fig1 illustrates a schematic diagram of a portion 100 of a simple process map showing substantially sequential stages ; “ stage 1 ” 102 , “ stage 2 ” 104 , “ stage 3 a ” 107 , “ stage 3 b ” 108 and “ stage 4 ” 111 of a process , and the inputs and outputs that connect them — starting with the initial measured input to the process 101 and sequentially : measured output 103 from “ stage 1 ” which is measured input to “ stage 2 ”, measured output 105 from “ stage 2 ” which is measured input to “ stage 3 a ”, measured output 106 from “ stage 2 ” which is measured input to “ stage 3 b ”, measured output from “ stage 3 a ” 109 which is measured input to “ stage 4 ”, measured output from “ stage 3 b ” 110 which is measured input to “ stage 4 ”, and measured output 112 from “ stage 4 ”. referring to fig1 , depicted is an example simple process map . the boxes in the diagram represent sequential stages in a portion of a typical process , and the arrows indicate the direction in which output from one stage flows as input to the next stage . often , this input or output is measured for purposes of process control . simply stated , process control generally relates to determining the optimal values for control parameters at a stage in a process to improve quality or quantity of yield at that stage in the process . stages 3 a and 3 b represent parallel stages , which can run simultaneously or in an alternating manner . for example , a process would utilize such stages when an operation carried out at stage 3 is slower in relation to actions carried out at other stages in the process . when a stage in a process is slower in relation to the rest of the process , it is advantageous to break down the slower stage into parallel stages as seen in fig1 ; to speed up process time at the that stage . another example of when parallel stages are used would be for one process that produces two types of output . such a process will elect which of the different operations will be carried out at the “ parallel stage ”. fig2 illustrates a schematic diagram of a process stage 200 , i . e . any single stage in a process ( such as those shown in fig1 ) represented here as “ stage x ” 203 . there is measured input from “ stage x − 1 ” 201 and measured output to “ stage x + 1 ” 205 . also included in the process are control parameters measured by actuators at “ stage x ” 202 and monitored parameters from sensors at “ stage x ” 204 . referring to fig2 , depicted is a typical stage of the process represented in fig1 , referred to in fig2 as stage x . to the left of stage x are shown control parameters for actuators in the process conducted in stage x ; to the right of stage x are shown sensors for monitoring parameters of the input to or output from stage x , or the process conducted during stage x . actuators perform actions in the context of a specific process step . sensors perform measurements relevant to the process . the data coming from sensors and control parameter data sent to actuators constitute fundamental dynamic aspects that are required for the purposes of process control . fig2 a illustrates a data set table 300 for a given stage in a process . there are four families of data represented for any given number of items or batches 305 produced in the process : measured inputs from previous stage 301 , outputs 302 , control parameters 303 , and monitored parameters 304 . “ arrow a ” 306 represents a data set for a process output o 2 at the given stage and “ arrow b ” 307 represents a data set for a control parameter cp c at the given stage . substantially associated with every set of control parameters , monitored parameters , input , and output at any given stage in a process are data sets , as illustrated in fig2 a . the control parameters , monitored parameters , input , and output at any given stage represent four families of data sets . within the control parameters , monitored parameters , and output families , there can be from 1 to any number of data sets . within the input family , there can be from 0 to any number of data sets . in fig2 a , the input family has a data sets ( where a is any integer greater than 0 ), the output family has b data sets ( where b is any integer greater than 1 ), the control parameters family has c data sets ( where c is any integer greater than 1 ), and the monitored parameters family has d data sets ( where d is any whole number greater than 1 ). a data set for the input or output families consists of quantities or measurements of that given input or output , for a given number of items or batches produced in a process . for example , arrow a in fig2 a represents a data set for o 2 , where o 2 is the length of an item that is output from the given process stage , and this data is recorded for 50 times that the item is produced on a given day , where e is 50 . a typical data set for the control parameters family consists of data for a given parameter setting for an operation occurring at the given process stage , for a given number of items or batches produced in a process . for example , arrow b in fig2 a represents a data set for cp c , where cp c is the pressure setting of an item produced at the given process stage , and this data is recorded for 50 times that the item is produced on a given day , where e is 50 . a data set for the monitored parameters family consists of data for a given parameter