Hybrid Learning Moth Search Algorithm for Solving Multidimensional Knapsack Problems

Abstract

The moth search algorithm (MS) is a relatively new metaheuristic optimization algorithm which mimics the phototaxis and Lévy flights of moths. Being an NP-hard problem, the 0–1 multidimensional knapsack problem (MKP) is a classical multi-constraint complicated combinatorial optimization problem with numerous applications. In this paper, we present a hybrid learning MS (HLMS) by incorporating two learning mechanisms, global-best harmony search (GHS) learning and Baldwinian learning for solving MKP. (1) GHS learning guides moth individuals to search for more valuable space and the potential dimensional learning uses the difference between two random dimensions to generate a large jump. (2) Baldwinian learning guides moth individuals to change the search space by making full use of the beneficial information of other individuals. Hence, GHS learning mainly provides global exploration and Baldwinian learning works for local exploitation. We demonstrate the competitiveness and effectiveness of the proposed HLMS by conducting extensive experiments on 87 benchmark instances. The experimental results show that the proposed HLMS has better or at least competitive performance against the original MS and some other state-of-the-art metaheuristic algorithms. In addition, the parameter sensitivity of Baldwinian learning is analyzed and two important components of HLMS are investigated to understand their impacts on the performance of the proposed algorithm.

1. Introduction

The multidimensional knapsack problem (MKP) [1,2] is a generalization of the 0–1 knapsack problem (0–1 KP) [3]. As a classical combinatorial optimization problem, numerous real-world applications can be modeled as MKP, such as capital budgeting [4,5], cutting stock [6], and loading problem [7,8].

Then the MKP is: given a set of n items and a set of m knapsacks (m < n), with c_j = profit of item j, b_i = capacity constraint of knapsack i,a_ij = resource consumption of item j in the ith knapsack. The goal of MKP is to select a subset of items so that the total profit of the selected items is a maximum while the total weights in each dimension i (i = 1, 2, …, m) do not exceed the corresponding capacity b_i. Formally,

$$max\begin{array}{cc}& z={\displaystyle \sum _{j=1}^n{c}_j}\end{array}{x}_j$$

(1)

$$s.t.\begin{array}{cc}& {\displaystyle \sum _{j=1}^n{a}_{ij}}\end{array}{x}_j\le b_i,\begin{array}{cc}& \forall \end{array}i\in \{1,2,\dots ,m\}$$

(2)

$$x_j\in \{0,1\},\begin{array}{cc}& \forall \end{array}j\in \{1,2,\dots ,n\}$$

(3)

where x_j (j = 1, 2, …, n) is a 0–1 decision variable, such that x_j = 1 if item j is assigned to a knapsack, x_j = 0 otherwise. When m = 1, MKP reduces to the 0–1 KP.

Known as the NP-hard combinatorial optimization problem of MKP, the conventional exact algorithms usually are unable to obtain a satisfactory solution in a reasonable time, especially for large-scale instances. It is because that combinatorial optimization problems often feature a combination of the explosion and then the search space grows exponentially with the expansion of the scale. In this case, metaheuristic algorithms are effective methods for MKP, which can obtain near-optimal solutions in a reasonable acceptable time. Representative metaheuristic algorithms for solving MKP include the two-phase tabu evolutionary algorithm (TPTEA) [9], binary particle swarm optimization (PSO) [10], quantum particle swarm optimization (QPSO) [11], diversity-preserving quantum particle swarm optimization (DQPSO*) [12], hybrid estimation of distribution algorithm (EDA) [13], memetic algorithm (MA) [14], binary grey wolf optimizer (GWO) [15], hybrid harmony search algorithm (HS) [16], binary multi-verse optimizer (MMVO) [17], cuckoo search (CS) [18], pigeon-inspired optimization algorithm (PIO) [19], binary moth search algorithm (MS) [20], sine cosine algorithm (SCA) [21], and binary slime mould algorithm (BSMA) [22]. For more information on evolutionary algorithms (EAs) [23] and exact methods for solving MKP problem, please refer to [24].

Recently, some novel metaheuristic algorithms have been proposed and used to solve various optimization problems, including continuous optimization problems and discrete optimization problems, such as differential evolution algorithm (DE) [25,26], cuckoo search algorithm (CS) [27], bat algorithm (BA) [28], krill herd (KH) [29,30], elephant herding optimization (EHO) [31], monarch butterfly optimization algorithm (MBO) [32,33], brain storm optimization algorithm (BSO) [34], fruit fly optimizer (FFO) [35], whale optimization algorithm (WOA) [36], wind driven optimization (WDO) [37], salp swarm algorithm (SSA) [38], Harris hawks optimization (HHO) [39], artificial Jellyfish Search (JS) optimizer [40], artificial rabbits optimization (ARO) [41], mountain gazelle optimizer (MGO) [42], moth-flame optimizer (MFO) [43], and many more.

The moth search (MS) [44,45] algorithm was recently developed by Wang and inspired by the phototaxis and Lévy flights of moths. Owing to its simplicity and high search efficiency, MS has been successfully applied to solve various optimization problems, such as the cloud task scheduling problem [46], drone placement problem [47], constrained optimization problems [48], discounted {0–1} knapsack problem [49], and set-union knapsack problem [50]. Previous studies have shown that MS is effective for solving various combinatorial optimization problems, especially for various variants of KP problems [49,51]. To the best of our knowledge, there are few literatures on applying MS to solve MKP. Hence, we concentrated on the MS algorithm for solving MKP.

In recent years, the learning-based metaheuristic algorithms have been extensively reported in literature because of their effectiveness and efficiency in solving various optimization problems. The core idea is that the metaheuristic algorithm combines specific learning operators or learning mechanisms to enhance itself some learning ability, and then owning better optimization behavior. There are many learning methods, for example, deep learning [52,53], reinforcement learning [54,55,56], transfer learning [57], Q-learning [58], information feedback [59], orthogonal learning [34], comprehensive learning [60], Baldwinian learning, and so on.

Inspired by Darwinian evolution, evolutionary computation mainly includes the random variation of individuals and fitness selection mechanism [61]. It seems time-consuming to search randomly for good genotypes without using phenotypes. One of the possible ways to surmount this deficiency is to integrate the learning mechanism into the evolutionary search. The learning mechanism can provide a more effective search path [62]. Hence, learning-based mechanisms have been extensive researched and applied in enhancing the search performances of evolution algorithms [14,31]. The Baldwin effect, sometimes called Baldwinian learning, was widely incorporated into a variety of evolutionary computing models to enhance their performance [61,63,64]. The above works motivate us to propose a Baldwinian learning MS to solve MKP.

Harmony search (HS) [65] is metaheuristic and imitates the musical improvization process to search for a perfect state of harmony. Since proposed, HS has always attracted much attention from researchers and has been successfully applied to deal with various optimization problems. An effective variant of HS, called global-best HS (GHS), was proposed by Omran [66]. Compared with HS and other variants, GHS alleviates the problem of tuning the parameter bw and can work efficiently on both continuous and discrete problems. The GHS algorithm, due to its easy implementation and quick convergence, has been applied to many fields. Xiang et al. [67] proposed a discrete GHS (DGHS) for solving 0–1 KP. Keshtegar et al. [68] proposed a Gaussian GHS (GGHS) algorithm for solving numerical optimization problems. EI-Abd et al. [69] proposed an improved GHS (IGHS) algorithm to solve continuous optimization. In essence, musical performances seek to find pleasing harmony, just like the process of finding the global optimal solution through continuous learning. Based on this, GHS, as another learning scheme, is integrated into the MS for a global search.

Aiming at resolving the above issues, we propose a hybrid learning MS (HLMS), by introducing the Baldwinian learning and GHS learning mechanism. In HLMS, a novel Baldwin learning strategy based on Cauchy distribution is proposed instead of Gaussian distribution in [70]. During the evolutionary process, each individual in the whole population performs GHS learning and Baldwinian learning successively with a certain probability to reduce the time-consumption. The beneficial combination and complementarity of these two mechanisms lead HLMS to evolve towards the global optimum. Intuitively, HLMS has better search performance than the original MS because of the good balance between the exploration capacity of GHS learning and the exploitation ability of the Baldwinian learning.

The novelty and main contributions of this work include.

• A novel Baldwinian learning strategy based on Cauchy distribution is proposed, the analysis and experiment of which show this strategy is more effective than the Baldwin learning strategy based on Gaussian distribution.
• Combined with Baldwin learning, GHS is an effective global search operator to enhance the exploration ability of HLMS. Meanwhile, the pitch adjustment process based on dimensional learning can generate a large jump in the search process.
• To reduce computation costs, the proposed HLMS triggers Baldwin learning and GHS learning with a certain probability in each iteration.
• Exploration and exploitation are two common and fundamental features of any optimization method. In the evolution of HLMS, Lévy flight and GHS learning are mainly responsible for exploration, whilst flight straightly and Baldwinian mainly implement exploitation.

The rest of this paper is organized as follows. Section 2 reviews the MS algorithm, the Baldwinian learning scheme, and the GHS algorithm. In Section 3, the proposed HLMS for the MKP is introduced in detail. Extensive experiments and comparisons are conducted in Section 4. Finally, conclusions and suggestions are provided in Section 5.

