A microbial kinetic optimization approach

2.1 Scene design of MDO algorithm

Suppose that there are N microbial populations living in the microbial culture system E, namely P₁, P₂,...,P _i ,...,P _N ; each population not only feeds on the culture fluid in E, but also feeds the other K populations in E for food. In the process of microbial cultivation, the prepared fresh medium is periodically injected into E; at the same time, the useless medium is continuously removed from E; the medium absorbed by the microbial population will not be converted into immediately new microbial population, in other words, there is a period of time in the process of transforming from the culture fluid to the microbial population; as the culture fluid is injected, harmful substances are also injected. In order to control the destructive effects of harmful substances on the population, it is necessary to continuously assess the degree of damage to the population by the harmful substances. The operation of the culture system can be achieved by differently controlling two factors: one is the concentration S of nutrients in the culture solution; the other is the flow rate Q of the culture solution. Changes in the concentration of S will cause changes in the growth of microbial populations.

The characteristics of the microbial population P _i can be represented by P _i  = {f_i,1,f_i,2,...,f_i,n}, where f_i,j are the characteristics j of P _i , n is the total number of characteristics of each population, i = 1∼N, j = 1∼n; the influence of the nutrient concentration of the culture medium in E on the population P _i is reflected in the random influence on the characteristics of the population P _i .

Suppose the optimization model of the required solution is Equation (1): { min f ( X ) X ∈ H ⊂ R n(1)

In the formula (1), R ⁿ is the n-dimensional Euclidean space, H is the search space; f(X) is the objective function; X is the n-dimensional solution vector, X = (x₁,x₂,...,x _i ,...,x _n ), and x _i is the solution component. N initial solutions are selected arbitrarily in H, namely H = {X₁,X₂,...,X _N }, where X _i  =  (x_i,1,x_i,2,...,x_i,n), i = 1∼N; culture system E corresponds to the search space H. The N initial solutions of the optimization model correspond to the N microbial populations in E, that is, X _i corresponds to P _i one to one; more precisely, the variables x_i,j of X _i correspond to the characteristics f_i,j of P _i one to one.

In summary, the initial solution and the microbial population are equivalent concepts and will not be distinguished later. After P _i takes in nutrients, its growth state will change. This change is projected to H, it is equivalent to transferring the initial solution Xi from the spatial position a to b. A spatial location is called a state and is described by its subscript.

Assuming that the current state of P _i is c, this is equivalent to the position X _c in H. If P _i evolves from state c to state d after taking in nutrients, this is equivalent to jumping from position X _c to position X _d in H. Calculate according to formula (1), if f(X _c )  > f(X _d ), indicating that position X _c is better than position X _d , then P _i is considered strong. Conversely, if f(X _c )≤f(X _d ), indicating that the position X _d is worse than the position X _c , or there is no difference (because f(X _c )  = f(X _d )), then P _i is considered weak. The strong population can continue to grow with a higher probability, while the weak population may not grow anymore.

The strength of P _i can be expressed by the microbial growth index (MGI). A good quality solution corresponds to a population with a higher MGI index, that is, a strong population. A poor quality solution corresponds to a population with a lower MGI index, that is, a weak population. For formula (1), the MGI index of Pi is calculated according to formula (2): MGI ( X i ) = { 1 1 + f ( X i ) , if f ( X i ) > 0 1 + | f ( X i ) | , if f ( X i ) ⩽ 0 , i = 1 ∼ N(2)

In E, the biological intelligence characteristics of microbial cultivation are reflected in the following aspects:

Various groups have mutual influence due to competition for culture solution. Such influence will be reflected in the random interaction between certain characteristics of the group. This is the “competitive intelligent characteristic”.

The characteristics of biological intelligence cultivated by microorganisms are finally embodied in the form of biological evolution operators. The interaction between all microbial populations is projected to H in Equation (1), that is, there is an interaction between the solution vectors.

The MDO algorithm uses such strategies to solve the global optimal solution of Equation (1).

