JPH07319848A - System and method for retrieving optimum solution - Google Patents

Abstract

PURPOSE:To shorten the time of retrieval processing and to reduce the volume of a memory necessary for the retrieval processing by utilizing already detected solution information. CONSTITUTION:A retrieval control information generating means 4 generates retrieval control information 5 indicating the priority directions of respective variables stored in solution information 3 generated by optimizing a combined optimizing problem 1 by an optimizing means 2. A retrieving means 7 judges the retrieval priority directions of respective variables based upon the information 5 and executes the retrieval of an optimum solution in the same or similar integer programming problem or mixed integer programming problem(MIP) 6 as/to the problem 1 by a branch and bound method. The retrieved result is outputted as solution information 8. The optimizing means 2 is not restricted by the branch and bound method. Thus the efficiency of retrieval in the case of repeatedly retrieving the same or similar problem can be improved.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an optimal solution search method and an optimal solution search method for an integer programming problem or a mixed integer programming problem, and more particularly to an optimal solution search method and an optimal solution search method for searching an optimal solution by a branch and bound method. Regarding

[0002]

2. Description of the Related Art Conventionally, it has been required to find an optimum value of a combination of resources existing in a production plan in a manufacturing industry and a distribution plan in a distribution industry in a wide range of technical and industrial fields. Each conditional expression of such a combinatorial optimization problem is expressed by a linear equality or a linear inequality regarding some variables, and the function to be minimized or maximized is expressed by a linear expression. Is called a linear programming problem, and what considers how to solve it is linear programming.

If the problem expressed by the first-order equality or the first-order inequality is a condition that all the values of each variable can take real values, the mathematical programming method, which is one of the operations research methods, is used. Known as the Medium Linear Programming Problem.

Regarding the processing method of the linear programming problem, the basic part of the solution method has been developed. This solution is one of linear programming methods called simplex method. This is because the constraint conditions are simultaneous temporary inequalities, and the point of maximizing (or minimizing) the objective function most within the region that satisfies each constraint condition is This is a method that utilizes the end points of a convex polyhedron. This is a method in which the objective function is finally maximized by sequentially tracing the end points of the convex polyhedron in the direction in which the objective function increases (in the case of the maximization problem), and a solution satisfying all the constraints can be obtained. This method has also been used mainly in actual computer systems for many years. In this case, the revised simplex method and interior point method have a fast and stable processing method.

However, when the condition that the value of each variable must be an integer variable is further added, an integer programming problem (all unknown variables used in simultaneous inequalities are integers, which is a development of the linear programming problem. There is a case where it is required to satisfy the condition) or a mixed integer programming problem (when any number of unknown variables used in simultaneous inequalities is required to satisfy the integer condition). Especially for industrial use, a method of presenting a solution in which a variable takes an integer value is required.

In the conventional processing methods for integer programming problems and mixed integer programming problems, the branch and bound method has been used as a representative and practical exact solution method. In the branch-and-bound method, the domain (domain) of a variable that is requested to take an integer value obtained when it is optimized as a linear programming problem is divided using the value of that variable, The solution in the specified area is sought. Among the obtained solutions, the one with the best value is the optimum solution. Such a method of subdividing an expression is called branching or branching processing, and a node tree is often used as a method of expressing a state of subdividing. Also, the initially given planning problem is called a parent node, and the planning problem generated by dividing each variable area of the parent node is called a child node.

Furthermore, as a method for more efficiently using the branch and bound method, a method called cutoff or branch picking is also used. This method uses the objective function value of the best optimal solution obtained so far. For example, if the objective function value of the optimum solution resulting from executing the convergence condition is softened and the cutoff value is applied to the problem with the next convergence condition changed, it is possible to generate no extra node. It is a thing. The process of stopping the search for such a node is called cutoff or cutoff by cutoff.

As an example of how the branch and bound method is actually used, consider the following example (this problem will be referred to as the track problem hereinafter). The shipping company has n types of equipment c _j
It can be assumed that an appropriate one is selected from (j = 1, ..., N), placed on a truck, and transported to a factory.
Here, the utility of 1 per individual gear c _j when carried the weight of each gear c _j a _j (kg), the plant (effectiveness)
Is b _j, and the weight limit of the truck is b (kg).
The respective values are as follows: equipment c ₁ has weight a ₁ = 36 and utility b ₁ = 54. The equipment c ₂ has a weight a ₂ = 24 and a utility b ₂ = 18. The equipment c ₃ has a weight a ₃ = 30 and a utility b ₃ = 60. The equipment c ₄ has a weight a ₄ = 32 and a utility b ₄ = 32. The equipment c ₅ has a weight a ₅ = 26 and a utility b ₅ = 13. Suppose

[0009] When the weight limit of the track is 91 kg, since the amount of equipment that can be Komu carry the factory is less weight limit, when the number carried in the track and x _j, the utility of the effect due to carry The number x _j of each piece of equipment c _j that maximizes the total sum C can be derived by the following simultaneous inequalities.

