1 KP. In addition, many algorithms have been flourished for solving the 0–1 KP such as genetic algorithm (GA), particle swarm optimiza-
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Complex Intell. Syst.
tion (PSO), artificial fish-swarm algorithm (AFSA), harmony search algorithm (HS), chemical reaction optimization based on greedy strategy (CROG), genetic mutation bat algorithm (GMBA), monarch butterfly optimization and hybrid cuckoo search based on harmony search [11–19]. Owing to the importance of the knapsack problem in the academic area and practical applications, developing new algorithms with more promising performance to solve large-scale types of the knapsack problem applications undoubtedly becomes a true challenge. Bat algorithm (BA) is one of the recent metaheuristic algorithms that are inspired by the echolocation behavior of micro-bats [20]. During flying, bats emit short and ultrasonic pulses to the environment and record their echoes. The recorded information from the echoes helps the bats to build an airtight image of their surroundings and locate precisely the distance, shapes and prey’s position. The ability of such echolocation of micro-bats is charming, as these bats can find their prey and distinguish different types of insects even in complete darkness [20]. The earlier applications showed that BA could solve different optimization problems and proved that its efficiency and robustness compared to different algorithms such as GA and PSO [20–22]. A new trend in bat algorithms is focusing on hybridizing BA with different strategies [23–31]. Fister et al. [23] developed a hybrid BA based on various evolution strategies for solving optimization tasks, while Baziar et al. [24] proposed a modified BA based on adaptive self-strategy. A hybrid BA based on harmony search for solving optimization problems was proposed by Wang and Guo [25]. Yilmaz and Kucuksille [26] developed an improved BA using some modifications, while Wang et al. [27] presented a modified BA through adjusting the flight speed and the flight direction adaptively. Fister et al. [28] introduced a new version of BA based on self-adaptation of control parameters. Further, binary versions of BA were developed in [29–31]. Mirjalili et al. [29] introduced a binary version of BA by employing a V-shaped transfer function to overcome the drawback of the sigmoid transfer function which keeps the positions unchanged during the iterations of the algorithm. In [30], authors developed an integrating version of the binary BA based on Naïve Bayes classifier for feature selection problem. In [31], a binary vision of BA is established based on the sigmoid transfer function for solving different optimization problems. Due to continuous nature of BA, it is still in its infancy for solving combinatorial optimization problems, so this is also the motivation behind this study. This paper is motivated by several features. First, incorporating the rough set with bat algorithm to solve large-scale 0–1 KP has not been yet studied. Second, many optimization algorithms suffer from entanglement in local optima when solving large-scale problems. Last, solving large-scale knapsack problems have not received adequate attention yet.
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Hence solving large-scale knapsack problems to optimality undoubtedly becomes a true challenge. In this paper, we propose a novel binary bat algorithm (NBBA) to solve 0–1 knapsack problems. In contrast to the binary version of BA in [29], the multi-V-shaped transfer function for generating the solutions, the inclusion of the rough set scheme (RSS) as a local search strategy (LSS) and updating the solution through one-to-one strategy are introduced. The proposed algorithm combines two important phases: binary bat algorithm (BBA) and local search scheme (LSS). The bat algorithm enables the bats to enhance exploration capability while LSS aims to boost the exploitation tendency and, therefore, it can prevent the BBA–LSS from the entrapment in the local optima. Moreover, the LSS starts its search from BBA found so far. By this methodology, the BBA–LSS enhances the diversity of bats and improves the convergence performance. The proposed algorithm is tested on different size instances from the literature. Computational experiments show that the BBA–LSS can promise alternative for solving large-scale 0–1 knapsack problems. The main contributions of this approach are to (1) introduce a novel binary bat algorithm (NBBA) for solving large-scale 0–1 knapsack problems, (2) integrate intelligently the merits of two phases, namely binary bat algorithm (BBA) and rough set scheme (RSS) as a local search scheme, so it can avoid the sucking in the local optima, (3) improve the exploration capabilities of the BBA phase to seek the overall search space while incorporating RSS phase as a counterpart to enhance the exploitation tendencies, (4) implement the injective (one-to-one) strategy for updating mechanism between the two phases such that the fit ones among two phases replace the worst ones based on feasibility rule and (5) to integrate BBA and RSS to improve the quality of solutions and speed up the convergence to the global solution. On the other hand, the proposed algorithm is effectively applied for small- , medium- and large-size problems. The experimental results demonstrated the superiority of the proposed algorithm in achieving a high quality of solutions. The simulation results affirm that the application of RSS may be an effective scheme to improve the performances of optimization algorithms. The novelty of the proposed approach is cleared regarding proposing the multi-V-shaped transfer function for generating the solutions in the BBA phase, and then this can provide more explorations in the search space. Further, adopting the RSS as a local search scheme and introducing the injective (One-to-One) strategy can pick the fit solutions quickly and avoid the running of the algorithm without any improvement in the solutions. The rest of this paper is organized as follows: In Sect. 2, we describe the preliminaries of the 0–1 knapsack problems.
Complex Intell. Syst.
In Sect. 3, the basics of both BA and rough set theory (RST) are reviewed. The proposed algorithm is explained in detail in Sect. 4. The numerical experiments are given in Sect. 5 to show the superiority of the proposed algorithm. Section 6 gives the conclusions and the further work.
