Patent Publication Number: US-2004059549-A1

Title: Method and apparatus for evaluating a proposed solution to a constraint problem

Description:
BACKGROUND OF THE INVENTION  
       [0001] Most problems encountered in engineering design are nonlinear by nature and involve the determination of system parameters that satisfy certain goals for the problem being solved. Such problems can be cast in the form of a mathematical optimization problem where a solution is desired that minimizes a system function or parameter subject to limitations or constraints on the system. Both the system function and constraints are comprised of system inputs (control variables) and system outputs, which may be either discrete or continuous. Furthermore, constraints may be equalities or inequalities. The solution to a given optimization problem has either or both of the following characteristics: 1) minimizes or maximizes a desired condition or conditions, thus satisfying the optimality condition and 2) satisfies the set of constraint equations imposed on the system.  
       [0002] With the above definitions, several categories of optimization problems may be defined. A Free Optimization Problem (FOP) is one for which no constraints exist. A Constraint Optimization Problem (COP) includes both constraints and a minimize (or maximize) condition(s) requirement. In contrast, a Constraint Satisfaction Problem (CSP) contains only constraints. Solving a CSP means finding one feasible solution within the search space that satisfies the constraint conditions. Solving a COP means finding a solution that is both feasible and optimal in the sense that a minimum (or maximum) value for the desired condition(s) is realized.  
       [0003] The solution to such a problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, results in improvement of an objective function. An objective function is a mathematical expression having parameter values of a proposed solution as inputs. The objective function produces a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another. Numerous search algorithms exist and differ in the manner by which the control variables for a particular problem are modified, whether a population of solutions or a single solution is tracked during the improvement process, and the assessment of convergence. However, these search algorithms rely on the results of an objective function in deciding a path of convergence. Examples of optimization algorithms include Genetic Algorithms, Simulated Annealing, and Tabu Search.  
       [0004] Within optimization algorithms, the critical issue of handling constraints for COPs and CSPs must be addressed. Several classes of methods exist for dealing with constraints. The most widespread method is the use of the penalty approach for modifying the objective function, which has the effect of converting a COP or CSP into a FOP. In this method, a penalty function, representing violations in the set of constraint equations, is added to an objective function characterizing the desired optimal condition. When the penalty function is positive, the solution is infeasible. When the penalty function is zero, all constraints are satisfied. Minimizing the modified objective function thus seeks not only optimality but also satisfaction of the constraints.  
       [0005] Objective functions take application specific forms, and therefore, each new problem or modification to a problem requires the construction of a new objective function. Furthermore, the objective function plays the important role of guiding an optimization algorithm to a possible best solution. Presumably, the better the objective function, the better the optimization result and/or the more efficient the optimization operation. Accordingly, a constant demand exists in the field of constraint based problems for improved objective functions.  
       SUMMARY OF THE INVENTION  
       [0006] The invention provides a systematic and general method and apparatus for defining an objective function for Constrained Optimization Problems (COPs), Constraint Satisfaction Problems (CSPs) and Free Optimization Problems (FOP), independent of the optimization search employed. The invention provides a generic definition of an objective function. Given the particular optimization problem (e.g., boiler water nuclear reactor core design, transportation scheduling, pressure water reactor core design, or any large scale, combinatorial optimization problem in discrete or continuous space), the objective function is configured following the generic definition.  
       [0007] Specifically, the generic definition of the objective function according to the present invention is a sum of credit components plus a sum of penalty components. Each credit component includes a credit term times an associated credit weight. Each penalty term includes a penalty term times an associated penalty weight. A credit term is a mathematical expression representing an optimization parameter, and a penalty term is a mathematical expression representing an optimization constraint.  
       [0008] Configuring an objective function involves establishing the number of credit and penalty components, establishing the mathematical expressions for the credit and penalty terms and establishing the initial weights of the credit and penalty weights. This is accomplished through user input or by accessing a previously stored configured objective function.  
       [0009] The configured objective function is then usable as part of an optimization process, or is usable a tool when a user assesses a candidate solution to an optimization problem. Because of the flexibility of the invention changes in optimality conditions, constraint term definitions, and weight factors are readily accommodated. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0010] The present invention will become more fully understood from the detailed description given herein below and the accompanying drawings, wherein like elements are represented by like reference numerals, which are given by way of illustration only and thus are not limiting on the present invention and wherein:  
     [0011]FIG. 1 illustrates an embodiment of an architecture according to the present invention for implementing the method of evaluating a proposed solution according to the present invention;  
     [0012]FIG. 2 illustrates a screen shot of a optimization configuration page used in selecting one or more optimization parameters associated with the optimization problem of boiler water reactor core design according to an embodiment of the present invention;  
     [0013]FIG. 3 illustrates a screen shot of a optimization constraints page listing optimization constraints associated with the optimization problem of boiler water reactor core design according to an embodiment of the present invention; and  
     [0014]FIG. 4 illustrates a flow chart of an optimization process employing the objective function of the present invention.  
