Abstract: An order quantity determining method includes: receiving restriction information used for calculating a profit based on an order quantity of a product; specifying a combination of order quantities of a k-th period to a (k+H)-th period for which the profit is the highest by using the restriction information, by a processor; and outputting the order quantity of the k-th period based on the specified combination.

Abstract: In this disclosure, equations to be solved in the model predictive control are transformed by using an off-line algebraic simplification method into a matrix operational expression representing a product of a coefficient matrix and a vector regarding solution inputs within a control horizon is equal to a function vector regarding target values of output states and the output states. The size of the coefficient matrix is reduced compared with the conventional matrix. Then, the matrix operational expression is solved in an online plant control apparatus with present output states and present target values of the output stats of a plant to be controlled, by the direct method, to output the solution to the plant.

Abstract: When model expressions of objective functions are generated at vertexes of a quadrilateral on a plane concerning P and N channels of transistors in SRAM, the initial number of times of simulation is allocated to each objective function at each designated vertex according to weight values set based on relationships presumed among the objective functions at each designated vertex. For each objective function at each designated vertex, first simulation is executed the allocated number of times. Furthermore, a model expression is generated from the first simulation result, and an evaluation indicator of an approximation accuracy of the model expression is calculated. Then, for each model expression, it is determined whether the corresponding model expression has influence on the yield, and based on the evaluation indicator of the corresponding model expression and presence or absence of the influence, it is determined whether additional simulation is required for the corresponding objective function.

Abstract: In this disclosure, equations to be solved in the model predictive control are transformed by using an off-line algebraic simplification method into a matrix operational expression representing a product of a coefficient matrix and a vector regarding solution inputs within a control horizon is equal to a function vector regarding target values of output states and the output states. The size of the coefficient matrix is reduced compared with the conventional matrix. Then, the matrix operational expression is solved in an online plant control apparatus with present output states and present target values of the output stats of a plant to be controlled, by the direct method, to output the solution to the plant.

Abstract: This method includes: generating a constraint equation from data of an approximate expression of a cost function representing a relationship between a plurality of design parameters and a cost, data of a route in a cost space and data of a search range in a design parameter space; obtaining a logical expression of a solution for the constraint equation from a quantifier elimination processing unit that carries out a processing according to a quantifier elimination method; substituting coordinates of each of a plurality of points within the search range in the design parameter space into the logical expression of the solution to determine, for each of the plurality of points, true or false of the logical expression of the solution; and displaying the design parameter space in which a display object including a first point for which true is determined is disposed at the first point.

Abstract: A control server according to an embodiment sorts a plurality of notebook PCs into a plurality of groups so that the total value of the remaining amounts is a value similar to the total value of the remaining amounts of the rechargeable batteries of a plurality of notebook PCs included in a different group. The control server according to the embodiment performs local search individually on the sorted groups, and generates a control plan for the individual notebook PCs.

Abstract: A design support apparatus for determining a plurality of objective functions for modeling an object having a plurality of elements, each of the elements providing variable geometrical parameters, the design support apparatus includes a memory for storing the variable geometrical parameters and a processor for executing a process including: determining boundary information associated with specified geometrical parameters of the elements which indicate a state of contact between the elements, dividing the variable geometrical parameters into a plurality of groups on the basis of the boundary information, and determining the plurality of objective functions for each of the groups by using the variable geometrical parameters.

Abstract: A method for solving an optimization problem includes: causing a cylindrical algebraic decomposition (CAD) processing unit to carry out a projection processing for a first expression that appears in a quantifier elimination problem equivalent to the optimization problem including objective functions and to generate second expressions that are projection factors of the first expression; calculating plural sets of values of the objective functions by generating plural value sets of variables in the objective functions and substituting the generated plural value sets of the variables into the objective functions; extracting points including non-dominated solutions in a space mapped by the objective functions, from plural points corresponding to the plural sets of values; calculating, for each second expression, an evaluation value concerning a distance between a corresponding second expression and each of the extracted points; and identifying a second expression whose evaluation value is minimum.

Abstract: A design support apparatus includes: a logical expression substitution unit to substitute a part of the logical expression, which includes a function expression of the design variables and a quantifier attached to the design variable, with a substitution variable; a quantifier elimination unit to generate a relational expression including the substitution variable and design variables without the quantifier by eliminating the design variable to which the quantifier is attached from the logical expression; a sampling point generation unit to generate a plurality of sampling points corresponding to the design variables and the substitution variable included in the relational expression; a possible range computation unit to compute, for each of the sampling points, a possible range that the relational expression may take, by calculating values of remaining design variables included in the relational expression based on the relational expression; and a possible range display unit to display the possible range.

Abstract: A first model expression having a first order and representing a relationship between evaluation indicators and the design parameters and a second model expression having a second order higher than the first order and representing a relationship between them are generated. Then, according to a quantifier elimination method, values of the design parameters, which realize an optimum solution of the first model expression, are calculated. And, a design parameter whose value is identical to an upper limit value or a lower limit value is identified, and the second model expression is transformed by substituting the upper limit value or the lower limit value, as a value of the identified design parameter, for the second model expression. Finally, according to the quantifier elimination method, values of the design parameters in the transformed second model expression, which realize an optimum solution of the transformed second model expression, is calculated.

Abstract: An apparatus receives input of sample sets, each including a set of values of design parameters and a set of values of objective functions; calculates objective function approximating equations; and selects, as initial candidates for an optimal design parameter set, some sets of values of design parameters corresponding to non-dominated solutions. The apparatus calculates one or more interpolating design parameter sets interpolating between two adjacent components in the candidates; and approximates values of the objective functions for each interpolating design parameter set. The apparatus selects an optimal interpolating design parameter set corresponding to a non-dominated solution in the cost evaluation for a pair of objective functions; and integrates it into the candidates. The apparatus repeats processes on the new candidates while determining the parameter distance between components of the new candidates.

