A column design method based on reverse elite Harris hawk optimization

Abstract

The application discloses a column design method based on reverse elite Harris hawk optimization, and comprises the following steps: S1, a mathematical model of the column design problem is established, and a fitness function is constructed; S2, the initial Harris hawk population position and selection probability parameters are initialized; S3, the Harris hawk population position is updated by using elite reverse learning, and the fitness value of the Harris hawk population position is calculated; S4, the selection probability and escape energy are updated, when the escape energy is greater than 1, S5 is executed, otherwise, S6 is jumped; S5, the Harris hawk population position is updated by using an exploration formula, and S7 is jumped; S6, the Harris hawk population position is updated by using elite crossover evolution or elite fusion evolution according to the probability selection; S7, whether the termination condition is met is judged, if yes, the parameters of the column design are output, otherwise, the step S3 is jumped. By introducing the elite reverse learning into the Harris hawk optimization algorithm, the population structure is optimized, the ability of the algorithm to jump out of the local optimum is enhanced, the convergence speed and the robustness of the algorithm are improved, and the optimization effect of the column design is improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of engineering technology, and particularly relates to a pipe column design method based on reverse elite Harris eagle optimization. BACKGROUND

[0002] The pipe column is a circular thin-walled hollow structure for transmitting load and upper structure weight into the foundation, and is widely used in the engineering field. It is of practical significance to design the pipe column size to reduce the pipe column weight and material consumption as far as possible under the premise of meeting the engineering needs.

[0003] The Harris eagle optimization algorithm is a new swarm intelligence optimization algorithm, has the advantages of simple principle, wide applicability, no gradient information and strong ability to avoid local optimum, and can be used to solve the multi-peak, discontinuous and non-differentiable engineering problems which are difficult to effectively solve by traditional optimization algorithms. When the Harris eagle optimization algorithm is applied to the pipe column design, there are problems of easy falling into local optimum, slow convergence speed and low convergence precision when the Harris eagle optimization algorithm is used to optimize the pipe column design. SUMMARY

[0004] The application provides a pipe column design method based on reverse elite Harris eagle optimization which improves the convergence speed, robustness and stability of the algorithm.

[0005] The application is achieved as follows: a pipe column design method based on reverse elite Harris eagle optimization, characterized in that the method comprises:

[0006] S1: a mathematical model of the pipe column design problem is established, the objective function, the value range of the to-be-solved parameters and the constraint conditions are determined, then the objective function and the constraint conditions are combined to construct an auxiliary function by using an external point penalty function method, and the auxiliary function is used as a fitness function;

[0007] S2: the initial positions of the Harris eagle population in the solution space determined by the value range of the to-be-solved parameters are randomly generated, and a selection probability parameter is initialized;

[0008] S3: the positions of all Harris eagles in the population are subjected to elite reverse learning to generate reverse positions, the fitness values of all Harris eagle positions and their reverse positions are calculated, and the Harris eagle positions are preferentially updated to their reverse positions;

[0009] S4: the selection probability is updated according to the previous Harris eagle position update, the escape energy of the Harris eagle is updated according to the current iteration number, step S5 is executed when the escape energy of the Harris eagle is greater than 1, otherwise step S6 is jumped to;

[0010] S5: the Harris eagle position is updated by using an optimized exploration formula, and step S7 is jumped to;

[0011] S6: randomly selecting the Harris hawk position updated by elite crossover evolution or elite fusion evolution according to the selection probability;

[0012] S7: judging whether the termination condition is met, outputting the optimal final column design parameter if yes, otherwise jumping to step S3 to continue optimization.

[0013] Preferably, the mathematical modeling of the column design problem in S1 is specifically operated as follows:

[0014] The objective function is:

[0015] f(X) = 9.82x1x2 + 2x1

[0016] The constraint condition is:

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] The value range of the to-be-solved parameter is:

[0024] 2 ≥ x1 ≥ 1 / 4

[0025] 0.2 ≥ x2 ≥ 0.8

[0026] The calculation formula of the fitness function is:

[0027] F(X) = f(X) + 10000(g1(X) + g2(X) + g3(X) + g4(X) + g5(X) + g6(X))

[0028] In the formula, X = (x1, x2) is the to-be-solved parameter vector, f(X) is the objective function, g1(X), g2(X), g3(X), g4(X), g5(X) and g6(X) are barrier functions.