Overview of bat algorithm (BA) and rough set theory (RST) This section is devoted to describing the basics of bat algorithm (BA) and rough set theory (RST). Real behavior algorithm
Preliminaries Problem description There are N items and the knapsack capacity is C · w j is the weight of the jth item, p j is the profit of the jth item. Then solve which items are let into the knapsack to make the total weight of the items no more than capacity of the knapsack and get the maximum total of the profit. Mathematical description The mathematical description of the 0–1 knapsack problem can be formulated as follows: 0–1 KP: Max f (x) =
N j=1
(1)
The binary decision variables x j are used to determine whether the item j is put in the knapsack or not. In large-scale instances, the total weights of the items that can be packed in the knapsack may violate the constraint, and this violation is unacceptable and must be handled. The prominent way to handle the constraint is the penalty function method. It imposes the penalty on unfeasible solutions and, therefore, it can evolve the unfeasible solutions until they move to candidate feasible regions. By use of penalty function, the 0–1 KP can be reformulated as follows: 0–1 KP: ⎛ Max f (x) =
j=1
s.t. :
Bat algorithm (BA) Velocity and position
pjxj,
⎧ N ⎫ ⎨ j=1 w j x j ≤ C, ⎬ s.t. : x j = {0, 1}, j = 1, 2, . . . , N ⎩ ⎭ p j > 0, w j ≥ 0, C > 0
N
Bat algorithm was established based on echolocation process of bats. In the echolocation process, pulses will be created by bats which are alive for 8–10 ms at a constant frequency and corresponding wavelength as given in Fig. 1. The features of bats which are exhibited for the development of bat algorithm are as follows: (i) Even without visibility, bats can sense and estimate the distance between food and the obstacles behind them, (ii) the bats are associated with velocity, position, fixed frequency, varying loudness and wavelength when they start flying to find their food and (iii) many strategies are attributed to change in values of loudness from a small constant value to a maximum positive value.
⎧ ⎨
⎜ p j x j − λ ⎝max 0, ⎩
x j = {0, 1}, j = 1, 2, . . . , N p j > 0, w j ≥ 0, C > 0,
N j=1
BA starts with the random initial population of bats in a n-dimensional search space where the position of the bat i denoted by xit and its velocity denoted by vit at time t. Therefore, the new positions xit+1 and new velocities vit+1 at time step t + 1 can be determined by αi = αmin + (αmax − αmin )β
(3)
vit+1 xit+1
(4)
= =
vit xit
+ (xit − + vit+1 ,
x
best
)α
(5)
⎫2 ⎞ ⎬ ⎟ wjxj − C ⎠ ⎭ (2)
where λ defines the penalty coefficient where it is set to 1010 for all test instances.
Fig. 1 Real behavior of bats
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Complex Intell. Syst.
where β is a random number in [0, 1] and x best represents the current global optimal solution. αi represents the pulse frequency emitted by bat iat the current moment, and αmin and αmax represent the minimum and maximum values of pulse frequency, respectively. Initially, the pulse frequency is assigned randomly for each bat which is elected uniformly from [αmax , αmin ]. In this scenario, a bat is chosen randomly from the bat population, and then the corresponding position of this bat is updated according to Eq. (6). This random walk can be comprehended as a process of local search that generates a new solution by the selected solution. xnew = xold + ε At
(6)
where xold represents a random solution chosen from the current best solutions, At is the loudness and ε is a random vector 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 the local and global moves, where the loudness is strong, and pulse emission is small at the beginning of the search process. Once the bat has got its prey, the loudness decreases while pulse emission gradually increases. A(i) And r (i) are updated according to Eqs. (7) and (8): r t+1 (i) = r 0 (i) × [1 − e−γ t ] A
t+1
(7)
(i) = δ A (i), t
(8)
where both δ and γ are constants. A(i) = 0 means that the bat has just found its prey and temporarily stopped emitting any sound. For any 0 < δ < 1 and γ > 0, we have At (i) → 0, r t (i) → r 0 (i),
as t → ∞
(9)
The implementation steps of bat algorithm Step 1: Set the basic parameters: population size (PS), attenuation coefficient of loudness δ, increasing coefficient of pulse emission γ , the maximum loudness A0 and maximum pulse emission r 0 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 new solutions by updating process for both velocity and current position by using Eqs. (4) and (5). Otherwise, generate new position of bat by making a random disturbance, and go to step 5.
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Step 6: If rand < Ai and f (xi ) < f (x best ), accept the new solutions and fly to the new position. Step 7: If f (xi ) < f min , replace the best bat and adjust A(i)and r (i)according to Eqs. (7) and (8). Step 8: Evaluate the bat population, and return the best bat and its position. Step 9: If the termination condition is met (i.e., satisfy the search accuracy condition or reach a maximum number of iterations), go to step 10; else, go to step 5, and perform the next search. Step 10: Get the output (i.e., global solution and the best fitness). where, rand is a uniform distribution in [0, 1]. Rough set theory (RST) The fundamental concept of the RST is the indiscernibility relation, which is produced by the information of interested objects [32]. Because of discerning knowledge is lacking, one cannot identify some objects based on the available information. The indiscernibility relation relies on the granules of indiscernible objects as a fundamental basis. Some relevant concepts 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 ), where U is a nonempty finite set of objects and A is a non-empty finite set of attributes. An information function f maps an object to its attribute, i.e., f a : U → Va for every a ∈ A, where Va is the value set for attribute a. A posteriori knowledge (denoted by d) is denoted by one distinguished attributed. A decision system is an IS with the form DT = (U, A ∪ {d}, f ), where d∈ / A is used as supervised learning. The elements of A are called conditional attributes. Definition 2 (Indiscernibility) For an attribute set B ⊆ A, the equivalence relation induced by B is called a B-indiscernibility relation, i.e., INDT (B) = {(x, y) ∈ U 2 |∀a ∈ B, f a (x) = f a (y)} The equivalence classes of the B-indiscernibility relation are denoted as I B (x). Definition 3 (Set approximation) Let X ⊆ U and B ⊆ A in an IS, the B-lower approximation of X is the set of objects that belongs to X with certainty, i.e., B X = {x ∈ U |I B (x) ⊆ X }. The B-upper approximation is the set of objects that possibly belongs to X , where B¯ X = {x ∈ U |I B (x) ∩ X = φ}. 1 , X 2 , . . . , X r are the deciDefinition 4 (Reducts) If X DT DT DT sion classes of DT, the set POS B (d) = B X 1 ∪ B X 2 ∪ · · · ∪ B X r is the B-positive region of DT. A subset B ⊆ A is a set of relative reducts of DT if and only if POS B (d) = POSC (d)
Complex Intell. Syst. 1 0.9
BNB(X)
0.8
POSB(X)
Sig. transfer function
NEGB(X)
0.7 0.6 0.5 0.4 0.3 0.2
Fig. 2 Definitions regarding rough set approximations
and POS B−{b} (d) = POSC (d), ∀b ∈ B. In the same way POS B (X ), B N B (X ) and NEG B (X ) are defined below (refer to Fig. 2). • POS B (X ) = B X ⇒ certainly member of X • NEG B (X ) = U − B¯ X ⇒ certainly nonmember of X • B N B (X ) = B¯ X − B X ⇒ possibly the member of X.
