^{2} Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, China;
^{3} Key Laboratory of Marine Technology and Control Engineering Ministry of Communications, Shanghai Maritime University, Shanghai 201306, China
Differential evolution (DE) algorithm, which was introduced by Storn and Price^{[1]}, is a simple yet competitive metaheuristic algorithm. Although DE has been widely utilized to deal with a large number of benchmark test functions and industrial application problems, the search behavior of DE is mainly determined by three main parameters (i.e., mutation control parameter F, crossover control parameter CR, and population size NP) and two strategies (i.e., mutation and crossover)^{[2, 3]}. To improve DE performance, researchers have proposed different empirical guidelines to set control parameters in the previous studies^{[4–6]}. However, the above guidelines lack sufficient experimental justifications^{[7]} since they are achieved through some particular experiments.
Subsequently, a large number of researchers focus on the adaptation or selfadaptation of control parameters and strategies to enhance the search capability of DE and reduce manual tuning. For example, Liu and Lampinen^{[8]} proposed a fuzzy adaptive DE, in which fuzzy logic controllers are utilized to produce F and CR. A DE with selfadaptive parameter control (jDE) is presented by Brest et al.^{[9]}. In the jDE, F and CR are generated by a normal distribution function. Moreover, each individual in the population has its own F and CR. In [7], a selfadaptive DE (SaDE) is introduced. In the SaDE, suitable mutation strategy and control parameters can be automatically achieved during the run. Zhang and Sanderson^{[10]} introduced a novel DE (JADE) which uses an improved DE/currenttobest/1 mutation strategy and adjusts control parameters in an adaptive fashion. In [11], an ensemble of mutation strategies and control parameters with DE algorithm (EPSDE) is proposed. In the EPSDE, some combinations of mutation strategy and control parameters are employed. Wang et al.^{[12]} presented a composite DE (CoDE) wherein three fixed parameter combinations were combined with three selected mutation strategies randomly. In [13], a novel DE (CoBiDE) is proposed to improve the performance of DE, in which the covariance matrix learning and the bimodal distribution parameter setting are used. Fan and Yan^{[14]} introduced a DE with selfadaptive mutation strategy and control parameters (SSCPDE), in which suitable control parameters and mutation strategy can be simultaneously achieved in different stages of the evolution. Recently, Fan and Yan^{[15]} proposed a selfadaptive DE with zoning evolution of control parameters and adaptive mutation strategy (ZEPDE), in which F and CR can be automatically generated in their own zones and the suitable mutation strategy can be selfadaptively obtained in different phases of the evolution. In [16], a DE with dynamic parameters selection (DEDPS) is proposed, in which the best performing control parameter combination can be automatically found in different evolution stages.
Besides the above studies, many methods have also been introduced to improve the search performance of the DE. For instance, in [17], an oppositionbased learning is utilized to enhance the global search capability of the DE, the experimental results demonstrate that the approach is effective. Gong et al.^{[18]} introduced a crossover rate repair method to enhance the performance of other selfadaptive DE algorithms. Recently, Gong et al.^{[19]} introduced an adaptive ranking mutation operator (ARMOR) for the DE algorithm, which is used to deal with the constrained optimization problems introduced in CEC2006 and CEC2010. Yang et al.^{[20]} introduced an autoenhanced population diversity (AEPD) that is employed to improve the population diversity when the population is stagnant and premature. Fan et al.^{[21]} introduced an autoselection mechanism (ASM) to automatically choose a suitable algorithm from an algorithm pool. The results indicate that the ASM can improve the algorithm robustness. In [22], an eigenvectorbased crossover strategy is used to improve the search ability of the DE. For more related studies, interested readers can refer to [23–31].
