JPH0744611A - Multipurpose optimizing problem solving method - Google Patents

Abstract

PURPOSE:To make it possible to determine an optimizing solution by determining the transition probability to a new solution for each of plural objective functions and determining whether the new solution is adopted or not. CONSTITUTION:A system is composed of a central processing unit(CPU) 10, a memory 11, an external storage device 12 such as a magnetic disk, for instance, and an external output device 13 such as a display, a printer and a communication line, etc. A program for conducting the search of a solution in accordance with an algorithm is stored in the external storage device 12 and the program is loaded to the memory 11 from the external storage device 12 when a search operation is started. Subsequently, based on the algorithm, the program is read by the CPU 10 and a processing is conducted. After the processing is conducted, the data showing the determined optimizing solution is outputted from the memory 11 to the external output device 13.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to operation optimization plans for factories and combinatorial optimization problems such as delivery problems in the field of logistics. More specifically, the present invention relates to a multi-objective optimization that seeks solutions that optimize each of a plurality of objective functions as much as possible. Regarding problem solving method.

[0002]

2. Description of the Related Art A considerable amount of research and development has been conducted in the fields of OR and AI regarding the solution of combinatorial optimization problems. Among these methods, the mountain climbing method will be described first.

In the hill climbing method, first, an appropriate solution S is generated. Next, a solution S ′ is generated by modifying a part of S. Here, if the objective function value of S'is better than the objective function value of S, S ← S '. If not, reject S'and recreate S '. This operation is continued until it becomes impossible to create S'that has an objective function better than S, and S at that time is output as a solution. Generally, when S'is made from S, only a small part of S is changed (for example, only the value of one variable is changed), so the solution obtained by the hill climbing method is only a local optimum solution.

The simulated annealing method is a method for solving this drawback and obtaining a strict optimum solution. Unlike the hill-climbing method, the simulated annealing method differs from the hill-climbing method in determining whether to move from the original solution S to the modified solution S ', even if the objective function value is S' Allow movement (of course when it gets better). However, the probability of moving (transitioning) to a solution that deteriorates the objective function value is reduced as the search progresses. It is known that a strict optimal solution can be obtained by properly controlling the method of changing this probability, but it takes a huge amount of time to do so, so it is usually necessary to obtain a strict optimal solution. Gives up and instead controls the probability so that the search ends in a reasonable time. Still, in normal problems we can get much better solutions than the hill-climbing method.

[0005]

The hill climbing method and the simulated annealing method described above have a problem that only one objective function can be set in the combinatorial optimization problem.

Therefore, when there are a plurality of objective functions, the plurality of objective functions are appropriately weighted to form a linear combination to form one virtual objective function, and the objective function is calculated. The work of optimizing was done. However, the optimization related to the virtual objective function created in this way does not always correspond to obtaining the solution desired by the user, and depending on the weight setting method in the linear combination, for example, The value of the objective function is very good, but there is a possibility that a solution may be obtained in which the values of other objective functions are very bad.

The present invention makes it possible to find a solution that optimizes each of a plurality of objective functions as much as possible in a combinatorial optimization problem in which a plurality of objective functions exist.

[0008]

FIG. 1 is a functional block diagram of the present invention. This figure is a functional block diagram of a multi-objective optimization problem solving method for obtaining a solution that optimizes each of a plurality of objective functions as much as possible.

In FIG. 1, first, one solution is generated at 1, that is, an initial solution S, and at 2 a search parameter for determining a transition probability as a probability that a new solution should be adopted instead of the solution S is determined. At 3, the neighborhood S ′ of the solution S is generated. This neighborhood S'is generated, for example, by changing the value of one of the plurality of variables for each of the plurality of objective functions described above.

Then, in 4, the transition probabilities to a new solution, that is, to the neighborhood S ', are obtained for each of the plurality of objective functions,
At 5, a new solution is calculated from the transition probabilities for each objective function, that is, the overall transition probabilities as probabilities that the neighborhood S ′ should be adopted. This overall transition probability is obtained, for example, as the product of the transition probabilities for each objective function.

