Abstract
Harmony Search Algorithm (HSA) is an evolutionary algorithm which mimics the process of music improvisation to obtain a nice harmony. The algorithm has been successfully applied to solve optimization problems in different domains. A significant shortcoming of the algorithm is inadequate exploitation when trying to solve complex problems. The algorithm relies on three operators for performing improvisation: memory consideration, pitch adjustment, and random consideration. In order to improve algorithm efficiency, we use roulette wheel and tournament selection in memory consideration, replace the pitch adjustment and random consideration with a modified polynomial mutation, and enhance the obtained new harmony with a modified β-hill climbing algorithm. Such modification can help to maintain the diversity and enhance the convergence speed of the modified HS algorithm. β-hill climbing is a recently introduced local search algorithm that is able to effectively solve different optimization problems. β-hill climbing is utilized in the modified HS algorithm as a local search technique to improve the generated solution by HS. Two algorithms are proposed: the first one is called PHSβ–HC and the second one is called Imp. PHSβ–HC. The two algorithms are evaluated using 13 global optimization classical benchmark function with various ranges and complexities. The proposed algorithms are compared against five other HSA using the same test functions. Using Friedman test, the two proposed algorithms ranked 2nd (Imp. PHSβ–HC) and 3rd (PHSβ–HC). Furthermore, the two proposed algorithms are compared against four versions of particle swarm optimization (PSO). The results show that the proposed PHSβ–HC algorithm generates the best results for three test functions. In addition, the proposed Imp. PHSβ–HC algorithm is able to overcome the other algorithms for two test functions. Finally, the two proposed algorithms are compared with four variations of differential evolution (DE). The proposed PHSβ–HC algorithm produces the best results for three test functions, and the proposed Imp. PHSβ–HC algorithm outperforms the other algorithms for two test functions. In a nutshell, the two modified HSA are considered as an efficient extension to HSA which can be used to solve several optimization applications in the future.
1 Introduction
As our society continues to advance, we face more complex problems in areas of science, engineering, economic and business. Proposing efficient techniques for resolving optimization problems is important for the aforementioned fields [57]. Metaheuristic algorithms are a class of approximation methods which cannot pledge to produce the global optima, but they can still help solve complex problems in a moderate amount of time. A metaheuristic-based algorithm tries to iteratively enhance the solution by exploring the problem search space. Such navigation is guided by an awareness about the problem with the anticipation of reaching a global optimum [20, 64].
Mainly, a metaheuristic is divided into local search-based algorithms and population-based algorithms. Population-based techniques firstly produce a group of randomly generated solutions. After that, the method combines the features from different solutions with the hope of generating better solutions [6, 11]. The population-based algorithms can concurrently look over large segments of the search space. Nevertheless, such algorithms may be unable to discover the local optima in each segment. As a result, the algorithm may not attain the optimal solution. Several population-based algorithms have been proposed and used to solve real-world problems such as genetic algorithm, bee colony optimization, and particle swarm optimization [45, 52, 62, 69, 70, 82].
Local search-based techniques initially begin with an individual solution which is repetitively modified using a neighborhood method. The algorithm stops when it reaches a local optimal solution. The search space can be decomposed into a group of smaller areas. Local search-based techniques work differently than population-based techniques, as they examine the search area intensely to obtain the local optima. Nevertheless, these techniques do not broadly investigate the search space.
Hill Climbing (HC) is considered the simplest local-search based technique. It is commonly used to boost the ability of population-based techniques in discovering the local optima in the currently searched segment [12]. Several techniques are introduced as an improvement of the HC to diminish the problem of being trapped in a local optima [20]. These techniques include Simulated Annealing (SA) [48], Tabu Search (TS) [37], Greedy Randomized Adaptive Search Procedure (GRASP) [29], Variable NeighborhoodSearch (VNS) [42], and Iterated Local Search (ILS) [50].
β-hill climbing is a recently introduced local search algorithm [7]. The algorithm has been used successfully to solve different optimization problems such as economic dispatch [8], text clustering [4], signal denoising [17, 18], multiple-reservoir scheduling [16], and feature selection [3]. Two operators are used in β-hill climbing to iteratively improve the solution which are neighborhood navigation (N-operator) and β-operator. The N-operator is used to reach randomly into a neighboring value. Then, some of the present solution values are selected or we choose a random value within the range using the β-operator. The β parameter is used to indicate the chances of using this operator.
Harmony Search Algorithm (HSA) [33] is a population-based algorithm which imitates the improvisation process of musicians. HSA has been applied to solving different optimization problems such as patient admission scheduling [24], face recognition [61], and many others [1, 5, 27, 31, 47, 67, 68, 76]. HSA begins by assigning values to harmonies stored in Harmony Memory (HM) within the range of the decision variables. After that, three operators are used to update the HM with a newly generated harmony. These operators are memory consideration, pitch adjustment, and random consideration. The memory consideration picks the value for the decision variable from the current values in HM. The pitch adjustment changes a current value in HM hoping to locally improve the solution. The random consideration selects a value randomly within the range of the decision variables to expand the solutions. Once the new harmony is created, it is compared to the worst harmony in the HM and replaces it if it is better. These steps are re-iterated until a stop criterion is met.