monitored at the given process stage , for a given number of items or batches produced in a process . fig3 ( generally referenced as ) 400 illustrates a schematic diagram of the process map from fig1 and now added to each stage in the process are the control parameters and monitored parameters that were depicted in stage x of fig2 . “ stage 1 ” 402 has control parameters measured by actuators 401 , monitored parameters from sensors 403 , initial measured input to the process 416 , and measured output 417 which is measured input to “ stage 2 ” 405 ; “ stage 2 ” 405 has control parameters measured by actuators 404 , monitored parameters from sensors 406 , and measured output 418 which is measured input to “ stage 3 a ” 408 ; “ stage 3 a ” 408 has control parameters measured by actuators 407 , monitored parameters from sensors 409 , and measured output 420 which is measured input to “ stage 4 ” 414 ; “ stage 3 b ” 411 has control parameters measured by actuators 410 , monitored parameters from sensors 412 , measured output 421 which is measured input to “ stage 4 ”; and “ stage 4 ” 414 has control parameters measured by actuators 413 , and monitored parameters from sensors 415 , and measured output 422 . referring to fig3 , depicted is the process map from fig1 , and now added to each stage in the process are the control parameters and monitored parameters that were depicted in stage x of fig2 . fig4 illustrates a schematic diagram of interrelationships and outside influences for stages in a process 500 . “ stage 1 ” 501 has an interrelationship 506 with “ stage 3 a ” 503 and an interrelationship 507 with “ stage 3 b ” 504 ; “ stage 2 ” 502 has an interrelationship 508 with “ stage 3 b ” 504 and an interrelationship 509 with 509 with “ stage 4 ” 505 . there is an outside influence 510 on “ stage 3 a ” 503 . referring to fig4 , depicted is an interrelationship and outside influences map for the stages in the process map of fig1 . an interrelationship between two stages exists when there is alleged or validated information that when a particular control parameter or parameters are set at an earlier stage x , a certain resulting output is received at a later stage x + n ( where n is any integer greater than 0 ). in fig4 , interrelationships exist between stage 1 and stages 3 a and 3 b , between stage 2 and stage 3 b , and between stage 2 and stage 4 . an outside influence exists when there is alleged or validated information that some factor outside of a process influences output at a given stage in the process . in fig4 , we see an outside influence exists on stage 3 a . fig4 a ( generally referenced as ) 600 presents two data set tables 601 and 604 for two types of outside influences . the first data set , represented by “ arrow a ” 607 , consists of data for any type of metric or discrete parameter 603 monitored in a time - dependent manner 602 . the second data set , represented by “ arrow b ” 608 , consists of data for any type of metric or discrete parameter 606 monitored according to another metric scale 605 . referring to fig4 a , like the relationships described in fig2 , outside influences also have data sets . data sets for outside influences can be one of two types . the first type is data for any type of metric or discrete parameter monitored in a time - dependent manner ( arrow a ), when the metric or discrete parameter is either suggested to affect or confirmed to affect output at a given stage . the second type is data for any type of metric or discrete parameter monitored according to another metric scale ( arrow b ), when the first metric or discrete parameter is either suggested to affect or confirmed to affect output at a given stage . fig5 ( generally referenced as ) 700 illustrates a schematic diagram of the process map from fig3 with the interrelationship and outside influences map of fig4 . “ stage 1 ” 702 has initial measured input to the process 721 , control parameters measured by actuators 701 , monitored parameters from sensors 703 , an interrelationship 717 with “ stage 3 a ” 708 , and an interrelationship 718 with “ stage 3 b ” 712 , and measured output 722 which is measured input to “ stage 2 ” 705 . “ stage 2 ” 705 which has control parameters measured by actuators at “ stage 2 ” 704 , and monitored parameters from sensors at “ stage 2 ” 706 , an interrelationship 719 with “ stage 3 b ” 712 , an interrelationship 720 with “ stage 4 ” 715 , measured output 723 which is measured input to “ stage 3 a ” 708 , and measured output 724 which is measured input to “ stage 3 b ” 712 ., “ stage 3 a ” 708 which has control parameters measured by actuators at “ stage 3 a ” 707 , monitored parameters from sensors at “ stage 3 a ” 709 , and an outside influence 710 , and measured output 725 which is measured input to “ stage 4 ” 715 . “ stage 3 b ” 712 which has control parameters measured by actuators at “ stage 3 b ” 711 , and monitored parameters from sensors at “ stage 3 b ” 713 , measured output 726 which is measured input to “ stage 4 ” 715 ; and “ stage 4 ” 715 with has control parameters measured by actuators at “ stage 4 ” 714 , and monitored parameters from sensors at “ stage 4 ” 716 , and measured output 727 . referring to fig5 , depicted is the process map of fig3 merged with the interrelationship and outside influences map of fig4 . the different relationships between and at stages in fig5 illustrate