2. Preliminaries

In this section, we summarize the core idea of MS, Baldwinian learning, and Global-best HS algorithm, which form the basis of the proposed HLMS framework.

2.1. Moth Search Algorithm

The moth search algorithm (MS) [44] is a new metaheuristic algorithm developed by Wang and inspired by the phototaxis and Lévy flights of moths. Based on these two behaviors, the flight straightly operator and Lévy flight operator of MS are derived, which can achieve good balance between the exploration capability and exploitation ability. Meanwhile, the whole population is subdivided into two subpopulations (named subpopulation1 and subpopulation2) based on the fitness of moth individuals. To this end, the position of offspring in subpopulation1 and subpopulation2 is updated by Lévy flight and flight straightly, respectively.

In the Lévy flight stage, the core mathematical formulation is:

$$X_i^{t+1}=X_i^t+\alpha L(s)$$

(4)

$$\alpha =S_{max}/t^2$$

(5)

$$L(s)=\frac{(\beta -1)\Gamma (\beta -1)\sin \left(\frac{\pi (\beta -1)}{2}\right)}{\pi s^{\beta }}$$

(6)

where X_i^t and X_i^t+1 denote the position vector of the ith moth at generation t and t + 1, respectively. $\alpha$ is the scale factor and S_max is the max walk step. L(s) represents the step drawn from Lévy distribition with $\beta$ = 1.5 and $\Gamma (x)$ is the gamma function.

In the flight straightly stage, the ith individual in subpopulation2 is considered to fly in a straight line towards the light source. The mathematical model of the flight straightly operator is formulated as follows:

$$X_i^{t+1}=\{\begin{array}{l}\lambda \times (X_i^t+\phi \times (X_{best}^t-X_i^t))\begin{array}{cc}& if\begin{array}{cc}rand>0.5& \end{array}\end{array}\\ \lambda \times (X_i^t+\frac{1}{\phi }\times (X_{best}^t-X_i^t))\begin{array}{cc}& else\end{array}\end{array}$$

(7)

where $\lambda$ is the scale factor, which is used to control the convergence speed of the algorithm and improve population diversity. $\lambda$ is set to a random number drawn by the standard uniform distribution. The acceleration factor $\phi$ is set to golden ratio (0.618). $X_{best}^t$ is the best moth individual at generation t. rand returns a random number that is uniformly distributed in (0, 1).

The pseudo code of MS is shown in Algorithm 1.

Algorithm 1. Moth search algorithm

Begin  Step 1: Initialization.  Set the maximum iteration number MaxGen and iteration counter G = 1; Initialize the parameters max walk step Smax, the index $\beta$, and acceleration factor $\phi$.  According to uniform distribution, the population with NP individuals is randomly initialized.  Step 2: Fitness calculation.  Compute the initial objective function values of each individual according to their position. Memory the best individual (denotes as Xbest).  Step 3: While (G < MaxGen) do  Divide the whole population into two subpopulations with equal size: subpopulation1 and subpopulation2, based on their fitness.  Update subpopulation1 by using Lévy flight operator (Equation (4)).  Update subpopulation2 by using flight straightly operator (Equation (7)).  Evaluate the objective function values of each individual and update Xbest.  G = G + 1.  Sort the population by fitness.  Step 4: End while  Step 5: Output: the best results.  End.

2.2. Baldwin Effect and Baldwinian Learning

The interactive way of learning and evolution was first proposed by Baldwin, known as the Baldwin effect. Hence, different Baldwinian learning models have been proposed based on the Baldwin effect. Hinton and Nowlan [62] found that it was difficult to find the optimal solution of more complex problems only by an evolutionary algorithm. However, when combined with Baldwinian learning, the performance of the hybrid algorithm can be effectively improved.

Generally speaking, Baldwinian learning is a type of local search strategy in an evolutionary algorithm. The Baldwinian learning mechanism was first combined with the clonal selection algorithm (CSA) by Gong et al. [61] to improve the performance of BCSA. Based on this, Peng et al. [70] proposed four Baldwinian learning strategies inspired by the trial vector generating strategy of differential evolution (DE).

In evolutionary computation, Cauchy mutation and Gaussian mutation are two popular and effective mutation techniques [71]. The characteristic of Gaussian mutation is to speed up the local convergence, and Cauchy mutation is better at escaping from local optimum. However, compared with Gaussian mutation, Cauchy mutation is insensitive to mutation step size and can achieve the acceptable performance.

According to the above analysis, the newly designed Baldwinian learning operator based on Cauchy distribution is proposed in HLMS. The mathematical expression is as follows:

$$Y_i=X_{r1}+c\cdot (X_{r2}-X_{r3})$$

(8)

where $Y_i$ is the donor vector for each moth individual $X_i$ from the current population after applying Baldwinian learning. $X_{r1}$, $X_{r2}$, and $X_{r3}$ are sampled randomly from the current population and r1, r2, and r3 are mutually exclusive integers randomly chosen from the range [1, NP], which are also different from the selected individual index i. The parameter c is the strength of Baldwinian learning and is a random number based on Cauchy distribution.

2.3. Global-Best HS Algorithm

GHS is a novel variant of HS and inspired by the concept of swarm intelligence of PSO [66]. The difference from the original HS is that the new harmony can mimic the best harmony in the harmony memory HM. Meanwhile, the parameter bw in HS is replaced and a social dimension is added to the GHS. In addition, the GHS dynamically updates the pitch adjusting rate PAR according to the following equation [72]:

$$PAR\left(t\right)=PA{R}_{min}+\frac{PA{R}_{max}-PA{R}_{min}}{NI}\times t$$

(9)

where PAR(t) is the pitch adjusting rate for generation t, $PA{R}_{min}$ is the minimum adjusting rate, and $PA{R}_{max}$ is the maximum adjusting rate. NI is the number of improvisations and t is the generation number.

The procedure of GHS is given in Algorithm 2.

Algorithm 2. The Global-best HS algorithm (GHS)

Begin  For each $i\in [1,n]$ do     /*n is the dimension of the problem*/  If U(0, 1) ≤ HMCR then   /* memory consideration*/  $x_i^h=x_i^j$, where $j\sim U(1,\dots ,HMS)$  If U(0,1) ≤ PAR (t) then /*pitch adjustment*/  $x_i^h=x_k^{best}$, where best is the best harmony in the HM and $k\sim U(1,n)$  Else         /*random selection*/  $x_i^h=L{B}_i+rand*(U{B}_i-L{B}_i)$  End.

Furthermore, it should be emphasized that, the concept of dimensional learning [73] is embodied in Algorithm 2, as shown below:

$$X_i^h=X_k^{best}$$

(10)

In Equation (7) of MS, the same dimension j is selected in $X_{best}^t-X_i^t$ for conducting the new solution. Under this dimension, if the component value of the ith individual is similar to the best individual, the difference $X_{best}^t-X_i^t$ will be very small, especially in the later stage of evolution. This means that such a step size is not conductive to $X_i$ jumping to a far position. If the best moth individual is local optimum, the solution hardly escapes from the local extremum. In Equation (10), the dimension index i of $X^h$ is not equal to the dimension index k of $X^{best}$. Generally, the difference between two different dimensions is large. Different dimensions can carry different information.

Based on the above analysis, dimension learning is embedding into Algorithm 2 and it should be an effective global search operator.

3. The Proposed HLMS for the MKP

The proposed HLMS algorithm for MKP is inspired from the studies [66,70] and distinguishes itself with two new features. First, GHS as a powerful global search operator is introduced to enhance the exploration ability of the algorithm. Second, a new Baldwinian learning strategy by replacing Gaussian distribution with Cauchy distribution is introduced on HLMS. The proposed algorithm framework and the main components of the MKP problem are described in the following subsection.

3.1. Population Initialization

In this stage, NP moth individuals are randomly generated in the search space. The swarm X = {X(1), X(2), …, X(NP)} is maintained and evolves, where each moth individual X(i) is a n-dimensional real-valued vector $X(i)=(x_1^i,x_2^i,\dots ,x_n^i)$ with $x_j^i\in \{-a,a\}\wedge j\in \{1,2,\dots ,n\}$ and n is the number of objects or items. Here, a takes the value 3 or 5 in this paper. Then, each moth individual X(i) is transformed into an n-dimensional binary vector by a mapping method, which is called discrete moth $Y(i)=(y_1^i,y_2^i,\dots ,y_n^i)$ with $y_j^i\in \{0,1\}\wedge j\in \{1,2,\dots ,n\}$.

3.2. Solution Representation

In HLMS, an n-bit binary string consisting of 0 and 1 is used to represent a candidate solution. If the item is selected, the bit is 1, otherwise it is 0. It should be noted that the MKP is a constrained optimization problem, so the solution generated in the evolution process may be infeasible.