The structure and how the operators work in the HSA is altered over time to improve the algorithm performance when solving different problems [15, 26, 32, 71]. This is done in different ways such as tuning HSA selection in memory consideration [2, 10, 25, 54], improving the neighborhood in pitch adjustment [22, 43], self adaptive choosing of the algorithm parameters [13, 36, 56], using a different harmony memory structure [9, 55], and hybridizing the algorithm with other optimization methods [60, 74, 77].
Recently, several researchers propose variations of HSA to improve its performance. Multi-population-based harmony search algorithm that saves the best solutions in an external archive was developed [72]. In order to improve the diversity of the population, it is divided into sub-populations. Each sub-population is used to introduce a new harmony which replaces the worst solution. A modified harmony search with intersect mutation operator was introduced by Yi et al. [79]. The harmony memory is divided into best and worst portions which are then used to develop the new harmony. Adaptive harmony search with best-based search strategy is proposed by [40] to improve algorithm search efficiency. The algorithm uses the best solutions in the pitch adjustment stage to generate the new solutions. In addition, the HMCR and PAR are tuned automatically during the search process based on the newly generated solutions. Another enhancement was introduced by Wang et al. [75] with an improved differential harmony search algorithm. The algorithm uses two differential evolution mutation operators to improve the algorithm exploitation. In addition, the pitch adjustment step uses the best solution to generate the new solution. Lastly, selective refining harmony search is introduced by Shabani et al. [63]. This algorithm takes the decision variable into consideration when developing the new harmony by changing only the variables with unwanted values from harmonies.
A significant shortcoming of the HSA is the inadequate exploitation when trying to solve complex problems [51]. Additionally, the algorithm needs setting the algorithm operators experimentally to find the suitable values that can improve the algorithm exploration and exploitation [34]. In order to improve the search ability of HSA this paper introduces a new HSA called best polynomial harmony search algorithm with best β–hill climbing with two variations. The first one is called PHSβ–HC and the second one is called Imp. PHSβ–HC. The proposed algorithm relies on replacing the random selection in memory consideration with roulette wheel and tournament selection as they provided better performance [10]. In addition, the original pitch adjustment and random consideration are replaced with a modified version of highly disruptive polynomial mutation presented in Hasan et al. [43]. The obtained new solution is then iteratively improved using a modified version of β-hill climbing as a local search. Such modification can help to maintain the diversity and enhance the convergence speed of the modified HS algorithm. The algorithm is evaluated using standard benchmark functions. The results demonstrate that the proposed algorithm outperforms other versions of HSA. In addition, the new algorithm exceeds the performance of other evolutionary algorithms for some of the benchmark functions.
The remainder of this paper is organized as follows: Section 2 gives an introduction on HSA. A description of the proposed method is presented in Section 3. Analysis and discussion of the algorithm evaluation are found in Section 4. Lastly, the paper concludes with possible future directions in Section 5.
2 The harmony search algorithm
Harmony Search algorithm (HSA) imitates how musicians improve their music iteratively [35]. During this process, each musician plays on an instrument to achieve the best harmony. In HSA, each decision variable represents a musician and the algorithm generates a value in order to find a global optimal solution.
The HSA consists of the following main steps [35]:
Step 1. Initialization of the problem and algorithm parameters
The optimization problem can be modeled as a function f to be minimized or maximized as follows:
Note that N is the decision variables count;
Step 2. Initialization of the harmony memory
In step 2, the solution variables (xi) are initialized randomly with feasible values picked from the interval between the upper and lower bounds
∀i = 1, 2, . . . , N and ∀j = 1, 2, . . . , HMS, and U(0, 1) generate a uniform random number between 0 and 1.
Step 3. Improvisation of a new harmony
A new harmony vector
Memory consideration. The memory consideration considers the value of the decision variable
Pitch adjustment. In the memory consideration, a neighboring rule is applied based on the probability of PAR where PAR ∈ (0, 1). The picked value in memory consideration for the decision variable
Random consideration. The decision variables that do not take values from the memory consideration will pick a feasible value randomly with a probability of (1-HMCR).
Step 4. Harmony memory update
In this step, the newly generated harmony vector in step 3 is evaluated to decide if it will be included in HM.
The objective function value for the vector
Step 5. Repeat until the stopping condition is met
The algorithm is terminated if it reaches the maximum number of improvisation (NI). Otherwise, steps 3 and 4 are repeated.