Suppose there are N species of microorganism populations in the culture system E, and each population feeds on the culture fluid and L other populations in E as food. In addition, toxic substances will poison all populations. At period t, the concentration of the population P _i is y _i (t), y _i (t) ≥0, i = 1∼N; the culture fluid flows into E at a flow rate Q, where the concentration of nutrients is S₀. This culture system has a time-lag effect on the mixed food chain microbial pulse culture dynamic model, as shown in formula (3) [14]: { dS ( t ) dt = - QS ( t ) - ∑ i = 1 N c 0 μ 0 S ( t ) K 0 + S ( t ) y i ( t ) , t ≠ kT dy i ( t ) dt = e - Q τ μ 0 S ( t - τ ) K 0 + S ( t - τ ) y i ( t + τ ) - ∑ s ∈ MS ( t ) α 0 γ 0 Y S ( t ) W 0 + Y S ( t ) - ( Q + rc ( t ) ) y i ( t ) , i = 1 , 2 , ⋯ , N , t ≠ kT dc ( t ) dt = - Qc ( t ) , t ≠ kT S ( t + ) = S ( t ) + S 0 , t = kT y i ( t + ) = y i ( t ) , i = 1 , 2 , ⋯ , N , t = kT c ( t + ) = c ( t ) + q 0 , t = kT S ( 0 + ) ⩾ 0 ; y i ( 0 + ) ⩾ 0 , i = 1 , 2 , ⋯ , N , c ( t + ) ⩾ 0 , t = kT(3)

In the formula (3), S(t) represents the nutrient concentration of the culture solution at period t; μ ₀ represents the relative growth coefficient of the population; C₀ represents the consumption rate of the culture solution by the population; MS(t) represents a collection of numbers of the K populations captured by the population P _i at period t; K₀, α₀, γ₀, W₀ represent constants related to population growth; y _i (t) represents the concentration of the population P _i at period t; c(t) represents the concentration of harmful substances, c(t)≥0; q₀ represents the quantity of harmful substances input in each culture medium feeding period; r represents the reduction rate of the population due to poisoning and death due to harmful substances, r > 0; T represents the feeding period of the culture fluid; τ represents the lag time of the transformation of the culture fluid to the microbial population; t⁺ is the time when the culture medium is put into use; k represents a positive integer, k≥1.

At period t, the proportion r _i (t) of the population P _i is calculated according to formula (4): r i ( t ) = y i ( t ) ∑ s = 1 N y s ( t ) , i = 1 ∼ N(4)

The larger the proportion r _i (t) of the population P _i , the greater the chance of being preyed by other populations. The values of the time t parameters C₀, Q, K₀, S₀, α₀, μ₀, r, γ₀, W₀, and q₀ are respectively C^t₀, Q^t, K^t₀, S^t₀, α^t₀, μ^t₀, r^t, γ^t₀, W^t₀, q^t₀. In order to facilitate the calculation, the formula (3) is changed to a discrete recursive form, with the following formula: { S ( t + 1 ) = S ( t ) - Q t S ( t ) - ∑ i = 1 N c 0 t μ 0 t S ( t ) K 0 + S ( t ) y i ( t ) , t ≠ kT c ( t + 1 ) = c ( t ) - Q t c ( t ) , t ≠ kT(5)

y i ( t + 1 ) = y i ( t ) + e - Q t τ c 0 t μ 0 t S ( t - τ ) K 0 + S ( t - τ ) y i ( t - τ ) - ∑ s ∈ MS ( t ) α 0 t γ 0 t y s ( t ) W 0 t + y s ( t ) - ( Q t + r t c ( t ) ) y i ( t ) , i = 1 , 2 , ⋯ , N , t ≠ kT(6) { S ( t + 1 ) = S ( t ) + S 0 t , t = kT c ( t + 1 ) = c ( t ) + q 0 t , t = kT(7) y i ( t + 1 ) = y i ( t ) , i = 1 , 2 , ⋯ , N , t = kT(8)

Refer to Table 1 for the meaning of each parameter and its value restriction conditions in formula (5) and formula (6).