In this paper, a simple and effective transfer function [50] is adopted and the function expression is as follows:

$$T(x)=x$$

(11)

The transfer method from a real-valued variable $x_i$ to a binary variable $y_i$ is calculated by:

$$y_i=\{\begin{array}{l}1,\begin{array}{c}if\begin{array}{c}T(x_i)>0\end{array}\end{array}\\ 0,\mathrm{else}\end{array}$$

(12)

3.3. Quick Repair Operator

Learning from previous research work [10,74], the HLMS algorithm also adopts a popular quick repair operator based on pseudo-utility which was proposed by Luo et al. [15]. In order to effectively apply the repair operator, the given MKP instance needs to be preprocessed. Specifically, all the items are renumbered in an ascending order based on their scaled pseudo-utility ratios ${\sigma }_j$ [75] defined as follows:

$${\sigma }_j=\frac{c_j}{{\displaystyle \sum _{i=1}^m\frac{a_{ij}}{b_i}}},\forall j\in \{1,2,\dots ,n\}$$

(13)

More exactly, the index values of all sorted items are stored in array J [1…n], such that ${\sigma }_{J[1]}\ge {\sigma }_{J[2]}\ge \dots \ge {\sigma }_{J[n]}$. The vectors $(c_1,c_2,\dots ,c_n)$ and $a_{i,j}$ (i = 1, 2, …, m, j = 1, 2, …, n) should be adjusted as well. The main framework of the quick repair procedure can be summarized in Algorithm 3.

Algorithm 3. The quick repair operator base on the scaled pseudo-utility

Begin  Step 1: Input. $X=\{x_1,x_2,\dots ,x_n\}\in {\{0,1\}}^n$.  Step 2: Calculation. Calculate the current total consumption of each resource.  $r_i={\displaystyle \sum _{j=1}^n{a}_{ij}}*x_j,\forall i=1,2,\dots ,m$.  Step 3: Repair process.  For j = n to 1 do  If $r_i\le b_i,\forall i=1,2,\dots ,m$ then  Break.  else if $x_{J[j]}$ = 1 then  Set $x_{J[j]}$= 0, and $r_i\leftarrow r_i-a_{iJ[j]},\forall i=1,2,\dots ,m.$  End  End  Step 4: Optimization process.  For j = 1 to n do  If $x_{J[j]}=0$ and $r_i+a_{iJ[j]}\le b_i,\forall i=1,2,\dots ,m$ then  Set $x_{J[j]}=1$ and $r_i\leftarrow r_i+a_{iJ[j]},\forall i=1,2,\dots ,m.$  End  End  Step 5: Output: $X=\{x_1,x_2,\dots ,x_n\}\in {\{0,1\}}^n$.  End.

Obviously, the quick repair operator of Algorithm 3 mainly consists of two phases. In the first phase, called the repair process, according to the ascending order of the scaled pseudo-utility ratios, the items are removed from the knapsack one by one until the solution is feasible. In the second phase, called the optimization process, for all the feasible solutions, greedily packed the items to be loaded into the knapsack based on the descending order of the scaled pseudo-utility ratios one by one. In this process, the feasibility of the solution needs to be maintained all the time. In brief, the first phase makes all the infeasible solutions become feasible, and the second phase enables the quality of feasible solutions better.

3.4. Procedure of HLMS for MKP

Based on the analysis above, the proposed HLMS algorithm for MKP is outlined in Algorithm 4. The algorithm framework includes the following main steps. (1) After initialization, the repaired population is divided into two subpopulations based on the fitness. (2) Subpopulation1 and subpopulation2 apply the Lévy flight operator and flight straightly operator, respectively. (3) GHS learning and Baldwinian learning are implemented in sequence to the whole population with a probability of 0.5. (4) The mapping from the real-valued vector to the binary vector is realized with a transfer function and then the repair of infeasible solutions and the optimization of feasible solutions are performed. (5) Evaluating the solution is based on the objective function and then the whole population is divided into two subpopulations. Steps (2)–(5) are repeated until the termination condition is reached.

Algorithm 4. Procedure of HLMS for MKP

Begin  Step 1: Initialization. Set the maximum iteration number MaxGen and iteration counter G = 1; Initialize the parameters max walk step Smax, the index $\beta$$\phi$, strength of Baldwinian learning c. According to uniform distribution, the population with NP individuals is randomly initialized. The transform function is used to discretize the real number vector to obtain the initial solution $X=\{x_1,x_2,\dots ,x_n\}\in {\{0,1\}}^n$. Repair the initial solution by Algorithm 3. Step 2: Fitness evaluation. Evaluate the initial solution using the objective function of MKP. Step 3: Whileg < MaxGen do  3.1 Divide the whole population into two subpopulations with equal size: subpopulation1 and subpopulation2, based on their fitness.     3.2 Update subpopulation1 by Lévy flight operator.     3.3 Update subpopulation2 by flight straightly operator.     3.4 GHS search      If U(0, 1) ≤ 0.5   Apply GHS algorithm on each individual X and generate the trial individual Y.        Choose the best one of X and Y to enter the next generation.     3.5 Baldwinian Learning      If U(0, 1) ≤ 0.5   Apply Baldwinin learning strategy on each individual X and generate the trial individual Y.        Choose the best one of X and Y to enter the next generation.     3.6 Apply transform function to obtain the potential solution of MKP.     3.7 Repair the potential solution by Algorithm 3.     3.8 Evaluate the fitness of the population and record the global best fitness.  G = G + 1.     3.9 Sort the population by fitness.    Step 4: End while    Step 5: Output: the best results.  End.

The algorithm framework is shown in Figure 1.

Figure 1. The framework of HLMS for MKP.

3.5. Computational Complexity of One Iteration of HLMS

The computational complexity of one iteration of HLMS based on Algorithm 4 is described as follows.

(1)

The initialization of HLMS requires O(NP × n) time, where NP denotes the population size, and n is the dimension of MKP (the number of the items).

(2)

The discretization process of NP moth individual costs O(NP × n) time.

(3)

The quick repair operator takes O(n × m) time, where m is the constraints of the MKP instance.

(4)

Fitness evaluation has O(NP) time.

(5)

Lévy flight operator has O(NP₁ × n) time, where NP₁ is the number of individuals of subpopulation1.

(6)

Flight straightly operator has O(NP₂ × n) time, where NP₂ is the number of individuals of subpopulation2.

(7)

GHS learning requires O(NP × n) time.

(8)

Baldwinian learning requires O(NP × n) time.

(9)

Sort the population based on Quick sort algorithm and it takes time O(NPlogNP).

In summary, the total computational complexity is O(NP × n) per generation for fixed m.

4. Experimental Studies

To comprehensively evaluate the performance of the proposed HLMS, large numbers of experiments are implemented on the benchmark instances commonly used in the literature and a comparative study is conducted between HLMS and several populations based on optimization algorithms.

4.1. Benchmark Test Functions

In this paper, four sets of well-known benchmark data for MKP are used to test the performance of HLMS. These instances are described in [74] and available at OR-Library (http://people.brunel.ac.uk/~mastjjb/jeb/orlib/mknapinfo.html, accessed on 1 July 2021).

Test set I contains 18 small-scale instances with m = 2 to 30 and n = 20 to 105, which are denoted as Sento [76], HP [77], PB [77], and Weing [4].

Test set II contains 30 medium-scale instances with m = 5 and n = 30 to 90, which are marked as Weish [7].

Test set III contains 30 large-scale instances. These instances are divided into two subsets. Subset I includes 15 instances with m = 15 and n $\in$ {100, 250, 500}, which are labeled as cb1, cb2, and cb3, respectively. Subset II also includes 15 instances with m = 15 and n $\in$ {100, 250, 500}, which are called as cb4, cb5, and cb6, respectively.

Test set IV contains 9 instances with m {15, 25, 50} and n $\in$ {100, 200, 500, 1000, 1500}, which were created by Glover and Kochenberger and then are marked as GK.

4.2. Experimental Environment and Parameters Setting

The proposed HLMS algorithm includes several important parameters, whose values are empirically set based on the preliminary experiments and the details are recorded in Table 1.

Table 1. Settings of parameters of HLMS.

Parameters	Section	Description	Values
Smax	2.1	The max step used in Equation (5)	1.0
$\phi$	2.1	The acceleration factor used in Equation (7)	0.618
$\lambda$	2.1	The scale factor used in Equation (7)	a random number of uniform distribution in [0, 1]
$\beta$	2.1	Parameter used in Equation (6)	1.5
c	2.2	The strength of Baldwinian learning	A random number based on Cauchy distribution
PARmax	2.3	Parameter used in Equation (9)	0.99
PARmin	2.3	Parameter used in Equation (9)	0.01
HMCR	2.3	Parameter used in Algorithm 2	0.9
NP	3.4	The population size	50
NFE	3.4	The maximal number of function evaluation	100,000

The HLMS algorithm was implemented in C language and compiled using the GNU GCC compiler. All the experiments were carried out on a computer with Intel (R) Core (TM) i7-7500 CPU (2.90 GHz and 8.00 GB RAM), running the Windows 10 operating system. HLMS was independently run 30 times for each instance to eliminate the unfairness brought by the stochastic characteristic.

To comprehensively evaluate the performance of the HLMS algorithm, fourteen MKP algorithms in the literature are selected as our comparative algorithms, which are listed as follows.