Then, the value of the overall transition probability is compared with a certain threshold value, and when the overall transition probability is larger, the neighborhood S'is adopted as a new solution S at 6 until the search termination condition is satisfied. The processes after the determination of the search parameters are repeated. Here, the search termination condition is defined as, for example, the number of times a transition to a new solution has occurred.

On the other hand, when the value of the overall transition probability is smaller than a certain threshold value, the above-mentioned search parameter is determined until the end condition of the search is satisfied without adopting the neighborhood S ′ as a new solution S at 7. Subsequent processing is repeated.

Further, in the present invention, after the generation of one solution S in 1 described above and the solution S is saved as one of the best solutions, the search parameter for determining the transition probability is determined in 2 And the neighborhood S ′ when the overall transition probability is larger than a certain threshold in 6 above is adopted as a new solution S, a value that is better than the values of the plurality of objective functions for the new solution is given to a plurality of objectives. It is determined whether a solution that takes all of the functions, that is, a solution that is clearly better than the new solution S, is already stored as the best solution, and if not, conversely, a value that is worse than the new solution is set for all objective functions. If there is a solution that takes for, the new solution is added as the best solution while the solution is removed from the saved state as the best solution, and the processing after the search parameter determination is performed until the above-mentioned search termination condition is satisfied. Repeated, also in the case of obvious solutions have already been stored, without changing the best solution,
The invention can be implemented so as to differ from the point that the processes after the determination of the search parameter are repeated until the search termination condition is satisfied.

Further, in the present invention, the number of solutions that can be stored as this best solution is limited to a fixed value or less, and the solution that takes good values for all of the above-mentioned plurality of objective functions is stored as the best solution. If not, the solution that takes a bad value for all objective functions is removed from the saved state as the best solution and a new solution is added as one of the best solutions.
As a result, when the number of solutions stored as the best solution exceeds the above-mentioned fixed value, the effectiveness of each stored solution is examined, and the solution with the worst effectiveness is excluded from the saved state as the best solution. It is also possible to implement the invention differently from the point that the processes after the determination of the search parameter are repeated until the above-mentioned condition for ending the search is satisfied.

As described above, in the present invention, the transition probability to a new solution is obtained for each of a plurality of objective functions, and the overall transition probability as the probability that a new solution should be adopted is, for example, for each objective function. Calculated as the product of transition probabilities,
The overall transition probability is compared to a threshold to determine whether to adopt a new solution.

[0016]

FIG. 2 is a block diagram showing the overall configuration of a computer system for implementing the multi-objective optimization problem solving method of the present invention. In the figure, the system is a central processing unit (CP
U) 10, a memory 11, an external storage device 12 such as a magnetic disk device, and an external output device 13 such as a display, a printer and a communication line.

A program for executing a search for a solution in accordance with an algorithm shown in a flowchart described later is stored in the external storage device 12 in FIG. 2, and is loaded from the external storage device 12 to the memory 11 at the start of the search operation. . After that, the program is read by the CPU 10 and the process is executed based on an algorithm described later. After the processing is executed, the data indicating the obtained optimum solution is output from the memory 11 to the external output device 13.

FIG. 3 is a processing flowchart of the first embodiment of the multi-objective optimization problem solving method of the present invention. In the figure, first, in step (S) 21, one suitable initial solution is obtained, and this is set as the solution S. Then, in S22, it is determined whether or not the search termination condition is satisfied.
The search termination condition is, for example, the condition of changing the solution S, that is, the number of transitions from the old solution to the new solution, or S ′ generated near the solution S as described later.
A condition such as the ratio of the number of successes adopted as a new solution to the number of failures not adopted as a new solution is used. Since the solution S is the initial solution here, the search end condition is not satisfied, and the process proceeds to S23.

In S23, a search parameter m for obtaining a transition probability as a probability that a new solution should be adopted.
_i is determined. The search parameters will be described later. Then, in S24, a neighborhood S'for the current solution S is generated. This neighborhood S'is generated, for example, by changing the value of one of the plurality of variables that determines the current S.