In Table 1, Rand(A,B) means generating a random number that obeys a uniform distribution in the interval [A,B]. A and B are constants, and A≤B.

At period t, suppose the current population is P _i , the generation strategy of the characteristic microbial population set is as follows:

Generate the set MS(t) of the eaten population. At period t, the population P _i feeds on another K populations, that is, K populations are randomly selected from N-1 populations with {r₁ (t), r₂ (t),...,r _N  (t)} as the probability distribution (But P _i cannot be selected), these populations form a set MS(t), let us set MS (t) = {i₁,i₂, ...  ,i _k , ...  ,i _K }, where i _k is the number of the population P _ik , k = 1∼K.

2.4 Design of microbial population evolution operator

The MDO algorithm uses the microbial cultivation dynamics model formula (3) to construct the evolution operator, so as to realize the information exchange between the cultivation system and the population and between the population and the population, and then realize the search of the search space H of the formula (1).

(1) Absorption operator. The absorption operator describes the nutrient supply relationship between the population P _i and the other K populations. Pass a randomly selected feature f_s,j,s∈MS(t) of the eaten population in the set MS(t) to the current population P _i , so that the population P _i can grow: v i , j ( t + 1 ) = η i x i , j ( t ) + ∑ s ∈ MS ( t ) β s x s , j ( t )(9)

In formula (9), x_s,j(t) represents the state value of the feature j of the population P _s at period t, that is, the value of the variable x_s,j at period t; v_i,j(t + 1) represents the state value of the feature j of the population P _i at period t + 1, that is, the value of the variable x_i,j in the period t + 1; Take β _s  = Rand(-0.95,0.95), η _i  = Rand(0.45,0.65).

(2) Snap operator. The snap operator describes that a population grabs the dominant characteristics of other populations to make it stronger. Let the characteristics f_sl,j, s _l ∈ {h₁,h₂, ...  ,h _L ; g₁,g₂, ...  ,g _L ; s₁,s₂, ...  ,s _L } correspond to the characteristics of the 3 L strong populations in FM ^t , GM ^t and HM ^t , f_sl,j The value is passed to the characteristic f_i,j of the population P _i , making P _i strong. Assuming that the current population is P _i and the feature for dominance grabbing is f_i,j, then there is formula (10): v i , j ( t + 1 ) = { { 0.9 x d 1 , j ( t ) + 0.5 ( x d 2 , j ( t ) - x d 3 , j ( t ) ) , | FM t | > 0 , 0 < Rand ( 0 , 1 ) ⩽ 1 2 x i , j ( t ) , | FM t | = 0 , 0 < Rand ( 0 , 1 ) ⩽ 1 2 { 0.9 x e 1 , j ( t ) + 0.5 ( x e 2 , j ( t ) - x e 3 , j ( t ) ) , | GM t | > 0 , 1 2 < Rand ( 0 , 1 ) ⩽ 3 4 x i , j ( t ) , | GM t | = 0 , 1 2 < Rand ( 0 , 1 ) ⩽ 3 4 { 0.9 x p 1 , j ( t ) + 0.5 ( x p 2 , j ( t ) - x p 3 , j ( t ) ) , | HM t | > 0 , 3 4 < Rand ( 0 , 1 ) ⩽ 1 x i , j ( t ) , | HM t | = 0 , 3 4 < Rand ( 0 , 1 ) ⩽ 1(10)

In the formula (10), d₁, d₂, d₃, e₁, e₂, e₃, p₁, p₂, and p₃ are randomly selected from the sets FM ^t , GM ^t , and HM ^t respectively, and they satisfy d₁≠d₂  ≠d₃, e₁≠e₂≠e₃, p₁≠p₂≠p₃.