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS  
     [0015] The Generic Objective Function  
     [0016] The present invention provides a generic definition of an objective function, which is applicable across a wide variety of constraint and optimization problems. Namely, the generic objective function is applicable to any large scale, combinatorial optimization problem in discrete or continuous space such as boiler water reactor core design, pressurized water reactor core design, transportation scheduling, resource allocation, etc. The generic objective function is defined as a sum of credit and penalty components. A penalty component includes a penalty term multiplied by an associated penalty weight. A credit component includes a credit term multiplied by an associated credit weight. The credit terms represent the optimality conditions for the problem. The penalty terms represent the constraints for the problem. Each credit term is a mathematical expression that quantifies an optimality condition. Each penalty term is a mathematical expression that quantifies a constraint. Mathematically, this can be expressed as follows:  
         F   obj     =         ∑   m            λ   m   credit          C   m         +       ∑   n            λ   n   penalty          P   n                         
 
     [0017] where,  
     [0018] F obj =objective function  
     [0019] C m =credit term m  
     [0020] P n =penalty term n  
     [0021]         λ   m   credit     =     weight  factor  credit  term     m               λ   m   penalty     =     weight  factor  penalty  term     n                     
     [0022] Credit and penalty terms may be defined by maximum (i.e. upper bounded) or minimum (i.e. lower bounded) values and can represent scalar or multi-dimensional values. The only requirements are: 1) the penalty terms must be positive for constraint violations and zero otherwise, and 2) in the absence of constraint violations, the credit terms are consistent with a minimization problem. Thus, minimizing the modified objective function solves the optimization problem.  
     [0023] As an example, consider an air-conditioning system where the optimization problem is to minimize the average air temperature within a room, yet assure that no region within the room exceeds a certain temperature. For this example, the credit would be the average air temperature within the room volume. The constraint would be a limit on the point-wise temperature distribution within the room, which, in the form of a penalty term, would be calculated as the average temperature violation. To obtain the average temperature violation one would sum the differences of actual and limiting temperature values for those points within the room that violate and divide by the total number of points. Alternatively, one could calculate the penalty term as the maximum value of the point-wise temperature violations within the room. The form of the generic objective function thus allows any number of credit and penalty terms to be defined in a general manner for the problem being solved.  
     [0024] Forms for the credit or penalty terms include, but are not limited to:  
     [0025] The maximum value within a data array;  
     [0026] The minimum value within a data array;  
     [0027] The average of values within a data array;  
     [0028] The integral of values within a data array;  
     [0029] The maximum of calculated differences between elements of a data array and the corresponding constraint limit, restricted to elements that violate;  
     [0030] The minimum of calculated differences between elements of a data array and the corresponding constraint limit, restricted to elements that violate;  
     [0031] The average of calculated differences between elements of a data array and the corresponding constraint limit, restricted to elements that violate; and  
     [0032] The integral of calculated differences between elements of a data array and the corresponding constraint limit, restricted to elements that violate.  
     [0033] A Generic Architecture for Implementing the Objective Function  
     [0034]FIG. 1 illustrates an embodiment of an architecture according to the present invention for implementing the method of evaluating a proposed solution according to the present invention. As shown, a server  10  includes a graphical user interface  12  connected to a processor  14 . The processor  14  is connected to a memory  16 . The server  10  is directly accessible by a user input device  18  (e.g., a display, keyboard and mouse). The server  10  is also accessible by computers  22  and  26  over an intranet  20  and the Internet  26 , respectively. The operation of the architecture shown in FIG. 1 will be discussed in detail below.  
     [0035] Configuring the Objective Function for an Optimization Problem  
     [0036] According to one embodiment, a configured objective function satisfying the above-described generic definition is already stored in the memory  16  of the server  10 . For example, the configured objective function could have been configured according to one of the embodiments described below. In this embodiment, the user instructs the server  10  to provide a list of the configured objective functions stored in the memory  16 , and instructs the server  10  to use one of the listed configured objective functions.  
     [0037] In another embodiment, a user via input  18 , computer  26  or computer  22  accesses the server  10  over the graphical user interface  12 . The user supplies the server  10  with a configured objective function meeting the definition of the above-described generic definition. In this embodiment, the user supplies the configured objective function using any well-known programming language or program for expressing mathematical expressions. Specifically, the user instructs the processor  14  via the graphical user interface  12  to upload a file containing the configured objective function. The processor  14  then uploads the file, and stores the file in memory  16 .  