0.1 0 -8
-6
-4
-2
0 2 Velocity range
4
6
8
Fig. 3 Sigmoid transfer function [34]
Therefore, the transfer function is responsible for the switching between “0” and “1” values. The traditional transfer function that has been used for binary particle swarm optimization (PSO) is defined as Eq. (10) and Fig. 3 [34]. 1
The proposed algorithm (IBBA-RSS)
Sig(vik (t)) =
In this section, we present the injective binary bat algorithm based rough set scheme (IBBA-RSS) to solve the KP. Different from the conventional BA, first, a discrete binary string is adopted to represent a solution; second the updating process of position using Eq. (4) cannot be used to handle the binary space directly; therefore, a new transfer function is introduced to map velocity values to probability values for updating process of the position; third the RSS is adopted to exploit the neighborhood in search process; fourth, after the binary BA procedures, the updating mechanism is implemented based on the injective (one-to-one) strategy, where the fit one replaces the worst one based on feasibility rule. By this methodology, the IBBA-RSS enhances the diversity of bats and improves the convergence performance. The details of the proposed algorithm are given below.
where Sig denotes the Sigmoid transfer function and vik (t) denotes the velocity of the bat i in kth dimension at iteration t. After calculating the transfer function values, the new position updating equation is necessary to update particles’ position as follows [34]:
Binary position scheme In this step, each bat of the population is a solution to the KP, where each bat is represented by the n-bit binary string, where n is the number of decision variables (items) in the KP. For example, considering that xi represents the bat bits, then its jth bit xi j = (xi1 , xi2 , . . . , xin ) is a binary variable, 0 or 1.
1 + e−vi (t)
xik (t
+ 1) =
k
0 1
,
if rand < S(vik (t + 1)) if rand ≥ S(vik (t + 1)),
(10)
(11)
where xik (t) indicates the position and vik (t) indicates the velocity of ith the bat at iteration t in kth dimension. But the drawback of the sigmoid transfer function is that the particles’ positions remain unchanged when their velocity values increase. To overcome this drawback, a multi V-shaped transfer function (see Fig. 4) is introduced to oblige the bats with high velocity to diverse their positions. A multi Vshaped transfer function and the new position are stated as in Eqs. (12) and (13), respectively.
Binary velocity scheme
Q π 2 k = arctan v (t) π 2 i k (xi (t)) if rand < V (vik (t + 1)) xik (t + 1) = xik (t) if rand ≥ V (vik (t + 1)),
In bat algorithm, the velocity of the bat is responsible for updating the position. To update the position, the transfer function is introduced to force bat to fly in a binary space.
where Q = 0.1 ∼ 3, xik (t) is the position and vik (t) is the velocity of the ith bat at iteration t in kth dimension and (xik (t)) is the complement of xik (t).
V (vik (t))
(12) (13)
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Complex Intell. Syst. 1 0.9
V- transfer function
0.8 0.7 0.6 0.5 0.4 Q=0.1 Q=0.5 Q=1 Q=2 Q=3
0.3 0.2 0.1 0 -8
-6
-4
-2
0
2
4
6
8
Velocity range
Fig. 5 An illustration of mutation process Fig. 4 Proposed multi V-shaped transfer function
Evaluation The distance estimation of the bat is related to objective function (fitness function). In this step, the fitness function is evaluated based on finding the maximum profit as in the Eq. (2). Therefore, the best estimation source with the highest fitness xbest is determined as follows: PS ) xbest = arg(Max{ f (xi )}i=1
items that included in NEG B (X ) and picking random value whether 0 or 1 for items that included in B N B (X ). If we have an empty B N B (X ) or zero values for D, then mutation strategy can be implemented by generated N neighbors around each solution as follows: for each string generated N -solutions randomly, K bits are selected randomly from xi of bat i; then the selected K bits are inverted. Figure 5 illustrates an example for this step with N = 3 and B = 2, and the colored bits denote the selected ones.
(14) Injective updating based on feasibility rule
Rough set scheme (RSS) In this step, the RSS is introduced to reduce the redundant bits. In this regard, the obtained population is assumed as an information system consisting of bats’ solutions where each bat is represented by a set of condition attributes and one decision attribute. For the bat i, xi j the condition attribute illustrates the selected item j, and the decision attribute demonstrates the feasibility of this bat. The term feasibility means that the candidate bat satisfies the knapsack capacity. When the candidate bat is feasible, the decision attribute takes one value; otherwise it takes 0 value. After that, all solutions are formulating as augmented matrix consisting of the PS , condition and decision attributes [xi1 , xi2 , . . . , xin |{D}]i=1 where D denotes decision attribute that takes 1 or 0 value. Therefore, D splits the population into two classes: members that picked value of one in D and members that picked value of zero in D. Let U be the set of objects (solutions) and X ⊆ U that contains the one values of D and B = {x1 , x2 , . . . , xn } is the set of condition attribute in an IS. Then according to Definition 4, the redundant items are eliminated where B X, B¯ X B N B (X ) and NEG B (X ) of X are obtained based on the process of attribute reduction. Afterward, the population is updated by putting one value for items that belong to the B X , picking zero value for
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Now the feasibility rule must be carried out to obtain a new population. In this step, the population is updated based on injective scheme (one-to-one), where the solution after implementing the rough set scheme is compared with corresponding one obtained by the bat procedures. The winner solution is selected for updated process based on the feasibility rule that was introduced by Deb [35]. These rules are defined as follows: 1. Through comparing two feasible solutions, the chosen is the 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 chosen is the one with the lower sum of constraint violation. The sum of constraint violation for a solution given by ⎞ ⎛ N w j x j − C, 0⎠ CV(x) = max ⎝
(15)
j=1
The flowchart of the proposed IBBA-RSS algorithm is shown in Fig. 6, where the changes are highlighted.