Although various adaptation techniques can improve the performance of the DE effectively, most of them are based on the population information. In most cases, this information may be deceptive or misleading in complex optimization environment^{[32]}, thus a wellknown empirical guideline is incorporated into the JADE (called TPCJADE) to improve the search efficiency of DE in the current study. In the TPCJADE, the parameter adaptation technique used in the JADE is employed to search the promising region in the early evolution stages, while the empirical guideline is utilized to enhance the search efficiency of the JADE in the later evolution stages. The main reason is that the fitness landscape of the current population may be not very complex in the later phases of the search. The performance of the proposed algorithm is compared with that of four DE variants on two widely used test suites, i.e., IEEE CEC2005^{[33]} and IEEE CEC2015^{[34]}. The experimental results reveal that the TPCJADE is competitive among all competitors.
Several problems in chemical/biochemical processes can be described as a set of nonlinear differential equations, which are named as dynamic optimization problems. To solve these dynamic optimization problems, three main approaches^{[35, 36]}, which include dynamic programming (DP) methods, indirect optimization methods, and direct optimization methods, have been introduced.
For the DP, it is based on Bellman′s optimality conditions^{[37]} and is a promising approach to solve dynamic optimization problems. However, DP is not always available for solving highdimensional dynamic optimization problems. Subsequently, Luus^{[38]} proposed an iterative dynamic programming (IDP) to overcome the defects of the DP, in which the coarse grid points and search region reduction strategy are utilized. The results indicate that the IDP is able to improve the solution accuracy and computation efficiency when compared with the DP.
In indirect methods, a dynamic optimization problem is converted into a boundary value problem (BVP), which can be addressed by some methods^{[36, 39, 40]}. In fact, these approaches are very effective, but they are difficult to be used in solving dynamic optimization problems due to the active inequality constraints^{[41]}.
For direct methods, a dynamic optimization problem can be usually transcribed into a nonlinear programming (NLP) problem and is addressed by parameterization, which can be classified into two different techniques^{[42]}, namely, control vector parameterization (CVP) and complete parameterization. In the CVP, only control variables are discretized, whereas complete parameterization discretizes both control variable and state variable. In the current study, the CVP and the TPCJADE are used to solve actual dynamic optimization problems. The results show that the TPCJADE is competitive when compared with the other approaches.
2 Differential evolutionA minimization problem can be expressed as follows:
$f({{{x}}^ * }) = \mathop {\min }\limits_{{x_i} \in \Omega } f({{{x}}_i}) ,\;{{{x}}_i} \in {{{P}}_0} = \prod\limits_{j = 1}^D {\left[ {{L_j},\;{U_j}} \right]}$  (1) 
where f denotes the objective function;
DE is a populationbased stochastic optimization approach.
The procedure of executing DE is described as follows^{[43]}:
1) Initialization: Determine F, CR, NP, and maximum number of generations
2) Mutation: After initialization operation, for each
“DE/rand/1”
${{v}}_i^G = {{x}}_{{r_1}}^G + F \times ({{x}}_{{r_2}}^G  {{x}}_{{r_3}}^G)$  (2) 
“DE/rand/2”
${{v}}_i^G = {{x}}_{{r_1}}^G + F \times ({{x}}_{{r_2}}^G  {{x}}_{{r_3}}^G) + F \times ({{x}}_{{r_4}}^G  {{x}}_{{r_5}}^G)$  (3) 
“DE/currenttobest/1”
${{v}}_i^G = {{x}}_i^G + F \times ({{x}}_{{\rm{best}}}^G  {{x}}_i^G) + F \times ({{x}}_{{r_1}}^G  {{x}}_{{r_2}}^G)$  (4) 
“DE/currenttobest/2”
${{v}}_i^G = {{x}}_i^G + F \times ({{x}}_{{\rm{best}}}^G  {{x}}_i^G) + F \times ({{x}}_{{r_1}}^G  {{x}}_{{r_2}}^G + {{x}}_{{r_3}}^G  {{x}}_{{r_4}}^G)$  (5) 
“DE/randtobest/1”
${{v}}_i^G = {{x}}_{{r_1}}^G + F \times ({{x}}_{{\rm{best}}}^G  {{x}}_i^G) + F \times ({{x}}_{{r_2}}^G  {{x}}_{{r_3}}^G)$  (6) 
where
3) Crossover: For each
$u_{ij}^G = \left\{ \begin{aligned}& v_{ij}^G,\;\;\;\;{\rm {if}}\;\;{R_j} \le CR \;\; {\rm or} \;\; j = {j_{rand}}\\& x_{ij}^G ,\;\;\;\;{\rm{otherwise}}\end{aligned} \right.\;\;\;\;j = 1,2,\cdots\!,\;D$  (7) 
where
4) Selection: The trial vector
${{x}}_i^{G + 1} = \left\{ \begin{aligned}& {{u}}_i^G,\;\;\;\;\;\;{\rm {if}}\;\;f({{u}}_i^G) \le f({{x}}_i^G)\\& {{x}}_i^G,\;\;\;\;\;\;{\rm{otherwise}}.\end{aligned} \right.$  (8) 
5) G = G + 1.