• Modified binary particle swarm optimization (MBPSO) [78];
• Chaotic binary particle swarm optimization with time-varying acceleration coefficients (CBPSOTVAC) [79];
• Binary PSO with time-varying acceleration coefficients (BPSOTVAC) [79];
• Modified multi-verse optimization (MMVO) algorithm [17];
• New binary particle swarm optimization with immunity-clonal algorithm (NPSOCLA) [80];
• Binary gravitational search algorithm (BGSA) [81];
• Binary hybrid topology particle swarm optimization (BHTPSO) [81];
• Binary hybrid topology particle swarm optimization quadratic interpolation (BHTPSO-QI) [81];
• New binary particle swarm optimization (NBPSO) [81];
• Binary version of PSO (BPSO) [82];
• Binary version of the Harris hawks algorithm (BHHA) [22,83];
• Binary version of the salp swarm algorithm (BSSA) [84];
• Binary version of the modified whale optimization algorithm (BIWOA) [85];
• Binary version of the sin-cosine algorithm (BSCA) [86].

It should be noted that the results of the comparative algorithms are compiled from the related papers. If the result of an algorithm for a MKP instance is not available, the result of the instance is ignored. In addition, considering that different comparative algorithms are written in different programming languages, or run on different computing platforms based on different termination conditions and algorithm parameters, we focus on comparing solution quality.

For this purpose, in this paper, eight typical statistical evaluation criteria are selected to evaluate the performance of all the comparative algorithms.

• Best value (Best):

$$Best=\max (f_i),\mathrm{for}\begin{array}{c}\forall i\in [1,t]\end{array}$$

(14)

where $f_i$ is the fitness value for $ith$ time. t is the total number of independent experiments.

• Worst value (Worst):

$$Worst=\min (f_i),\mathrm{for}\begin{array}{c}\forall i\in [1,t]\end{array}$$

(15)

• Mean value (Mean):

$$Mean=\frac{1}{t}{\displaystyle {\sum }_{i=1}^t{f}_i}$$

(16)

The mean value characterizes the centralized trend of the values of random variables. The larger the mean value, the more concentrated the results of multiple runs of the algorithm will be.

• Standard deviation (Std):

$$Std=\sqrt{\frac{1}{t}{\displaystyle {\sum }_{i=1}^t(f_i-mean)}}$$

(17)

Standard deviation describes the degree of dispersion of random variable values relative to the mean value. Meanwhile, standard deviation reflects the fluctuation in the value of a random variable. In other words, stability is an important evaluation criterion of a stochastic algorithm. If the Std value is high, the stability of the algorithm is poor, otherwise, its performance is good.

• Success rate (SR):

$$SR=\frac{st}{t}$$

(18)

where st denotes the success times, that is, the number of the known theoretical optimal solution is obtained. The high success rate indicates that the algorithm has good stability and optimization performance.

• Percent deviation (PDev):

$$PDev=\frac{Opt-Mean}{Opt}\ast 100$$

(19)

where Opt represents the optimal or the best-known solution. PDev reflects the degree to which the mean value deviates from the known theoretical optimal solution when the algorithm solves a single instance.

• Average error (AE):

$$AE=\frac{1}{N}\ast {\displaystyle {\sum }_{i=1}^N\frac{(Opt-profit)}{Opt}}\ast 100$$

(20)

Average error is an indicator that reflects the general level of error between random variables and Opt. Here, profit can be Best, Worst, or Mean. N is the number of benchmark instances. Clearly, the smaller AE value indicates that the algorithm has better performance. AE indicates the overall performance of the algorithm for solving a set of MKP instances.

• Percentage gap (Gap):

$$Gap=\frac{Opt-Best}{Opt}\ast 100$$

(21)

Similar to AE, for the maximization problem, the smaller the Gap is, the better the performance of the algorithm is. Gap investigates the performance of the algorithm to solve a single MKP instance.

Moreover, to determine whether there are significance differences between HLMS and other algorithms, the p-value based on the nonparametric Wilcoxon signed ranks test at the 95% confidence level is reported as well. Note that a p-value less than 0.05 represents that there exists a significant difference between the paired compared results. All the statistical results have been performed by the statistical software R language.

4.3. Comparisons on the Small-Scale Test Set

The proposed HLMS is first substantiated based on the 18 small-scale instances of Test set I and the experimental result is listed in Table 2, along with the available results of the comparative algorithms. In Table 2, the first three columns record the names of instances, the dimension information (n is the number of items, and m is the number of knapsacks), and the known optimum results (Opt), respectively. In addition, aggregate data are recorded at the bottom of the table. #Opt shows the number of the best-known solution obtained by the corresponding algorithm. #SR and #Std represent the number of instances for which the corresponding algorithm obtained a better result in terms of SR and Std among the comparative algorithms. MSR is the mean value of success rate and the ranks in descending order on the MSR are provided. Besides, the best findings among the comparison results are indicated in bold.

Table 2. The results of HLMS with 3 comparative algorithms for Test set I.

Prob.	n × m	Opt.	MS		HLMS		MBPSO		CBPSOTVAC
			SR	Std	SR	Std	SR	Std	SR	Std
Sento1	60 × 30	7772	0.17	54.46	0.90	33.73	0.16	43.23	0.39	357.78
Sento2	60 × 30	8722	0.00	27.79	0.47	4.72	0.03	18.80	0.20	101.03
HP1	28 × 4	3418	0.13	23.11	0.40	19.96	0.10	25.52	0.38	10.69
HP2	35 × 4	3186	0.00	26.47	0.40	32.70	0.11	39.15	0.59	21.35
PB1	27 × 4	3090	0.03	30.89	0.43	17.16	0.11	24.32	0.40	10.52
PB2	34 × 4	3186	0.00	19.95	0.70	13.83	0.16	39.31	0.51	18.73
PB4	29 × 2	95,168	0.00	894.68	0.30	1521.73	0.27	1803	0.84	875.1
PB5	20 × 10	2139	0.17	23.88	0.70	21.26	0.08	24.36	0.80	6.83
PB6	40 × 30	776	0.43	24.27	0.80	17.28	0.28	29.12	0.54	40.17
PB7	37 × 30	1035	0.03	3.86	0.50	5.83	0.05	16.29	0.40	24.25
Weing1	28 × 2	14,1278	0.90	214.94	0.93	89.77	0.82	250.43	0.92	281.98
Weing2	28 × 2	130,883	0.30	5731.34	0.97	29.21	0.65	314.08	0.88	545.50
Weing3	28 × 2	95,677	0.13	3767.42	0.80	1996.82	0.11	876.78	0.75	672.42
Weing4	28 × 2	119,337	0.73	1329.64	0.37	711.43	0.76	1270.80	0.97	378.58
Weing5	28 × 2	98,796	0.23	2671.33	1.00	0.00	0.52	1923.5	0.94	572.82
Weing6	28 × 2	130,623	0.23	164.95	0.97	71.20	0.36	322.40	0.97	343.45
Weing7	105 × 2	1,095,445	0.00	482.74	0.00	2872.29	0.02	1130.60	0.00	30,020.00
Weing8	105 × 2	624,319	0.33	1966.37	0.03	1135.87	0.03	4704.30	0.20	75,169.00
#Opt			13		17		18		17
#SR			0		12		1		5
#Std			1		10		0		7
MSR			0.21		0.54		0.26		0.59
Rank of MSR			4		2		3		1
p-value			0.001	0.029			0.001	0.663	0.641	0.896

From Table 2, the proposed HLMS is able to obtain the known optimum solution for almost all the 18 instances except for Weing7. However, MS can only reach the known optimum solution for 13 instances. Considering the #SR, HLMS performs much better than the competing algorithms. In terms of MSR, HLMS is slightly worse than CBPSOTVAC and ranks two. Moreover, the clear superiority of HLMS is established in comparison with MS in terms of all evaluation criteria. Therefore, we can conclude that it is beneficial to use the GHS global search algorithm combined with the Baldwinian learning strategy. In terms of the p-value of SR, the difference between HLMS and MS, HLMS and MBPSO is statistically significant (p-value < 0.05). However, there are no significant differences in Std among the latter two groups (p-value > 0.05).

The related box plots are given in Figure 2 in terms of SR. As can be seen from Figure 2, the difference of SR among four algorithms is very obvious. Although the distributions of the SR value of MS and MBPSO are more uniform than that of HLMS and CBPSOTVAC, the interquartile ranges of the former are worse than that of the latter. Moreover, outliers exist in MS and MBPSO on the SR value. In addition, we also observe the maximum, upper quartile, the mean value of HLMS is equal to or close to 1.0, 0.8, and 0.6, respectively.

Figure 2. Boxplot for 4 comparative algorithms on SR for Test set I.

Based on the above analysis, we can draw a conclusion that HLMS can obtain the best-known solution of 18 small-scale instances with a high success rate.

4.4. Comparisons on the Medium-Scale Test Set

In the second experiment, HLMS is used to solve medium-scale test instance (Test set II) to verify the performance of algorithms. The results are reported in Table 3, together with the results of other six state-of-the-art MKP algorithms, including MS, BIWOA, BMMVO, BSCA, BHHA, and BSSA. Note that these six algorithms are all novel swarm intelligence algorithms proposed in recent years and it is meaningful to select them for comparative study of MKP.