Abstract: The disclosed method includes: generating a problem for a quantifier elimination (QE) method from a cost function representing a relationship between a parameter set and a cost; executing, by a module that executes a processing for the QE method by term substitution, a processing for the problem, to obtain a first processing result; when the first processing result includes a first partial problem for which the term substitution is impossible, executing, by a module that execute a numerical analysis processing, a processing to minimize the cost for the first partial problem, to obtain a second processing result; when the first processing result includes a logical expression as a solution of a second partial problem for which the term substitution has been completed, generating data to identify a minimum cost value from the logical expression; and generating a minimum cost for the problem from at least the second processing result.

Abstract: An embodiment relates to a multi-objective optimization design supporting technique to reduces the computational complexity of QE/CAD. When the input logical expression generated by a logical-expression-with-qualifier generation unit is satisfied in regard to the sample point included in a certain piece of cell information for each value of the same design parameter, a first cell processing unit does not evaluate the input logical expression on the cell information including other sample points having a value equal to or smaller than the value of a predetermined design variable (for example, a design variable indicating a yield) corresponding to the sample point above, but selects it as the cell information for an output of a logical expression without a qualifier.

Abstract: An objective function can be mathematically approximated using a prescribed number of sample sets of design parameters and sets of a plurality of objective functions computed corresponding to them. A logical expression indicating a relation between or among arbitrary two or three objective functions of the plurality of mathematically approximated objective functions is computed as an inter-objective-function logical expression and a region that the arbitrary objective function values can take is displayed as a feasible region in an objective space corresponding to the arbitrary objective functions. Furthermore, a point or area in a design space corresponding to arbitrary design parameters corresponding to a point or area specified by a user in the displayed feasible region is displayed.

Abstract: The disclosed method includes: generating, from model expressions, each representing a relationship between input parameters and output evaluation indicators and a range of residuals for each model expression, at least one of a first problem to calculate a region that can be feasible with the model expressions, value ranges of the input parameters and the range of the residuals for each model expression and a second problem to calculate a region that is always feasible with the model expressions, the value ranges of the input parameters and the influence of the aforementioned range; calculating a feasible region(s) for a generated problem(s) to obtain data of the feasible region(s); and generating visualization data of the feasible region(s) to output the visualization data.

Abstract: In this disclosure, equations to be solved in the model predictive control are transformed by using an off-line algebraic simplification method into a matrix operational expression representing a product of a coefficient matrix and a vector regarding solution inputs within a control horizon is equal to a function vector regarding target values of output states and the output states. The size of the coefficient matrix is reduced compared with the conventional matrix. Then, the matrix operational expression is solved in an online plant control apparatus with present output states and present target values of the output stats of a plant to be controlled, by the direct method, to output the solution to the plant.

Abstract: A method for solving an optimization problem includes: causing a cylindrical algebraic decomposition (CAD) processing unit to carry out a projection processing for a first expression that appears in a quantifier elimination problem equivalent to the optimization problem including objective functions and to generate second expressions that are projection factors of the first expression; calculating plural sets of values of the objective functions by generating plural value sets of variables in the objective functions and substituting the generated plural value sets of the variables into the objective functions; extracting points including non-dominated solutions in a space mapped by the objective functions, from plural points corresponding to the plural sets of values; calculating, for each second expression, an evaluation value concerning a distance between a corresponding second expression and each of the extracted points; and identifying a second expression whose evaluation value is minimum.

Abstract: This method includes: generating a constraint equation from data of an approximate expression of a cost function representing a relationship between a plurality of design parameters and a cost, data of a route in a cost space and data of a search range in a design parameter space; obtaining a logical expression of a solution for the constraint equation from a quantifier elimination processing unit that carries out a processing according to a quantifier elimination method; substituting coordinates of each of a plurality of points within the search range in the design parameter space into the logical expression of the solution to determine, for each of the plurality of points, true or false of the logical expression of the solution; and displaying the design parameter space in which a display object including a first point for which true is determined is disposed at the first point.

Abstract: When model expressions of objective functions are generated at vertexes of a quadrilateral on a plane concerning P and N channels of transistors in SRAM, the initial number of times of simulation is allocated to each objective function at each designated vertex according to weight values set based on relationships presumed among the objective functions at each designated vertex. For each objective function at each designated vertex, first simulation is executed the allocated number of times. Furthermore, a model expression is generated from the first simulation result, and an evaluation indicator of an approximation accuracy of the model expression is calculated. Then, for each model expression, it is determined whether the corresponding model expression has influence on the yield, and based on the evaluation indicator of the corresponding model expression and presence or absence of the influence, it is determined whether additional simulation is required for the corresponding objective function.

Abstract: A shape optimization method includes: obtaining parameter data including data concerning a first relational expression that causes coordinate values of plural vertexes in at least a portion of an object to be changed together and includes a parameter capable of setting values from outside; determining, according to a predetermined algorithm, a value of the parameter in the parameter data; calculating coordinates values of the plural vertexes from a second relational expression determined by the first relational expression and the determined value of the parameter; generating shape data including coordinate values of first vertexes to define the shape of the object from initial coordinate values and the calculated coordinate values; causing to execute cost calculation of the shape defined by the shape data; and outputting shape data in case of a best result of the cost calculation after repeating the aforementioned processing based on a result of the cost calculation.