6) Implement Steps 2 to 5 repeatedly until the number of generations is equal to G_{max}.
3 JADE using twophase parameter control schemeFor most evolutionary algorithms, it is generally believed to be a good idea to encourage the global search in the early stages of the evolution and ensure the local search in the later evolution phases^{[44]}, thus the above view can be considered as the empirical guideline. For the DE algorithm, a large value of F can provide good exploration capability, whereas a small value of F can accelerate the convergence speed^{[3]}. Meanwhile, a large value of CR can provide good exploitation capability^{[45]}. In the current study, this empirical guideline can be used to improve the search efficiency of the DE in the later evolution stages. Namely, the value of F gradually decreases and the value of CR gradually increases during the entire evolutionary process.
The adaptation technique can enable control parameters to make timely adjustment when compared with the empirical guideline. In other words, the adaptability of the control parameters produced by the adaptation approach is better than that of the control parameters generated by the empirical guideline. Additionally, the fitness landscape of the population is usually very complex in the early stages of the search since individuals in the current population may be distributed in the entire search space, but individuals may gather in a small search region in the later phases of the evolution. Based on the above introduction, the parameters of JADE are adjusted through two different approaches in the current study. Namely, the parameter adaptation method proposed in JADE^{[10]} is used in the early evolution stages and the empirical guideline is utilized to guide the evolution of control parameters in the later phases of the evolution. The main target is to take advantages of the two different parameter control approaches.
3.1 Evolution of control parametersIf G <
$\begin{split} F_i^{G + 1} = \;& Cauchy \left(\left(0.6  0.5 \times \frac{{G  gs \times {G_{\max }}}}{{(1  gs) \times {G_{\max }}}}\right),\;\sigma_1\right)\\& i = 1, \cdots\!,\;NP\end{split}$  (9) 
where Cauchy is a Cauchy distribution function,
If G <
$\begin{split}CR_i^{G + 1} =\; & N \left(\left(1  0.5 \times \left(1  \frac{{G  gs \times {G_{\max }}}}{{(1  gs) \times {G_{\max }}}}\right)\right),\;\sigma_2\right)\\ & i = 1, \cdots\!,\;NP\end{split}$  (10) 
where N is a normal distribution function,
Based on the above descriptions, it can be observed that the empirical guideline can provide a good exploitation capability for the DE algorithm. Therefore, it is useful to improve the search efficiency of DE in the later search process. Additionally, the parameter adaptation approach used in the early evolution stages can help to locate the promising search region. This is because the adaptation technique based on the population information can automatically adjust the control parameters when the fitness landscape of the population is very complex.
3.2 Overall implementation of TPCJADE1) Initialization
Determine the population size NP and maximal number of generation G_{max}. Initialize a population
2) Population evolution
Mutation operation: For each
${{v}}_i^G = {{x}}_i^G + {F_i} \times ({{x}}_{pbest}^G  {{x}}_i^G) + {F_i} \times ({{x}}_{{r_1}}^G  {{x}}_{{r_2}}^G)$  (11) 
where
Crossover operation: Equation (7) is used to generate a trial vector
3) Control parameters adaptation
The more detailed descriptions can be seen in Section 3.1.