Then, in S25, the transition probability is obtained for each of the plurality of objective functions. The details of this transition probability will be described later. After the transition probabilities for the respective objective functions are obtained, the transition probabilities Ptot of the solution as a whole are calculated based on the values of the respective transition probabilities p _{i in} S26.
al is calculated. The overall transition probability is obtained by, for example, the product of the transition probabilities p _i for each objective function calculated in S25.

After that, in S27, one of the random numbers between 0 and 1 is set to r, and in S28, it is obtained in r and S26.
The values of Ptotal are compared, and when r is larger, the processing from S22 is repeated. On the other hand, when r is smaller, the neighborhood S ′ is newly adopted as the solution S in S29, and then the processing of S22 and thereafter is repeated. When it is determined in S22 that the search termination condition is satisfied, the solution S is output in S30, and the process ends.

With respect to the first embodiment shown in FIG. 3, the details of the processing will be described first using a simple example. First, generally, a plurality of objective functions are F ₁ (S), F ₂ (S), ... F
_N (S) N pieces and N pieces of objective functions F ₁ , F ₂ .
... The transition probabilities from the solution S ₁ to the solution S ₂ corresponding to F _N are P ₁ (m ₁ , e ₁₁ , e ₁₂ ), P ₂ (m ₂ , e)
₂₁ , e ₂₂ ), ..., P _N (m _N , e _N1 , e _N2 ). The search parameter m _i corresponds to the degree of progress of the search, and e _i1 and e _i2 are F _i (S ₁ ) and F _i , respectively.
(S ₂ ) (i = 1, 2, ..., N).

As a simple example, the following combination optimization problem of two variables (X, Y) (two objective functions)
think of. That is, the solution is represented by a combination of the values of two variables.

Example ○ Variable region Xε {1,2,3,4,5,6,7,8,9,10} Yε {1,2,3,4,5,6,7,8, 9,10} ○ Objective function F1 ([X, Y]) = least common multiple of X and Y F2 ([X, Y]) = XY (difference between X and Y) To solve such a problem First, the functions P1 and P2 that give transition probabilities are defined as follows (both aim at minimization).

[0025]

[Equation 1]

Thus, the value of the transition probability is the solution S before the transition.
₁The value of the objective function for (eg E ₁₁) Is the solution S after transition
₂The value of the objective function for (E₁₂) When above, reverse
If less than, the search parameter (m₁) E₁₁-E
₁₂(Difference in objective function value) raised to the power.

First, S = [5,4] is set as an initial solution for performing a search in S21 of FIG. Further, the search parameters are determined in S23 as follows. m ₁ = m ₂ = 2 The objective function values of the initial solution are as follows.

F ₁ (S) = F ₁ ([5,4]) = 20 F ₂ (S) = F ₂ ([5,4]) = 1 Next, in S24, a part of the initial solution S is changed. By doing so, the neighborhood S ′ is generated. For example, the value of Y is increased by 1 to 5. At this time, the objective function value of S ′ is F ₁ (S ′) = F ₁ ([5,5]) = 5 F ₂ (S ′) = F ₂ ([5,5]) = 0. When the transition probabilities for each objective function are calculated in S25 from P1 and P2 given above, P ₁ = P ₁ (2,20,5) = 1 P ₂ = P ₂ (2,1,0) = 1 The transition probability P _{total in} S26 by taking the product of
Then, P _total = P ₁ ^* P ₂ = 1 Now, it is determined whether or not to transit from S to S ′ by a random number of 0 or more and less than 1. When 0.3 was obtained as the time being random number _r in S27, it performs the S ← S 'in S29 because r <P _{to tal} is established at S28.

Then, it is determined in S22 whether or not the search is completed.
Here, the search is continued. As the update of the search parameter in S23, for example, both m1 and m2 are increased by one. That is, m1 = m2 = 3. Next, in S24, the neighborhood S'is generated again. Here again, the value of Y is increased by 1 to 6. As a result, F ₁ (S ′) = F ₁ ([5,6]) = 30 F ₂ (S ′) = F ₂ ([5,6]) = − 1. When the transition probabilities for each objective function are obtained in S25 from P ₁ and P ₂ given above, p ₁ = P ₁ (3,5,30) = 3 ⁻²⁵ p ₂ = P ₂ (3,0, − 1) = 1. When the transition probability p _total is obtained in S26 by taking these products, p _total = p ₁ ^* p ₂ = 3 ⁻²⁵ Now, it is determined whether or not to transit from S to S ′ by a random number of 0 or more and less than 1. To do. For the time being, in S27, the random number r
If 0.1 is obtained, r <p _total is obtained in S28.
Is not established, S'is rejected.

