New binary bat algorithm for solving 0–1 knapsack problem
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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 largescale 0–1 knapsack problems.
Keywords
Bat algorithm Local search scheme Knapsack problemIntroduction
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, portfolio optimization, investment decisionmaking, project selection, resource distribution, and so on. Unfortunately, KP is nonpolynomial (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 largescale 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 goodquality 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 multimutation mechanism to solve KP. Zhang et al. [8] introduced amoeboid organism algorithm to solve 0–1 KP. Bhattacharjee and Sarmah [9] proposed a shuffled frogleaping 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 optimization (PSO), artificial fishswarm 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, 12, 13, 14, 15, 16, 17, 18, 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 largescale 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 microbats [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 microbats 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, 21, 22]. A new trend in bat algorithms is focusing on hybridizing BA with different strategies [23, 24, 25, 26, 27, 28, 29, 30, 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 selfstrategy. 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 selfadaptation of control parameters. Further, binary versions of BA were developed in [29, 30, 31]. Mirjalili et al. [29] introduced a binary version of BA by employing a Vshaped 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 largescale 0–1 KP has not been yet studied. Second, many optimization algorithms suffer from entanglement in local optima when solving largescale problems. Last, solving largescale knapsack problems have not received adequate attention yet. Hence solving largescale 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 multiVshaped 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 onetoone 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 largescale 0–1 knapsack problems.
The main contributions of this approach are to (1) introduce a novel binary bat algorithm (NBBA) for solving largescale 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 (onetoone) 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 largesize 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 multiVshaped 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 (OnetoOne) 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. 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.
Preliminaries
Problem description
There are N items and the knapsack capacity is \(C\cdot w_{j}\) 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:
In largescale 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:
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
Bat algorithm (BA)
Velocity and position
Loudness and pulse emission
The implementation steps of bat algorithm
Step 1: Set the basic parameters: population size (PS), attenuation coefficient of loudness \(\delta \), increasing coefficient of pulse emission \(\gamma \), the maximum loudness \(A^{0}\) and maximum pulse emission \(r^{0}\) and the maximum number of iterations T.
Step 2: Define objective function \(f(x_{i}),i=1,2,\ldots ,\text {PS}\).
Step 3: Initialize pulse frequency \(\alpha _i \in [\alpha _{\min }, \alpha _{\max }]\);
Step 4: Initialize the bat population x and v.
Step 5: Start the main loop. If \(\hbox {rand}<r_i\), 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.
Step 6: If \(\hbox {rand}<A_i\) and \(f(x_{i})<f(x^{\mathrm{{best}}})\), accept the new solutions and fly to the new position.
Step 7: If \(f(x_{i})<f_{\min }\), 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 nonempty finite set of attributes. An information function f maps an object to its attribute, i.e., \(f_a:U\rightarrow V_a \) for every \(a\in A\), where \(V_a \) 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 \(\text {DT}=(U,A\cup \{d\},f)\), where \(d\notin A\) is used as supervised learning. The elements of A are called conditional attributes.
Definition 2
(Indiscernibility) For an attribute set \(B\subseteq A\), the equivalence relation induced by B is called a Bindiscernibility relation, i.e., \(\mathrm{{IND}}_\mathrm{T} (B)=\{(x,y)\in U^{2}  \forall a\in B, f_a (x)=f_a (y)\}\) The equivalence classes of the Bindiscernibility relation are denoted as \(I_B(x)\).
Definition 3
(Set approximation) Let \(X\subseteq U\) and \(B\subseteq A\) in an IS, the Blower approximation of X is the set of objects that belongs to X with certainty, i.e., \({\underline{B}}X=\{{x\in U  I_B (x)\subseteq X}\}\). The Bupper approximation is the set of objects that possibly belongs to X, where \(\bar{{B}}X=\{{x\in U  I_B (x)\cap X\ne \phi }\}\).
Definition 4
(Reducts) If \(X_{\mathrm{{DT}}}^1,X_{\mathrm{{DT}}}^2,\ldots ,X_{\mathrm{{DT}}}^r \) are the decision classes of \(\text {DT}\), the set \(\text {POS}_B (d)=\underline{B}X^{1}\cup \underline{B}X^{2}\cup \cdots \cup \underline{B}X^{r}\) is the Bpositive region of \(\text {DT}\). A subset \(B\subseteq A\) is a set of relative reducts of \(\text {DT}\) if and only if \(\text {POS}_B (d)=\text {POS}_C (d)\) and \(\text {POS}_{B\{b\}} (d)\ne \text {POS}_C (d), \forall b\in B\). In the same way \(\text {POS}_B (X)\), \(BN_B (X)\) and \(\text {NEG}_B (X)\) are defined below (refer to Fig. 2).