[0010]

## EQU1 ## 36x ₁ + 24x ₂ + 30x ₃ + 32x ₄ + 26x ₅ ≤91 ... (1)

[0011]

[Number 2] _{54x 1 + 18x 2 + 60x 3} + 32x 4 + 13x 5 = total C · · · · (2) utility that is, while satisfying the inequality (1), the sum of utilities represented by formula (2) A method of determining a combination of resources for maximizing C is required. The inequality in (1) is called a constraint condition, and the expression in (2) that finds the total sum C of utilities is called an objective function. Further, it is assumed that x _j can be an integer for the number of trucks to carry, and that only one piece of equipment is required. That is, 0 ≦ x ₁ ≦ 1, 0 ≦ x ₂ ≦ 1, 0
The variable conditions of ≦ x ₃ ≦ 1, 0 ≦ x ₄ ≦ 1, 0 ≦ x ₅ ≦ 1 must be satisfied.

In order to facilitate the description below, the following abbreviations are used. MIP Problem: Mixed Integer Programming Problem, We use this term to include integer programming problems as well. MIP variable: A variable that requires a discrete condition such as having to take an integer value among the variables forming the MIP problem. MIP condition: A discrete condition such as an integer value required for a MIP variable. Continuous relaxation solution: A solution that is linearly optimized as a linear programming problem by not having the required MIP condition in the MIP problem. MIP possible solution: A solution that satisfies the MIP condition.

In the branch and bound method, the condition that the variable x _j is an integer is first ignored and considered, and the problem is treated as a linear programming problem to find a continuous relaxation solution. In the shipping company example above, the solution is 139.0, and the variable from which the solution is obtained is x ₁
= 1, x ₂ = 0, x ₃ = 1, x ₄ = 0.78, x ₅ = 0
Is. This solution, because x ₄ is not an integer, x ₄ ≦ 0 in the case of, x ₄ variable conditions as for ≧ 1 are given further 2
Search for a linear programming problem that is bifurcated into streets. In this way, branching is performed one after another.

FIG. 11 shows the optimum solution search by the conventional branch and bound method.
It is a figure which shows the node tree created by the search method. this
In the figure, each node 21 to 33 has a linear
The optimal solution (OBJ) obtained as a planning problem and the
Variable (x₁, X₂, X₃, X _Four, X_Five) Value or
Is the value on the left side (RO
W1) is shown and the respective values are as follows. Parent
The first node 21, which is the node, has OBJ = 139.0,
x₁= 1, x₂= 0, x₃= 1, x_Four= 0.78, x_Five
= 0. The second node 22 has OBJ = 135.5, x₁
= 0.81, x₂= 0, x₃= 1, x_Four= 1, x_Five=
0. The third node 23 has OBJ = 132, x₁= 1,
x₂= 0, x₃= 0.77, x_Four= 1, x_Five= 0. Fourth
The node 24 of ROW1 is 98. Fifth node 25
Is OBJ = 103.25, x₁= 1, x₂= 0.9
6, x₃= 0, x_Four= 1, x_Five= 0. Sixth node 26
Is ROW1 = 92. The seventh node 27 has OBJ = 9
7.5, x₁= 1, x₂= 0, x₃= 0, x_Four= 1, x
_Five= 0.88. The eighth node 28 has ROW1 = 94.
The ninth node 29 has OBJ = 86, x₁= 1, x₂=
0, x₃= 0, x_Four= 1, x_Five= 0. Tenth node 3
0 is OBJ = 132.5, x₁= 1, x₂= 1, x₃
= 1, x_Four= 0, x_Five= 0.04. Eleventh node 31
Is OBJ = 132, x₁= 1, x₂= 1, x₃= 1,
x_Four= 0, x_Five= 0. The twelfth node 32 is OBJ =
125.5, x₁= 0.97, x₂= 0, x₃= 1, x
_Four= 0, x_Five= 1. The thirteenth node 33 has OBJ = 1
12.5, x₁= 0, x₂= 1, x₃= 1, x_Four= 1,
x_Five= 0.19.