Complex Intell. Syst.
End Start
Return the global best solution Yes
Initialization: xi = rand (0,1) ,
vi = 0 ∀i and initialize
No
Is the iteration satisfied?
α
Survival Population Initialize pulse rates ri and loudness Ai Update v and adjust
Injective strategy based on feasibility rule
α
Calculate transfer function value using equation (12)
Population after rough set phase
Mutation strategy
Replace the items of BX with 1, NEGB ( X ) with 0, BN B ( X ) with (0 or 1), values
Update positions using equation (13)
No BN B ( X ) = φ
Select a solution among Yes the best solutions and change some dimensions of solution randomly
Yes
rand > ri
Find BX , BX , No Generate a new solution by flying randomly
BN B ( X ), NEGB ( X )
Select some knowledge randomly from D as X ⊆ D
Evaluate the fitness of all bats Formulate the information system matrix ([POP. D]) Accept the new solutions and then increase ri and reduce Ai
Yes
rand < Ai &
If f ( xi ) < f ave then D=1 else D=2 (for feasible solutions) and D=0 for unfeasible solutions
f ( xi ) < f ( x best ) No Rank the bats and update global best solution
Calculate f ( xi ) and the average ( f ave ) for all solutions
Population of bats (POP)
Fig. 6 Flowchart of the proposed IBBA-RSS approach
Experimental results and analysis In this section, the performance of the IBBA-RSS algorithm is extensively investigated by a large number of experimental studies. Ten low-dimensional, ten medium size and twelve large-scale instances are considered to validate the robustness of the proposed IBBA-RSS algorithm. The algorithm is coded in MATLAB 7, running on a computer with an Intel
Core I 5 (1.8 GHz) processor and 4 GB RAM memory and Windows XP operating system. Low-dimensional 0–1 knapsack problems In this section, the performance of proposed algorithm is investigated to solve ten low-dimensional 0–1 knapsack problems, where these instances are taken from [36,37]. The
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Complex Intell. Syst. Table 1 The parameters, dimension and optimum of ten test 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 dimension and parameters is listed in Table 1. The maximum number of iterations is set to 400 iterations for each instance with 30 bats for the population size, where each instance is tested with 30 independent algorithm runs. To completely evaluate the IBBA-RSS performance statistical measures such as success rate (SR) among all runs in reaching the appointed Optima, “Best”, “Median”, “Worst”, “Mean” and standard division (Std.) are calculated. On the other hand, the performance of the proposed IBBARSS algorithm is compared with six different algorithms that are reported in [37]: NGHS1 [36], SBHS [37], BHS [38], DBHS [39], ABHS [40] and ABHS1 [41]. Table 2 shows the comparisons between the proposed algorithm and six algorithms, where best results are highlighted in bold. The obtained results showed that the proposed algorithm could achieve the optima for the low-dimensional knapsack problems, where the proposed IBBA-RSS algorithm is competitive with SBHS, ABHS and DBHS and outperforms BHS, NGHS1 and ABHS1. Medium size 0–1 knapsack problems This section is devoted to investigate the performance of the proposed algorithm to solve medium size 0–1 knapsack prob-
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lems. Ten instances are taken from [42], where sizes of these instances include 30, 35, 40, 45, 50, 55, 60, 65, 70 and 75 items. The information about these instances such as dimension, parameters and optimum solution are listed in Table 3. Extensive experimental tests were carried out to adjust the maximum numbers of iterations. Based upon these tests, the maximum numbers of iterations are accordingly set to 400 iterations for KP10 –KP15 and 500 iterations for KP16 –KP20 . The proposed IBBA-RSS is run 30 times for each instance with 30 bats for the population size. To demonstrate the effectiveness and robustness of the proposed IBBA-RSS, it is implemented and compared with the BBA phase. The statistical measures for the each instance is obtained using BBA and IBBA-RSS and reported in Table 4 where best results are highlighted in bold. The statistical measures such as the best, median, worst, mean values and standard deviations are determined. The proposed IBBA-RSS is compared with BBA, Cohort Intelligence (CI) and Branch and Bound method (B&B) as in Table 4. From these Tables, we can see that the proposed IBBA-RSS is statistically superior to other algorithms for the most KP instances and similar for the some KP instances. It can be perceived from Table 4 that the proposed IBBARSS is competent to obtain very competitive solutions with other algorithms. For KP15, KP16, KP18, KP19 and KP20
Complex Intell. Syst. Table 2 Comparisons of the small sizes KP KP1
KP2
KP3
KP4
KP5
KP6
KP7
KP8
KP9
KP10
TSR
BHS SR
1
0.96
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
0.81
1.64
0.99
0
7.59
0.51
4.5
Std
0.78
0.92
0.98
0.9
0.56
0.82
1.52
0.98
1.7
0.94
1
1.44
DBHS SR
1
1
1
1
1
1
1
1
1
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
295
1024
35
23
481.07
52
107
9767
130
1025
Mean
295
1024
35
23
481.07
52
107
9767
130
1025
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0.96
1
0.94
1
1
Std
10
NGHS1 SR 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
0
0
0
0
0
0.2
0
0.48
0
0
Std
8
ABHS SR
1
1
1
1
1
1
1
1
1
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
295
1024
35
23
481.07
52
107
9767
130
1025
Mean
295
1024
35
23
481.07
52
107
9767
130
1025
0
0
0
0
0
0
0
0
0
0
Std
10
ABHS1 SR
0.86
0.96
1
Best
295
1024
35
Median
295
1024
35
Worst
293
1018
35
Mean
294.72
Std
0.98 23
0.98
0.84
0.48
1
1
9767
130
1025
105
9767
130
1025
96
9762
130
1025
481.07
52
107
23
481.07
52
22
475.48
49
0.82
1023.76
35
22.98
480.96
51.68
105.18
9766.44
130
1025
0.7
1.19
0
0.14
0.8
0.82
2.95
1.33
0
0
1
1
1
1
1
1
1
1
1
1
3
SBHS SR 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
0
0
0
0
0
0
0
0
0
0
Std
10
123
Complex Intell. Syst. Table 2 continued KP1
KP2
KP3
KP4
KP5
KP6
KP7
KP8
KP9
KP10
TSR
SR
1
1
1
1
1
1
1
1
Best
295
1024
35
23
481.07
52
107
9767
1
1
10
130
1025
Median
295
1024
35
23
481.07
52
107
9767
130
1025 1025
IBBA-RSS
Worst
295
1024
35
23
481.07
52
107
9767
130
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 Problem
Parameter
Dimension
Optimum
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 it is capable of outperforming the BBA phase. The solutions of the proposed IBBA-RSS are also superior to the results of the other evaluated techniques in the most of the test cases.