3 Proposed Method: best polynomial harmony search algorithm with best β-hill climbing
3.1 Best β-hill climbing
In this section, the β-hill climbing optimizer [7] and its modified version are explained. β-hill climbing starts with a set of solutions randomly produced X = (X1, . . . , Xn). Note that n is the number of decision variables where its selected value is feasible (i.e., within the variable range). The generated initial solution will be iteratively modified in the hope of obtaining the optimal solution. The β-hill climbing optimizer relies on using two operators to undergo updating the solution which are: N-operator and β-operator.
The N-operator is used for exploitation in the algorithm by applying a small change in the current solution. The operator is used to update the variable Xi, i ∈ [1, 2, . . . , n] for a randomly picked value from the solution as follows:
On the other hand, the β-operator is responsible for exploration by applying a uniform mutation to the current solution. The decision variable is selected to be changed using the β parameter as follows:
Note that β is how often the uniform mutation is applied. The r operator is a random number between 0 and 1. The pseudo code of the β-hill climbing is presented in algorithm 1.
Algorithm 1
Original β–hill climbing pseudo-code
xi = LBi + (UBi − LBi)× U(0, 1), ∀i = 1, 2, 3, and 4 {The initial solution x}
Calculate f (x)
itr = 0
while (itr ≤ Max_Itr) do
x′ = improve( N (x))
for i = 1, · · · , N do
if (r ≤ β) then
end if { r ∈ [0, 1]}
end for
if f (x′) ≤ f (x) then
x = x′
end if
itr = itr + 1
end while
The proposed best β-hill climbing algorithm is shown in algorithm 2. We add the tournament selection [11, 38] in addition to the random selection in the β-operator to improve the algorithm exploration. Furthermore, the N-operator is replaced with information from the global best to improvise the exploitation using the following modified equation adapted from [23, 41]:
where r1 and r2 are two distinct integers from the range of available solutions, and v is the picked value in the β-operator.
3.2 Selection in memory consideration of harmony search
The selection method used in HSA has an effect on improving the diversity of the generated new harmonies [10] and as a result, affects the ability of the algorithm to escape from premature convergence. The selection schemes can be categorized into static and dynamic [19]. The dynamic selection changes the chance of picking a solution at each iteration [10] such as roulette wheel selection. On the other hand, the static selection chance of picking a solution stays fixed during searching such as tournament selection.
The proportionate (or Roulette wheel) selection scheme has been used in genetic algorithms [46]. This selection method relies on using the fitness value of any solution and compared against other solutions fitness values. The steps of this selection scheme are presented in algorithm 3.
As proposed in Albetar et al. [10], the tournament selection method picks randomly k solutions from the HM and after that chooses the best fitness solution. Algorithm 4 shows the procedure of tournament selection.
A comparison of six different selection methods for HSA in Albetar et al. [10] recommends the use of roulette wheel and tournament selection as they outperform other methods in most of the evaluated functions. The proposed HSA replaces the random memory consideration in the original HSA with roulette wheel or tournament selection based on a specific probability.
Algorithm 2
The proposed best β–hill climbing pseudo-code
itr = 0
while (itr ≤ Max_Itr) do
i=random number from the range of available solutions
if (r ≤ SelectionRate) then
else
end if
for j = 1, · · · , N do
if (r ≤ β) then
Pick two distinct integers from the range of available solutions r1 and r2
end if { r ∈ [0, 1]}
end for
if f (x′) ≤ f (x) then
x = x′
end if
itr = itr + 1
end while
Algorithm 3
Pseudocode for the roulette wheel selection method
Set r ∼ U(0, 1).
Set found = False
Set sum_prob = 0.
Set k = 0.
while (i ≤ HMS) AND NOT(found) do
sum_prob = sum_prob + pi
if ( sum_prob ≥ r ) then
k = i
found = True
end if
i = i + 1
end while
Return(k)
Algorithm 4
Pseudo code for the tournament selection method
Set t=tournament size
Set T= randomly picked t solutions from the HM
Return index of the best solution from T
3.3 Polynomial mutation
One of the Genetic Algorithms (GAs) operators is mutation which is used to modify some features of the newly generated chromosome [65]. The aim of applying such operation is to diversify the newly created solution to escape from being stuck in local optima. Several mutation schemes are presented in Hasan et al. [43] to alter the original HSA. The evaluation results when comparing six different mutation schemes prove that highly disruptive polynomial mutation (shown in algorithm 5) gives better results in most of the cases. Our proposed HSA applies a modified version of polynomial mutation (shown in algorithm 6) in both random consideration and pitch adjustment operators. We call this version best polynomial mutation.
Algorithm 5
Highly disruptive polynomial mutation
i ←from the range of available solutions
r ← U[0, 1]
if (r ≤ Pm) then
r ← U[0, 1]
xi ← xi + δq .(UBi − LBi)
end if
Note that n is the number of decision variables, Pm is the mutation probability and ηm is the distribution index presented as a non-negative value. Each decision variables xi has an upper bound UBi and a lower bound LBi.