In this node tree, the ninth node 29 and the eleventh node 31 detect the MIP possible solution. Since the MIP feasible solution is detected, it is not necessary to perform the subsequent search. And the MIP detected at the eleventh node
The possible solution has the maximum value (OBJ = 132), and this is the optimal solution of this problem.

The fourth node 24 and the sixth node 2
The sixth and eighth nodes 28 cannot be executed as a linear programming problem because the value of ROW1 exceeds 91.
Subsequent searches are unnecessary. Further, the twelfth node 32 and the thirteenth node 33 respectively have OBJ = 125.5,
OBJ = 112.5 and already the eleventh node 31
Since the value is lower than the MIP possible solution (OBJ = 132) detected in, the subsequent search is unnecessary and the cutoff is performed.

If, before solving this problem, it was known that the optimal solution was near 132, then the value of "cutoff =
By optimizing as “around 132”, generation of extra nodes can be suppressed. As a result, the node 25, which should be originally subjected to the branch search, does not generate a better solution than the cutoff value, so that the subsequent search can be terminated. Therefore, the sixth node 26 to the ninth node 29 can be cut off.

Further, there is a method of obtaining a practical local optimum solution, which is not an optimum solution by an approximate solution method but an exact solution method such as the branch and bound method. Furthermore, Toshihide Ibaraki, Japan Society for Operations Research, 30th Symposium New Development of Modern Heuristic Technique 1994 pp.
According to "Recent topics on combinatorial optimization methods" published in 1-10 and 1994, the modern heuristic method (as a method for obtaining an approximate solution that is faster than the exact solution method and more accurate than the simple approximate solution method) It is usually called meta-heuristic technique or meta-strategy).

[0019]

However, in the exact solution method represented by the branch and bound method, since a finite number of variables have a finite domain, theoretically, it is generated by a finite number of branch operations. By optimizing a finite number of nodes as a linear programming problem, it is possible to find the combination of variables that maximizes (or minimizes) the objective function, but it is not possible to perform all searches in a practical time. difficult. further,
The amount of data to be handled becomes enormous, and the memory capacity during data processing and the storage capacity of a storage device such as a hard disk when saving are very large.

There is a rationale that many of these are essentially difficult in terms of computational complexity, as typified by NP-hardness (there is no algorithm that can be solved in polynomial time). . That is, regarding these problems, it is considered that there is no effective solution except for enumerating a considerable part of the solution space (exponentially increasing number of solutions). Therefore, in the exact solution method for obtaining an accurate optimum solution, how to reduce the search area is a big problem under the theoretical limit.

Further, when the cutoff is performed, when solving a similar problem, the value of the solution when optimized may be different. In this case, depending on the value of the cutoff, the optimum value may be obtained. There is a risk that the solution cannot be detected. (Similar problems here have little effect on the value of the solution, such as differences in specific values of the objective function, constraint conditions, or variable conditions. I say the problem that only the part is different). In this case, there is no influence on the selection method of the variable to be divided and the processing method of linearly optimizing the divided child node, and in addition to the objective function value, the variables of each variable Although the value has been calculated,
The variable value of each variable is not used effectively.

Further, the approximate solution method is applied to the actual problem individually for each method, and there is a case where the application of the approximate solution method is tried only by the approximate solution method and the solution accuracy cannot be put to practical use. Here, if an attempt is made to improve the accuracy of the approximate solution, the processing time increases (again, in this case, the accuracy guarantee of optimality may not theoretically exist depending on the approximate solution used).

However, the construction of the business system accompanied by the optimization of the MIP problem is a work carried out by the applied company for the purpose of reducing the production cost, reducing the distribution cost and increasing the profit of the company. For this reason, the objective function value obtained as a result of optimizing the problem by computer software should be as close as possible to the objective function value obtained in a strictly optimal state that should exist theoretically and ideally. Although it varies slightly depending on the content of the problem, in most cases, it is required to search for the objective function value within a numerical range of 3% at worst and 1-2% in operation. Therefore, 0.01% must be adopted as a standard value in order to sufficiently meet the demands of system users. In order to obtain such a better solution, the system is often improved and tuned. In such a case, it is necessary to gradually change the convergence condition of the system and solve the same problem many times.