(3) Hybrid operator. The confounding operator describes the interaction between the population and the population. For the current population P _i , the state value of which the features f_sl,j, s _l ∈{a₁,a₂, ...  ,a _L ; b₁,b₂, ...  ,b _L ;c₁,c₂, ...  ,c _L } of the 3 L populations in AS ^t , BS ^t , CS ^t correspond to is passed to the characteristic f_i,j of the population P _i after being mixed and fused, then there is formula (11): v i , j ( t + 1 ) = { λ i x i , j ( t ) + ∑ s ∈ AS t ρ s x s , j ( t ) , j = 1 ∼ n , 0 < Rand ( 0 , 1 ) ⩽ 1 6 λ i x i , j ( t ) + ∑ s ∈ AS t ρ s x s , j ( t ) , j = 1 ∼ n , 1 6 < Rand ( 0 , 1 ) ⩽ 1 3 λ i x i , j ( t ) + ∑ s ∈ BS t ρ s x s , j ( t ) , j = 1 ∼ n , 1 3 < Rand ( 0 , 1 ) ⩽ 1 2 λ i x i , j ( t ) + ∑ s ∈ BS t ρ s x s , j ( t ) , j = 1 ∼ n , 1 2 < Rand ( 0 , 1 ) ⩽ 2 3 λ i x i , j ( t ) + ∑ s ∈ AS t ρ s x s , j ( t ) , j = 1 ∼ n , 2 3 < Rand ( 0 , 1 ) ⩽ 5 6 λ i x i , j ( t ) + ∑ s ∈ AS t ρ s x s , j ( t ) , j = 1 ∼ n , 5 6 < Rand ( 0 , 1 ) ⩽ 1 λ i = Rand ( 0.65 , 0.85 ) , ρ s = Rand ( 0.75 , 0.85 )(11)

(4) Toxin operator. The toxin operator describes the toxic effects of toxic substances in the external system on microorganisms, and the toxic effects can be transmitted between microbial populations. For the current population P _i , there is formula (12): v i , j ( t + 1 ) = c ( t ) ( x g 1 , j ( t ) + x g 2 , j ( t ) ) + ( 1 - c ( t ) ) x g 3 , j ( t )(12)

In formula (12), g₁, g₂, g₃ are randomly selected from {1, 2, . . . , N}, and they satisfy g₁≠g₂≠g₃.

(5) Growth operator. The definition of the growth operator is as formula (13):

X i ( t + 1 ) = { V i ( t + 1 ) , if , if MGI ( V i ( t + 1 ) ) > MGI ( X i ( t ) ) X i ( t ) , if , if MGI ( V i ( t + 1 ) ) ⩽ MGI ( X i ( t ) ) , i = 1 ∼ N(13)

In formula (13), X _i (t)  =  (x_i,1(t),x_i,2(t),...,x_i,n(t)); V _i (t + 1)  =  (v_i,1(t + 1), v_i,2(t + 1),...,v_i,n(t + 1)); Equation (2) is used to calculate MGI(V _i (t + 1)) and MGI(X _i (t)).

(S1) Initialization: ➀ Let t = 0, the number of evolutionary periods G = 8000∼60000, the error requirement ɛ  = 10^–5∼10^–10, N = 50∼500, the highest probability of predation of the microbial population E₀ = 1/1000∼1/100, K = 2∼10, T = 1∼10, τ=1∼10, L = 2∼10. ➁ Randomly determine {X₁ (0), X₂ (0),...,X _N  (0)}, {y₁(0), y₂(0),...,y _N (0)},S (0).

(S2) Perform the following operations:

For t = 0 to G

Calculate C ^t ₀, Q ^t , K ^t ₀, S ^t ₀, α ^t ₀, μ ^t ₀, r ^t , γ ^t ₀, W ^t ₀, q ^t ₀ according to the strategy given in Table 1.

Calculate r₁(t), r₂(t),...,r _N (t) according to formula (4).