By repeating such an operation, a suboptimal solution can be obtained for both of the two objective functions. INDUSTRIAL APPLICABILITY As described above, the present invention can be used for general combinatorial optimization problems such as a planning problem and a design problem such as a factory operation plan. Other planning problems include an operation schedule creation problem for trains and airplanes, a delivery problem in the field of logistics, and a timetable creation problem in schools. The design problem is a problem that decides what kind of parts should be combined and how to make a certain product, such as a mechanical design problem, an LSI logic design,
There are problems such as LSI layout design.

In practical use, in order to solve these problems, the purpose of satisfying constraint conditions is required in addition to the purpose of optimization. In such a case, in a neighborhood search method such as a simulation annealing method, optimization is performed by appropriately defining a function that represents the degree of constraint violation. Therefore, as an example of use of the present invention, considering the operation planning problem in a factory, the process in the first embodiment will be described in more detail with reference to the flowchart of FIG.

The operation plan problem of a factory is such that when a set of orders (representing which products need to be shipped by what time) is created, an operation plan that satisfies them as much as possible is created. The purpose is to decide what machine is used to make what from what time to what time. At this time, it is necessary to satisfy various restrictions, and it is necessary to reduce the operating cost of the entire factory as much as possible. There are various variations in the operation planning problem, but here we consider the following problem as an example.

A factory consisting of two machines (machine 1 and machine 2) prepares an operation plan for 5 days. According to the plan, production shall be carried out on a daily basis. That is, to make an operation plan is to decide which product each machine produces each day. In this factory, three kinds of products (Product 1, Product 2, and Product 3) are produced. In Machine 1, Product 1 and Product 2 can be produced, and in Machine 2, all products can be produced. Instead, it is assumed that the machine 1 has a larger production capacity, that is, the amount of production per day. An order is given for each product, and the delivery date of all orders is at the end of production on the fifth day, that is, the order is the production requirement amount in five days. Satisfying this requirement (producing at least the required quantity) is a constraint on this problem.

Further, when a product different from the previous day is made on a certain machine, it is necessary to perform a work called switching. This work is expensive and we want to minimize it. This is the objective function of this problem. As a constant characterizing this problem in FIG. 4, the required amount R [p] (product p
Required amount), the production capacity of each machine MA [m, p]
(Production capacity of machine m for product p), switching cost SC
[M] (switching cost of machine m) is shown.

At this time, the original objective function, that is, the total switching cost f ₁ is as follows.

[0036]

[Equation 2]

However, δ (u, v) is 1 when u = v,
It is a function that becomes 0 when u ≠ v. Where x [m,
d] (m = 1, 2 d = 1, 2, 3, 4, 5) is a variable that represents the product produced by the machine m on the day d, and the value of this variable can be determined. To make an operation plan. Moreover, the constraint that the required quantity must be produced is as follows.

[0038]

[Equation 3]

The following function can be used as a function representing the degree of violation of this constraint.

[0040]

[Equation 4]

Since this function is obtained by adding the production shortage amount from the required amount for each product, the smaller the value is, the smaller the insufficient amount is. If it becomes 0, the requirement is satisfied. Show.

In the first embodiment whose flow chart is shown in FIG. 3, this operation planning problem is solved by finding a solution that minimizes the two functions f ₁ and f ₂ . For that purpose, it is first necessary to define transition probabilities for each objective function. Here, when transitioning from a certain solution S (the solution is given by a set of x [m, d]) to its neighborhood S ′,
transition probabilities p ₁ and with respect to f _1, gives the transition probability p ₂ for f ₂ as follows

[0043]

[Equation 5]

Here, m ₁ and m ₂ are objective functions f ₁ and f
It is a parameter indicating the progress of the search in ₂ and its initial value is 2 in both cases. Next, it is necessary to give an initial solution for the search. This initial value may be randomly generated or a “good” initial solution may be generated by some means. Here, it is assumed that the initial solution as shown in FIG. 5 is input in S21 of FIG.