As described above, the problem that a company puts into practical use is a planning problem for a certain period, and the problem is repeatedly executed as long as the activity of the company continues. In this case, most of the same problems are optimized for very similar problems with some conditions changed. Nevertheless, the conventional method does not effectively utilize the data of the solution of the problem previously solved, so that the processing time cannot be shortened and the storage device such as a memory for storing various data cannot be shortened. A huge amount of memory was required.

The present invention has been made in view of the above point, and by utilizing the solution information that has already been detected, it is possible to reduce the search processing time and the memory amount required for the search processing. The purpose is to provide an optimal solution search method that

Another object of the present invention is to provide an optimal solution search method that can shorten the search processing time and the memory amount required for the search processing by utilizing the solution information that has already been detected. Is to provide.

[0027]

FIG. 1 is a principle diagram of an optimum solution search system of the present invention for achieving the above object. In the present invention, the search control information generation means 4 for creating the search control information 5 indicating the priority direction of each variable from the solution information 3 of the combination optimization problem 1 that has already been detected, and the search control information 5 for each variable. And a search means 7 for determining an optimal solution of the integer programming problem or the mixed integer programming problem 6 by deciding the priority direction when branching the variable condition and generating the node.

FIG. 2 is a flow chart showing the processing procedure of the optimum solution searching method of the present invention which achieves the above object. From the solution information 3 of the combination optimization problem 1 that has already been detected, search control information 5 indicating the preferential direction of each variable is created (step 2), and the variable condition of each variable is branched by the search control information 5 and the node is set. Judge the priority direction when generating,
The optimum solution of the integer programming problem or the mixed integer programming problem 6 is searched (step 3).

[0029]

The search control information generating means 4 creates the search control information 5 indicating the priority direction of each variable from the solution information 3 of the combination optimization problem 1 that has already been detected. The search means 7 is
Based on the search control information 5, the variable condition of each variable is branched to determine the priority direction when generating a node, and the optimum solution of the integer programming problem or the mixed integer programming problem 6 is searched.

In the processing procedure, first, the search control information 5 indicating the priority direction of each variable is created from the solution information 3 of the combination optimization problem 1 that has already been detected (step 2). Then, based on the search control information 5, the variable condition of each variable is branched to determine the priority direction when the node is generated, and the optimum solution of the integer programming problem or the mixed integer programming problem 6 is searched (step 3).

As a result, the solution information 3 of the combinatorial optimization problem 1 that has already been detected can be effectively used, and the branching can be efficiently performed when searching for the optimum solution of the integer programming problem or the mixed integer programming problem 6. it can.

[0032]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to the drawings. FIG. 1 is a principle diagram of the optimum solution search method of the present invention. The search control information generation means 4 is based on the solution information 3 generated by the optimization means 2 optimizing the combination optimization problem 1, and the search control indicating the preferential direction of each variable stored in the solution information 3. Information 5 is generated. The search means 7 determines the priority direction when searching each variable based on the search control information 5, and determines the optimum solution of the integer programming problem or the mixed integer programming problem (MIP problem) 6 that is the same as or similar to the combinatorial optimization problem 1. The search is performed by the branch and bound method. The result of the search is the solution information 8
Output as. The optimizing means 2 need not be the branch and bound method.

FIG. 2 is a flow chart showing the processing procedure of the optimum solution search method of the present invention. [S1] It is determined whether or not the solution information 3 is used. If the solution information 3 is used, the process proceeds to step 2. If not, the process proceeds to step 3. [S2] The solution structure is read from the solution information 3 to generate search control information. This is a function performed by the search control information generation means 4. [S3] The integer programming problem and the mixed programming problem are optimized by the branch and bound method. At this time, if there is search control information, the priority direction of the search is determined based on the search control information. If a search strategy such as cutoff is specified, it is used. [S4] The solution information 8 is created. Note that steps 3 and 4 are functions performed by the search means 7.

Next, the contents of the solution information 3 will be described.
FIG. 3 is a diagram showing header information of solution information. In this header information, data of a creation date, a creation time, and an objective function value are stored as a solution. One piece of this header information is held for each solution information. The header information is not always necessary data.

FIG. 4 is a diagram showing the variable information of the solution information 3. In this variable information, the variable name, the variable type, the value of the variable, the lower limit value of the variable and the upper limit value of the variable are stored in the solution information. It is sufficient for the variable name and variable type to be able to determine whether the corresponding variable is required to take an integer value. Also, the lower limit value of the variable and the upper limit value of the variable are
It is not necessary data because it can be extracted from the problem.