[0029] Preferably, in S2, the random generation formula of the initial position of the Harris hawk is:

[0030] X = r(UB-LB) + LB

[0031] In the formula, r is a random number in the interval (0, 1), and UB and LB are the upper and lower bounds of the solution space, respectively.

[0032] Preferably, the specific operation of S3 is as follows:

[0033] The elite reverse learning is performed with the elite center as the symmetric center, and the definition of the elite center is:

[0034]

[0035] In the formula, X o1 , X o2 , X o3 are three optimal positions in the Harris hawk population position in the last iteration, and r is a random number in the interval (0, 1);

[0036] The elite reverse learning is performed with the elite center as the symmetric center to obtain the symmetric position of the Harris hawk position, and the Harris hawk reverse position is randomly generated in the solution space with the symmetric position as the center and the distance from the Harris hawk position to the symmetric center as the radius, and the calculation formula is:

[0037]

[0038] In the formula, is the Harris hawk reverse position, and r1 and r2 are random numbers in the interval (0, 1);

[0039] First, the fitness value of the Harris hawk reverse position is calculated, and if it is less than the fitness value of the optimal position of the Harris hawk in the iteration process, the Harris hawk position is updated to the reverse position; otherwise, the Harris hawk position is not updated and its fitness value is calculated; the Harris hawk position update formula in the elite reverse learning is:

[0040]

[0041] In the formula, is the Harris hawk reverse position, and fitness is the fitness value of the optimal position of the Harris hawk in the iteration process.

[0042] Preferably, the specific operation of S4 is as follows:

[0043] First, the fitness value of the Harris hawk position calculated in step S3 is compared with the fitness value of the Harris hawk historical optimal position, and it is judged whether the Harris hawk historical optimal position is updated in the last iteration, if the Harris hawk historical optimal position is not updated after the last iteration, the selection probability pr is updated, and the update formula is:

[0044]

[0045] where the selection probability pr is a number in the interval [0, 1], and its initial value is 0.5; the parameter n is the number of continuous iteration rounds of the Harris hawk historical optimal position; the parameter diff is the difference between the number of times of updating the Harris hawk position using the elite crossover evolution and the elite fusion evolution in the last iteration round; if the value of the selection probability pr after updating exceeds the interval [0, 1], the value is corrected to the nearest boundary value, and the counter n is set to 0;

[0046] The escape energy is randomly changed with the iteration number, and the updating formula is as follows:

[0047]

[0048] where the initial escape energy E0 is a random number in the interval (0, 1), t is the current iteration round number, and T is the maximum iteration round number.

[0049] Preferably, the optimization exploration formula in S5 is as follows:

[0050]

[0051] where r1, r2, r3, r4, r5, r6 and q are random numbers in the interval (0, 1), X e1 is the Harris hawk optimal position, X r is the position of a Harris hawk randomly selected from the Harris hawk population, and X m is the average position of all Harris hawks in the Harris hawk population, and UB and LB are the upper and lower bounds of the search space.

[0052] Preferably, the specific operation of S6 is as follows:

[0053] The elite crossover evolution obtains a new position by crossing and combining three historical optimal Harris hawk positions, and obtains the final Harris hawk position by performing a small range mutation on the new position. The related formula of the elite crossover evolution is as follows:

[0054] p = 0.2 x r x (1-t / T)

[0055]

[0056] X c2 = X c1 +N(0, 0.1)|X c1 -X|

[0057] where p is the probability of obtaining a new Harris hawk position by combining different dimensions of different Harris hawk positions, r is a random number in the interval (0, 1), X e1 , X e2 and X e3 are the Harris hawk optimal positions, and satisfy F(Xe1 )≤F(X e2 )≤F(X e3 ), represents that different dimensional combinations of different Harris hawk positions are directly obtained to get new positions, represents that different dimensions of the Harris hawk position are used to combine new positions with a certain probability, and N is a random number in a normal distribution with a mean of 0 and a standard deviation of 0.1;

[0058] The elite fusion evolution obtains a new position by fusing the search space center and the optimal position of the Harris hawk, and then fuses the new position and the optimal position of the Harris hawk to obtain the final Harris hawk position. The formula related to the elite fusion evolution is:

[0059] q=1-t / T

[0060] X f1 =CL+N|CL-X e1 |

[0061] X f2 =(1+r1×q)×X e1 +r2×q×X f1

[0062] In the formula, X e1 is the optimal position of the Harris hawk, q is a parameter for controlling the fusion ratio of different positions varying with the iteration round, wherein N is a random number in a standard normal distribution with a mean of 0 and a standard deviation of 1, and r1 and r2 are random numbers in the interval (-1, 1);

[0063] The formula for updating the Harris hawk position by the elite crossover evolution and the elite fusion evolution according to the selection probability is:

[0064]

[0065] In the formula, X c2 is the Harris hawk position updated by the elite crossover evolution, X f2 is the Harris hawk position updated by the elite fusion evolution, r is a random number in the interval (0, 1), and pr is the selection probability.

[0066] Preferably, the pipe column is selected to bear a compression load of P=2500kgf, and the diameter x1 and the thickness x2 vary in the intervals of 2-14cm and 0.2-0.8cm, respectively.

[0067] The parameters of the pipe column composition material are: yield stress σ y =500kgf / cm 2 , elastic modulus E=0.85×10 6 , and the pipe column length is 250cm.

[0068] The beneficial effects of the present application: 1. By introducing elite reverse learning into Harris hawk optimization algorithm, the population structure is optimized, the ability to jump out of local optimum is enhanced, the convergence speed of the algorithm is improved, and the robustness of the algorithm is enhanced; thereby improving the optimization effect of the pipe column design.

[0069] 2. By introducing elite evolution strategy into Harris hawk optimization algorithm, the local development ability of the algorithm is enhanced, the convergence speed and convergence accuracy of the algorithm are improved, the selection probability is dynamically adjusted, and the stability of the algorithm is enhanced, thereby further improving the optimization effect of the pipe column design, reducing the weight of the pipe column, and reducing material consumption. BRIEF DESCRIPTION OF DRAWINGS

[0070] Figure 1 The flowchart of the method of the present application. DETAILED DESCRIPTION

[0071] The present application will be further described below in conjunction with the drawings.

[0072] As shown in the figure, a pipe column design method based on reverse elite Harris hawk optimization, the specific steps are: Figure 1 S1: The pipe column design problem is mathematically modeled, the objective function, the parameter value range to be solved and its constraint conditions are determined, then the objective function and the constraint conditions are combined to construct an auxiliary function using the external point penalty function method, and the auxiliary function is used as the fitness function, which specifically includes:

[0073] The design goal is to manufacture a uniform pipe column with the smallest cost, which can withstand a compression load of P=2500kgf, the diameter x1 and the thickness x2 vary in the interval of 2-14cm and 0.2-0.8cm respectively. The parameter settings of the pipe column material are: yield stress σ y =500kgf / cm 2 , elastic modulus E=0.85×10 6 , the length of the pipe column is 250cm, and the mathematical modeling is as follows:

[0074] The objective function is:

[0075] f(X)=9.82x1x2+2x1

[0076] The constraint condition is:

[0077]

[0078]

[0079]

[0080]

[0081] ​

[0082]

[0083]

[0084] The value range of the to-be-solved parameter is:

[0085] 2≥x1≥14

[0086] 0.2≥x2≥0.8

[0087] The calculation formula of the fitness function is:

[0088] F(X)=f(X)+10000(g1(X)+g2(X)+g3(X)+g4(X)+g5(X)+g6(X))

[0089] In the formula, X=(x1, x2) is a to-be-solved parameter vector, f(X) is a target function, g1(X), g2(X), g3(X), g4(X), g5(X), and g6(X) are barrier functions.

[0090] S2: Randomly generate the initial position of the Harris eagle population in the solution space with the value range of the to-be-solved parameter determined, and initialize the selection probability parameter, specifically including:

[0091] The random generation formula of the initial position of the Harris eagle is:

[0092] X=r(UB-LB)+LB

[0093] In the formula, r is a random number in the interval (0, 1), and UB and LB are the upper and lower bounds of the solution space, respectively.