\(\text {POS}_B (X)={\underline{B}}X\Rightarrow \) certainly member of X

\(\text {NEG}_B (X)=U\bar{{B}}X\Rightarrow \) certainly nonmember of X

\(BN_B (X)={\bar{B}}X{\underline{B}}X\Rightarrow \) possibly the member of X.
The proposed algorithm (IBBARSS)
In this section, we present the injective binary bat algorithm based rough set scheme (IBBARSS) 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 (onetoone) strategy, where the fit one replaces the worst one based on feasibility rule. By this methodology, the IBBARSS enhances the diversity of bats and improves the convergence performance. The details of the proposed algorithm are given below.
Binary position scheme
In this step, each bat of the population is a solution to the KP, where each bat is represented by the nbit binary string, where n is the number of decision variables (items) in the KP. For example, considering that \(x_{i}\) represents the bat bits, then its jth bit \(x_{ij} =(x_{i1},x_{i2},\ldots ,x_{in})\) is a binary variable, 0 or 1.
Binary velocity scheme
Evaluation
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 \(i,x_{ij}\) 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 condition and decision attributes \([{x_{i1},x_{i2},\ldots ,x_{in}{\{D\}}}]_{i=1}^{\mathrm{PS}}\), 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 \(X\subseteq U\) that contains the one values of D and \(B=\{x_1,x_2,\ldots ,x_n\}\) is the set of condition attribute in an IS. Then according to Definition 4, the redundant items are eliminated where \({\underline{B}}X,{\bar{B}}X\) \(BN_B (X)\) and \(\text {NEG}_B (X)\) of X are obtained based on the process of attribute reduction.
The parameters, dimension and optimum of ten test instances
Problem  Parameter  Dimension  Optimum 

KP\(_{1}\)  \(w=(95,4,60,32,23,72,80,62,65,46),\) \(C=269,\) \(p=(55,10,47,5,4,50,8,61,85,87)\)  10  295 
KP\(_{2}\)  \(w= (92,\) 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 
KP\(_{3}\)  \(w=(6,5,9,7),\) \(C =20,\) \(p=(9,11,13,15)\)  4  35 
KP\(_{4}\)  \(w=(2,4,6,7),\) \(C =11,\) \(p=(6,10,12,13)\)  4  23 
KP\(_{5}\)  \(w= (56.358531,\) 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 
KP\(_{6}\)  \(w=(30,25,20,18,17,11,5,2,1,1),\) \(C =60,\) \(p=(20,18,17,15,15,10,5,3,1,1)\)  10  52 
KP\(_{7}\)  \(w=(31,10,20,19,4,3,6),\) \(C =50,\) \(p=(70,20,39,37,7,5,10)\)  7  107 
KP\(_{8}\)  \(w=(983, 982,\) 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 
KP\(_{9}\)  \(w=(15,20,17,8,31),\) \(C=80,\) \(p=(33,24,36,37,12)\)  5  130 
KP\(_{10}\)  \(w= (84, 83,\) 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 
Comparisons of the small sizes KP
KP\(_1\)  KP\(_2\)  KP\(_3\)  KP\(_4\)  KP\(_5\)  KP\(_6\)  KP\(_7\)  KP\(_8\)  KP\(_9\)  KP\(_{10}\)  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  
IBBARSS  
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 
The parameters, dimension and optimum of ten test problems
Problem  Parameter  Dimension  Optimum 

KP\(_{11}\)  \(w=[46,\) 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 
KP\(_{12}\)  \(w=[7,\) 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 
KP\(_{13}\)  \(w=[28,\) 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 
KP\(_{14}\)  \(w=[18,\) 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 
KP\(_{15}\)  \(w=[15,\) 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 
KP\(_{16}\)  \(w=[27,\) 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 
KP\(_{17}\)  \(w=[7,\) 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 
KP\(_{18}\)  \(w=[47,\) 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 
KP\(_{19}\)  \(w=[4,\) 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 
KP\(_{20}\)  \(w=[24,\) 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 
Comparison results for the medium sizes KP (KP\(_{11}\)–KP\(_{20})\)
Fun  Algorithm  Obtained solution  Best  Mean  Worst  Std 