FIG. 5 is a flowchart showing the procedure from the solution information 3 to the generation of the search control information 5. This is a function performed by the search control information generation means 4. [S11] One variable is read from the file containing the solution information. [S12] The variable type of the read variable is checked. [S13] Determine whether the variable type is a MIP variable,
If it is the MIP variable, the process proceeds to step 14, and if it is not the MIP variable, the process proceeds to step 18. [S14] Determine whether the value is at the lower limit,
If the value has the lower limit value, the process proceeds to step 15. If the value has not the lower limit value, the process proceeds to step 16. [S15] Downward priority instruction data is generated for the variable. At this time, if necessary, the selection priority flag is set. [S16] Determine whether the value is at the upper limit,
If the value has the upper limit value, the process proceeds to step 17, and if the value has not the upper limit value, the process proceeds to step 18. [S17] Upward priority instruction data is generated for the variable. At this time, if necessary, the selection priority flag is set. [S18] It is determined whether or not all variables have been checked, and if there is a variable that has not been checked yet, the process proceeds to step 1, and if all variables have been checked, the process ends.

In this way, the search control information 5 is generated. In other words, the solution information generated by optimizing the target problem by some method is processed so that it can be used as input information for the search. In step 14 and step 16 described above, the determination as to whether the lower limit value or the upper limit value is taken is made within a certain range (integer tolerance). This integer tolerance can be increased to 0.5 in operation.

FIG. 6 is a diagram showing the contents of the search control information. In the figure, each variable name (x ₁ x ₂ ...) Has direction indication data and a selection priority flag.
The direction instruction data is either an upward instruction, a downward instruction, or no instruction. The selection priority flag is either "1" with priority instruction or "0" without priority instruction. An optimum solution is searched using this search control information.

FIG. 7 is a flowchart showing the first half procedure of the optimum solution search using the search control information. This is a function performed by the search means 7. [S21] The MIP problem is continuously optimized as a linear programming problem. [S22] It is determined whether or not the cutoff value is set and the value of the continuous optimum solution is cutoff. If cutoff, the process proceeds to step 29 (FIG. 8), and if not cutoff, the process proceeds to step 23. . [S23] It is determined whether or not the continuous optimum solution satisfies all MIP conditions. If satisfied, the process proceeds to step 24. If not satisfied, the process proceeds to step 26. That is, if all MIP variables satisfy the MIP conditions, the subsequent branch processing is unnecessary. [S24] The information of the MIP possible solution is saved. [S25] The objective function value of the possible solution is set as the cutoff value. [S26] It is determined whether or not the linear programming problem can be executed. If the linear programming problem can be executed, the process proceeds to step 27, and if not, the process proceeds to step 29 (FIG. 8). The case where the linear programming problem cannot be executed is, for example, the case where there is no searchable region within the range satisfying the conditions of each variable. [S27] One MIP variable to be a node branch target is determined. At this time, the variable for which the selection priority flag is set is given priority for branching. [S28] Whether to branch the selected variable upward or downward is determined by using the search control information, and the newly generated node that is branched is registered in the waiting node, and the process proceeds to step 29 of FIG. move on. In other words, branching newly creates child nodes in the upward direction and the downward direction. Which node is to be preferentially searched is determined, and the node to be preferentially searched is registered in the waiting node. Then, another node recognizes that the node exists without being searched.

FIG. 8 is a flow chart showing the latter half procedure of the optimum solution search using the search control information. [S29] It is determined whether or not there is an unsearched node. If there is an unsearched node, the process proceeds to step 30, and if there is no unsearched node, the process proceeds to step 32. The search at this time is
Search all nodes regardless of node priority. [S30] The node to be searched next is selected. At this time, in step 28 of FIG. 7, the node determined to be the node to be preferentially searched is preferentially selected. [S31] After storing the selected node as having been searched,
Proceed to step 21 in FIG. [S32] Solution information is created.

Further, the procedure for determining the branching direction of the variable, which is performed in step 28, will be described in detail. FIG. 9 is a flowchart showing a procedure for determining the branch direction of a variable. [S41] Search for search control information for the selected MIP variable. [S42] It is determined whether or not there is an upward priority instruction, and if there is an instruction, the operation proceeds to step 43, and if not, the operation proceeds to step 44. [S43] A node in the direction of updating the lower limit is selected.
That is, control is added to preferentially select a child node whose lower limit value is updated. [S44] It is determined whether or not there is a downward priority instruction, and if there is an instruction, the operation proceeds to step 45, and if there is no instruction, the operation ends. [S45] A node in the direction of updating the upper limit value is selected.
That is, control is added to preferentially select a child node whose upper limit value is updated.