various factors that may be involved in process control . fig6 ( generally referenced as ) 800 illustrates a schematic diagram of a process stage “ stage x ” 804 with all the various process control relationships that “ stage x ” participates in . these relationships include : measured input from “ stage x − 1 ” 803 , measured output to “ stage x + 1 ” 807 , an outside influence 801 , control parameters measured by actuators at “ stage x ” 805 , monitored parameters from sensors at “ stage x ” 806 , an interrelationship with a process stage before “ x ” 802 , and an interrelationship with a process stage after “ x ” 808 . referring to fig6 , depicted is one stage in a given process , stage x , with all of the various relevant process relationships that have been depicted in fig1 - 5 . stages prior to stage x are hereinafter referred to as stages x − 1 , x − 2 , . . . , x − n , etc . stages following stage x are hereinafter referred to as stages x + 1 , x + 2 , . . . , x + n , etc . stages which may be substantially simultaneous to stage x can be arbitrarily assigned the nomenclature x − 1 or x + 1 . fig6 illustrates various process control relationships in which stage x participates . fig7 ( generally referenced as ) 900 illustrates a schematic diagram of “ stage x ” 904 with process control relationships that are relevant to applying process control only at “ stage x ”. these relationships include : measured input from “ stage x − 1 ” 903 , measured output to “ stage x + 1 ” 907 , an outside influence 901 , control parameters measured by actuators at “ stage x ” 905 , monitored parameters from sensors at “ stage x ” 906 , an interrelationship with a process stage before “ x ” 902 , and an interrelationship with a process stage after “ x ” 908 . fig8 ( generally referenced as ) 1000 illustrates a schematic diagram of a process control algorithm standard frame of reference for stage “ x ” in a process 1000 . the relevant factors herein are : control parameter ( s ) 1002 , input from “ stage x − 1 ” 1003 , interrelationship ( s ) with stage ( s ) before “ x ” 1004 , interrelationship ( s ) with stage ( s ) after “ x ” 1005 , outside influence ( s ) 1006 , with the resultant output to “ stage x + 1 ” 1007 , and monitored parameters from sensors 1008 . referring to fig7 , fig7 depicts stage x with relationships relevant to applying process control only at stage x . referring to fig8 , fig8 takes the relationships depicted in fig7 and presents them from the perspective of how to implement an algorithm for standard process control at stage x , and what factors must be considered when implementing that algorithm . the following figures more specifically relate to benefits of the present invention above those that can be achieved from the vantage of the prior art . for example , it will be appreciated , from the forthcoming figures and description , that the method of the present invention relates to process control for a juncture in a process wherein there are no a priori notions regarding predominant influences to the process at that juncture . furthermore , that process control recipes can be found , calculated or chosen using decision criteria that could not be directly articulated using prior art methods . more specifically , the present invention leads to the development of a well - defined search - space or ensemble of process control vectors whereby multiple goal optimizations may be realized . fig9 ( generally referenced as ) 2000 illustrates the relative importance according to classical thinking , per se , of relationships relevant to applying process control at stage x 2002 . these relationships include : control parameter ( s ) 2011 which are “ internal ” 2014 ; input from “ stage x − 1 ” 2010 which is “ near ” 2013 ; interrelationship ( s ) with stage ( s ) before “ x ” 2007 , interrelationship ( s ) with stage ( s ) after “ x ” 2005 , and outside influence ( s ) 2001 which are all “ far ” 2012 ; monitored parameters 2004 which are “ internal ” 2015 ; and output to “ stage x + 1 ” 2008 which is “ near ” 2016 . polygons “ a ” 2009 , “ b ” 2003 , and “ c ” 2006 ” depict the standard types of relationships that are examined when applying standard process control at stage x . the polygons : “ a ” 2009 , “ b ” 2003 , “ c ” 2006 show types of relationships between input from “ stage x − 1 ”, control parameter ( s ), monitored parameters from sensors 2004 ( of internal 2015 ) and the output to “ stage x + 1 ” 2008 ( of near 2016 ) which are examined when applying process control at “ stage x ”. fig9 illustrates the relative importance according to classical thinking , per se , of relationships relevant to applying process control at stage x . relative importance is classically determined by the proximity of a relationship to stage x . that is , the closer the relationship , the greater significance is attributed to it when implementing process control at stage x . relationships labeled internal ( control parameters ) have the greatest significance , relationships labeled near ( input from stage x − 1 ) have the next greatest significance , and relationships labeled far ( interrelationships with stages before x , interrelationships with stages after x , and outside influences ) have the least significance . within the far relationships , interrelationships with stages before x have the greatest significance , interrelationships