123
Further, the convergence behavior for each instance is depicted in Fig. 7, where the KP11 is depicted in Fig. 7a, the KP12 is depicted in Fig. 7b and so on. As shown in theses graphs the proposed IBBA-RSS gives better results than BBA, and consequently, the profit for each instance is improved.
Complex Intell. Syst. Table 3 continued Problem
Parameter
Dimension
Optimum
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-RSS algorithm, twelve large-scale 0–1 knapsack instances were utilized. The sizes of these instances include 100, 200, 300, 500, 700, 1000, 1200, 1500, 1800, 2000, 2600 and 3000 items. Each large-scale KP (KP11 –KP22 ) is generated as follows: the volume of each item is randomly chosen from 0.5 and 2 and its corresponding profit is randomly set between 0.5 and 1. The maximal volume capacity of the knapsack is limited to 0.75 times of the sum volumes of the items generated following the above procedure. It is worth noting that these instances are created only once using a random generator and kept constant for all the experiments. Extensive experimental tests were carried out to adjust the maximum numbers of iterations. Based upon these tests, the maximum numbers of iterations are accordingly set to 300, 600, 600, 1000, 1800, 1800, 2500, 10,000, 10,000, 16,000, 18,000 and 18,000 respectively. The proposed IBBA-RSS is run 30 times for each instance with 30 bats for the population size. The proposed IBBA-RSS and BBA phase are compared with the V-shaped binary bat algorithm (V-BBA) which was developed in [29]. The statistical measures for each instance using the proposed IBBA-RSS and other comparative algorithms are presented in Tables 5 and 6 while best results are highlighted in bold. The statistical measures such as the best, median, worst, mean values and standard deviations are determined where the success rate (SR) results are not reported because the optimal profits of KP21 –KP32 are unknown. The proposed IBBA-RSS is compared with 16 different algorithms as in Tables 5 and 6. From these tables, we can see that the proposed IBBA-RSS outperforms the other algorithms for all KP instances (KP21 –KP32 ). Also, the proposed algorithm saves the commotional time, where it is consumed a small number of iterations compared with the other algorithms [37].
Further, the convergence behavior for each instance is depicted in Fig. 8, where the KP21 is depicted in Fig. 8a, the KP22 is depicted in Fig. 8b and so on. As shown from these graphs, the proposed IBBA-RSS achieves better simulation results than the BBA phase and V-BBA. Consequently, the profit for each instance has improved significantly. The improved 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, when comparing IBBA-RSS with BBA phase, while the improved ratio obtained by comparing IBBA-RSS with V-BBA is equivalent to 5.4714, 6.6404, 2.4526, 2.6791, 1.7067, 1.7673, 2.2394, 0.9915, 0.5675, 1.6160, 1.4548 and 3.2288%, respectively. Further, comparing BBA with V-BBA achieves the following 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 ratios seem small for some instances, it is very significant from the practical point of view for large-scale problems. Based on the above-improved ratios, it can be concluded that proposed IBBA-RSS algorithm has better ratios. Therefore, the proposed IBBA-RSS is robust approach and has powerful searching quality. Regarding overall results on Tables 2, 4, 5 and 6, amongst the evaluated optimizers, IBBA-RSS could achieve the best performance. The main reason for the superior performance of the proposed IBBA-RSS lies behind the multi-V-shaped transfer function. The multi-V-shaped transfer function helps the proposed algorithm to preserve the diversity of the solutions and thus refine the convergence rate of the proposed algorithm. Also incorporating of RSS-based approximations can efficiently redistribute the search bats to enhance their diversity and to the emphasis on more explorative steps in case of convergence to the local optimum. The new transfer function and RSS strategies have improved the searching capacities and quality of the solutions of the proposed algorithm. Therefore, these strategies can assist the proposed
123
Complex Intell. Syst.