In this way, the selected child node is optimized as a linear programming problem, and it is determined whether all MIP variables satisfy the MIP condition. If not satisfied, determine whether all possibilities have been examined.
If all MIP variables satisfy the MIP condition, the information is saved as a MIP feasible solution. When a MIP possible solution is detected, an objective function value of this MIP possible solution can be set and used as a cutoff value.
Of these detected MIP possible solutions, the one with the best value is the optimum solution.

A description will be given of the case where the optimum solution of the above-mentioned track problem is searched for in the system having the solution guidance system having the above-mentioned configuration. In this example, the solution information obtained in advance has an objective function value of 132, and variable values at that time are x ₁ = 1, x ₂ = 1 and x ₃ =, respectively.
It is assumed that 1, x ₄ = 0 and x ₅ = 0.

First, the search control information generating means 5 uses this solution.
Generate search control information from the information. Where x₁And x₂
Are variable conditions (0 ≦ x₁≦ 1, 0 ≦ x₂≦ 1)
X is the upper limit of₃, X_Four, And x_FiveIs a variable
Condition (0 ≦ x₃≦ 1, 0 ≦ x _Four≦ 1, 0 ≦ x_Five≦ 1)
Taking the lower limit. Therefore, the search system for each variable
The content of your information is x₁And x₂Is an upward direction priority instruction,
x₃, X_Four, And x _FiveIs a downward priority instruction. That
Then, the search means 7 uses the generated branch and bound method to find the optimal solution.
To explore. At this time, the priority set in the search control information
Create a node according to the forward direction.

FIG. 10 is a diagram showing a node tree created using the search control information. In this figure, each node 11 to 15 has an optimal solution (OBJ) obtained from the node as a linear programming problem and variables (x ₁ , x ₂ ,
x _3, x _4, x ₅ values each value are shown in) is as follows. The first node 11 that is the parent node is an OB.
_{J = 139.0, x 1 = 1} , x 2 = 0, x 3 = 1, x 4
= 0.78, x ₅ = 0. The second node 12 has x ₄ ≤0
And OBJ = 132.5, x ₁ = 1, x ₂ = 1, x
_{3 = 1, x 4 = 0} , x 5 = 0.04. Third node 13
Is x ₅ ≦ 0, OBJ = 132, x ₁ = 1 and x ₂
= 1, x ₃ = 1, x ₄ = 0, x ₅ = 0. 4th node 1
4 is x ₅ ≧ 1, OBJ = 125.5, x ₁ =
_{0.97, x 2 = 0, x} 3 = 1, x 4 = 0, x 5 = 1.
The fifth node 15 has x ₄ ≧ 1 and OBJ = 13
_{5.5, x 1 = 0.81, x} 2 = 0, x 3 = 1, x 4 =
1, x ₅ = 0. The sixth node 16 has x ₁ ≧ 1,
_{OBJ = 132, x 1 = 1} , x 2 = 0, x 3 = 0.7
7, x ₄ = 1 and x ₅ = 0. The seventh node 17 has x ₁ ≦
0, OBJ = 112.5, x ₁ = 0, x ₂ = 1,
_{x 3 = 1, x 4 =} 1, x 5 = 0.19.

In this node tree, the MIP possible solution (OBJ = 132) is detected at the third node 13. Since the MIP feasible solution is detected, it is not necessary to perform the subsequent search. The fourth node 13, the sixth node 16, and the seventh node 17 have OBJ = 125.5,
OBJ = 132, OBJ = 112.5, and the MIP possible solution (OBJ) already detected by the fourth node 14
= 132) or less, the subsequent search is unnecessary,
Cut off.

In this way, when searching for the same problem or a similar problem, the value of each variable value of the existing solution can be used to efficiently solve the problem without the user being aware of it. Be able to optimize. This requires a much smaller number of nodes than conventional methods.

The existing solution for generating the search control information does not have to be the solution solved by the same method. It is also possible to efficiently obtain an accurate solution by the branch and bound method by using the variable value of each variable derived from the result obtained by a completely different method such as the metaheuristic method. As a result, the solution of the combinatorial optimization problem requires a very long processing time to cause combinatorial explosion, but it can be processed in a short time. Moreover, the use of the approximate solution can be promoted by creating a smooth flow to the exact solution.