[0094] S3: Perform elite reverse learning on the positions of all Harris eagles in the population to generate reverse positions, calculate the fitness values of all Harris eagle positions and their reverse positions, and preferentially update the Harris eagle positions to their reverse positions, specifically including:

[0095] The elite reverse learning performs reverse learning with the elite center as the symmetric center, and the definition of the elite center is:

[0096]

[0097] In the formula, X o1 , X o2 , and X o3 are three optimal positions (current optimal position) in the Harris eagle population position in the last iteration, and r is a random number in the interval (0, 1).

[0098] Then, the elite inverse learning obtains the symmetric position of the Harris hawk position with the elite center as the symmetric center, and generates the Harris hawk inverse position in the solution space with the symmetric position as the center and the distance from the Harris hawk position to the symmetric center as the radius, and its calculation formula is:

[0099]

[0100] In the formula, is the Harris hawk inverse position, and r1 and r2 are random numbers in the interval (0, 1);

[0101] Finally, the fitness value of the Harris hawk inverse position is calculated first, and if it is less than the fitness value of the optimal position of the Harris hawk in the iteration process, the Harris hawk position is updated to the inverse position; otherwise, the Harris hawk position is not updated and its fitness value is calculated. The Harris hawk position update formula in the elite inverse learning is:

[0102]

[0103] In the formula, is the Harris hawk inverse position, and fitness is the fitness value of the optimal position of the Harris hawk in the iteration process.

[0104] S4: Update the selection probability according to the previous Harris hawk position update, and update the escape energy of the Harris hawk according to the current iteration number. When the escape energy of the Harris hawk is greater than 1, step S5 is executed, otherwise, step S6 is jumped to, and specifically comprising:

[0105] First, the fitness value of the Harris hawk position calculated in step S3 is compared with the fitness value of the Harris hawk historical optimal position, and it is judged whether the Harris hawk historical optimal position is updated in the last iteration. If the Harris hawk historical optimal position is not updated after the last iteration, the selection probability pr is updated, and the update formula is:

[0106]

[0107] In the formula, the selection probability pr is a number in the interval [0, 1], and its initial value is 0.5; the parameter n is the number of continuous iterations in which the Harris hawk historical optimal position is not updated; and the parameter diff is the difference between the number of times that the Harris hawk position is updated using the elite crossover evolution and the elite fusion evolution in the last iteration. If the value of the selection probability pr is updated beyond its value interval [0, 1], it is corrected to the nearest boundary value, and the counter n is set to 0;

[0108] The escape energy is randomly changed based on the iteration number, and its update formula is:

[0109]

[0110] where E0 is a random number in the interval (0, 1), t is the current iteration round, and T is the maximum iteration round.

[0111] S5: updating the Harris hawk position with the optimized exploration formula and jumping to step S7, specifically including:

[0112] The optimized exploration formula is:

[0113]

[0114] where r1, r2, r3, r4, r5, r6, and q are random numbers in the interval (0, 1), X e1 is the optimal position of the Harris hawk (historical optimal position), X r is the position of a randomly selected Harris hawk in the Harris hawk population, and X m is the average position of all Harris hawks in the Harris hawk population, and UB and LB are the upper and lower bounds of the search space.

[0115] S6: randomly selecting the Harris hawk position updated by elite crossover evolution or elite fusion evolution according to the selection probability, specifically including: elite crossover evolution obtains a new position by crossing and combining three historical optimal Harris hawk positions, and performs small-range mutation on the basis of the new position to obtain the final Harris hawk position, and the related formula of elite crossover evolution is:

[0116] p = 0.2 * r * (1-t / T)

[0117]

[0118] X c2 = X c1 +N(0, 0.1)|X c1 -X|

[0119] where p is the probability of obtaining a new Harris hawk position by combining different dimensions of different Harris hawk positions, r is a random number in the interval (0, 1), X e1 , X e2 , and X e3 are the optimal positions of the Harris hawk (historical optimal positions), and satisfy F(X e1 ) ≤ F(X e2 ) ≤ F(X e3 ), represents directly obtaining a new position by combining different dimensions of different Harris hawk positions, represents the probability of using different dimensions of the Harris hawk position to combine a new position, and N is a random number in the normal distribution with a mean of 0 and a standard deviation of 0.1;

[0120] The elite fusion evolution obtains a new position by fusing the search space center and the Harris hawk optimal position, and obtains the final Harris hawk position by fusing the new position and the Harris hawk optimal position. The elite fusion evolution related formula is as follows:

[0121] q = 1 - t / T

[0122] X f1 = CL + N|CL - X e1 |

[0123] X f2 = (1 + r1 x q) x X e1 + r2 x q x X f1

[0124] In the formula, X e1 is the Harris hawk optimal position, q is a parameter for controlling the fusion proportion of different positions varying with the iteration round number, wherein N is a random number in a standard normal distribution with a mean of 0 and a standard deviation of 1, and r1 and r2 are random numbers in the interval (-1, 1);

[0125] The formula for updating the Harris hawk position by the elite crossover evolution and the elite fusion evolution according to the selection probability is as follows:

[0126]

[0127] In the formula, X c2 is the Harris hawk position updated by the elite crossover evolution, X f2 is the Harris hawk position updated by the elite fusion evolution, r is a random number in the interval (0, 1), and pr is the selection probability.

[0128] S7: Determine whether the termination condition is met. If yes, output the final column design parameter, otherwise jump to step S3 to continue optimization.

[0129] When the Harris hawk population size is 30 and the maximum iteration number is 500, the Harris hawk optimization algorithm (HHO) and the reverse elite Harris hawk optimization algorithm (OEHHO) are independently used to solve the column design problem 30 times, and the diameter x1 and the thickness x2 when the fitness value is minimum, and the maximum value, the minimum value, the average value and the variance of the fitness value are recorded. The experimental results are shown in Table 1:

[0130] Table 1 is a comparison table of HHO and OEHHO independently solving the column design

[0131] x1 / cm x2 / cm max_F(x) min_F(x) mean_F(x) std_F(x) HHO 5.4602 0.2928 26.8527 26.5345 26.6169 0.0820 OEHHO 5.4512 0.2920 26.5314 26.5313 26.5313 9.07e-06

[0132] As shown in Table 1, the maximum value, the minimum value, the average value, and the variance of the fitness value of the column design problem solved by the reverse elite Harris hawk optimization are all less than those of the Harris hawk optimization algorithm, that is, the diameter x1 and the thickness x2 of the column solved by the reverse Harris hawk optimization make the column weight smaller.

[0133] The above merely illustrates the embodiments of the present application but is not intended to limit the present application. The present application can have various modifications and changes for those skilled in the art. Any modification, equivalent replacement, improvement, etc. within the spirit and principle of the present application shall be included in the scope of claims of the present application.

Claims

1. A method for column design based on reverse elite Harris hawk optimization, characterized in that: The method comprises the following steps: S1: mathematically modeling the pipe column design problem, determining a target function, a to-be-solved parameter value range and a constraint condition, then combining the target function and the constraint condition to construct an auxiliary function by using an external point penalty function method, and taking the auxiliary function as a fitness function; The specific operation of mathematically modeling the pipe column design problem is as follows: The target function is: ； The constraint condition is: ； ； ； ； ； ； The to-be-solved parameter value range is: ； ； The calculation formula of the fitness function is: ； wherein is the parameter vector to be determined, is the objective function, , and is the barrier function; The pipe string is selected to withstand compressive loads with diameters and thicknesses varying within and ranges, respectively; The parameters of the tubular string composition material are: yield stress , elastic modulus , tubular string length ; S2: randomly generating initial positions of a Harris eagle population in a solution space determined by the to-be-solved parameter value range, and initializing a selection probability parameter; S3: generating reverse positions by elite reverse learning of all Harris eagle positions in the population, calculating fitness values of the Harris eagle positions and the reverse positions, and preferentially updating the Harris eagle positions to the reverse positions; S4: updating the selection probability according to the previous Harris eagle position updating condition, and updating the escape energy of the Harris eagle according to the current iteration number; when the escape energy of the Harris eagle is greater than 1, step S5 is executed; otherwise, step S6 is jumped to; S5: updating the Harris eagle positions by using an optimized exploration formula, and jumping to step S7; S6: randomly selecting, according to the selection probability, whether to update the Harris eagle positions by elite crossover evolution or elite fusion evolution; S7: judging whether a termination condition is met; if yes, outputting optimal final pipe column design parameters; otherwise, jumping to step S3 to continue optimization.

2. The pipe column design method based on reverse elite Harris eagle optimization according to claim 1, characterized in that: In S2, the random generation formula of the initial positions of the Harris eagles is: ； In the formula, is a random number in the interval, , are the lower and upper bounds of the solution space, respectively.