If T is not divisible by t, calculate c(t + 1) and S(t + 1) according to formula (5);

If T can divide t, calculate c(t + 1) and S(t + 1) according to formula (7).

For i = 1 to N

With r₁(t), r₂(t),...,r _N (t) as the probability distribution, K populations are randomly selected, but P _i cannot be selected, forming a population number set MS(t).

If T is not divisible by t, calculate y _i (t + 1) according to formula (6);

If T can divide t, calculate y _i (t + 1) according to formula (8).

For j = 1 to n

p = Rand(0,1), p is the probability that the growth of the population will be affected by osmotic mixing, dominance grabbing and feature absorption;

        If p < E₀ then

q = Rand(0,1), where q is the probability that the absorption operator, the grab operator and the hybrid operator are executed;

        If q≤1/4 then

          Execute the absorption operator according to formula (9) to obtain v_i,j(t + 1);

        Else if 1/4 < q≤2/4 then

          Execute the grab operator according to formula (10), and get v_i,j(t + 1);

        Else if 2/4 < q≤3/4 then

          Execute the confounding operator according to formula (11) to obtain v_i,j(t + 1);

        Else if 3/4 < q≤1 then

          Execute the toxin operator according to formula (12) to obtain v_i,j(t + 1);

        End if

      Else

        v_i,j(t + 1)  = x_i,j(t);

    End if

        End for

          Execute the growth operator according to formula (13) to obtain X_i(t + 1);

    End for

    If |X^*t +1-Y^*t +1| < ɛ then

        Goto step (S3);

    Else

        Save the optimal solution X^*t +1;

        Y^*t +1 = X^*t +1;

    End if

End for

(S3) End

2.6 Algorithm feature analysis

4.1 Measurement methods of penetration ability and expansion ability

In the solution process of the MDO algorithm, the penetration ability and the expansion ability are used to describe the ability to escape from the trap after falling into the local optimal solution trap and converge to the global optimal solution with a certain accuracy. Penetration ability is used to describe the ability of MDO algorithm to quickly escape from traps or quickly converge to the global optimal solution when solving optimization problems; while expansion ability is used to describe the ability of MDO algorithm to tentatively jump out of local optimization pitfalls, or tentatively improve the ability of optimal solution accuracy when solving optimization problems. The penetration and expansion capabilities of the MDO algorithm can be described by the behavior of microbial populations. When a population is searching, its state can be divided into penetrating state and expanding state. A population can only stay in one state at a time: either in a penetrating state or in an expanding state. In order to emphasize the behavior of the population entering the local optimal solution trap, the expansion state is divided into two states, one is called the expansion state and the other is called the sticky state.

When the population is searching, if the current position of the population remains unchanged for a long time, it is considered that the population has fallen into a sticky state. Two situations may lead to a sticky state: one is that the population falls into a local optimal solution trap; the other is that the current position of the population cannot be updated by information thrown by other species.

When all populations fall into a sticky state, the search will automatically stop, and the global optimal solution will always remain unchanged. On the other hand, if some optimization problems have high condition numbers, it is always difficult to improve the accuracy of the global optimal solution when the population is prone to fall into a sticky state.

When the MDO algorithm solves the optimization problem, if the penetration state is always dominant, it means that the search will either quickly escape from the trap or quickly approach the global optimal solution. Therefore, higher penetration capability is a good thing for the MDO algorithm; If the expansion state is always dominant, it means that the search is tentatively jumping out of the local optimal solution trap, or is tentatively improving the accuracy of the current optimal solution, but it may consume huge computing time, so the expansion ability of the algorithm should be controlled to a certain extent; if the sticky state is dominant, it means that the search is almost stopped, so the sticky state must be completely avoided.

The evaluation of penetration and expansion capabilities can be based on four convergence curves: (1) the convergence curve of any population, tracking the behavior of any population to search for the optimal solution; (2) the convergence curve of the best population, which tracks the best population search behavior of the optimal solution; (3) The average convergence curve of all populations, which tracks the average behavior of all populations when searching for the optimal solution; (4) The best convergence curve, which tracks all populations during the search process, and finds the current optimal Solution behavior.