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
IBBA-RSS
11011111111010111111101101110111111
1689
1689
1689
0
BBA
11011111111010111111101101110111111
1689
1689
1689
0
KP12
KP13
KP14
KP15
KP16
KP17
KP18
KP19
KP20
CI
NA
1689
1686.5
1679
3.8188
B&B
NA
1689
NA
NA
NA
IBBA-RSS
0011110011111011111101011111001111111111
1821
1821
1821
0
BBA
0011110011111011111101011111001111111111
1821
1821
1821
0
CI
NA
1816
1807.5
1791
9.604
B&B
NA
1821
NA
NA
NA
IBBA-RSS
11110100111111011111011111111111101011111 1001
2033
2033
2033
0
BBA
11110100111111011111011111111111101011111 1001
2033
2030.3333
2016
6.0988
CI
NA
2020
2017
2007
4.749
B&B
NA
2033
NA
NA
NA
IBBA-RSS
11111001111111110110111111111010111111011101111111
2448
2448
2448
0
BBA
11111001111111110110111111111010011111011111111111
2440
2439.633333
2435
1.1591
CI
NA
2440
2436.166
2421
6.841
B&B
NA
2440
NA
NA
NA
IBBA-RSS
1110011111011111011111101001111111111001101101110101111
2643
2642.6000
2632
2.0103
BBA
1111011111011111011111101001111111111001101101111101110
2642
2640.4000
2614
5.5930
CI
NA
2643
2605
2581
22.018
B&B
NA
2440
NA
NA
NA
IBBA-RSS
1111101011011111011001111111110111111111011 11011111111101111
2917
2917
2917
0
BBA
111110101101111101100111111111011111111101111011111111101111
2917
2915
2893
6.1923
CI
NA
2917
2915
2905
4.472
B&B
NA
2917
NA
NA
NA
IBBA-RSS
11110101111011101111101111111111111 001011111100111011111111101010
2818
2817.6333
2814
1.0661
BBA
11111101111011101101101111111111111 001011111100111111111111101010
2809
2808.3333
2802
1.881549
CI
NA
2814
2773.66
2716,
18.273
B&B
NA
2818
NA
NA
NA
IBBA-RSS
111110111010110111011111110101110101111 1111100111111111011011111111111
3223
3222.6000
3219
1.1017
BBA
11111011101011011001111111010111110111111 11111111011111011011111111111
3213
3212.9000
3209
1.4936
CI
NA
3221
3216
3211
4.3589
B&B
NA
3223
NA
NA
NA
IBBA-RSS
01101111101100101111111111101111110011110111 1111011111111001111111111101101
3614
3613.2333
3605
2.4166
BBA
0110111110110010111111111110111111001111011 11111111111110100111111111101101
3602
3600.3793
3588
4.1611
CI
NA
3614
3603.8
3591
8.035
B&B
NA
3614
NA
NA
NA
123
Complex Intell. Syst. Table 5 Comparisons of the large-size KP IBBA-RSS
BBA
SBHS
IHS
GHS
SAHS
EHS
NGHS
NDHS
KP21 Best
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
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
Worst
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
0.45
KP22
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
Worst
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
Worst
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
Worst
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
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
Worst
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
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
Worst
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
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
KP23
KP24
KP25
1.23
KP26
1.9
KP27
1.95
1.33
1.41
2.14
123
Complex Intell. Syst. Table 5 continued IBBA-RSS
BBA
SBHS
IHS
GHS
SAHS
Worst
965.5550
962.6020
964.7
Mean
966.0164
962.661
964.85
Std
0.1099
0.0152
EHS
NGHS
NDHS
891.26
895.58
897.62
900.63
899.04
937.07
938.31
923.45
903.83
940.72
941.97
0.06
2.68
1.77
926.62
2.54
1.68
1.7
2
KP28
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
Worst
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
KP29
KP30 Worst
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
Worst
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
3.33
2.93
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
KP31
3.2
2.71
3.68
KP32 Worst
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
0.06
3.81
2.78
3.09
2.26
4.06
1.323
algorithm to switch between exploration and exploitation behaviors more effectively. Performance assessment Regarding the assessment, the performance of the proposed algorithm is investigated through using the Wilcoxon signed ranks (WSRs) test for a better comparison [43]. WSRs test is a nonparametric test that utilized in a hypothesis testing situation involving a design with two samples [42]. It is a pair-wise test that aims to find out significant differences between the behaviors of two algorithms. WSRs test is working as follows: First, the difference between the scores of the two algorithms on ith of n problems and the differences are ranked according to their absolute values. Second R + and R − are determined, where R + is the sum of positive ranks, while R − is the sum of negative ranks; then the minimum of R + and R − is obtained. If the result of the test is returned in p < 0.05 (i.e. p-value is the probability of the null
123
4
hypothesis being true) indicates a rejection of the null hypothesis, while p > 0.05 indicates a failure to reject the null hypothesis. Therefore, we apply the WSRs test for the proposed IBBA-RSS algorithm against the different algorithms that appear in Table 2 and the obtained results for WSRs test is reported in Table 7. Also, the WSRs test is employed for the results of the medium size KP instances that depicted in Table 4, where obtained results for WSRs test for the medium size instances are reported in Table 8. Also the WSRs test is employed for the results of the largescale KP instances that depicted in Tables 5 and 6, where obtained results for WSRs test for the large-scale instances are reported in Table 9. From Tables 7, 8 and 9, it can be concluded that the proposed IBBA-RSS has superior characteristics both in the high quality of the solution and robustness of the results. Also, it can keep a significant balance between the global exploration and the local exploitation.
Complex Intell. Syst.
(a) 1450
(b)
1700
1400 1600
Profit (KP12)
Profit (KP11)
1350 1300 1250
1500
1400
1200 1300
1100
BBA IBBA-RSS
BBA IBBA-RSS
1150
1200
0
50
100
150
200
250
300
350
400
0
50
Iteration Convergence behavior for LKP11
250
300
350
400
2100 2000
1700
1900
Profit (KP14)
Profit (KP13)
(d)
1600 1500
1800 1700 1600
1400 1300
1500
BBA IBBA-RSS 0
50
100
150
200
250
300
350
1400
400
BBA IBBA-RSS 0
Iteration Convergence behavior for LKP13
50
100
150
200
250
300
350
400
Iteration
Convergence behavior for LKP14
(e) 2500
(f)
2700 2600
2400
2500
Profit (KP16)
2300
Profit (KP15)
200
Iteration
1800
2200 2100 2000 1900
2400 2300 2200 2100
1800
0
50
100
150
200
250
300
350
BBA IBBA-RSS
2000
BBA IBBA-RSS
1700 1600
150
Convergence behavior for LKP12
(c) 1900
1200
100
400
1900
0
100
300
400
500
Iteration
Iteration
Convergence behavior for LKP15
200
Convergence behavior for LKP16
Fig. 7 The Convergence behavior for medium KP (KP11 –KP20 )
Most of the p values reported in Tables 7, 8 and 9 are less than 0.05 (5% significance level) which is a robust evidence against the null hypothesis, concluding that the obtained results by the proposed approach are statistically better and they have not happened by chance.