with stages after x have the next greatest significance , and outside influences have the least significance . polygons a , b , and c in fig9 depict the standard types of relationships that are examined when applying standard process control at stage x . that is , standard process control at stage x looks for a correlation between the input from stage x − 1 and control parameters at stage x ( polygon a ), a correlation between control parameters at stage x and the monitored parameters at stage x ( polygon b ), and a correlation between monitored parameters at stage x and the output to stage x + 1 ( polygon c ). at this juncture , some additional notes about nomenclature are appropriate . constants relate to measurements that reflect physical aspects measured in a process ; such as weight of an object coming into a process juncture , height of an object coming out of a process juncture , etc . in this sense , constants are constraints that are outside of the scope of control of the process control engineer . therefore , even though these aspects are called constants , on an item - to - item basis they may contribute to describing a broad distribution of values . for example , each egg that arrives at an egg - sorting machine has a specific size and weight that is constant for that egg . nevertheless , one can describe a distribution of weights and sizes for eggs arriving at the sorter . variables relate to aspects of the process at the process juncture that may be elected by command of a process control engineer , or by his proxy in the form of process control software ( such as that of the present invention ) setting process control parameters at the process juncture . fig1 ( generally referenced as ) 3000 illustrates a simple schematic diagram of the various process control input and output relationships of fig8 and 9 in terms of constants and variables , which is the way an embodiment of the present invention addresses process control at “ stage x ” 3001 . an interconnection cell ( according to an embodiment of the present invention ) 3000 at process “ stage x ” 3001 , and its constant / variable relationships . these relationships include : input constant ( s ) 3002 , input variables 3004 , output constants 3005 , and monitored parameters 3003 ; and by calculating a correlation coefficient between it and an output constant ( s ) 3005 . as well , ideally , there should be no discrepancy between input variable ( s ) i . e . control parameters 3004 and their output values i . e . monitored parameter ( s ) 3003 . referring to fig1 , fig1 displays the various process control input and output relationships of fig8 and 9 in terms of constants and variables , which is the way an embodiment of the present invention addresses process control at stage x . in the present invention , there are no a priori notions regarding predominant influences at stage x . the input constants in fig1 consist of interrelationships that stage x has with other stages ( before or after stage x ), outside influences , and input from stage x − 1 . input variables consist of control parameters at stage x . constants and variables can further be understood in terms of data sets . that is , each constant or variable is one of the types of data sets described in fig2 a and 4 a . the influences on stage x numbered 2 - 5 in fig9 are depicted in fig1 as having equal potential significance until empirical weighting proves otherwise . it should be noted that this facilitates a significant departure from the normal means of standard process control , where relative importance is classically determined by the proximity of a relationship to stage x as depicted in fig9 . in the context of the present invention , the process stage and constant / variable relationships depicted in fig1 are referred to as an interconnection cell . at the highest level of resolution and in the preferred conception , the interconnection cell represents a single process stage , hereinafter a tool . an interconnection cell can also represent two ( or more ) contiguous stages in a process , hereinafter a module . further , an interconnection cell can represent a plurality of discontiguous stages , hereinafter a virtual module , or a plurality of contiguous or discontiguous modules . according to an embodiment of the present invention , an algorithm is used to determine validity of input constants by calculating a correlation coefficient between an input constant with an output constant . this algorithm may be any well - known and accepted algorithm for calculating a correlation coefficient between two data sets , or any algorithm that produces a substantially equivalent result . we will use the convention that a high correlation coefficient ( i . e . a number with an absolute value close to 1 on the scale of 0 to 1 ) means that the input constant has a relatively valid relationship with an output constant at stage x and should be considered when implementing process control at stage x for that output constant . likewise , we will use the convention that a low correlation coefficient means that there is relatively no valid relationship between the input constant and the output constant at stage x . it is often desirable in process control to consider the most valid relationships to process stages . the choice of how many relationships and which relationships is partially determined