     [0038] In still another embodiment, configuring the objective function is interactive between the user and the server  10 . Here, the user instructs the processor  14  to start the process for configuring an objective function. The processor  14  then requests the user to identify the number of credit components and the number of penalty components. For each credit component, the processor  14  requests that the user provide a mathematical expression for the credit term and an initial weight for the associated credit weight. For each penalty component, the processor  14  requests that the user provide a mathematical expression for the penalty term and an initial weight for the associated penalty weight. In supplying the mathematical expression, the processor  14  via the graphical user interface  12  accepts definitions of mathematical expressions according to any well-known programming language or program.  
     [0039] In another embodiment, the server  10  is preprogrammed for use on a particular constraint or optimization based problem. In this embodiment, the server  10  stores possible optimization parameters and possible constraint parameters associated with the particular optimization or constraint problem. When a user instructs the processor  14  via the graphical user interface  12  to configure an objective function, the processor  14  accesses the possible optimization parameters already stored in the memory  16 , and provides the user with the option of selecting one or more of the optimization parameters for optimization.  
     [0040]FIG. 2 illustrates a screen shot of a optimization configuration page used in selecting one or more optimization parameters associated with the optimization problem of boiler water reactor core design according to this embodiment of the present invention. As shown, the optimization parameters  40  of optimize rod patterns, optimize core flow, and optimize sequence intervals are available for selection by the user as optimization parameters. Optimize rod patterns means making an optimal determination of individual control rod positions within a control rod grouping (called a sequence), for the duration of time during the operating cycle when the given sequence is being used to control the reactor. Rod positions affect the local power as well as the nuclear reaction rate. Optimize core flow means making an optimal determination of reactor coolant flow rate through the reactor as a function of time during the operating cycle. Flow rate affects global reactor power as well as the nuclear reaction rate. Optimize sequence intervals means making an optimal determination of the time duration a given sequence (i.e., control rod grouping) is used to control the reactor during the operating cycle. Sequence intervals affect local power as well as the nuclear reaction rate.  
     [0041] Using the data input device  18 , computer  22  or computer  26 , each of which includes a display and a computer mouse, the user selects one or more of the optimization parameters by clicking in the selection box  42  associated with an optimization parameter  40 . When selected, a check appears in the selection box  42  of the selected optimization parameter. Clicking in the selection box  42  again de-selects the optimization parameter.  
     [0042] The memory  16  also stores constraint parameters associated with the optimization problem. The constraint parameters are parameters of the optimization problem that must or should satisfy a constraint or constraints. FIG. 3 illustrates a screen shot of a optimization constraints page listing optimization constraints associated with the optimization problem of boiler water reactor core design according to this embodiment of the present invention. As shown, each optimization constraint  50  has a design value  52  associated therewith. Each optimization constraint must fall below the specified design value. The user has the ability to select optimization parameters for consideration in configuring the objective function. Using the data input device  18 , computer  22  or computer  26 , each of which includes a display and a computer mouse, the user selects an optimization constraint by clicking in the selection box  54  associated with an optimization constraint  50 . When selected, a check appears in the selection box  54  of the selected optimization constraint  50 . Clicking in the selection box  54  again de-selects the optimization constraint.  
     [0043] Each optimization parameter has a predetermined credit term and credit weight associated therewith stored in memory  16 . Similarly, each optimization constraint has a predetermined penalty term and penalty weight associated therewith stored in memory  16 . In the embodiment shown in FIG. 3, the penalty term incorporates the design value, and the user can change (i.e., configure) this value as desired. Additionally, the embodiment of FIG. 3 allows the user to set an importance  56  for each optimization constraint  50 . In the importance field  58  for an optimization constraint, the user has pull down options of minute, low, nominal, high and extreme. Each option correlates to an empirically predetermined penalty weight such that the greater the importance, the greater the predetermined penalty weight. In this manner, the user selects from among a set of predetermined penalty weights.  
     [0044] Once the above-selections have been completed the processor  14  configures the objective function according to the generic definition discussed above and the selections made during the selection process. The resulting configured objective function equals the sum of credit components associated with the selected optimization parameters plus the sum of penalty components associated with the selected optimization constraints.  
     [0045] Additionally, the embodiment provides for the user to select a method of handling the credit and penalty weights. For example, the user is supplied with the possible methodologies of static, death penalty, dynamic, and adaptive for the penalty weights; is supplied with the possible methodologies of static, dynamic and adaptive for the credit weights; and the methodology of relative adaptive for both the penalty and credit weights. The well-known static methodology maintains the weights at their initially set values. The well-known death methodology sets each penalty weight to infinity. The well-known dynamic methodology adjusts the initial weight value during the course of the objective function&#39;s use in an optimization search based on a mathematical expression that determines the amount and/or frequency of the weight change. The well-known adaptive methodology is also applied during the course of an optimization search. In this method, penalty weight values are adjusted periodically for each constraint parameter that violates the design value. The relative adaptive methodology is disclosed in U.S. Application No. unknown, titled METHOD AND APPARATUS FOR ADAPTIVELY DETERMINING WEIGHT FACTORS WITHIN THE CONTEXT OF AN OBJECTIVE FUNCTION, by the inventors of the subject application, concurrently filed herewith.  