Table 3. The results of HLMS with 6 comparative algorithms for Test set II.

Prob.	n × m	Opt.		MS	HLMS	BIWOA	BMMVO	BSCA	BHHA	BSSA
Weish01	30 × 5	4554	Best	4554	4554	4554	4554	4554	4554	4554
			Worst	4477	4534	4554	4554	4534	4554	4554
			PDev	0.38	0.18	0.00	0.00	0.09	0.00	0.00
Weish02	30 × 5	4536	Best	4536	4536	4536	4536	4536	4536	4536
			Worst	4440	4504	4536	4536	4536	4536	4536
			PDev	0.38	0.02	0.00	0.00	0.00	0.00	0.00
Weish03	30 × 5	4115	Best	4106	4115	4106	4106	4106	4106	4106
			Worst	4106	4106	4106	4106	4106	4106	4106
			PDev	0.22	0.19	3.97	3.05	0.00	2.24	3.97
Weish04	30 × 5	4561	Best	4561	4561	4561	4561	4561	4561	4561
			Worst	4505	4531	4561	4561	4561	4561	4561
			PDev	0.37	0.09	0.00	0.00	0.00	0.00	0.00
Weish05	30 × 5	4514	Best	4514	4514	4514	4514	4514	4514	4514
			Worst	4514	4514	4514	4514	4514	4514	4514
			PDev	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Weish06	40 × 5	5557	Best	5557	5557	5557	5557	5557	5557	5557
			Worst	5502	5542	5542	5542	5542	5544	5557
			PDev	0.40	0.09	0.12	0.14	0.14	0.02	0.00
Weish07	40 × 5	5567	Best	5567	5567	5567	5567	5567	5567	5567
			Worst	5360	5542	5567	5567	5567	5567	5567
			PDev	0.53	0.03	0.00	0.00	0.00	0.00	0.00
Weish08	40 × 5	5605	Best	5605	5605	5605	5605	5605	5605	5605
			Worst	5478	5603	5605	5603	5603	5605	5605
			PDev	0.30	0.01	0.00	0.03	0.01	0.00	0.00
Weish09	40 × 5	5246	Best	5246	5246	5246	5246	5246	5246	5246
			Worst	5185	5246	5246	5246	5246	5246	5246
			PDev	0.08	0.00	0.00	0.00	0.00	0.00	0.00
Weish10	50 × 5	6339	Best	6339	6339	6323	6303	6303	6303	6303
			Worst	6255	6280	6303	6303	6303	6303	6303
			PDev	0.55	0.04	0.31	0.56	0.56	0.56	0.56
Weish11	50 × 5	5643	Best	5643	5643	5643	5643	5643	5643	5643
			Worst	5592	5643	5643	5643	5643	5643	5643
			PDev	0.03	0.00	0.00	0.00	0.00	0.00	0.00
Weish12	50 × 5	6339	Best	6339	6339	6302	6301	6302	6302	6302
			Worst	6090	6304	6302	6301	6301	6301	6301
			PDev	0.94	0.07	0.58	0.59	0.59	0.59	0.59
Weish13	50 × 5	6159	Best	6159	6159	6159	6159	6159	6159	6159
			Worst	6025	6025	6159	6159	6159	6159	6159
			PDev	0.98	0.23	0.00	0.00	0.00	0.00	0.00
Weish14	60 × 5	6954	Best	6954	6954	6923	6923	6923	6923	6923
			Worst	6769	6902	6900	6900	6900	6923	6923
			PDev	0.80	0.28	0.71	0.57	0.66	0.44	0.44
Weish15	60 × 5	7486	Best	7486	7486	7486	7486	7486	7486	7486
			Worst	7199	7442	7453	7449	7486	7486	7486
			PDev	0.84	0.08	0.08	0.11	0.00	0.00	0.00
Weish16	60 × 5	7289	Best	7289	7289	7289	7289	7289	7289	7289
			Worst	6942	7221	7288	7281	7281	7281	7281
			PDev	0.96	0.14	0.01	0.10	0.09	0.10	0.10
Weish17	60 × 5	8633	Best	8633	8633	8633	8624	8633	8633	8633
			Worst	8141	8633	8575	8497	8506	8633	8633
			PDev	0.31	0.00	0.39	0.96	0.42	0.00	0.00
Weish18	70 × 5	9580	Best	9540	9580	9560	9456	9573	9580	9573
			Worst	8857	9525	9461	9318	9451	9521	9527
			PDev	1.42	0.11	0.65	1.92	0.62	0.27	0.17
Weish19	70 × 5	7698	Best	7698	7698	7698	7698	7698	7698	7698
			Worst	7448	7674	7632	7629	7698	7698	7698
			PDev	0.85	0.01	0.38	0.35	0.00	0.00	0.00
Weish20	70 × 5	9450	Best	9450	9450	9450	9445	9450	9450	9450
			Worst	9306	9408	9400	9365	9433	9445	9450
			PDev	0.49	0.03	0.23	0.57	0.02	0.01	0.00
Weish21	70 × 5	9074	Best	9074	9074	9074	9074	9074	9074	9074
			Worst	8922	9008	9016	8969	9033	9074	9074
			PDev	0.44	0.05	0.19	0.64	0.03	0.00	0.00
Weish22	80 × 5	8947	Best	8790	8929	8909	8886	8909	8912	8912
			Worst	7904	8708	8908	8886	8886	8909	8912
			PDev	5.51	0.58	0.43	0.68	0.63	0.39	0.39
Weish23	80 × 5	8344	Best	8170	8344	8303	8250	8344	8344	8344
			Worst	7246	8154	8245	8233	8245	8250	8303
			PDev	5.74	0.85	0.66	1.16	0.89	0.71	0.26
Weish24	80 × 5	10,220	Best	10,189	10,220	10,189	10,058	10,215	10,202	10,220
			Worst	9807	10,091	10,053	9787	10,042	10,134	10,132
			PDev	1.70	0.16	1.23	3.14	0.86	0.57	0.48
Weish25	80 × 5	9939	Best	9922	9939	9885	9844	9939	9939	9939
			Worst	9703	9885	9808	9710	9885	9915	9915
			PDev	1.02	0.03	0.94	1.63	0.20	0.21	0.11
Weish26	90 × 5	9584	Best	9581	9584	9575	9575	9575	9575	9575
			Worst	8904	9514	9477	9439	9476	9488	9575
			PDev	1.66	0.19	0.69	1.17	0.92	0.24	0.09
Weish27	90 × 5	9819	Best	9764	9819	9778	9589	9764	9764	9764
			Worst	9319	9764	9773	9487	9631	9678	9764
			PDev	2.09	0.50	0.45	2.53	0.88	0.61	0.56
Weish28	90 × 5	9492	Best	9492	9492	9454	9400	9454	9454	9454
			Worst	9034	9438	9411	9183	9400	9400	9400
			PDev	1.54	0.19	0.49	1.70	0.78	0.62	0.45
Weish29	90 × 5	9410	Best	9369	9410	9369	9369	9369	9369	9369
			Worst	8927	9369	9369	9135	9369	9369	9369
			PDev	1.45	0.40	0.43	1.75	0.43	0.43	0.43
Weish30	90 × 5	11,191	Best	11,148	11,187	11,121	11,025	11,169	11,169	11,169
			Worst	10,808	11,155	10,979	10,790	10,948	11,135	11,154
			PDev	1.44	0.08	1.23	2.49	0.61	0.27	0.20
#Opt				20	27	16	14	18	19	19
#Worst				2	9	12	10	12	18	23
The mean of PDev				1.11	0.15	0.47	0.86	0.31	0.28	0.29
p-value (PDev)				0.000		0.011	0.000	0.033	0.328	0.848

From Table 3, the results demonstrate that HLMS reaches the optimum solutions on 27 out of 30 instances, while the six comparative algorithms obtain the optimum solutions only on 20, 16, 14, 18, 19, and 19, respectively. In terms of #Worst, BSSA outperforms the other six algorithms on 23 instances. PDev measures the deviation between the mean and the best-known solution. The small mean value of PDev also confirms the superiority of HLMS. Comprehensively speaking, MS has the worst performance among all the comparative algorithms. Moreover, there are significant differences (p-value < 0.05) between the comparisons of the first four groups concerning PDev.

Figure 3 presents the box plots of the PDev values for all the comparative algorithms. The span of each box implicitly reflects the stability of the algorithm. The smaller the span is, the better the stability of the algorithm is. As can be seen from Figure 3, HLMS has a significant advantage over the other six algorithms since the span of the box for HLMS is obviously smaller than that of the other comparative algorithms. It should be noted that the dot in Figure 3. represent outliers.

Figure 3. Boxplot for 7 comparative algorithms on PDev for Test set II.

In summary, the results in Table 3 and Figure 3 indicate that our HLMS algorithm is very competitive compared to the other six MKP algorithms. The findings are based on the fact that the GHS learning scheme enhances the global search ability of HLMS. On this basis, Baldwinian learning can effectively adjust the shape of search space and thereby provides good search paths towards the best solutions.