This solution is such that the machine 1 produces the product 1 every day, and the machine 2 produces the product 2 in the first 2 days and the product 3 in the remaining 3 days. First, let S be this initial solution. The value of the objective function in the initial solution is as follows.

F ₁ (S) = 1 f ₂ (S) = 4 Next, in S24, a part of this solution is changed so that the neighborhood of S '
To generate. There are various methods for neighborhood generation,
Here, as the most primitive method, it is assumed that one variable is selected and its value is changed to generate a neighborhood. As the first neighborhood S ′, the variable x
It is assumed that a solution in which the value of [2,5] is changed to 2 is generated. At this time, the objective function value in S'is f1 (S ') = 2 f2 (S') = 2. From this, the transition probabilities p ₁ and p ₂ for each objective function in S25 are as follows.

P ₁ = P ₁ (2,1,2) = 2 ⁻¹ = 0.5 P ₂ = P ₂ (2,4,2) = 1 By taking the product of these two probabilities, the total of When the transition probability P _total is obtained, P _total = P ₁ ^* P ₂ = 0.5. Next, a random number r of 0 or more and 1 or less is used to determine whether to move from S to S '. Here, it is assumed that r = 0.35 in S27. At this time, in S28, r <P
_{Since total} holds, S ← S ′ is set in S29 (moves to S ′).

Next, in S22, the search parameters m ₁ and m ₂ are updated assuming that the search end condition is not satisfied. Here, both are incremented by 1, that is, both are set to 3 (There are also various variations in this updating method. For example, instead of multiplying by a constant or constantly changing the value, once every several times. In some cases, it may be changed, and in other cases, it may be decided whether or not to change depending on the number of movements).

Next, a new S neighborhood S'is generated. At this time, by changing the value of x [1,1] to 2,
It is assumed that S'is created in S24. At this time, the objective function value in S'is as follows.

F ₁ (S ') = 4 f ₂ (S') = 0 Therefore, the transition probabilities for each objective function in S25 are as follows.

P ₁ = p ₁ (3,2,4) = 3 ⁻² = 1/9 P ₂ = p ₂ (3,2,0) = 1 Therefore, the _total transition probability P _{total in} S 26 is 1 /
Become 9. Here, the random number is used again to determine whether to move to S '. If the random number r is 0.72 in S27, r ≧ P _total in S28, and therefore the process does not move to S ′.

Thereafter, the same procedure is continued until the search end condition is satisfied, and when the end condition is satisfied, S at that time is output as a solution and the process ends. FIG. 6 is a processing flowchart of the second embodiment of the present invention. In the figure, the processes of steps S21 to S29 are the same as those of FIG. 4 showing the flowchart of the first embodiment.

In FIG. 6, S3 is inserted between S21 and S22.
The determination process of 2 is added, and the determination process of S32 is performed after the neighborhood S ′ is adopted as the new solution S in S29. In S32, the solution S ″ having a smaller value of the corresponding objective function than the respective values of the N objective functions Fi (S) (i = 1, 2, ..., N) for the new solution S is stored as the best solution. The best solution storage section stores the solution to be output in S35, which will be described later, as the best solution, that is, in S32, it is set as a new solution in S29. It is determined whether a solution that is clearly better than S is stored in the best solution storage unit. If stored, it is not necessary to store this solution S in the best solution storage unit. Move to processing.

In S32, if a solution that is obviously better than the new solution S is not stored in the best solution storage unit, the process proceeds to S33. Contrary to S32, in S33, a solution that is obviously worse than the new solution S among the solutions stored in the best solution storage unit is searched for, and if there is such a solution, the solution S ″ is deleted. That is, if a solution S ″ having a larger value of the corresponding objective function than each value of the objective function for the new solution S is stored in the best solution storage unit, the solution S ″ is deleted from the best solution storage unit. Note that in S32, S ″ clearly indicates a good solution, whereas in S33, the solution S ″ clearly indicates a bad solution.