     [0046] Optimization Using the Objective Function  
     [0047]FIG. 4 illustrates a flow chart showing one of the many uses for the function of the present invention. Specifically, FIG. 4 illustrates a flow chart of an optimization process employing the objective function of the present invention. For the purposes of explanation only, the optimization process of FIG. 4 will be described as being implemented by the architecture illustrated in FIG. 1. As shown, in step S 10  the objective function is configured as discussed above in the preceding section, then the optimization process begins. In step S 12 , the processor  14  retrieves from memory  16  or generates one or more sets of values for input parameters (i.e., system inputs) of the optimization problem based on the optimization algorithm in use. For example, for the optimization problem of boiler water reactor core design, some of the input parameters would be placement of fresh and exposed fuel bundles within the reactor, selection of the rod groups (sequences) and placement of the control rod positions within the groups as a function of time during the cycle, core flow as a function of time during a cycle, reactor coolant inlet pressure, etc. Each input parameter set of values is a candidate solution of the optimization problem. The processor  14  runs a simulated operation and generates a simulation result for each input parameter set of values. For example, for boiler water reactor core design, a well-known simulation program for boiler water reactor operation is run using an input parameter set. The simulation result includes values (i.e., system outputs) for the optimization parameters and optimization constraints. These values, or a subset of these values, are values of the variables in the mathematical expressions of the objective function.  
     [0048] Then, in step S 14 , the processor  14  uses the objective function and the system outputs to generate an objective function value for each candidate solution. In step S 16 , the processor  14  assesses whether the optimization process has converged upon a solution using the objective function values generated in step S 14 . If no convergence is reached, then in step S 18 , the input parameter sets are modified, the optimization iteration count is increased and processing returns to step S 12 . The generation, convergence assessment and modification operations of steps S 12 , S 16  and S 18  are performed according to any well-known optimization algorithm such as Genetic Algorithms, Simulated Annealing, and Tabu Search. When the optimization problem is boiler water reactor core design, the optimization algorithm can be, for example, one of the optimization processes as disclosed in U.S. application Ser. No. 09/475,309, titled SYSTEM AND METHOD FOR OPTIMIZATION OF MULTIPLE OPERATIONAL CONTROL VARIABLES FOR A NUCLEAR REACTOR filed Dec. 30, 1999 or U.S. application Ser. No. 09/683,004, tilted SYSTEM AND METHOD FOR CONTINUOUS OPTIMIZATION OF CONTROL-VARIABLES DURING OPERATION OF A NUCLEAR REACTOR, filed Nov. 7, 2001.  
     [0049] The Objective Function as a Tool  
     [0050] Before the advent of fairly successful optimization tools, generating viable solutions to an optimization problem fell on the shoulders of highly experienced individuals, who through years of practice and experience in a particular field, developed a set of skills for generating possible solutions to the optimization problem. Even today such practices continue. However, these individuals still need a reliable method for assessing their solutions. The objective function of the present invention provides such a tool.  
     [0051] Referring again to FIG. 1 for the purposes of explanation only, an individual desiring to apply an objective function according to the present invention accesses the server  10  via input  18 , computer  26  or computer  22 . The user then configures an objective function as described above; for example, the user accesses a previously stored configured objective function or configures the objective function through data entry. The user then supplies the values for the input variables of the objective function, and receives an objective function result. The user an perform this operation for different candidate solutions to obtain figures of merit regarding the solutions. In this manner, the user treats the objective function as a tool in determining a solution to the optimization problem.  
     [0052] The invention provides a systematic and general method for defining an objective function for Constrained Optimization Problems (COPs), Constraint Satisfaction Problems (CSPs) and Free Optimization Problems (FOP), independent of the optimization search employed. The invention provides a mechanism within the context of the penalty function approach for addressing: 1) any number of constraints and optimality conditions, 2) different mathematical forms for the credit and penalty terms and 3) any number of dimensions in the candidate solution data and constraints. The invention provides a prescription for encoding objective function definitions within an optimization search through a software data module. Because of the flexibility of the invention, changes in optimality conditions, constraint term definitions, and weight factors are readily accommodated within the calling program simply by changing the data passed to the software data module.  
     [0053] The technical effort of the invention is a computer system that provides for configuring an objective function for a given optimization problem; a computer system that generates a possible solution for a particular optimization problem; and a computer system that allows internal and external users to measure the performance of their possible solutions to an optimization problem.  
     [0054] The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.