Algorithm 6
The proposed best polynomial mutation
i ←taken from the range of available solutions
r ← U[0, 1]
if (r ≤ Pm) then
r ← U[0, 1]
y = (LBi + (UBi − LBi)× U(0, 1))/xi
end if
3.4 Best polynomial harmony search algorithm with best β-hill climbing
In order to improve the exploitation process of HSA, we embed the proposed best β-hill climbing as a local search method to dig deeper into the search space. The best β-hill climbing will be used after we generate the new harmony and it will iteratively work to improve it. In addition, we replace the techniques used in the three operators of the original HSA with efficient techniques that will help in a better exploration of the search space. The random selection in the memory consideration operator is replaced with roulette wheel
Algorithm 7
The proposed PHSβ–HC algorithm
Set HMCR, PAR, NI, HMS, SelectionRate, β–SelectionRate, β–Rate, β, β–NI.
itr = 0
while (itr ≤ NI) do
for i = 1, · · · , N do
if (U(0, 1) ≤ HMCR) then
if (U(0, 1) ≤ SelectionRate) then
else
end if
if (U(0, 1) ≤ PAR) then
end if
else
end if
end for
The proposed best β–hill climbing shown in algorithm 2 improvise the generated new harmony x′
if
Include x′ to the HM.
Exclude xworst from HM.
end if
itr = itr + 1
end while
or tournament selection schemes shown previously in algorithm 3 and 4. The random consideration and the pitch adjustment use the proposed best polynomial mutation presented in algorithm 6. The proposed best polynomial harmony search algorithm with best β–hill climbing (PHSβ–HC) steps are shown in algorithm 7. The steps of the algorithm are presented in Figure 1.
The flowchart of the proposed PHSβ–HC.
After that, two new concepts are introduced to enhance the algorithm:
In the β–HC algorithm, we improvise the best solution found so far, not the newly generated harmony vector.
The updated algorithm is called improved PHSβ–HC algorithm (Imp. PHSβ–HC) and is shown in algorithm 8. The steps of the algorithm are presented in Figure 2.
The flowchart of the proposed Imp. PHSβ–HC.
4 Experimental Results
In this section, the proposed PHSβ–HC algorithm is experimentally evaluated on 13 classic benchmark functions shown in Table 1 [14, 39, 41, 78]. These functions cover a wide range of characteristics (i.e., unimodal, multimodal, separable, and non-Separable) as shown in Table 1. The experiments were executed on an Intel CORE i7 vPro 7th Gen with 16 GB of RAM where the proposed algorithm is coded using MATLAB R2016a under Windows 10.
The 13 classical test functions, Characteristics, U: Unimodal, M: Multimodal, S: Separable, N: Non-Separable
| Function | Name | Characteristics | Range | fmin |
|---|---|---|---|---|
| f1 | Sphere problem | US | [− 100, 100]D | 0 |
| f2 | Schwefel’s problem 2.22 | UN | [− 10, 10]D | 0 |
| f3 | Schwefel’s problem 1.2 | UN | [− 100, 100]D | 0 |
| f4 | Schwefel’s problem 2.21 | US | [− 100, 100]D | 0 |
| f5 | Rosenbrock’s function | UN | [− 30, 30]D | 0 |
| f6 | Step function | US | [− 100, 100]D | 0 |
| f7 | Quatric function with noise | US | [− 1.28, 1.28]D | 0 |
| f8 | Rastrigin’s function | MS | [− 5.12, 5.12]D | 0 |
| f9 | Ackley’s function | MN | [−32, 32]D | 0 |
| f10 | Griewank function | MN | [− 600, 600]D | 0 |
| f11 | Penalized function 1 | MN | [− 50, 50]D | 0 |
| f12 | Penalized function 2 | MN | [− 50, 50]D | 0 |
| f13 | Alpine function | MS | [− 10, 10]D | 0 |
The dimensionality of all the test functions are set to D=30 which is similar to what has been done by the comparative methods. An extensive testing of different parameter settings is performed to study their effect on the convergence behaviour of the proposed methods. These parameters are set according to intensive investigation with guidance from literature [6, 44, 54]. The parameter settings for the proposed PHSβ– HC algorithm are as follows: HMS=100, HMCR=0.9, PAR=0.3, NI=50,000, SelectionRate=0.7, β–Rate=0.8, β– SelectionRate=0.8, β=0.3, β–NI=200, and ηm = 10. On the other hand, The parameter settings for the proposed improved PHSβ–HC algorithm are the same except HMCR=(U(0.9, 1), PAR=(U(0.1, 0.3).