3. The column design method based on reverse elite Harris hawk optimization according to claim 1, characterized in that: The specific operation of S3 is as follows: Elite reverse learning performs reverse learning with an elite center as a symmetric center, and the definition of the elite center is: ； wherein , , are the three best positions in the Harris' hawks population positions at the previous iteration, is the random number in the interval [0, 1], is the random number in the interval [0, 1]. Elite reverse learning obtains symmetric positions of the Harris eagle positions with the elite center as the symmetric center, and randomly generates reverse positions of the Harris eagles in a solution space with the symmetric positions as centers and distances from the symmetric center to the Harris eagle positions as radii, and the calculation formula is: ； wherein is the Harris Falcon reverse position, , is a random number in the interval; First, the fitness value of the reverse position of the Harris eagle is calculated; if the fitness value is less than the fitness value of the optimal position of the Harris eagle in the iteration process, the Harris eagle position is updated to the reverse position; otherwise, the Harris eagle position is not updated and the fitness value is calculated; The Harris eagle position updating formula in the elite reverse learning is: ； wherein is the Harris hawk inverse position, is the fitness value of the Harris hawk optimal position during the iteration process.

4. The column design method based on reverse elite Harris hawk optimization according to claim 1, wherein: The specific operation of S4 is as follows: The fitness value of the Harris hawk position calculated in step S3 is compared with the fitness value of the Harris hawk historical optimal position, and it is determined whether the Harris hawk historical optimal position is updated in the last iteration. If the Harris hawk historical optimal position is not updated after the last iteration, the selection probability of the Harris hawk historical optimal position is updated, and the selection probability of the Harris hawk position calculated in step S3 is updated. The update formula is: ； In the formula, the selection probability is a number in the interval , and the initial value is ; the parameter is the number of iterations of the optimal position of the continuous and non-updated Harris hawk history; the parameter is the difference between the number of times of updating the Harris hawk position using elite crossover evolution and elite fusion evolution in the last iteration; if the value of the selection probability exceeds the interval after being updated, the value is corrected to the nearest boundary value, and the counter is set to 0. The escape energy is randomly changed based on the iteration number, and the updating formula is: ； where the initial escape energy is a random number in the interval is a random number in the interval is the current iteration number, is the maximum iteration number.

5. The column design method based on reverse elite Harris hawk optimization according to claim 1, characterized in that: The optimized exploration formula in S5 is: ； wherein , , , , , and are random numbers in the interval [0, 1], is the position of the Harris hawk with the best fitness, is the position of a randomly selected Harris hawk in the population of Harris hawks, is the average position of all Harris hawks in the population of Harris hawks, and are the upper and lower bounds of the search space.

6. The column design method based on reverse elite Harris hawk optimization according to claim 1, characterized in that: The specific operation of S6 is as follows: Elite crossover evolution obtains a new position by crossing and combining three historical optimal Harris eagle positions, and obtains a final Harris eagle position by performing a small range mutation on the new position, and the related formula of the elite crossover evolution is: ； ； ； wherein is the probability of obtaining a new Harris hawk position from different dimensional combinations of different Harris hawk positions, is a random number within the interval , , and is the optimal Harris hawk position, and satisfies , represents directly obtaining a new position from different dimensional combinations of different Harris hawk positions, represents the probability with which different dimensions of the Harris hawk position are used to combine a new position, is a random number in a normal distribution with a mean of and a standard deviation of . Elite fusion evolution obtains a new position by fusing a search space center and an optimal Harris eagle position, and obtains a final Harris eagle position by fusing the new position and the optimal Harris eagle position, and the related formula of the elite fusion evolution is: ； ； ； wherein is the Harris hawk's optimal position, is a parameter that controls the fusion ratio of different positions with the iteration round number, wherein is a random number in a standard normal distribution with a mean of and a standard deviation of , , is a random number in the interval . The formula for randomly selecting, according to the selection probability, the elite crossover evolution and the elite fusion evolution to update the Harris eagle position is: ； In the formula, The position of the Harris Eagle is updated through elite cross-evolution. The position of the Harris Eagle is updated through elite fusion and evolution. It is an interval Random numbers within, It is the probability of choice.

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