by computational capacity and partially determined by data availability . those input constants that were found to have high correlation coefficients with output constants at stage x , together with the input variables at stage x , serve as an input set combination for carrying out process control optimization at stage x with preferred embodiments of the present invention . this input set combination is the basis for carrying out process control optimization at stage x with the preferred embodiments of the present invention . in addition , according to an embodiment of the present invention , an algorithm is used to determine correlation coefficients for input variables ( i . e . control parameters at stage x ) with monitored parameters at stage x . ideally , the correlation coefficient should have a value of 1 . 0 , meaning that there is no discrepancy between control parameters and their monitored output values . a deteriorating correlation coefficient for an input variable with its corresponding monitored parameter means that intended values for control parameters are not properly received or implemented at stage x and that serious correction is needed . while the many embodiments of the present invention can be used for improving the yield of stage x according to relationships between input variables and monitored parameters , the immediately following discussion will deal with the issue of process control for stage x without addressing the issues of discrepancies between input variables and monitored parameters . it is an object of stage x to achieve a maximum measurable output yield . in order to do this , embodiments of the method of the present invention can be implemented in one of two general manners . the more straightforward embodiment focuses on assigning boundary values to the input constants and variables of the input set combination at stage x to optimize yield for a given output constant at stage x yield . another , more convoluted embodiment first assigns boundary values to output constants , then uses an analysis according to output boundary values in order to assign boundary values to the input set combination at stage x constants . the more convoluted embodiment first considers assigning at least one boundary value to at least one dividing data for data set of output constant data set and thereby forming at least two discrete sub - sets into at least two discrete sub - sets , and then discovering how to assign boundary values to the input set combination at least one input data set by analyzing portions of the input set ( s ) one or more data sets in the input set combination that contributed to each of the discrete sub - sets or to more than one of the discrete subsets . once the contributing portions of the input data sets have been discovered , then the more simple straightforward methods , for assignment of boundary values to input constants and variables , may be applied . a simple example should suffice to disclose this convolution . a data set of an output constant ( for a process juncture ) shows a clear bimodal distribution of measured articles being manufactured at that juncture . accordingly , a boundary value is assigned to the minima between the two apparent modes of the distribution . a data set for an input constant associated with this same juncture shows a relatively uniform “ flat ” distribution of variation . when the two modes of the output set are “ reverse projected ” back from this input set , one discloses three discrete regions . a first region that resulted in the first output mode , a second region that resulted in the second output mode , and a third region that resulted from elements contributed from both modes . accordingly , one resultant idea for assigning boundary values to this input set combination would be according to these three disclosed regions . the following discussion will deal with the more simple straightforward methods for assignment of boundary values to input constants and variables in the input set combination , and with uses thereof . there are also variations of the more straightforward embodiment for optimizing output yield , wherein boundary values are assigned to : 1 ) input constants only , 2 ) input constants and monitored parameters together , 3 ) input constants , input variables , and monitored parameters together 4 ) input variables only , 5 ) monitored parameters only , and 6 ) input variables and monitored parameters together . likewise , there are variations of the more convoluted embodiment , wherein boundary values are assigned to : 1 ) output constants and input variables , 2 ) output constants and monitored parameters , and 3 ) output constants , input variables , and monitored parameters . furthermore , monitored values for some purposes are considered a preliminary output value , for other purposes it is a supplement to input values . however , most often monitored values are used as a corrective feedback to maintain desired input variable values . boundary values can be determined and assigned according to various criteria . more specifically , the selection of boundary values may significantly affect the quality of results achieved by using the method of the present invention . in some circumstances , selection of a boundary value is done subjectively by a process control engineer according to his judgment and experience . other times , boundary values are selected according to specifications inherent to the actual goals of the process . more often , boundary values are set according to measurable characteristics of the population of values in a given data set . fig1 a ( generally referenced as ) 4000 illustrates the assignment of boundary values to three inputs represented by diagrams 10 a . 