The convergence curve of any population may cause huge computational burden and computer memory because there are many populations in the ecosystem; in addition, the behavior of the populations is too random to control them. The convergence curve of the best population is difficult to grasp, because the best population may change over time. The average convergence curve of all populations is similar to the convergence curve of any population, which may not only cause a huge computational burden and computer memory, but also increase the MGI value of the bad population and compress the MGI value of the good population. The best convergence curve can produce the smallest computational burden and computer memory between the four convergence curves, and the curve does not focus on any single population, but the best behavior produced by all populations.

The penetration and expansion capability evaluation of the MDO algorithm is based on the optimal convergence curve, which is a simple and intuitive method to judge the penetration and expansion capability of the MDO algorithm.

Figure 2 is the best convergence curve when the MDO algorithm solves a certain optimization problem. Point C is the starting position of the evolution, and the corresponding best objective function value is F₀, and the CPU time t = 0; when the evolution reaches time t = t₁, the current best position is at A, the corresponding best objective function value is F₁; when the evolutionary arrival time t = t₂, the current best position is at B, and the corresponding best objective function value is F₂. The penetration and expansion ability of the population can be identified by the slope of the straight line AB, so the following formula: slope ( t 1 ) = β = 180 π ardtan | F 2 - F 1 t 2 - t 1 |

OPEN IN VIEWER

Fig. 2

Evaluation of penetration and expansion capacity of MDO algorithm.

Obviously, if the straight line AB is steeper, the penetrating ability will be higher; if the straight line AB evolves from steep to gentle slope, the expansion ability will change from good to bad, especially when slope(t₁) is close to 0, the expansion ability becomes worse, a sticky state appears. Therefore, two thresholds λ _Ps and λ _Es must be defined to illustrate these situations: λ _Ps is used to distinguish between penetration state and expansion state, and λ _Es is used to distinguish between expansion state and sticky state.

If slope(t₁) > λ _Ps , it means that some populations stay in the penetrating state.

Now the penetration and expansion capabilities of the population are considered in the time interval [t _A , t _B ]. Let CountPs(t _A ,t _B ) denote the number of positions in the interval [t _A , t _B ], t _A ≤t≤t _B that satisfies slope(t) > λ _Ps ; CountEs(t _A ,t _B ) denote the number of positions in the interval [t _A , t _B ] that satisfies λ _Es  < slope _i (t)≤λ _Ps ; CountSs(t _A ,t _B ) represents the number of positions that satisfy slope(t₁)≤λ _Es in the interval [t _A , t _B ]. There are the following formulas: CountPs ( t A , t B ) = Count ( i | slope i ( t ) > λ Ps , t A ⩽ t < t B ) CountEs ( t A , t B ) = Count ( i | λ Es ⩽ slope i ( t ) ⩽ λ Ps , t A ⩽ t < t B ) CountSs ( t A , t B ) = Count ( i | slopei ( t ) < λ Es , t A ⩽ t < t B )

Wherein, Count (.) is a function to calculate the position that meets the given conditions.

The measurement method of the coordination between penetration and expansion defined in the time interval [t_A, t_B] is as follows: C P s / E s ( t A , t B ) = 1 + CountPs ( t A , t B ) 1 + CountEs ( t A , t B )

Wherein, C_Ps/Es(t _A ,t _B ) is called penetration-expansion coordination coefficient.

If C_Ps/Es(t _A ,t _B )  > C _Ps , it means that the search is in a penetration dominant state within the time interval [t _A ,t _B ].

Similarly, the measurement method of the degree of coordination between expansion and viscosity in the time interval [t _A , t _B ] can be defined as follows: C E s / S s ( t A , t B ) = 1 + CountEs ( t A , t B ) 1 + CountSs ( t A , t B )

Wherein, C_Es/Ss(t _A ,t _B ) is called the expansion-viscosity coordination coefficient.