KP\(_{11}\)  IBBARSS  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  
KP\(_{12}\)  IBBARSS  11011111111010111111101101110111111  1689  1689  1689  0 
BBA  11011111111010111111101101110111111  1689  1689  1689  0  
CI  NA  1689  1686.5  1679  3.8188  
B&B  NA  1689  NA  NA  NA  
KP\(_{13}\)  IBBARSS  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  
KP\(_{14}\)  IBBARSS  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  
KP\(_{15}\)  IBBARSS  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  
KP\(_{16}\)  IBBARSS  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  
KP\(_{17}\)  IBBARSS  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  
KP\(_{18}\)  IBBARSS  11110101111011101111101111111111111001011111100111011111111101010  2818  2817.6333  2814  1.0661 
BBA  11111101111011101101101111111111111001011111100111111111111101010  2809  2808.3333  2802  1.881549  
CI  NA  2814  2773.66  2716,  18.273  
B&B  NA  2818  NA  NA  NA  
KP\(_{19}\)  IBBARSS  1111101110101101110111111101011101011111111100111111111011011111111111  3223  3222.6000  3219  1.1017 
BBA  1111101110101101100111111101011111011111111111111011111011011111111111  3213  3212.9000  3209  1.4936  
CI  NA  3221  3216  3211  4.3589  
B&B  NA  3223  NA  NA  NA  
KP\(_{20}\)  IBBARSS  011011111011001011111111111011111100111101111111011111111001111111111101101  3614  3613.2333  3605  2.4166 
BBA  011011111011001011111111111011111100111101111111111111110100111111111101101  3602  3600.3793  3588  4.1611  
CI  NA  3614  3603.8  3591  8.035  
B&B  NA  3614  NA  NA  NA 
Comparisons of the largesize KP
IBBARSS  BBA  SBHS  IHS  GHS  SAHS  EHS  NGHS  NDHS  

KP\(_{21}\)  
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  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 
KP\(_{22}\)  
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 
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 
KP\(_{23}\)  
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 
KP\(_{24}\)  
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 
KP\(_{25}\)  
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  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 
KP\(_{26}\)  
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  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 
KP\(_{27}\)  
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  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 
KP\(_{28}\)  
Worst  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 
KP\(_{29}\)  
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 
KP\(_{30}\)  
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 
KP\(_{31}\)  
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  3.2  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 
KP\(_{32}\)  
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  1.323  0.06  3.81  2.78  4  3.09  2.26  4.06 
Injective updating based on feasibility rule
 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.
Experimental results and analysis
In this section, the performance of the IBBARSS algorithm is extensively investigated by a large number of experimental studies. Ten lowdimensional, ten medium size and twelve largescale instances are considered to validate the robustness of the proposed IBBARSS 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.
Lowdimensional 0–1 knapsack problems
In this section, the performance of proposed algorithm is investigated to solve ten lowdimensional 0–1 knapsack problems, where these instances are taken from [36, 37]. The 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 IBBARSS 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 lowdimensional knapsack problems, where the proposed IBBARSS 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 problems. 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 KP\(_{10}\)–KP\(_{15}\) and 500 iterations for KP\(_{16}\)–KP\(_{20}\). The proposed IBBARSS is run 30 times for each instance with 30 bats for the population size.
To demonstrate the effectiveness and robustness of the proposed IBBARSS, it is implemented and compared with the BBA phase. The statistical measures for the each instance is obtained using BBA and IBBARSS 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 IBBARSS 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 IBBARSS 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 the solutions of the proposed IBBARSS demonstrate that it is capable of outperforming the BBA phase. The solutions of the proposed IBBARSS are also superior to the results of the other evaluated techniques in the most of the test cases.
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 IBBARSS gives better results than BBA, and consequently, the profit for each instance is improved.
Largescale 0–1 knapsack problems
To further prove the proficiency of the proposed IBBARSS algorithm, twelve largescale 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 largescale KP (KP\(_{11}\)–KP\(_{22})\) 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 IBBARSS is run 30 times for each instance with 30 bats for the population size.
The proposed IBBARSS and BBA phase are compared with the Vshaped binary bat algorithm (VBBA) which was developed in [29]. The statistical measures for each instance using the proposed IBBARSS 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 KP\(_{21}\)–KP\(_{32}\) are unknown.
The proposed IBBARSS is compared with 16 different algorithms as in Tables 5 and 6. From these tables, we can see that the proposed IBBARSS outperforms the other algorithms for all KP instances (KP\(_{21}\)–KP\(_{32}\)). 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 IBBARSS achieves better simulation results than the BBA phase and VBBA. 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 IBBARSS with BBA phase, while the improved ratio obtained by comparing IBBARSS with VBBA 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 VBBA 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 largescale problems. Based on the aboveimproved ratios, it can be concluded that proposed IBBARSS algorithm has better ratios. Therefore, the proposed IBBARSS is robust approach and has powerful searching quality.
Comparisons for the large sizes KP (continued)
PSFHS  BHS  DBHS  NGHS1  ABHS  ABHS1  ITHS  VBBA  