1 ( generally referenced as ) 4001 , 10 a . 2 ( generally referenced as ) 4009 , and 10 a . 3 ( generally referenced as ) 4014 . diagram 10 a . 2 ( generally referenced as ) 4009 in fig1 a ( generally referenced as ) 4000 illustrates the assignment of one boundary value x 2 4011 to a population of data measurements for an input , where the population of data measurements divides according to a bimodal distribution curve 4010 . a 2 4012 represents the lower boundary value range for this input and b 2 4013 represents the upper boundary value range for this input . a simple example of an input constant with one assigned boundary value is an egg - sorting machine that sort eggs according to egg diameter . in such a case , a boundary value of 2 . 50 cm is set as the minimum diameter for eggs to be cleaned and boxed for the consumer market . an egg with a diameter of less than 2 . 50 cm will be sent to the frozen egg line to be frozen and used for powdered eggs . another example of an input constant or variable with one boundary value is illustrated in diagram 10 a . 2 of fig1 a . the population of data measurements from some process is shown in the figure . this population substantially divides according to an obvious bimodal distribution . line x 2 indicates the minimum value between two modes . this minimum value is a boundary value for the input constant , and a 2 represents the range of input values less than the boundary value represented by line x 2 the range of input values represented by a 2 is referred to as a boundary value range . similarly , b 2 represents the boundary value range of input values greater than or equal to the boundary value represented by line x 2 diagram 10 a . 1 ( generally referenced as ) 4001 in fig1 a ( generally referenced as ) 4000 illustrates the assignment of two boundary values x 1 4003 and y 1 4004 to a population of data measurements for an input , where the population of data measurements divides according to normal distribution curve 4002 . the average of this normal distribution is represented by x 4005 . a 1 4006 represents the lower boundary value range for this input , b 1 4007 represents the in - specification boundary value range for this input , and c 1 4008 represents the upper boundary value range for this input . an example of an input variable with two boundary values could be the percentage of salinity in sardines . consumer engineers determine that a range of 7 %- 9 % salinity in sardines is an acceptable range for consumer tastes . thus , 7 % and 9 % are the assigned boundary values . sardines produced with a salinity percentage that falls within that range are considered acceptable , and percentages below or above that range are respectively unacceptable . another example of an input constant or variable with two boundary values is illustrated in diagram 10 a . 1 of fig1 a . the population of data measurements from some process is shown in the figure . this population substantially divides according to an obvious normal distribution . line x 1 represents a lower boundary value of x − 2σ and line y 1 represents an upper boundary value of x + 2σ . a 1 represents the boundary value range of input values less than x − 2σ , b 1 represents the boundary value range of input values between x − 2σ and x + 2σ inclusively , and c 1 represents the boundary value range of input values greater than x + 2σ . input values within the b 1 range are considered within a specification range , input values within the a 1 range are considered below the specification range , and input values within c 1 range are considered above the specification range . diagram 10 a . 3 ( generally referenced as ) 4014 in fig1 a ( generally referenced as ) 4000 illustrates the assignment of three boundary values x 3 4016 , y 3 4017 , and z 3 4018 to a population of data measurements for an input , where the population of data measurements divides according to a decaying distribution curve 4015 . a 3 4019 represents the lowest boundary value range for this input , b 3 4020 represents the next lowest boundary value range for this input , c 3 4021 represents the second highest boundary value range for this input , and d 3 4022 represents the highest boundary value range for this input . diagram 10 a . 3 in fig1 a depicts an input constant or variable with three boundary values . the population of data measurements from some process is shown in the figure . this population substantially divides according to an obvious decaying distribution . according to one judgment , the boundary values have been determined according to 25 % of the total population in the distribution ( x 3 ), 50 % of the total population in the distribution ( y 3 ), and 75 % of the total population in the distribution ( z 3 ). for example , if 1 , 000 measurements are represented in diagram 10 a . 