4.5. Comparisons on the Large-Scale Test Set

In the third experiment, the performance of HLMS is verified by solving large-scale problems, and the comparative results on Test set III and Test set IV are reported in Table 4 and Table 5. In order to make a fair comparison with different classical algorithms using appropriate evaluation criteria, the experiment is divided into three groups on different scale instances.

Table 4. The results of HLMS with 5 comparative algorithms for Test set III (cb1-cb3).

Prob.	n × m	Opt.	Profit	MS	HLMS	MMVO	BGSA	BHTPSO	BHTPSO-QI
cb1-1	100 × 5	24,381	Best	24,253	24,381	24,192	24,152	24,169	24,301
			Mean	24,004	24,301	24,050	23,835	23,822	23,821
			Worst	23,311	24,238	23,920	23,175	23,415	23,287
cb1-2	100 × 5	24,274	Best	24,258	24,274	24,274	23,986	24,109	23,944
			Mean	23,934	24,231	24,274	23,536	23,657	23,688
			Worst	23,366	24,116	24,274	23,177	22,953	23,375
cb1-3	100 × 5	23,551	Best	23,538	23,551	23,538	23,386	23,435	23,418
			Mean	23,272	23,521	23,520	23,041	23,072	23,073
			Worst	22,953	23,468	23,494	22,543	22,678	22,621
cb1-4	100 × 5	23,534	Best	23,256	23,503	23,288	23,172	23,253	23,192
			Mean	23,024	23,420	23,120	22,863	22,928	22,923
			Worst	22,542	23,288	23,042	22,468	22,507	22,234
cb1-5	100 × 5	23,991	Best	23,845	23,966	23,947	23,755	23,815	23,774
			Mean	23,567	23,937	23,900	23,459	23,473	23,527
			Worst	23,062	23,836	23,799	23,106	23,155	23,053
cb2-1	250 × 5	59,312	Best	58,084	59,063	58,473	57,565	57,814	57,800
			Mean	57,369	58,862	58,240	56,554	56,874	56,685
			Worst	55,984	58,653	58,112	55,191	54,935	55,255
cb2-2	250 × 5	61,472	Best	60,248	61,295	60,692	60,057	59,982	59,767
			Mean	59,386	61,051	60,390	58,613	58,588	58,680
			Worst	58,167	60,870	60,194	57,707	56,807	56,821
cb2-3	250 × 5	62,130	Best	61,212	61,767	61,702	59,936	60,630	60,524
			Mean	59,922	61,552	61,330	58,975	59,234	59,186
			Worst	57,885	61,303	61,158	57,723	57,435	57,278
cb2-4	250 × 5	59,463	Best	58,386	59,140	58,441	57,970	57,736	57,884
			Mean	57,752	58,922	58,300	56,744	56,773	56,584
			Worst	56,763	58,710	58,163	55,371	55,589	55,164
cb2-5	250 × 5	58,951	Best	57,755	58,605	58,082	56,959	57,378	57,550
			Mean	56,929	58,390	58,300	55,961	56,129	56,361
			Worst	56,326	58,088	58,163	54,637	54,364	53,929
cb3-1	500 × 5	120,148	Best	116,296	119,101	119,978	111,206	114,493	114,438
			Mean	115,444	118,457	119,900	108,930	111,017	111,469
			Worst	114,634	117,842	119,810	106,951	106,454	107,005
cb3-2	500 × 5	117,879	Best	113,732	116,227	115,634	108,522	112,821	112,147
			Mean	112,257	115,704	115,400	106,631	109,276	109,247
			Worst	111,381	115,053	115,222	104,519	100,118	104,696
cb3-3	500 × 5	121,131	Best	117,666	119,990	119,156	111,271	114,774	116,099
			Mean	116,367	119,468	118,900	109,430	112,035	112,001
			Worst	115,160	119,054	118,651	107,683	106,406	104,627
cb3-4	500 × 5	120,804	Best	116,454	119,015	119,124	111,283	115,828	114,327
			Mean	115,396	118,386	118,900	109,062	112,200	111,671
			Worst	114,100	117,572	118,623	107,061	106,222	107,578
cb3-5	500 × 5	122,319	Best	117,900	120,918	121,141	112,391	115,889	117,242
			Mean	116,767	120,278	120,800	110,564	112,253	113,364
			Worst	115,062	119,519	120,401	108,670	102,820	103,910
#Best				0	12	4	0	0	0
#Mean				0	11	4	0	0	0
#Worst				0	8	7	0	0	0
AE of Best				1.91%	0.57%	0.98%	3.97%	2.70%	2.72%
AE of Mean				3.08%	0.92%	1.23%	5.63%	4.81%	4.75%
AE of Worst				4.81%	1.36%	1.48%	7.49%	8.29%	8.09%
Rank of AE of Best				3	1	2	6	4	5
p-value (Mean)				0.000		0.201	0.000	0.000	0.000

Table 5. The results of HLMS with 5 comparative algorithms for Test set III (cb4–cb5).

Prob.	n × m	Opt.	Profit	MS	HLMS	BGSA	BHTPSO	BHTPSO-QI
cb4-1	100 × 10	23,064	Best	22,753	23,055	22,836	22,905	22,876
			Mean	22,459	22,914	22,334	22,425	22,449
			Worst	22,080	22,753	21,975	21,980	21,999
cb4-2	100 × 10	22,801	Best	22,611	22,743	22,441	22,573	22,408
			Mean	22,255	22,629	21,991	22,047	22,017
			Worst	21,622	22,407	21,435	21,322	21,454
cb4-3	100 × 10	22,131	Best	21,886	22,131	21,849	21,797	21,949
			Mean	21,466	21,908	21,313	21,342	21,461
			Worst	20,841	21,855	20,957	20,958	20,886
cb4-4	100 × 10	22,772	Best	22,319	22,717	22,325	22,418	22,376
			Mean	21,992	22,528	21,961	22,037	22,029
			Worst	21,465	22,016	21,488	21,228	21,533
cb4-5	100 × 10	22,751	Best	22,440	22,751	22,168	22,215	22,254
			Mean	22,132	22,603	21,840	21,822	21,903
			Worst	21,738	22,272	21,271	21,362	21,339
cb5-1	250 × 10	59,187	Best	57,757	58,903	56,928	57,530	57,036
			Mean	56,708	58,477	55,759	55,854	55,960
			Worst	55,510	58,182	54,217	53,570	53,381
cb5-2	250 × 10	58,781	Best	57,363	58,346	56,337	56,568	56,490
			Mean	56,793	58,098	55,455	55,443	55,708
			Worst	56,126	57,792	53,739	53,274	52,907
cb5-3	250 × 10	58,097	Best	56,690	57,674	55,573	56,426	55,982
			Mean	56,024	57,417	54,638	54,793	54,727
			Worst	55,281	57,044	53,516	52,871	52,714
cb5-4	250 × 10	61,000	Best	59,930	60,505	58,595	59,030	59,077
			Mean	58,934	60,282	57,766	58,057	57,721
			Worst	57,765	59,870	56,701	56,254	53,774
cb5-5	250 × 10	58,092	Best	56,863	57,468	56,186	56,217	56,204
			Mean	56,066	57,220	54,850	54,941	54,872
			Worst	55,182	56,869	53,612	51,850	50,832
cb6-1	500 × 10	117,821	Best	113,362	116,015	108,487	110,996	111,669
			Mean	112,541	115,379	105,760	107,698	108,367
			Worst	111,397	114,509	102,725	104,239	103,802
cb6-2	500 × 10	119,249	Best	115,022	117,778	109,569	114,262	113,001
			Mean	114,250	117,102	106,775	108,648	109,197
			Worst	112,596	116,418	103,478	100,740	100,764
cb6-3	500 × 10	119,215	Best	115,419	117,345	109,705	113,987	112,419
			Mean	114,372	116,842	106,853	108,576	109,004
			Worst	113,495	116,115	104,565	102,439	103,703
cb6-4	500 × 10	118,829	Best	115,038	117,281	108,628	112,476	112,198
			Mean	113,444	116,446	105,679	107,692	107,796
			Worst	112,405	115,872	102,679	101,860	99,470
cb6-5	500 × 10	116,530	Best	112,971	114,909	106,972	109,567	109,287
			Mean	111,707	114,183	104,509	106,217	106,212
			Worst	110,156	113,533	102,665	100,836	100,509
#Best				0	15	0	0	0
#Mean				0	15	0	0	0
#Worst				0	14	0	0	0
AE of Best				2.30%	0.76%	4.58%	3.25%	3.52%
AE of Mean				3.57%	1.33%	6.58%	5.92%	5.78%
AE of Worst				5.20%	2.12%	8.77%	9.64%	10.11%
Rank of AE of Best				2	1	5	3	4
p-value (Mean)				0.000		0.000	0.000	0.000

4.5.1. Performance Comparison on Test Set III (cb1–cb3)

Table 4 summarizes the experimental results of the first group large-scale benchmarks. From Table 4, it can be seen clearly that the proposed HLMS still keeps the best performance in terms of all six evaluation criteria. Specifically speaking, HLMS outperforms the other comparative algorithms. In addition, the p-value indicates that there is significant difference between HLMS and MS, BGSA, BHTPSO, and BHTPSO-QI in terms of mean. However, the p-value is 0.201 for MMVO (p-value > 0.05) and then reject the null hypothesis. Hence, insignificant difference can be detected between HLMS and MMVO.