4) G = G + 1.
5) Steps 2 to 4 are repeated until the maximum number of generations is equal to G_{max}.
The framework of the proposed algorithm is shown in Fig. 1.
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Fig. 1. Framework of the proposed algorithm 
4 Experimental results
In this section, two famous test suites (i.e., IEEE CEC2005 and IEEE CEC2015) are utilized to assess the performance of the TPCJADE. Moreover, the performance of the TPCJADE is compared with that of four stateoftheart DE variants, namely, jDE^{[9]}, SaDE^{[7]}, JADE^{[10]}, and CoBiDE^{[13]}. All the compared algorithms are coded in Matlab (Matlab R2012a) and run on Windows 7 operating system (64 bit). For each test function, the maximum numbers of fitness evaluations are set to 300 000 for 30dimensional functions and 500 000 for 50dimensional functions. Since each DE variant has its own suitable NP, recommended NP from their literatures are utilized, namely, 100 for jDE, JADE, and TPCJADE, 50 for SaDE, and 60 for CoBiDE. Furthermore, to compare the performance of all the algorithms, the Wilcoxon′s rank sum test^{[46]} at the 0.05 significance level and the Friedman′s test^{[47]} are utilized. For the Wilcoxon′s rank sum test, the “+”, “−” and “≈” signs denote that the search performance of the proposed algorithm is significantly better than, worse than, and almost similar to that of the other compared algorithms in a statistically significant way, respectively.
4.1 Comparison with four DE variants on 30dimensional CEC2005 functionsIn this part, twentyfive 30dimensional CEC2005 test functions are used to evaluate the performance of the TPCJADE. Each test function is independently run 30 times. The results are presented in Table 1. For unimodal test functions F1_{CEC2005}−F5_{CEC2005}, as shown in Table 1, the overall performance of TPCJADE is better than that of jDE, SaDE, and CoBiDE since TPCJADE uses a greedy mutation strategy which is the same as JADE. JADE significantly outperforms TPCJADE on three test functions. The main reason may be that the evolution information produced by unimodal test function is less misleading. Therefore, the adaptation technique is more effective than the empirical guideline. For multimodal test functions F6_{CEC2005}−F14_{CEC2005}, jDE and JADE cannot perform better than TPCJADE on any test functions. Moreover, SaDE and CoBiDE perform better than TPCJADE on one and two test functions, respectively. However, the performance of TPCJADE is significantly better than that of SaDE and CoBiDE on six and four test functions, respectively. Therefore, the average performance of TPCJADE is the best among five compared algorithms on multimodal test functions. For hybrid composition functions F15_{CEC2005}−F25_{CEC2005}, jDE, SaDE, and JADE cannot outperform TPCJADE on any test functions since the adaptation technique can help TPCJADE to adapt complex environment in the early stages of the evolution and the empirical guideline can improve the search efficiency of the DE in the later evolution phases. Moreover, the CoBiDE performs better than TPCJADE on one test function. However, the performance of TPCJADE is significantly better than that of CoBiDE on six test functions.
Based on the above comparisons, it can be observed from Table 1 that the average performance of TPCJADE is the best among five compared DE variants, the adaptation technique can assist DE to find optimal/near search region in the early evolution stages, while the empirical guideline can enhance the local search capability of DE at later phases of the search.