In addition to the meta-heuristic method, the current approximate solution method includes genetic algorithm, tab search, simulated annealing, constraint logic, expert method, greedy method, local search, LK method,
There are neural networks and evolutionary algorithms.

As described above, according to the present invention, in the optimum solution search of the integer programming problem and the mixed integer programming problem, each variable value of the solution information already obtained is used, and search control is performed based on this. The following effects are generated because the configuration is adopted. The processing time required for the optimum solution search can be shortened. It is possible to reduce the amount of memory required for optimum solution search. Then, the disk capacity required for the optimum solution search can be reduced.

These effects are particularly effective under the following conditions. When solving the same problem again by changing the numerical values (coefficient value, upper limit value, lower limit value). When the convergence condition of the search for the same problem is slightly changed and the solution is performed again. To solve similar problems. To restart a problem that was interrupted and solve it. Then, the exact solution is obtained by using the approximate solution detected by another method. As long as the problem is similar, the solution of the linear programming problem can be used, but the use of the solution of the MIP problem can be expected to have a great effect.

When repeatedly solving the MIP problem,
Even when the number of variables increases or decreases, it can be easily dealt with as follows. When the number of variables is reduced, the MIP variable of the target MIP problem is iteratively processed, and thus the reduced variable is not searched. When the number of variables increases, no branch condition is specified for the increased variables.
By preferentially branching a variable for which the selection priority flag is set, the effect can be enhanced depending on the meaning of the increased variable.

When the same problem is executed under the same conditions, the following applications are possible. For example, a large M
When optimizing an IP problem, there is a possibility of interruption due to some factor during processing. In such a case, a large effect can be obtained by repeatedly solving the same problem under the same conditions.

When the interruption is performed, the solution information detected so far is saved, and the process using the best possible solution detected is restarted. Considering the case of saving the information at the time of interruption by the conventional save function, each MIP
“Variable name of MIP variable, lower limit value at interruption point, upper limit value at interruption point” is stored for the variable. If the size of each information is 8 bytes, a large storage capacity of “8 bytes × 3 × the number of MIP variables” corresponds to the number of nodes is required. However, since solution information is usually always generated at the time of interruption, the addition of an area for storing necessary information is 0 bytes.

As a result, even when the processing is interrupted, the node tree used at the time of searching in the save function is saved, as in the conventional processing of saving the search state and restoring the search information. It is not necessary to save the constituent information memory in a file and restore the saved information from the file to the memory by the restore function. As the information that constitutes the node tree, the configuration information of the node tree at the time of interruption and the MIP at each node
Information on variable conditions (variable name of MIP variable, upper limit value and lower limit value at the time of interruption) is required, and a huge amount is required as the problem becomes larger.

In other words, it is not necessary to provide a large storage area required for saving various data, it is possible to reduce the amount of information required for resuming the interruption process, and at the time of interruption. It is possible to reduce the processing time for storing information.

In the above explanation, the case of searching for the solution of the track problem has been explained, but in addition to this, a class formation problem (for example, when classifying new students at a university, put all of them in the first choice) If it is not possible to maximize the satisfaction of individual students and add conditions so as not to exceed the number of students in each class), mix problems (for example, in the feed purchase plan of pig farmers, How to combine multiple types of commercially available feeds will produce the lowest cost and will be expressed by adding the conditions that meet the intake requirement of pigs for one month), and determine the optimal price. Problem, schedule planning problem, or knapsack problem (when hikers choose suitable ones from multiple types of mountaineering equipment and pack them in the knapsack, go hiking equipment The greatest the sum total of use, and added conditions to fit within budget expressed in) utilized in such freezing can.

[0058]

As described above, according to the present invention, the search control information is generated by using the variable value of each variable possessed by the solution information that has already been detected, and the branch control is performed by the search control information. Since the optimum solution is searched by the method, the branching at the time of the search can be effectively performed, the search processing time at the time of the optimum solution search can be shortened, and the memory etc. required for the search processing can be reduced. It is possible to reduce the required amount of storage device.

[Brief description of drawings]

FIG. 1 is a principle diagram of an optimum solution search method of the present invention.

FIG. 2 is a flowchart showing a processing procedure of an optimum solution search method of the present invention.