The two proposed algorithms (PHSβ– HC and Imp. PHSβ–HC) are compared against the algorithms presented in Table 2 in terms of mean and standard deviation. The mean and the standard deviation for the tested
Key to comparative methods for the first group experiment with multiple HS algorithms.
| No. | Abbr | Name | Citation |
|---|---|---|---|
| 1 | HS | Harmony Search algorithm | [35] |
| 2 | ACHS | HS with the adaptive control parameters strategy | [73] |
| 3 | GHS | Global best HS | [54] |
| 4 | OHS | HS with opposition-based learning | [30] |
| 5 | IGHS | Improved global-best HS | [28] |
Algorithm 8
The proposed Imp. PHSβ–HC algorithm
Set HMCR, PAR, NI, HMS, SelectionRate, β–SelectionRate, β–Rate, β, β–NI.
itr = 0
while (itr ≤ NI) do
HMCR = U(0.9, 1)
PAR = U(0.1, 0.3)
for i = 1, · · · , N do
if (U(0, 1) ≤ HMCR) then
if (U(0, 1) ≤ SelectionRate) then
else
end if
if (U(0, 1) ≤ PAR) then
end if
else
end if
end for
if (f (x′) < f (xworst)) then
Include x′ to the HM.
Exclude xworst from HM.
end if
The proposed best β–hill climbing shown in algorithm 2 improvise the best solution found so far
itr = itr + 1
end while
benchmark functions using is shown in Table 5. The best algorithm with the best result for each function is shown in bold text font. IGHS produces the best results for the five functions f1, f3, f4, f6, and f10. The IGHS algorithm uses improved global best technique which picks the best individuals of the members in the generated solutions, which makes a balance between exploration and exploitation when solving (unimodal, separable) and (unimodal, non-separable) functions.
On the other hand, the proposed Imp. PHSβ–HC generates the best solution for the four functions f2, f5, f6, and f7. The algorithm escapes the problem of the local optimum. It is remarkable that the performance of Imp. PHSβ–HC obtains the best results on four functions which are (unimodal , non-separable) and (unimodal, separable). This is because the proposed algorithm is a hybrid technique that uses a population-based technique empowered by a local search method.
In addition, the proposed algorithm PHSβ–HC generates the best results for the three functions f 11, f 12, and f13. This algorithm explores the search space deeply and diversifies the selected individuals in each iteration. As a result the algorithm obtains the best results for (multi-modal , non-separable) and (multi-modal, separable) functions. From the results, we can conclude that the two proposed algorithms are competitive when compared against the original HS and the other versions of HS in terms of obtaining the best solution for the different function.
The proposed algorithms provide a tradeoff between exploration and exploitation. The algorithms are capable of finding better solutions by adding diversity to the population which allows more exploration of the search space. Furthermore, the local search used allows more focused exploitation to find better solutions. The proposed Imp. PHSβ–HC algorithm has the ability to get out of a local minimum and can be efficiently used for unimodal, non-separable and unimodal, separable functions. On the other hand, the proposed PHSβ– HC algorithm is quite successful in optimizing multimodal, non-separable and multimodal, separable functions.
In order to compare the performance of the proposed algorithms (i.e., PHSβ–HC and Imp. PHSβ–HC) with multiple HS algorithms on all the benchmark function, the Friedman test is applied. Table 3 presents the average rankings of the two proposed algorithms against five other HS algorithms. The results show that the two proposed algorithms ranked 2nd and 3rd. On the other hand, IGHS obtains the best ranking.
Average rankings of HS, ACHS, GHS, OHS, IGHS, PHSβ–HC, Imp. PHSβ–HC on the 13 test functions gained by the Friedman test
| Algorithm | Mean Rank |
|---|---|
| HS | 5.81 |
| ACHS | 5.42 |
| GHS | 4.58 |
| OHS | 5.73 |
| IGHS | 3.27 |
| PHSβ–HC | 4.73 |
| Imp. PHSβ–HC | 3.81 |
Two-tailed t-test at a 0.05 significance level is applied to see if there is significant evidence that the mean of the proposed method is different than other methods means. The obtained results show that the proposed method PHSβ– HC mean is different than the methods HS, NGHS, and OHS. In addition, the proposed method Imp. PHSβ–HC mean is different than the methods HS, ACHS, GHS, NGHS, OHS, and IGHS.
4.1 Comparisons with other evolutionary algorithms (EAs)
The two proposed algorithms are compared with similar EAs including particle swarm optimization (PSO) and differential evolution (DE) algorithms presented in Table 4. Table 6 compares between the two proposed algorithms (PHSβ–HC and Imp. PHSβ–HC) and four versions of PSO. The four versions of PSO are HPSO-TVAC, CLPSO, FPSO, and OLPSO-G. The results are shown in Table 6 show that the proposed PHSβ–HC algorithm generates the best result for the functions f6, f11, and f12. In addition, the proposed Imp. PHSβ–HC algorithm is able to overcome the other algorithms for functions f6 and f8. The CLPSO algorithm obtains the best result for four functions f5, f6, f7, and f10.