4.2 Comparison with four DE variants on 50dimensional CEC2005 functions
In this section, twentyfive 50dimensional CEC2005 test functions are employed to verify the performance of the proposed algorithm. For each test function, the setting of run times is the same as in Section 4.1. The simulation and statistical analysis results are shown in Table 2. For unimodal test functions F1_{CEC2005}−F5_{CEC2005}, Table 2 indicates that the TPCJADE outperforms all the compared algorithms except for JADE. However, JADE performs better than our proposed algorithm on only one test function. Based on the results shown in Table 1, we can find that the local search capability of TPCJADE is not getting worse with the increasing of the dimension when compared with JADE. For multimodal test functions F6_{CEC2005}−F14_{CEC2005}, as shown in Table 2, jDE cannot perform better than the proposed algorithm on any test functions. SaDE, JADE, and CoBiDE perform better than TPCJADE on one test function, respectively. However, TPCJADE performs better than SaDE, JADE, and CoBiDE on seven, five, and six test functions, respectively. Therefore, the optimization performance of TPCJADE is the best among these selected algorithms on the multimodal test functions. For hybrid composition functions F15_{CEC2005}−F25_{CEC2005}, JADE and SaDE cannot perform better than TPCJADE on any test functions. The performance of jDE and CoBiDE is significantly better than that of TPCJADE on one and four test functions, respectively. However, TPCJADE significantly performs better than jDE and CoBiDE on four and five test functions, respectively. Therefore, the average performance of TPCJADE is similar as that of CoBiDE on complex functions and is better than that of jDE, SaDE, and JADE.
Based on the above analyses, the statistical analysis results shown in Table 2 indicate that the overall performance of TPCJADE is better than that of other competitors on twentyfive 50dimensional CEC2005 functions.
4.3 Comparison with four DE variants on 30dimensional CEC2015 functions
In this experiment, fifteen 30dimensional functions introduced in IEEE CEC2015 are employed to demonstrate the performance of TPCJADE. For each function, the run times are set to be 30, and the mean and standard deviation values are presented in Table 3. Additionally, the statistical analysis results obtained by the Wilcoxon′s rank sum test are also shown in Table 3.
According to the results summarized in Table 3, it can be observed that the performance of TPCJADE is significantly better than that of jDE, SaDE, JADE, and CoBiDE on ten, twelve, nine, and seven functions, respectively. Note that jDE, SaDE, and JADE cannot provide significantly better results than TPCJADE on any functions. We can find that the control parameters guided by the empirical guideline can improve the search efficiency of JADE. It can be also seen from Table 3 that CoBiDE significantly outperforms TPCJADE on four functions. This is because the covariance matrix learning is used in the DE. Overall, TPCJADE provides the best average performance among all compared algorithms on 15 30dimensional IEEE CEC2015 functions.
5 Parameter analysis
In this part, 25 30dimensional CEC2005 test functions are utilized to investigate the effect of σ, which is selected from the set (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8). The maximum number of function evaluations and run times are set to be 300 000 and 30, respectively.
The statistical analysis results achieved by the Friedman′s test are plotted in Fig. 2. As shown in Fig. 2, we can find that the overall performance of TPCJADE is susceptible to the value of sigma and is the best when σ = 0.6. Therefore, σ = 0.6 is chosen in our proposed algorithm.
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Fig. 2. Rankings obtained by Friedman′s test under different values of sigma 
6 Case study
A typical dynamic optimization problem can be formulated as follows^{[39, 48]}:
$\begin{split}& \min \;\;\;J({{u}}(t)) = G(x({t_{{f}}})) + \int_{{t_0}}^{{t_{\rm{f}}}} {F({{x}},{{u}}){\rm d}t} \\& {\rm s.t.}\;\;\;\frac{{{\rm d}{{x}}(t)}}{{{\rm d}t}} = f({{x}}(t),{{u}}(t)),\;\;{{x}}(0) = {{{x}}_0}\\&\;\;\;\;\;\;\;\; 0 \le t \le {t_{{f}}},\;{{{u}}_{{\rm{min}}}} < {{u}} < {{{u}}_{{\rm{max}}}}\end{split}$  (12) 
where J denotes the performance index; x and u are the state and control vectors, respectively; G and F are the terminal time performance index and integrated performance, respectively; t_{0} is the initial time and t_{f} is the final time; u_{min} and u_{max} are the lower and upper bounds of u, respectively.