3 , then a 3 represents the boundary value range of the lowest 250 measurements below a boundary value x 3 , b 3 represents the boundary value range of the next 250 measurements between boundary values x 3 and y 3 , c 3 represents the boundary value range of the next 250 measurements between boundary values y 3 and z 3 , and c 3 represents the boundary value range of the 250 highest measurements above the boundary value z 3 . according to another judgment , the boundary values in diagram 10 a . 3 have been determined according to 25 % of the total of the possible domains of measurements with equidistant segments in the distribution ( x 3 ), 50 % of the total of the possible domains of measurements with equidistant segments in the distribution ( y 3 ), and 75 % of the total of the possible domains of measurements with equidistant segments in the distribution ( z 3 ). for example , the domain allows measurements between 0 % and 100 % mixtures of two materials . therefore , the measurements can never have a value less than 0 % or greater than 100 %. while it is understood that this value has a significant effect on the process , there is no a priori understanding of what the preferred range for this value should be in order to optimize the yield from the process . therefore , the measurement scale is divided into the following boundary value ranges : a 3 representing the range from 0 - 25 %, b 3 representing the range from 25 - 50 %, c 3 representing the range from 50 - 75 %, and d 3 representing the range from 75 - 100 %. it should be clear that the use of boundary values and boundary value ranges are used in the context of the present invention as a means to create discrete values for data with values that fall within a continuous range . however , certain types of input constants and variables by their inherent nature have values that fall within a discrete range . for example , parallel tools in a given process are discrete values and can be considered as input . assume that a given process has three parallel tools , where process input flows into any one of the three tools and the same types of input constants and variables exist at each tool . for the purposes of process control , it has been decided to monitor into which tool process input flows and that the tool location at this parallel stage is to be considered as an input to the process . the three parallel tools are therefore categorized by the discrete values a , b , and c . for the purposes of the present invention , these discrete values are applied in the same manner as boundary value ranges . in the event that there are large numbers of values in a discrete range , then the equivalent of boundary values can be applied ; thereby forming discrete groups . it should be noted that the ability for this embodiment of the present invention to use input data whose values fall into either continuous or discrete ranges is a versatile feature ; as methods of standard process control today are primarily designed to handle input data whose values fall only within a continuous range , while methods for handling input data whose values fall only within a discrete range are less common . however , methods of process control for dealing with both input values in a continuous range and input values in a discrete range such as this embodiment of the present invention are potentially of great use in countless diverse applications . an embodiment of the present invention uses the boundary value ranges assigned to input constants and variables to form data vectors for a given stage in a process . each input constant or variable is a component entry of the vector . given the inputs depicted in the three diagrams in fig1 a and their respective boundary values , it is seen that the following 24 data vectors exist for the output produced by the inputs depicted in fig1 a : {( a 1 , a 2 , a 3 ), ( a 1 , a 2 , b 3 ), ( a 1 , a 2 , c 3 ), ( a 1 , a 2 , d 3 ), ( a 1 , b 2 , a 3 ), ( a 1 , b 2 , b 3 ), ( a 1 , b 2 , c 3 ), ( a 1 , b 2 , d 3 ), ( b 1 , a 2 , a 3 ), ( b 1 , a 2 , b 3 ), ( b 1 , a 2 , c 3 ), ( b 1 , a 2 , d 3 ), ( b 1 , b 2 , a 3 ), ( b 1 , b 2 , b 3 ), ( b 1 , b 2 , c 3 ), ( b 1 , b 2 , d 3 ), ( c 1 , a 2 , a 3 ), ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , c 3 ), ( c 1 , a 2 , d 3 ), ( c 1 , b 2 , a 3 ), ( c 1 , b 2 , b 3 ), ( c 1 , b 2 , c 3 ), ( c 1 , b 2 , d 3 )} referring again to fig1 a , for the sake of example , assume that 10 a . 1 represents an input constant , and that 10 a . 2 and 10 a . 3 represent input variables at a given stage in a process . the boundary values for 10 a . 1 are x 1 = 24 . 98 mm and y 1 = 25 . 02 mm , where b 1 is the preferred boundary value range for values between 24 . 98 - 25 . 02 mm inclusively , a 1 is the boundary value range for values less than 24 . 98 mm , and c 1 is the boundary value range for values greater than 25 . 02 mm . for 10 a . 2 , there is one boundary value x 2 = 10 . 00 mm . a 2 is the boundary value range for values less than or equal to 10 . 00 mm , and b 2 is the boundary value range for values greater than 10 . 00 mm . for 10 a . 3 , there are three possible boundary values , denoted x 3 , y 3 , and z 3 a 3 , b 3 , c 3 , and d 3 represent four possible boundary value ranges for the height of an item . the possible range of the height of the item varies from 0 . 00 mm to 10 . 00 mm . a 3 is the boundary value range for values greater than 0 . 00 mm and up to and including 2 . 50 mm , b 3 is the boundary value range for values greater than 2 . 50 mm and up to and including 5 . 