Key to comparative particle swarm optimization (PSO) and differential evolution (DE) algorithms.
| No. | Abbr | Name | Citation |
|---|---|---|---|
| 1 | HPSO-TVAC | Self-organizing hierarchical particle swarm optimizer | [59] |
| 2 | CLPSO | Comprehensive learning particle swarm optimizer | [49] |
| 3 | FPSO | Fully informed particle swarm | [53] |
| 4 | OLPSO-G | Orthogonal learning particle swarm optimization | [80] |
| 5 | jDE | Self adaptive differential evolution | [21] |
| 6 | JADE | Adaptive differential evolution with optional external archive | [81] |
| 7 | SaDE | Differential evolution algorithm with strategy adaptation | [58] |
In addition, the two proposed algorithms are compared with four variations of DE which are DE [66], jDE, JADE, and SaDE. It is clear from Table 7 that the proposed PHSβ–HC algorithm produces the best results for the functions f6, f11, and f12. Furthermore, the proposed Imp. PHSβ–HC algorithm outperforms the other algorithms for functions f6 and f8. The JADE algorithm outperform other algorithms in the five functions f1, f2, f3, f7, and f9.
Two-tailed t-test at a 0.05 significance level is applied to see if there is significant evidence that the mean of the proposed method is different than other methods means. The results show that the proposed method PHSβ– HC mean is different than the methods HPSOTVAC, CLPSO, FPSO, and OLPSOG. Furthermore, the proposed method Imp. PHSβ–HC mean is different than the methods HPSOTVAC, CLPSO, and OLPSOG. On the other hand, the proposed methods PHSβ– HC and Imp. PHSβ–HC mean is not different than the methods DE, jDE, JADE, and SaDE.
Experimental results of HS, ACHS, GHS, OHS, IGHS, PHSβ–HC , and Imp. PHSβ–HC over 30 independent runs on the 15 test functions
| HS | ACHS | GHS | OHS | IGHS | PHSß–HC | Imp. PHSß–HC | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |
| f1 | 2.81E–04 | 1.25E–05 | 5.95E–05 | 2.10E–06 | 1.00E–05 | 0.00E+00 | 2.93E–04 | 3.67E–05 | 9.65E–08 | 8.76E–09 | 1.29E–01 | 4.27E–02 | 7.16301E–07 | 1.03435E–06 |
| f2 | 6.40E–02 | 6.82E–04 | 1.98E–02 | 3.27E–03 | 7.28E–02 | 0.00E+00 | 6.16E–02 | 1.76E–03 | 1.29E–03 | 9.28E–05 | 1.97E+00 | 1.90E–01 | 9.67193E–05 | 7.64385E–05 |
| f3 | 8.30E+02 | 4.46E+02 | 4.32E+02 | 4.91E+01 | 5.83E+00 | 2.80E+00 | 1.87E+02 | 4.43E+01 | 4.33E–07 | 1.46E–07 | 4.23E–01 | 1.03E–01 | 3.85E+02 | 3.10E+02 |
| f4 | 7.09E–01 | 1.77E–01 | 6.29E–01 | 1.99E–01 | 2.77E+00 | 1.23E+00 | 8.82E–01 | 2.30E–01 | 1.42E–04 | 1.25E–05 | 3.29E–01 | 3.11E–02 | 3.24E–02 | 1.50E–02 |
| f5 | 3.29E+01 | 2.88E+01 | 3.38E+01 | 2.80E+01 | 4.97E+01 | 1.28E+01 | 3.10E+01 | 3.16E+01 | 6.08E+01 | 2.80E+01 | 3.07E+01 | 3.41E+01 | 1.61E+01 | 1.67E+01 |
| f6 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |
| f7 | 6.01E–03 | 9.74E–04 | 4.21E–03 | 1.21E–03 | 1.71E–03 | 1.38E–03 | 5.37E–03 | 4.13E–04 | 2.24E–03 | 6.52E–04 | 5.40E+00 | 1.43E+00 | 1.99E–02 | 6.23E–03 |
| f8 | 4.47E–05 | 4.92E–03 | 1.11E–02 | 2.87E–03 | 8.63E–03 | 0.00E+00 | 5.11E–02 | 7.26E–03 | 1.77E–05 | 3.53E–06 | 6.50E+00 | 7.66E–01 | 2.07893E–07 | 2.78854E–07 |
| f9 | 4.47E–05 | 9.94E–05 | 5.94E–03 | 3.63E–04 | 2.09E–02 | 0.00E+00 | 1.27E–02 | 8.64E–04 | 2.21E–04 | 1.27E–06 | 4.64E–01 | 9.81E–02 | 1.16E–04 | 1.03E–04 |