The performance comparison of six methods based on AE is plotted in Figure 4. It is evident that the axis of HLMS on radar charts has a point nearer to the center in comparison with the other five algorithms when considering AE of Best, AE of Mean, and AE of Worst, which indicates that it is more effective with respect to quality of solutions. It can be considered that HLMS obtained optimal or near-optimal for most of the instances in terms of Best, Mean, and Worst, and could beat all the other competing algorithms.

Figure 4. The performance comparison of 6 methods based on AE for Test set III (cb1–cb3).

To observe the stability of MS and HLMS more intuitively, the error bar based on variance and the trends plot based on Std are shown in Figure 5 and Figure 6, respectively. It can be observed clearly from Figure 5 that the variances of HLMS are apparently smaller than that of MS for all benchmarks. Moreover, the variances will increase with the growth of the scale of instances. It is clear from Figure 6 that the trend lines of HLMS are located in the relatively low area for ten instances of cb1 and cb2. However, the curve of cb3 has an upward trend. In brief, the curve of MS is higher than that of HLMS, which indicates that HLMS has more stable performance than MS.

Figure 5. The error bars (Variance) for MS and HLMS on Test set III (cb1–cb3).

Figure 6. The trends of Std for MS and HLMS on Test set III (cb1–cb3).

4.5.2. Performance Comparison on Test Set III (cb4–cb6)

Table 5 summarizes the experimental results of the second group large-scale benchmarks. Table 5 shows that HLMS is also very efficient for 15 large instances with m = 10. Moreover, HLMS is superior to other five algorithms in absolute advantage, which is confirmed by the small p-values (0.000 ≤ 0.05).

Similarly, radar charts are plotted to visualize three evaluation criteria, AE of Best, AE of Mean, and AE of Worst in Figure 7. From Figure 7, the phenomenon is almost consistent with Figure 4. The point on the HLMS axis is very close to the center point. By implication, HLMS has smaller AE value compared with other algorithms.

Figure 7. The performance comparison of 5 methods based on AE for Test set III (cb4–cb6).

The error bar based on variance for HLMS and MS is illustrated in Figure 8, which is to assess the stability of algorithms. As can be seen from Figure 8, the variance of HLMS is almost unaffected by the scale of MKP. However, with the expansion of the scale, the variance of MS is increasing gradually.

Figure 8. Error bars (Variance) for MS and HLMS on Test set III (cb4–cb6).

The trend plot of Std for HLMS and MS is given in Figure 9. It can be seen from Figure 9 that the trend curve of MS is significantly higher than that of HLMS, which further indicates that HLMS has better stability than MS.

Figure 9. The trends of Std for MS and HLMS on Test set III (cb4–cb6).

4.5.3. Performance Comparison on Test Set IV

Table 6 summarizes the experimental results of the third group large-scale benchmarks. Overall, HLMS still outperforms all other comparative algorithms. In terms of #Best, #Worst, and #Mean, HLMS obtains a better result respectively on 5, 1, and 3 out of 9 instances. The results of BIWOA are respectively on 2, 2, and 2 out of 9 instances. BSSA with better performance obtains 2, 6, and 5 out of 9 instances, respectively. For the significance, the p-values for BMMVO and BSCA are both smaller than 0.05 concerned with Mean, except for MS, BIWOA, BHHA, and BSSA, which indicates that the difference between HLMS and most comparative algorithms is not significant when facing Test set IV.

Table 6. The results of HLMS with 6 comparative algorithms for Test set IV.

Prob.	n × m	Opt.		MS	HLMS	BIWOA	BMMVO	BSCA	BHHA	BSSA
GK01	100 × 15	3766	Best	3732	3752	3743	3698	3725	3746	3744
			Worst	3714	3722	3731	3666	3690	3728	3738
			Mean	3721	3742	3736	3678	3705	3734	3742
GK02	100 × 25	3958	Best	3920	3948	3949	3885	3913	3929	3939
			Worst	3900	3928	3924	3859	3883	3915	3924
			Mean	3908	3938	3931	3871	3897	3919	3934
GK03	150 × 25	5656	Best	5585	5610	5613	5561	5563	5580	5606
			Worst	5545	5575	5584	5507	5533	5554	5594
			Mean	5564	5596	5596	5519	5543	5568	5598
GK04	150 × 50	5767	Best	5702	5733	5712	5651	5678	5695	5712
			Worst	5677	5658	5690	5628	5652	5672	5696
			Mean	5688	5710	5701	5638	5664	5683	5704
GK05	200 × 25	7561	Best	7479	7502	7499	7365	7411	7463	7495
			Worst	7422	7413	7476	7344	7375	7426	7477
			Mean	7452	7474	7485	7353	7391	7443	7488
GK06	200 × 50	7680	Best	7617	7611	7607	7522	7551	7578	7617
			Worst	7561	7568	7584	7492	7520	7562	7598
			Mean	7591	7592	7597	7505	7532	7569	7611
GK07	500 × 25	19,220	Best	19,066	19,151	19,110	18,738	18,783	19,005	19,100
			Worst	19,005	18,890	19,093	18,635	18,689	18,961	19,048
			Mean	19,033	19,067	19,102	18,668	18,734	18,983	19,087
GK08	500 × 50	18,806	Best	18,612	18,642	18,641	18,385	18,462	18,601	18,646
			Worst	18,557	18,498	18,607	18,335	18,395	18,597	18,637
			Mean	18,582	18,594	18,627	18,361	18,428	18,598	18,640
GK09	1500 × 25	58,087	Best	57,753	57,886	57,868	56,746	56,932	57,547	57,346
			Worst	57,636	56,725	57,830	56,519	56,624	56,699	56,615
			Mean	57,676	57,547	57,843	56,619	56,719	57,719	56,959
#Best				1	5	2	0	0	0	2
#Worst				0	1	2	0	0	0	6
#Mean				0	3	2	0	0	0	5
AE of Best				0.11%	0.07%	0.08%	0.24%	0.19%	0.12%	0.09%
AE of Mean				0.16%	0.18%	0.11%	0.29%	0.27%	0.17%	0.13%
AE of Worst				0.13%	0.10%	0.10%	0.27%	0.22%	0.13%	0.11%
Rank on AE of Best				4	1	2	7	6	5	3
p-value (Mean)				0.327		0.172	0.001	0.001	0.069	0.327

Graphically, Figure 10 shows the AE of Best, AE of Mean, and AE of Worst obtained by seven methods. In terms of Best, HLMS outperforms the other comparative algorithms in absolute small AE value. However, HLMS is slightly worse than BIWOA in terms of AE of Mean.

Figure 10. The performance comparison of 7 methods based on AE for Test set IV.

In summary, the above experimental results and comparisons show that the proposed HLMS also has excellent optimization performance in solving large-scale MKP instances, in terms of solution quality, convergence, and stability of algorithms. This is mainly due to GHS learning and Baldwinian learning being able to effectively balance exploration and exploitation in the evolution process. Compared with the original MS, the hybrid learning strategy focuses more on discovering and utilizing useful information from the whole population and whole search experience, rather than the experience of some random local individuals.

4.6. Sensitivity Analysis on the Positional Parameter and the Scale Parameter

As is known to all, Gaussian distribution and Cauchy distribution are two important distributions, which have been integrated into many algorithms and proved an effective strategy to enhance the ability of an elaborate search. The strength of Baldwinian learning in [47] is random real number obeying Gaussian distribution. However, previous studies have revealed that Cauchy mutation possesses more power in escaping local optima and converging to the global optimum. Hence, the parameters used in [47] for Gaussian distribution and the positional parameter x and the scale parameter y for Cauchy distribution are investigated in this subsection. It is noted that, to eliminate the influence of GHS, HLMS only adopted the Baldwinian learning strategy in this experiment. We tested HLMS for Gaussian distribution and different combinations for (x, y) of Cauchy distribution: (0, 1), (0, 0.5), (0, 2), and (−2, 1). Table 7 summaries the Mean and SR for 18 instances of Test set I. In addition, boxplot for five parameter combinations on SR is plotted in Figure 11.

Table 7. The results of HLMS using different parameters of two distributions for Test set I.