If C_Es/Ss(t _A ,t _B ) > C _Es , it means that the search is in a expansion dominant state within the time interval [t _A ,t _B ].

The Michalewicz function is chosen to illustrate the penetration and expansion capabilities of the MDO algorithm. The form of the Michalewicz function is as follows: min E 1 ( X ) = - ∑ i = 1 n sin ( x i ) [ sin ( ix i 2 π ) ] 2 m , ( m = 15 )

This function has the characteristics shown in Table 9. The Michalewicz function has n! local extremum points. When n = 80, there are 7.156 9×10¹¹⁸ local extremum points, so it is extremely difficult to obtain the optimal solution for this function.

Figure 3 shows the distribution of penetration, expansion and viscous state when MDO algorithm solves the Michalewicz function in the time interval [0,694]. The calculation parameters are n = 80, E₀ = 0.005, K = 3, T = 4, τ= 3, L = 3, N = 200, G = 80000. It can be seen from Fig. 3 that when the MDO algorithm solves the Michalewicz function, CountPs(0,694) = 103, CountEs(0,694) = 297, CountSS(0,694) = 40, it takes 23.41% of the time to penetrate, 67.50% for expansion, there is only 9.09% probability of falling into a sticky state.

OPEN IN VIEWER

Fig. 3

Distribution of penetration, expansion and viscosity(ɛ  = 1.0E-11,λ _Ps  = 1.0,λ _Es  = 1.0E-10).

When the MDO algorithm solves the Michalewicz function, Fig. 4 shows the number of penetration, expansion, and viscous states in the time interval [0,694]. It can be seen from Fig. 4 that when the MDO algorithm solves the Michalewicz function, the penetration and expansion states alternate from 0 to 250 s, but the penetration state is always dominant from 0 to 250 s. Only the expansion state occurs between 250 s and 350 s; after 350 s, the expansion and the viscous state alternately appear, but the probability of the occurrence of the viscous state increases, and the expansion state always prevails after 350 s. This means that penetration and expansion are carried out sequentially from 0 to 250 s; between 250 s and 350 s, the search is close to the global optimum, and only expansion is required; when the search is carried out for 350 s, the sticky state occasionally starts to appear.

OPEN IN VIEWER

Fig. 4

Number of times of penetration, expansion and viscous state λ _Ps  = 1.0,λ Es = 10^-10).

When the MDO algorithm solves the Michalewicz function, Fig. 5 shows the coordination between penetration and expansion in the time interval [0,694]. It can be seen from Fig. 5 that when the MDO algorithm solves the Michalewicz function, the penetration state is dominant from 0 to 250 s; when t = 250 s, the penetration state is dominant and reaches the maximum value; after 250 s, the penetration state is dominant. The superiority begins to decrease; on the contrary, the probability of the inflated state begins to increase. This conclusion is consistent with the conclusion obtained from Fig. 4.

OPEN IN VIEWER

Fig. 5

Variation of coordination coefficient of penetration and expansion with time (C _Ps  = 1.0).

When the MDO algorithm solves the Michalewicz function, Fig. 6 shows the coordination between expansion and stickiness in the time interval [0,694]. It can be seen from Fig. 6 that when the MDO algorithm solves the Michalewicz function, when t = 350 s, the expansion state dominance reaches the maximum value, and then the number of occurrences of the dominance state begins to decrease. On the contrary, the number of occurrences of the sticky state begins to increase, but the number of occurrences of the sticky state is always not dominant.

OPEN IN VIEWER

Fig. 6

Variation of coordination coefficient of expansion and viscosity with time (C _Es  = 1.0).

From the above analysis, it can be seen that when solving optimization problems with a large number of local optimal solutions and optimization problems with high condition numbers, the MDO algorithm shows a good global optimization ability, and the balance between the local optimization ability and the global optimization ability is well control.