| f10 | 9.88E–03 | 8.05E–03 | 3.26E–02 | 3.31E–02 | 1.02E–01 | 0.00E+00 | 3.68E–02 | 1.40E–02 | 3.29E–03 | 4.65E–03 | 3.00E–02 | 2.90E–02 | 2.08E–02 | 2.45E–02 |
| f11 | 2.12E–06 | 1.68E–07 | 4.81E–07 | 7.96E–08 | 8.03E–07 | 0.00E+00 | 2.05E–06 | 8.13E–08+ | 9.48E–10 | 1.03E–10+ | 1.57054E–32 | 5.5674E–48 | 2.79E–02 | 1.54E–02 |
| f12 | 3.17E–05 | 2.55E–06 | 3.67E–03 | 5.18E–03 | 1.13E–05 | 0.00E+00 | 3.55E–05 | 2.02E–06 | 1.50E–08 | 2.73E–09 | 1.34978E–32 | 5.5674E–48 | 2.94E–09 | 4.74355E–09 |
| f13 | 7.70E–03 | 1.10E–03 | 1.21E–02 | 3.37E–03 | 0.00E+00 | 0.00E+00 | 6.82E–03 | 1.37E–03 | 1.54E–04 | 5.57E–06 | 1.42749E–13 | 4.28212E–13 | 1.31E–05 | 1.02552E–05 |
Comparisons between PHSβ–HC, Imp. PHSβ–HC, and PSO algorithms on 30-dimensional test functions
| HPSO-TVAC | CLPSO | FPSO | OLPSO-G | PHSβ–HC | Imp. PHSβ–HC | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |
| f1 | 2.83E–33 | 3.19E–33 | 1.58E–12 | 7.70E–13 | 2.40E–16 | 2.00E–31 | 4.12E–54 | 6.34E–54 | 1.29E–01 | 4.27E–02 | 7.16E–07 | 1.03E–06 |
| f2 | 9.03E–20 | 9.58E–20 | 2.51E–08 | 5.84E–09 | 1.58E–11 | 1.03E–22 | 9.85E–30 | 1.01E–29 | 1.97E+00 | 1.90E–01 | 9.67E–05 | 7.64E–05 |
| f5 | 2.39E+01 | 2.65E+01 | 1.13E+01 | 9.85E+00 | 2.81E+01 | 2.31E+02 | 2.15E+01 | 2.99E+01 | 3.07E+01 | 3.41E+01 | 1.61E+01 | 1.67E+01 |
| f6 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |
| f7 | 9.82E–02 | 3.26E–02 | 5.85E–03 | 1.11E–03 | 4.16E–03 | 2.40E–06 | 1.16E–02 | 4.10E–03 | 5.40E+00 | 1.43E+00 | 1.99E–02 | 6.23E–03 |
| f8 | 9.43E+00 | 3.48E+00 | 9.09E–05 | 1.25E–04 | 7.38E+01 | 3.70E+02 | 1.07E+00 | 9.92E–01 | 6.50E+00 | 7.66E–01 | 2.08E–07 | 2.79E–07 |
| f9 | 7.29E–14 | 3.00E–14 | 3.66E–07 | 7.57E–08 | 2.17E–09 | 1.71E–18 | 7.98E–15 | 2.03E–15 | 4.64E–01 | 9.81E–02 | 1.16E–04 | 1.03E–04 |
| f10 | 9.75E–03 | 8.33E–03 | 9.02E–09 | 8.57E–09 | 1.47E–03 | 1.28E–05 | 4.83E–03 | 8.63E–03 | 3.00E–02 | 2.90E–02 | 2.08E–02 | 2.45E–02 |
| f11 | 2.71E–29 | 1.88E–28 | 6.45E–14 | 3.70E–14 | 5.51E–18 | 1.45E–34 | 1.59E–32 | 1.03E–33 | 1.57E–32 | 5.57E–48 | 2.79E–02 | 1.54E–02 |
| f12 | 2.79E–28 | 2.18E–28 | 1.25E–12 | 9.45E–12 | 1.37E–17 | 3.42E–32 | 4.39E–04 | 2.20E–03 | 1.35E–32 | 5.57E–48 | 2.94E–09 | 4.74E–09 |
Comparisons between PHSβ–HC, Imp. PHSβ–HC, and DE algorithms on 30-dimensional test functions
| DE | jDE | JADE | SaDE | PHSβ–HC | Imp. PHSβ–HC | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |
| f1 | 9.80E–14 | 8.40E–14 | 1.46E–28 | 1.78E–28 | 1.32E–54 | 9.22E–54 | 8.43E–06 | 3.63E–20 | 1.29E–01 | 4.27E–02 | 7.16E–07 | 1.03E–06 |
| f2 | 1.60E–09 | 1.10E–09 | 9.02E–24 | 6.01E–24 | 3.18E–25 | 2.05E–24 | 3.51E–25 | 2.74E–25 | 1.97E+00 | 1.90E–01 | 9.67E–05 | 7.64E–05 |
| f5 | 2.10E+00 | 1.50E+00 | 1.30E+01 | 1.40E+01 | 3.20E–01 | 1.10E+00 | 2.10E+01 | 7.80E+00 | 3.07E+01 | 3.41E+01 | 1.61E+01 | 1.67E+01 |
| f6 | 4.70E+03 | 1.10E+03 | 6.13E+02 | 1.72E+02 | 5.62E+00 | 1.87E+00 | 5.07E+01 | 1.34E+01 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |
| f7 | 4.70E–03 | 1.20E–03 | 3.35E–03 | 8.68E–04 | 6.14E–04 | 2.55E–04 | 4.86E–03 | 5.21E–04 | 5.40E+00 | 1.43E+00 | 1.99E–02 | 6.23E–03 |