Convergence analysis To analyze the convergence analysis of the proposed algorithm, statistical measures, Wilcoxon signed ranks (WSRs) test and improvement ratio were developed. Tables 2, 4, 5
123
Complex Intell. Syst.
(g)
(h)
3000
2900 2800 2700
Profit (KP18)
Profit (KP17)
2800
2600
2400
2500 2400 2300 2200
2200
2000
2600
BBA IBBA-RSS
2000
0
100
200
300
400
BBA IBBA-RSS
2100
500
0
100
Iteration
3000
3400
2800
2600
100
200
300
400
3200 3000 2800
BBA IBBA-RSS 0
500
3800 3600
Profit (KP20)
Profit (KP19)
(j)
3200
2200
400
Convergence behavior for LKP18
3400
2400
300
Iteration
Convergence behavior for LKP17
(i)
200
500
Iteration
Convergence behavior for LKP19
2600
BBA IBBA-RSS 0
100
200
300
400
500
Iteration Convergence behavior for LKP20
Fig. 7 continued
and 6 demonstrated the superiority of the proposed approach regarding optimality. Further, the nonparametric WSRs is employed to offer the winner algorithm, where Tables 7, 8 and 9 show that the proposed algorithm outperforms the other comparative algorithms regarding the obtained p value. Also, the improvement ratio for the large-scale test instances is recorded as 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, when comparing IBBA-RSS with BBA phase, while the improved ratio obtained by comparing IBBA-RSS with V-BBA is equivalent to 5.4714, 6.6404, 2.4526, 2.6791, 1.7067, 1.7673, 2.2394, 0.9915, 0.5675, 1.6160, 1.4548 and 3.2288%, respectively. Further, comparing BBA phase with 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 on the above-improved ratios, it can be concluded that proposed IBBA-RSS algorithm has better ratios. From the practi-
123
cal point of view for large-scale problems, these ratios are very significant. Despite the high dimensionality of the KP problems, it is noteworthy that the proposed algorithm is competent to provide very significant results in a small number of iterations compared to the other algorithms. Regarding presented analyses, it can be concluded that the inherent characteristic of this improvement is contained in incorporating the RSS as a local search strategy that accelerates the convergence behavior and avoids the systematic running of the algorithm without any improvements in the outcomes. It can be concluded that the proposed IBBARSS has a significant performance and that the immature convergence inaccuracies of BBA phase are mitigated, efficiently. In this subsection, a comparative study has been carried out to evaluate the performance of the proposed IBBA-RSS algorithm concerning the binarization strategy and hybridization. In this respect, a new strategy is introduced based on
Complex Intell. Syst. Table 6 Comparisons for the large sizes KP (continued) PSFHS
BHS
DBHS
NGHS1
ABHS
ABHS1
ITHS
V-BBA
KP21 Best
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
1.82
0.1
0.43
0.17
0.09
0.1
0.07
Std Best
0.84
106.52
129.27
118.24
128.41
129.31
129.29
129.24
122.4199
99.89
129.06
115.95
127.72
129
128.95
128.88
121.1979
Worst
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
2.92
0.13
1.12
0.21
0.18
0.19
0.60
Median KP22
Std
0.6
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
Worst
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
0.64
0.24
0.34
0.36
KP23
Std
2.71
0.25
1.27
Best
234.23
311.85
257.61
310.1
312.51
310.28
310.94
307.8299
2.54
Median
224.64
310.6
252.58
308.39
311.92
309.32
309.85
307.7595
KP24 Worst
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
1.86
0.83
0.48
0.61
0.61
Std
4.26
0.6
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
Worst
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
KP25
Std
1.02
0.51
0.93
0.83
Best
453.2
4.29
626.04
0.75
482.59
2.8
626.27
632.38
620.31
624.04
628.0995
3.84
Median
431.71
623.09
475.73
623.07
630.33
618.12
621.68
628.0995
Worst
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
6.68
1.25
3.15
1.64
1.02
1.28
1.18
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
KP26
Std
4.30
KP27 Worst
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
Best
659.05
938.36
700.81
944.36
950.7
1.11
927.6
1.36
937.62
1.59
956.4658
4.72
Median
631.94
935.21
693.91
941.18
949.42
924.15
933.82
956.4548
Worst
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
KP28
123
Complex Intell. Syst. Table 6 continued PSFHS Std
BHS
DBHS
NGHS1
ABHS
ABHS1 1.64
ITHS 1.8
V-BBA
8.37
1.29
3.87
2
1
Best
780.89
1118.83
833.43
1129.81
1140.69
1106.12
1121.58
1150.6657
2.15
Median
755.93
1115.78
819.96
1127.63
1136.71
1102.06
1115.3
1150.5952
Worst
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
9.97
1.68
4.59
1.93
2.57
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
KP29
Std
1.8
1.6
0.07
KP30 Worst
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
5.44
1.71
2.07
2.26
Best
1092.87
9.72
1579.8
1.85
1148.13
1613.95
1623.3
1.54
1559.19
1585.94
1644.1290
1.05
Median
1061.59
1577.19
1140.14
1609.36
1618.89
1553.33
1579.92
1644.1290
Worst
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
KP31
Std
10.58
1.74
4.17
2.17
2.09
2.71
3.43
Best
1269.54
1830.65
1332.06
1875.71
1879.12
1803.16
1839.01
1865.5547
17.00
Median
1222.61
1826.22
1314.44
1871.05
1874.11
1797.45
1830.55
1865.5547
Worst
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
12.96
2.38
2.28
2.65
2.75
3.88
KP32
Std
7
the multi-V-shaped transfer function that has effective exploration than the sigmoid transfer function. On the one hand, hybridization mechanism is implemented through incorporating the RSS as a local search step can avoid the trapping on undesirable values. Also, this hybridization can accelerate the convergence and also preserve its searching efficiency and effectiveness. On the other hand, the proposed algorithm is highly competitive when comparing it with the other methods regarding calculating the statistical measures and Wilcoxon signed ranks test. So the use of the hybrid approach has a great potential for solving large-scale knapsack problems. Moreover, it can be recognized from the obtained results on 32 test instances that the proposed IBBA-RSS is capable of attaining satisfactory solutions with an appropriate exploitation potential. The reason is that the proposed RSS-embedded mechanisms can stimulate the exploitation tendency of the BBA phase, effectively. Careful observation will reveal the following achievements of the proposed IBBA-RSS algorithm: (a) IBBA-RSS performs better on all large-scale KP problems while the other algorithms often miss the better results.