Prob.		$\mathit{\mu }=0.5,\mathit{\delta }=0.3$	x = 0, y = 1	x = 0, y = 0.5	x = 0, y = 2	x = −2, y = 1
Sento1	Mean	7729	7740	7749	7753	7749
	SR	0.43	0.67	0.63	0.67	0.60
Sento2	Mean	8701	8713	8708	8712	8713
	SR	0.13	0.33	0.10	0.07	0.27
HP1	Mean	3391	3376	3371	3385	3367
	SR	0.43	0.20	0.03	0.07	0.10
HP2	Mean	3115	3125	3124	3135	3119
	SR	0.10	0.03	0.07	0.00	0.00
PB1	Mean	3045	3058	3060	3057	3055
	SR	0.13	0.13	0.33	0.17	0.20
PB2	Mean	3145	3145	3142	3148	3145
	SR	0.07	0.13	0.10	0.10	0.10
PB4	Mean	92,093	92,664	92,808	93,229	92,457
	SR	0.00	0.10	0.07	0.03	0.03
PB5	Mean	2098	2118	2112	2115	2106
	SR	0.17	0.50	0.47	0.57	0.37
PB6	Mean	753	757	757	756	758
	SR	0.43	0.53	0.53	0.53	0.53
PB7	Mean	1024	1026	1025	1025	1026
	SR	0.10	0.13	0.13	0.07	0.13
Weing1	Mean	139,551	139,615	139,397	140,199	138,803
	SR	0.33	0.07	0.23	0.20	0.13
Weing2	Mean	130,307	130,370	126,292	129,613	129,989
	SR	0.27	0.40	0.23	0.20	0.60
Weing3	Mean	92,362	94,546	93,373	93,033	94,489
	SR	0.27	0.50	0.50	0.30	0.60
Weing4	Mean	118,384	117,864	117,199	117,489	117,544
	SR	0.23	0.20	0.13	0.03	0.13
Weing5	Mean	96,465	95,421	97,209	95,372	94,898
	SR	0.43	0.33	0.67	0.40	0.37
Weing6	Mean	129,533	128,954	129,753	129,983	129,732
	SR	0.37	0.17	0.40	0.37	0.47
Weing7	Mean	1,048,378	1,038,366	1,043,595	1,035,059	1,035,752
	SR	0.00	0.00	0.00	0.00	0.00
Weing8	Mean	622,201	621,720	622,515	621,778	622,127
	SR	0.00	0.00	0.03	0.03	0.00
#Mean		2	5	3	6	3
#SR		5	7	6	5	6
MSR		0.22	0.25	0.26	0.21	0.26
p-value (Mean)			1.000	0.556	0.727	0.635

Figure 11. Boxplot for 5 parameter combinations on SR for Test set I.

From Table 7, all HLMS with four different combinations of x and y find better results than that of HLMS combined with Gaussian distribution. In terms of #Mean, #SR, and MSR, HLMS with four groups (x, y) shows similar results. It can be observed from Figure 11 that HLMS-C2 shows excellent comprehensive performance. HLMS-C2 has the best maximum and three-quarter quantile. Hence, considering all of the parameter combinations, we concluded that the setting x = 0 and y = 0.5 for the HLMS is an appropriate choice.

We can draw a conclusion from this experiment that it is better to use a random real obeying Cauchy distribution than Gaussian distribution as the Baldwinian learning strength. The reason may be that Cauchy mutation has stronger ability to jump from local optimum than Gaussian mutation.

4.7. The Effectiveness of the Two Components in HLMS

As mentioned above, HLMS includes two learning strategies: Baldwinian learning strategy and GHS learning strategy. The aim of this subsection is to investigate the effectiveness of these two learning strategies. Therefore, one additional experiment is conducted on Test set I and the results are summarized in Table 8. HLMS, which only adopted the Baldwinian learning strategy, is denoted as HLMS-B. HLMS, which only adopted the GHS learning strategy, is denoted as HLMS-H.

Table 8. Comparisons of MS, HLMS-B, HLMS-H, and HLMS on Test set I.

Prob.		MS	HLMS-B	HLMS-H	HLMS
Sento1	Mean	7677	7749	7763	7762
	SR	0.17	0.63	0.87	0.90
Sento2	Mean	8695	8708	8717	8719
	SR	0.00	0.10	0.53	0.47
HP1	Mean	3370	3371	3392	3399
	SR	0.13	0.03	0.37	0.40
HP2	Mean	3098	3124	3135	3158
	SR	0.00	0.07	0.30	0.40
PB1	Mean	3036	3060	3075	3074
	SR	0.03	0.33	0.50	0.43
PB2	Mean	3139	3142	3172	3178
	SR	0.00	0.10	0.60	0.70
PB4	Mean	92,312	92,808	93,111	93,063
	SR	0.00	0.07	0.17	0.30
PB5	Mean	2091	2112	2117	2125
	SR	0.17	0.47	0.57	0.70
PB6	Mean	751	757	767	768
	SR	0.43	0.53	0.83	0.80
PB7	Mean	1023	1025	1028	1030
	SR	0.03	0.13	0.20	0.50
Weing1	Mean	141,207	139,397	141,227	141,260
	SR	0.90	0.23	0.83	0.93
Weing2	Mean	130,760	126,292	130,877	130,877
	SR	0.30	0.23	0.97	0.97
Weing3	Mean	90,866	93,373	95,355	94,801
	SR	0.13	0.50	0.83	0.80
Weing4	Mean	116,487	117,199	118,883	118,956
	SR	0.73	0.13	0.50	0.37
Weing5	Mean	95,802	97,209	98,384	98,796
	SR	0.23	0.67	0.87	1.00
Weing6	Mean	129,176	129,753	130,429	130,610
	SR	0.23	0.40	0.63	0.97
Weing7	Mean	1,069,121	1,043,595	1,073,467	1,073,783
	SR	0.00	0.00	0.00	0.00
Weing8	Mean	620,483	622,515	622,905	622,775
	SR	0.33	0.03	0.00	0.03
#Mean		0	0	6	13
#SR		1	2	7	13
MSR		0.21	0.26	0.53	0.54
p-value		0.0003	0.0003	0.155

As can be seen from Table 8, compared with HLMS-B, HLMS-H, and HLMS, MS shows the worst performance in terms of #Mean, #SR, and MSR. The results further reveal that two learning strategies are effective in the search process. Additionally, it is noted that HLMS-B shows similar performance with MS while the performance difference between HLMS-H and MS is significant. However, the performance of HLMS integrated with two learning strategies is obviously better than that of any one.

Moreover, the convergence graphs of the average objective function values obtained by four algorithms are plotted in Figure 12 and Figure 13 for four representative instances: Sento1, Sento2, Weing7, and Weing8. As seen from Figure 12 and Figure 13, MS has the slowest convergence speed, while HLMS-H and HLMS have similar convergence speed and converge both faster than MS and HLMS-B.

Figure 12. Convergence graphs of MS, HLMS-B, HLMS-H, and HLMS on Sento1 and Sento2.

Figure 13. Convergence graphs of MS, HLMS-B, HLMS-H, and HLMS on Weing7 and Weing8.

In summary, we can draw a conclusion that the above two learning strategies can realize complementary advantages to enhance the performance of MS. Indeed, only using Baldwinian learning is not sufficient for exploitation. The conclusion is that the beneficial combination of the two strategies is significant for improving the performance of the algorithm.

4.8. Discussion

Comparison results demonstrate that the Baldwinian learning and GHS learning strategies can really improve the performance of HLMS, thereby making it better than the original MS and most other comparative algorithms on the majority of MKP instances, in terms of solution accuracy, convergence speed, and algorithm stability. The advantage of HLMS mainly lies in that GHS is used to guide the global search and dimensional learning can achieve a large jump to help solutions escape local extremum. The other reason is that Baldwinian learning as a local search strategy has the effect of changing the fitness landscape. This interaction between learning and evolution is very beneficial. Accordingly, both Baldwinian learning and GHS learning are more efficient and effective than MS alone.

In fact, based on the previous experimental results, we can find that considering small-scale MKP instances, medium-sized MKP instances, and large-scale MKP instances, the HLMS algorithm combining Baldwinian learning and GHS learning is an effective algorithm for solving MKP problems. Besides that, balancing exploration and exploitation is an important factor in metaheuristic algorithms by maintaining adequate diversity in swarm individuals so that reducing the probability of trapping in local optimal locations. In HLMS, GHS learning and Baldwinian learning respectively play the roles of exploration and exploitation, making the optimization performance of the algorithm better.

5. Conclusions and Future Work

This paper proposed a hybrid learning moth search algorithm (HLMS) inspired by the idea that the learning strategy could direct the evolutionary process. The framework proposed in this work includes two learning strategies: Baldwinian learning and GHS learning. In the search process, the two learning strategies play the role of local exploitation and global exploration, respectively.

HLMS is verified by solving the NP-hard 0–1 multidimensional knapsack problems. The experimental results on the 87 instances commonly used in literature showed that HLMS performs competitively in comparison with MS and other state-of-the-art meta-heuristics algorithms. Sensitivity analysis of Gaussian distribution and Cauchy distribution on Baldwinian learning is provided. The results proved that Cauchy mutation is more effective than Gaussian mutation as learning length. The effectiveness of two important learning strategies of HLMS is investigated. The results demonstrated that Baldwinian learning and GHS learning both play a major role in improving the performance of HLMS. MS with two learning strategies surpasses MS and MS with a single strategy. It confirms the effectiveness of our proposed learning strategies.

Future research on MS can be divided into two main directions: research on more real-world applications and research on improvements of the algorithms. Concerning the application of MS, it has the potential to solve more combinatorial optimization problems, such as maximum diversity problems (MDP), multi-objective knapsack problems (MOKP), and multi-demand multidimensional knapsack problems (MDMKP). In terms of the improvements of algorithms, more effective learning based on strategies can be adopted to enhance the search ability of the algorithm, such as orthogonal learning, reinforcement learning, and adaptive learning.