6.1 Feedingrate optimization for LeeRamirez bioreactorIn this section, TPCJADE is utilized to solve a dynamic optimization problem proposed by Lee and Ramirez^{[49]}. Additionally, the direct method adopted in [50] is employed. This dynamic optimization problem can be described as follows:
$\begin{array}{l} \frac{{{\rm{d}}{x_1}}}{{{\rm{d}}t}} = {u_1} + {u_2},\;\;\;\;\;\;\;\;\;\;\;\;{x_1}(0) = 1\\ \frac{{{\rm{d}}{x_2}}}{{{\rm{d}}t}} = \mu {x_2}  \frac{{{u_1} + {u_2}}}{{{x_1}}}{x_2},\;\;\;\;{x_2}(0) = 0.1\\ \frac{{{\rm{d}}{x_3}}}{{{\rm{d}}t}} = \frac{{{u_1}}}{{{x_1}}}{C_{nf}}  \frac{{{u_1} + {u_2}}}{{{x_1}}}{x_3}  {Y^{  1}}\mu {x_2},\;\;\;{x_3}(0) = 40\\ \frac{{{\rm{d}}{x_4}}}{{{\rm{d}}t}} = {R_{fp}}{x_2}  \frac{{{u_1} + {u_2}}}{{{x_1}}}{x_4},\;\;\;\;{x_4}(0) = 0\\ \frac{{{\rm{d}}{x_5}}}{{{\rm{d}}t}} = \frac{{{u_2}}}{{{x_1}}}{C_{if}}  \frac{{{u_1} + {u_2}}}{{{x_1}}}{x_5},\;\;\;\;{x_5}(0) = 0\\ \frac{{{\rm{d}}{x_6}}}{{{\rm{d}}t}} =  {k_1}{x_6},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x_6}(0) = 1\\ \frac{{{\rm{d}}{x_7}}}{{{\rm{d}}t}} = {k_2}(1  {x_7}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x_7}(0) = 0\\ 0 \le {u_1},\;{u_2} \le 0.01 \end{array}$  (13) 
where
In (13), some parameters can be defined as follows:
$\mu = \frac{{0.407{x_3}}}{{\displaystyle\frac{{0.108 + {x_3} + x_3^2}}{{14\;814.8}}}}\left( {{x_6} + \frac{{0.22}}{{0.22 + {x_3}}}{x_7}} \right)$ 
${k_1} = {k_2} = \frac{{0.09{x_5}}}{{0.034 + {x_5}}}$ 
${R_{fp}} = \frac{{0.095{x_3}}}{{\displaystyle\frac{{0.108 + {x_3} + x_3^2}}{{14\;814.8}}}}\left( {\frac{{0.005 + {x_5}}}{{0.022 + {x_5}}}} \right)$ 
${C_{{{if}}}} = 4, \; {C_{{{nf}}}} = 100, \; {{Y = 0}}{\rm{.51}}.$ 
For the above fedbatch production of induced foreign protein, the objective is to maximize the economic benefit of this fedbatch system. Therefore, the objective function can be defined as
${\rm{max }}J{\rm{(}}{u_1}{\rm{,}}{u_2}{\rm{) = }}{x_1}({t_{{f}}}){x_4}({t_{{f}}})  Q\int_0^{{t_{\rm{f}}}} {{u_2}(t)} {\rm{d}}t$  (14) 
where Q = 5 and t_{f} = 10 h.
For this dynamic optimization problem, researchers have introduced various methods to obtain satisfactory results, which are 0.814 9^{[51]}, 0.816 7^{[52]}, 0.815 8^{[53]} and 0.816 4^{[54]}.