Following the processing of S33, a new solution S is added to the solution set stored in the best solution storage unit in S34, and the processing of S22 and thereafter is repeated. And S
When it is determined in 22 that the search termination condition is satisfied,
In S35, the solution set stored in the best solution storage unit is output, and the process ends.

Next, regarding the above-mentioned operation planning problem, the first actual
The processing in the second embodiment is centered on the difference from the embodiment.
explain. First of all,
Initialize the set of solutions to the empty set. The following is the first embodiment
And proceed with the search. First, the initial solution of Fig. 5 is generated.
It As described above, the objective function value is f₁= 1, f₂= 4
Is. The best solution store will still store any solution.
The set of solutions stored in the best solution store
In S34, an initial solution is added. In the solution to move to next
Is f₁= 2, f₂= 2 (See the description of the first embodiment.
See). This solution is the solution stored in the best solution store.
That is, compared with the initial solution, f₁Is getting worse,
f₂Is better, so save it in the best solution saver
It is added to the set of existing solutions in S34. Ie the best solution
Two solutions are stored in the storage unit. It
And the above, and the objective function in the next solution moved to
The value is f₁= 5, f₂= 3, this solution is
The second solution (f stored in the best solution storage unit₁= 2,
f₂= 2), f₁, F₂Because both are bad,
It is not saved in the good answer storage section. Next solution eye moved to
The function value is f₁= 1, f₂= 3. Do
And compare it with the stored solution
f₁, F₂There is no good one, so I added this solution as well.
Be done. Not only that, the initial solution is f
₁, F₂Since both are bad, the initial solution is saved in S33 as the initial solution.
Deleted from the department. Thus, every time the solution is updated
Updating the set of solutions stored in the best solution store
If the search termination condition is satisfied, the best solution is maintained in S35.
Output the set of solutions stored in the existing section and end.

FIG. 7 is a processing flowchart of the third embodiment of the present invention. When this figure is compared with the flowchart of the second embodiment of FIG. 6, the only difference is that the processing of S36 is added between S34 and S22. That is, S3
After the new solution S is added to the best solution store in step 4,
When the number of solutions stored in the best solution storage unit in 6 is greater than the number given in advance as the upper limit, the effectiveness of each solution is calculated, and the solution with the lowest effectiveness is stored in the best solution storage unit. It is deleted, and the process proceeds to S22. The definition of the effectiveness of the solution will be described later.

The process for the operation plan problem in the third embodiment will be described in the same way as for the second embodiment. In the third embodiment, if the number of solutions stored in the best solution storage unit is smaller than the upper limit given in advance, the operation is exactly the same as in the second embodiment. The operation after is equal to the upper limit will be described.

The upper limit of the number of solutions that can be stored is 2. At the stage where the search has advanced to some extent, the _two solutions of f ₁ = 1 and f ₂ = 4 (hereinafter referred to as S1) and f ₁ = 2 and f ₂ = 2 (hereinafter referred to as S2) are the best. It is assumed that the solution is stored in the solution storage section. Here, it is assumed that the next step is to move to a solution of f ₁ = 3 and f ₂ = 0 (hereinafter referred to as S3). At this time, as compared with S3, a solution having better f ₁ and f ₂ is not stored in the solution storage unit, so S3 is added to the set of solutions stored in the best solution storage unit. Then, the number of solutions stored in S36 becomes 3, which exceeds the upper limit value 2. At this time, with respect to the stored solution, the effectiveness of a predefined solution is calculated, and the worst solution is deleted.
For example, if the effectiveness of the solution is defined as f ₁ + f ₂ (the smaller the value, the more effective), the effectiveness of S1 is 5, the effectiveness of S2 is 4, and the effectiveness of S3 is 3. Therefore, S1 is deleted, and two solutions S2 and S3 are stored in the best solution storage unit. In this way, by proceeding with the search so that the number of solutions stored in the best solution storage unit during search is below the upper limit, the required memory is reduced and the solution is stored in the best solution storage unit. The total search time can be shortened by reducing the time for determining whether or not to do it.