| f8 | 1.80E+02 | 1.30E+01 | 3.32E–04 | 6.39E–04 | 1.33E–01 | 9.74E–02 | 2.43E+00 | 1.60E+00 | 6.50E+00 | 7.66E–01 | 2.08E–07 | 2.79E–07 |
| f9 | 1.10E–01 | 3.90E–02 | 2.37E–04 | 7.10E–05 | 3.35E–09 | 2.84E–09 | 3.81E–06 | 8.26E–07 | 4.64E–01 | 9.81E–02 | 1.16E–04 | 1.03E–04 |
| f10 | 2.00E–01 | 1.10E–01 | 7.29E–06 | 1.05E–05 | 1.57E–08 | 1.09E–07 | 2.52E–09 | 1.24E–08 | 3.00E–02 | 2.90E–02 | 2.08E–02 | 2.45E–02 |
| f11 | 1.20E–02 | 1.00E–02 | 7.03E–08 | 5.74E–08 | 1.67E–15 | 1.02E–14 | 8.25E–12 | 5.12E–12 | 1.57E–32 | 5.57E–48 | 2.79E–02 | 1.54E–02 |
| f12 | 7.50E–02 | 3.80E–02 | 1.80E–05 | 1.42E–05 | 1.87E–10 | 1.09E–09 | 1.93E–09 | 1.53E–09 | 1.35E–32 | 5.57E–48 | 2.94E–09 | 4.74E–09 |
| f13 | 2.30E–04 | 1.70E–04 | 6.08E–10 | 8.36E–10 | 2.78E–05 | 8.43E–06 | 2.94E–06 | 3.47E–06 | 1.43E–13 | 4.28E–13 | 1.31E–05 | 1.03E–05 |
Table 8 present a summarization of the best obtained results between the comparative methods. The proposed methods are shown in bold font.
Best obtained results between comparative methods
| function | Best result HS | Best result PSO | Best result DE |
|---|---|---|---|
| f1 | IGHS | HPSO-TVAC | JADE |
| f2 | Imp. PHSβ–HC | OLPSO-G | JADE |
| f3 | IGHS | - | - |
| f4 | IGHS | - | - |
| f5 | Imp. PHSβ–HC | CLPSO | JADE |
| f6 | All functions | All functions | Imp. PHSβ–HC; PHS-β–HC |
| f7 | GHS; Imp. PHSβ–HC | CLPSO | JADE |
| f8 | Imp. PHSβ–HC | Imp. PHSβ–HC | Imp. PHSβ–HC |
| f9 | HS | OLPSO-G | JADE |
| f10 | IGHS | CLPSO | SaDE |
| f11 | PHS-β–HC | PHS-β–HC | PHS-β–HC |
| f12 | PHS-β–HC | textbfPHS-β–HC | PHS-β–HC |
| f13 | PHS-β–HC | - | PHS-β–HC |
5 Conclusion and Future Work
This paper presented two new variations of harmony search with local search and updated harmony selection consideration, pitch adjustment and random consideration. The harmony selection utilized roulette wheel and tournament selection. The pitch adjustment and random consideration used a modified version of polynomial mutation called best polynomial mutation. These changes were introduced to improve the exploration of the algorithm. A recently proposed local search method called β–HC is used to improve the exploitation of the algorithm. The first proposed algorithm is called PHSβ–HC. In this algorithm, the HMCR and PAR are fixed and the local search is applied to the generated new harmony to improve it. The second proposed algorithm is called Imp. PHSβ–HC. This algorithm uses a randomly picked HMCR and PAR from an identified range based on the literature suggestion. The local search is applied to the best solution currently in the harmony memory.
The experimental results demonstrated the competitiveness of the two new proposed algorithms when compared against variations of HS, PSO, and DE. In the future, we will use the two proposed algorithms to solve economic dispatch and job scheduling problems. Another future direction can be evaluating the algorithm performance on the functions with other smaller or larger dimensions.
Acknowledgement
This research project is partially funded by the Dartmouth College and American University of Kuwait (Dartmouth-AUK) fellowship program.
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