In this experiment, NP and
1) The value 0.816 43 is achieved by TPCJADE when D is set to be 10. It can be seen that our result is better than the results achieved by Roubos et al.^{[51]}, Zhang et al.^{[53]}, and Fan et al.^{[54]} respectively. However, the result of TPCJADE is slightly worse than the result obtained by Sarkar and Modak^{[52]}. The main reason may be that Sarkar and Modak^{[52]} consume much more computational resources and the discrete time length is smaller than our proposal. In addition, the curves of two control variables (i.e., u_{1} and u_{2}) are plotted in Figs. 3 and 4, respectively.
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Fig. 3. Optimal control profile of glucose feed rate (D = 10) 
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Fig. 4. Optimal control profile of inducer feed rate (D = 10) 
2) Let D = 20, the value 0.816 47 is achieved by using TPCJADE. Clearly, the result achieved by TPCJADE is better than the other competitors except for the result obtained by Sarkar and Modak^{[52]}. This is because Sarkar and Modak consume more computational resources when compared with the proposed algorithm. Moreover, the obtained result is better than the result obtained by TPCJADE when D = 10. It means that the solution precision is directly influenced by the discrete time degree. Additionally, the curves of the optimal glucose feed rate and inducer feed rate are presented in Figs. 5 and 6, respectively.
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Fig. 5. Optimal control profile of glucose feed rate (D = 20) 
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Fig. 6. Optimal control profile of inducer feed rate (D = 20) 
3) In this experiment, D is set to be 30. The value 0.816 48 is achieved by the proposed algorithm. Although the result is slightly worse than Sarkar and Modak^{[52]}, the obtained result is better than the results reported by Roubos et al.^{[51]}, Zhang et al.^{[53]} and Fan et al.^{[54]}, respectively. The obtained result is also better than the results achieved by TPCJADE when D = 20. It can be observed from the obtained result that TPCJADE is a very competitive optimization tool for solving a complex dynamic optimization problem. The curves of optimal u_{1} and u_{2} are shown in Figs. 7 and 8, respectively.
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Fig. 7. Optimal control profile of glucose feed rate (D = 30) 
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Fig. 8. Optimal control profile of inducer feed rate (D = 30) 
Based on the above simulation results, it can be concluded that the solution precision is highly influenced by the dimensionality of the problem and the TPCJADE can perform well when the dimensionality of the problem is increasing. It should be noted that the value 0.816 7 is achieved by Sarkar and Modak^{[52]} when the number of function evaluations is more than 2 000 000. For our proposal, the maximum number of function evaluations is 300 000; therefore, the proposed algorithm is very competitive for solving this dynamic optimization problem. Additionally, from Figs. 3–8, it can be observed that the nutrient can be neglected when Q = 5. This is because glucose feed rate is maintained at zero during the entire period of operation.
7 ConclusionsIn the current study, a selfadaptive DE using twophase parameter control scheme (TPCJADE) is proposed to improve the search efficiency of DE. In the TPCJADE, the adaptation technique and empirical guideline are used to produce the control parameters in the early evolution phases and later evolution stages, respectively. Therefore, TPCJADE can adapt to the complex optimization environment and improve the search efficiency of DE in two different stages of the search. The search performance of TPCJADE is compared with that of jDE, SaDE, JADE, and CoBiDE on two wellknown test suites. Moreover, the Wilcoxon′s rank sum test is utilized to distinguish the performance difference between TPCJADE and its competitors. The simulation and statistical analysis results indicate that TPCJADE performs better than jDE, SaDE, JADE, and CoBiDE on these selected test functions, and the empirical guideline can enhance the search efficiency of DE.
The sensitivity of sigma in TPCJADE is analyzed by twentyfive 30dimensional CEC2005 test functions. The results indicate that the average performance of TPCJADE is influenced by the value of σ, i.e., a large value of σ can provide better performance when compared with a small value of sigma. Additionally, TPCJADE is employed to solve the dynamic optimization problem, the experimental results present that TPCJADE is an effective and efficient tool in dealing with complex dynamic optimization problems.
AcknowledgementsThis work was supported by National Natural Science Foundation of China (Nos. 61603244 and 41505001) and Fundamental Research Funds for the Central Universities (No. 222201717006).
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