KP\(_{21}\)  
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 
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 
KP\(_{22}\)  
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 
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 
KP\(_{23}\)  
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 
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 
KP\(_{24}\)  
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 
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 
KP\(_{25}\)  
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 
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 
KP\(_{26}\)  
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 
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 
KP\(_{27}\)  
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  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 
KP\(_{28}\)  
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 
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 
KP\(_{29}\)  
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 
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 
KP\(_{30}\)  
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  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 
KP\(_{31}\)  
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 
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 
KP\(_{32}\)  
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 
Std  12.96  2.38  7  2.28  2.65  2.75  3.88  3.0132 
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 pairwise 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. pvalue is the probability of the null 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 IBBARSS 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 largescale instances are reported in Table 9. From Tables 7, 8 and 9, it can be concluded that the proposed IBBARSS 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.
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 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 largescale 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 IBBARSS with BBA phase, while the improved ratio obtained by comparing IBBARSS with VBBA 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 VBBA 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 aboveimproved ratios, it can be concluded that proposed IBBARSS algorithm has better ratios. From the practical point of view for largescale 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.
Wilcoxon test for comparison results in Table 2
Compared methods  Solution evaluations  

The proposed  Compared algorithms  \(R^{}\)  \(R^{+}\)  p Value  Winner 
IBBARSS  BHS  45  0  0.007686  IBBARSS 
IBBARSS  DBHS  0  0  –  – 
IBBARSS  NGHS1  3  0  0.179712  IBBARSS 
IBBARSS  ABHS  0  0  –  – 
IBBARSS  ABHS1  28  0  0.017960  IBBARSS 
IBBARSS  SBHS  0  0  –  – 
Wilcoxon test for comparison results in Table 4
Compared methods  Solution evaluations  

The proposed  Compared algorithms  \(R^{}\)  \(R^{+}\)  p Value  Winner algorithm 
IBBARSS  BBA  15  0  0.043114  IBBARSS 
IBBARSS  CI  12.5  8.5  0.674987  IBBARSS 
IBBARSS  B&B  6  4  0.715001  IBBARSS 
Compared methods  Solution evaluations  

The proposed  Compared algorithms  \(R^{}\)  \(R^{+}\)  p Value  Winner algorithm 
IBBARSS  BBA  78  0  0.002218  IBBARSS 
IBBARSS  SBHS  78  0  0.002218  IBBARSS 
IBBARSS  IHS  78  0  0.002218  IBBARSS 
IBBARSS  GHS  78  0  0.002218  IBBARSS 
IBBARSS  SAHS  78  0  0.002218  IBBARSS 
IBBARSS  EHS  78  0  0.002218  IBBARSS 
IBBARSS  NGHS  78  0  0.002218  IBBARSS 
IBBARSS  NDHS  78  0  0.002218  IBBARSS 
IBBARSS  PSFHS  78  0  0.002218  IBBARSS 
IBBARSS  BHS  78  0  0.002218  IBBARSS 
IBBARSS  DBHS  78  0  0.002218  IBBARSS 
IBBARSS  NGHS1  78  0  0.002218  IBBARSS 
IBBARSS  ABHS  78  0  0.002218  IBBARSS 
IBBARSS  ABHS1  78  0  0.002218  IBBARSS 
IBBARSS  ITHS  78  0  0.002218  IBBARSS 
IBBARSS  VBBA  78  0  0.002218  IBBARSS 
 (a)
IBBARSS performs better on all largescale KP problems while the other algorithms often miss the better results.
 (b)
IBBARSS can obtain a relatively stable and better result on the whole regarding the statistical measures.
 (c)
IBBARSS combines the merits of the BBA and RSS to obtain an elevated performance.
 (d)
IBBARSS 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 multidimensional KP and quadratic KP.
Conclusions and future work
This paper presented a novel injective binary bat algorithmbased rough set scheme (IBBARSS) 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 (onetoone) strategy, where the fit one replaces the worst one based on feasibility rule. The performance of the proposed algorithm has been extensively investigated through using smallscale, medium scale and largescale 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 betterquality solutions, and it outperforms as the best amongst the other compared algorithms. Convergence trend for average best results for IBBARSS is preferable to equivalent curves for BBA. Also, nonparametric statistical tests also affirm that the optimality of solutions is enriched, significantly. The results reveal that IBBARSS is competent to provide very competitive results compared to BBA and other investigated algorithms. Regarding presented analyses, it can be concluded that IBBARSS 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 IBBARSS algorithm to solve different forms of KP problems like multidimensional 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 bilevel KP.
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