[0060]

As described above in detail, according to the present invention, in a multi-objective optimization problem having a plurality of objective functions, a plurality of objective functions are not sacrificed without sacrificing optimization. It is possible to obtain a sub-optimal solution that optimizes the objective function as much as possible, and it greatly contributes to the solution of design problems and planning problems.

[Brief description of drawings]

FIG. 1 is a functional block diagram of the present invention.

FIG. 2 is a configuration block diagram of a computer system for realizing the present invention.

FIG. 3 is a processing flowchart of the first embodiment of the present invention.

FIG. 4 is a diagram showing constants that characterize the operation planning problem.

FIG. 5 is a diagram showing an initial solution of x [m, d].

FIG. 6 is a processing flowchart of a second embodiment of the present invention.

FIG. 7 is a processing flowchart of a third embodiment of the present invention.

[Explanation of symbols]

 10 Central Processing Unit (CPU) 11 Memory 12 External Storage Device 13 External Output Device

Claims (4)

[Claims] 1. A multi-objective optimization problem solving method for obtaining a solution for optimizing a plurality of objective functions, wherein one solution S is generated (1) and a new solution is adopted in place of the solution S as a probability. The search parameter for obtaining the transition probability is determined (2), the neighborhood S ′ of the solution S is generated (3), and the neighborhood S ′ having a possibility of becoming a new solution for each of the plurality of objective functions is obtained. The transition probability is calculated (4), and the transition probability to the neighborhood S ′ as a whole is calculated from the transition probability to the neighborhood S ′ for each of the plurality of objective functions (5). When it is larger, the neighborhood S ′ is adopted as a new solution S, and the processing after the determination of the search parameter (2) is repeated (6) until the search termination condition for finding the solution is satisfied (6), and the entire transition When the probability is smaller than the threshold, the neighborhood S
Is set as a new solution S, and the processing after the determination of the search parameter (2) is repeated (7) until the termination condition of the search is satisfied, and each of the plurality of objective functions is optimized. Multi-objective optimization problem solving method. 2. Following the generation (1) of the one solution S, the transition probability as a probability that a new solution should be adopted in place of the solution S after the solution S is stored as one of the best solutions. (2) determining a search parameter for obtaining, and when the overall transition probability is larger than a certain threshold value, after adopting the neighborhood S ′ as a new solution S, values of a plurality of objective functions for the new solution It is determined whether a solution that takes a better value for all of the plurality of objective functions is stored as the best solution, and when not stored, among the solutions stored as the best solution, the The solution that takes a value that is worse than the values of the objective functions for the new solution for all of the objective functions is removed from the saved state as the best solution, and the new solution is added as one of the best solutions, Conditions for ending the search The processing after the determination (2) of the search parameter is repeated until it is satisfied, and when the result of the determination is stored, the termination condition of the search is satisfied without changing the solution stored as the best solution. 2. The multi-objective optimization problem solving method according to claim 1, wherein the processing after the determination (2) of the search parameter is repeated until the above-mentioned steps are performed. 3. When the number of solutions that can be stored as the best solution is limited to a certain value or less and the judgment result indicates that the solution is not stored, the new solution among the solutions stored as the best solution is used. The solution that takes a value worse than the values of the plurality of objective functions for the solution for all of the plurality of objective functions is removed from the saved state as the best solution, and the new solution is added as one of the best solutions, so that If the number of solutions stored as the best solution exceeds the fixed value, the effectiveness of each stored solution is examined, and the solution with the lowest effectiveness is removed from the storage state as the best solution. 3. The multi-objective optimization problem solving method according to claim 2, wherein the processing after the determination (2) of the search parameter is repeated until the termination condition of the search is satisfied. 4. The calculation of the overall transition probability (5), after obtaining one random number in the range of 0 or more and less than 1, the value of the one random number is used as the certain threshold value. The multi-objective optimization problem solving method according to item 1.