FIG. 3 is a diagram showing header information of solution information.

FIG. 4 is a diagram showing variable information of solution information.

FIG. 5 is a flowchart showing a procedure from generation of search control information from solution information.

FIG. 6 is a diagram showing the contents of search control information.

FIG. 7 is a flowchart showing a first half procedure of searching for an optimum solution using search control information.

FIG. 8 is a flowchart showing a latter half procedure of searching for an optimum solution using search control information.

FIG. 9 is a flowchart showing a procedure for determining a variable branch direction.

FIG. 10 is a diagram showing a node tree created using search control information.

FIG. 11 is a diagram showing a node tree created by a conventional optimal solution search method based on a branch and bound method.

[Explanation of Codes] 1 Combinatorial optimization problem 2 Optimization means 3 Solution information 4 Search control information generation means 5 Search control information 6 Integer programming problem or mixed integer programming problem (MIP problem) 7 Search means 8 Solution information

Claims (16)

[Claims] 1. In an optimal solution search method for an integer programming problem or a mixed integer programming problem by a branch and bound method, from the solution information (3) of the already detected combinatorial optimization problem (1), the priority direction of each variable is obtained. And a search control information generating means (4) for generating search control information (5) indicating that the search control information (5) and the search control information (5) determine the priority direction when branching the variable condition of each variable to generate a node, An optimal solution search method comprising: a search means (7) for searching an optimal solution of a programming problem or a mixed integer programming problem (6). 2. The search control information generation means (4) determines the priority direction of each variable from the information as to whether the variable value of the solution information (3) has an upper limit value or a lower limit value. The optimal solution search method according to claim 1, wherein the determination is made. 3. The search control information generation means (4) determines that the variable value of the solution information (3) is an upper limit value within a range given by an integer tolerance with respect to the upper limit value, and the lower limit value. 3. The optimum solution search method according to claim 2, wherein the value is a lower limit if it is within a range given by integer tolerance. 4. The optimum solution search method according to claim 1, wherein the combination optimization problem (1) is a mixed integer programming problem. 5. The optimum solution search method according to claim 1, wherein the combinatorial optimization problem (1) is an integer programming problem. 6. The optimum solution search method according to claim 1, wherein the solution information (2) is a solution solved by an approximate solution method. 7. The optimum solution search method according to claim 1, wherein the integer programming problem or the mixed integer programming problem (6) is the same problem as the combinatorial optimization problem (1). 8. The optimum solution search method according to claim 1, wherein the integer programming problem or the mixed integer programming problem (6) is a problem similar to the combinatorial optimization problem (1). 9. The optimum solution according to claim 1, wherein the search control information (5) has a selection priority flag for preferentially selecting a variable to be branched. Search method. 10. The search means (7) is required if a variable not required in the integer programming problem or the mixed integer programming problem (6) is included in the search control information (5). The optimum solution search method according to claim 1, wherein a search for a variable that does not occur is not performed. 11. The searching means (7), if a variable required for the integer programming problem or mixed integer programming problem (6) is not included in the search control information (5), the necessary variable. The optimum solution search method according to claim 1, wherein is treated as if there is no instruction of a priority direction. 12. An uncompleted process solution information generating means for generating solution information (3) of the best solution detected at the time of interruption when the optimal solution search process is interrupted. The optimum solution search method according to claim 1. 13. The searching means (7) uses the objective function value of the best solution satisfying the discrete condition as a cutoff value when a solution satisfying the discrete condition is detected. The optimum solution search method according to claim 1. 14. A method of searching for an optimal solution of an integer programming problem or a mixed integer programming problem by a branch and bound method, from the solution information (3) of the already detected combinatorial optimization problem (1), to the preferential direction of each variable. Is created (step 2), the variable direction of each variable is branched based on the search control information (5), and the priority direction in generating a node is determined, and an integer programming problem or a mixed integer is generated. An optimal solution search method characterized by performing an optimal solution search for the planning problem (6) (step 3). 15. When the search control information (5) is created, whether the variable value of the solution information (3) has an upper limit value,
15. The optimum solution search method according to claim 14, wherein the priority direction of each variable is determined from information indicating whether a lower limit value is set. 16. When creating the search control information (5), if the variable value of the solution information (3) is within a range given by integer tolerance with respect to the upper limit value, the upper limit value is determined,
15. The optimum solution search method according to claim 14, wherein the lower limit value is a lower limit value if it is within a range given by integer tolerance.