123
3.0132
(b) IBBA-RSS can obtain a relatively stable and better result on the whole regarding the statistical measures. (c) IBBA-RSS combines the merits of the BBA and RSS to obtain an elevated performance. (d) IBBA-RSS gives a promising improvement in the problem profit and can avoid the trapping in local optima. (e) The proposed methodology opens up numerous research directions for solving the different variants of KP problems such as multi-dimensional KP and quadratic KP.
Conclusions and future work This paper presented a novel injective binary bat algorithmbased rough set scheme (IBBA-RSS) for solving 0/1 knapsack problems. To overcome the BBA’s problem of being converged to local optima and to improve its exploration and exploitation tendencies, it is hybridized with the RSS phase. Furthermore, the survival process of bats is achieved based on the injective (one-to-one) strategy, where the fit one replaces the worst one based on feasibility rule. The performance of
Complex Intell. Syst. 64
(b)
135
62
130
60
125
58
120
Profit (KP22)
21
Profit (KP )
(a)
56 54 52
110 105 100
50 48
95
BBA IBBA-RSS V-BBA
46 44
115
0
50
100
150
200
250
BBA IBBA-RSS V-BBA
90 85
300
0
100
200
300
Iteration
500
600
Convergence behavior for LKP22
Convergence behavior for LKP21
(c)
400
Iteration
200
(d)
320
190 300
Profit (KP24)
Profit (KP23)
180 170 160 150
280
260
240
140 BBA IBBA-RSS V-BBA
130 120
0
100
200
300
400
500
BBA IBBA-RSS V-BBA
220
200
600
0
200
400
Convergence behavior for LKP23
(e)
600
800
1000
Iteration
Iteration
460
Convergence behavior for LKP24
(f)
650
440 420
600
Profit (KP )
380
26
Profit (KP25)
400
360 340
550
500
320 300
BBA IBBA-RSS V-BBA
280 260
0
200
400
600
800
1000 1200 1400 1600 1800
Iteration
Convergence behavior for LKP25
450
400
BBA IBBA-RSS V-BBA 0
200
400
600
800
1000 1200 1400 1600 1800
Iteration
Convergence behavior for LKP26
Fig. 8 The Convergence behavior for large scale KP (KP21 –KP32 )
123
Complex Intell. Syst.
(g)
(h) 1000
800
980 750
960 940
Profit (KP28)
Profit (KP27)
700
650
600
920 900 880 860 840
550
500
BBA IBBA-RSS V-BBA 0
500
1000
1500
800 2500
2000
BBA IBBA-RSS V-BBA
820 0
2000
4000
Iteration
Convergence behavior for LKP26
(i)
6000
8000
10000
Iteration
Convergence behavior for LKP27
(j)
1200
1300
1150
1200
1100
Profit (KP30)
Profit (KP29)
1050 1000 950 900 850 800
BBA IBBA-RSS V-BBA
750 700
0
2000
4000
6000
8000
1100
1000
900 BBA IBBA-RSS V-BBA
800
700
10000
0
2000
4000
6000
8000 10000 12000 14000 16000
Iteration
Iteration
Convergence behavior for LKP28
Convergence behavior for LKP29
(k) 1700
(l)
2000 1900
1600
1800
Profit (KP32)
Profit (KP31)
1500 1400 1300 1200
1600 1500 1400 1300
BBA IBBA-RSS V-BBA
1100 1000
1700
0
2000 4000 6000 8000 10000 12000 14000 16000 18000
Iteration Convergence behavior for LKP30 Fig. 8 continued
123
BBA IBA-RSS V-BBA
1200 1100
0
2000 4000 6000 8000 10000 12000 14000 16000 18000
Iteration
Convergence behavior for LKP31
Complex Intell. Syst. Table 7 Wilcoxon test for comparison results in Table 2
Table 8 Wilcoxon test for comparison results in Table 4
Table 9 Wilcoxon test for comparison results in Tables 5 and 6
Compared methods
Solution evaluations
The proposed
Compared algorithms
R−
R+
p Value
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
–
–
Compared methods
Winner
Solution evaluations R−
R+
p Value
Winner algorithm
BBA
15
0
0.043114
IBBA-RSS
CI
12.5
8.5
0.674987
IBBA-RSS
4
0.715001
IBBA-RSS
The proposed
Compared algorithms
IBBA-RSS IBBA-RSS IBBA-RSS
B&B
6
Compared methods
Solution evaluations
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 investigated through using small-scale, medium- scale and large-scale instances of 0–1 KP. The proposed algorithm is compared with several algorithms from the literature. Based on statistical measures, the proposed algorithm can explore/exploit better-quality solutions, and it outperforms as the best amongst the other compared algorithms. Convergence trend for average best results for IBBA-RSS is preferable to equivalent curves for BBA. Also, non-parametric statistical tests also affirm that the optimality of solutions is enriched, significantly. The results reveal that IBBA-RSS is competent to provide very competitive results compared to BBA and other investigated algorithms. Regarding presented analyses,
it can be concluded that IBBA-RSS has a desirable performance and the immature convergence inaccuracies of the BBA phase is mitigated, efficiently. For future works, it is possible to investigate the proposed IBBA-RSS algorithm to solve different forms of KP problems like multi-dimensional KP and quadratic KP. Further research on using other metaheuristic algorithms such as krill herd, monarch butterfly optimization (MBO), earthworm optimization algorithm (EWA), elephant herding optimization (EHO) and moth search (MS) algorithm need to be developed to solve different forms of KP problems. Finally, I hope to design a new version of KP that is bi-level KP.
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21. 22.
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