00 mm , c 3 is the boundary value range for values greater than 5 . 00 mm and up to and including 7 . 50 mm , and d 3 is the boundary value range for values greater than 7 . 50 mm and up to and including 10 . 00 mm . fig1 b illustrates a data arrays table of data arrays 5000 for a given stage in a process . the table is composed of a column for the number of the process run 5001 , columns for process input 5002 , and a column for a given process output constant 5006 . the inputs at this process stage are input constant 10 a . 1 5003 , input variable 10 a . 2 5004 , and input variable 5005 . values for these inputs corresponding to the data vector ( b 1 , a 2 , d 3 ) are received at the second process run 5007 , the e th process run 5008 , the e + 1 th process run , and the f th process run . the value for the given output constant for the second process run is o 2 5011 , the value for the given output constant for the e th process run is o e 5012 , the value for the given output constant for the e + 1 th process run is o e + 1 5013 , and the value for the given output constant for the f th process run is o f 5014 ; also ith inputs 5002 : input constant “ 10 a . 1 ” 5003 , input variables “ 10 a . 2 ” 5004 and “ 10 a . 3 ” 5005 and their respective output values 5006 make up a data array for any given process run ( s ) 5001 . if we received values of 25 . 01 mm for 10 a . 1 , 9 . 98 mm for 10 a . 2 , and 8 . 00 mm for 10 a . 3 , this data corresponds to the vector ( b 1 , a 2 , d 3 ), according to the assigned boundary values . referring to fig1 b , assume that the process is executed n times , and that after assigning boundary values to the data received for 10 a . 1 , 10 a . 2 , and 10 a . 3 , values corresponding to the vector ( b 1 , a 2 , d 3 ) are received for process runs 2 , e , e + 1 , and f where e is an integer whole number greater than 3 and f is an whole number integer greater than e + 1 and less than or equal to n . the values o 2 , o e , o e + 1 , and of represent the output values received for a given output constant for process runs 2 , e , e + 1 , and f respectively at the given stage in the process . the data received for any given process run , such as the value for input constant 10 a . 1 at run 2 , the values for input variables 10 a . 2 and 10 a . 3 at run 2 , and the value o 2 for the output at run 2 , are referred to as a data array . fig1 c ( generally referenced as ) 6000 illustrates a sample vector in a vector look - up table . the table is composed of columns for the data vector 6001 and columns for the given output constant data 6002 . the entries of the vector for the sample vector depicted in this table are input constant 10 a . 1 6003 , input variable 10 a . 2 6004 , and input variable 10 a . 3 6005 . the types of output constant data recorded in this vector look - up table are average 6006 , standard deviation 6007 , and population 6008 . the sample vector is vector ( b 1 , a 2 , d 3 ) 6009 . the average value for this vector for the given output constant is ō 6010 , the standard deviation is σ ( o ) 6011 , and the population number is 4 6012 . after a number of runs deemed sufficient by statistical criteria have been executed , the data arrays are sorted according to the data vectors they correspond to , and various meaningful statistical calculations are performed on the output data . for example , in fig1 b , data arrays corresponding to the vector ( b 1 , a 2 , d 3 ) were received for process runs 2 , e , e + 1 , and f . in fig1 c the output data for these four process runs is taken and the average and standard deviation of these four output values is calculated . the average value ō , the standard deviation σ ( o ), and the population number 4 are then entered in the vector look - up table in fig1 c by vector ( b 1 , a 2 , d 3 ). this output data is used by embodiments of the present invention for optimization of the given output constant . in addition to average , standard deviation , and population number ; other types of meaningful statistical calculations are performed on output data by embodiments of the present invention , such as determining the output constant &# 39 ; s process capability ratio ( c pk ), and the results of these calculations are used for process control optimization of that output . however , for the purposes of illustration , the examples that follow here refer to calculation and use of only standard deviation , average , and population number of output constant data . in many process control situations , it is understood that not all possible combinations of boundary value ranges for input constants and variables represent actual valid process input . therefore , for those vector input combinations that represent invalid input combinations for which the given process cannot be carried out , there will be no corresponding output data in the vector lookup table . the conventions of assigning boundary values to input data and sorting input data into data vectors enable detection of problematic input combinations and detection of input combinations that were assumed to yield output that is out of process specification standards and actually yield output that is within process specification standards . when problematic or unusual input combinations are detected , embodiments of the present invention provide appropriate system responses . one of these responses is a self - adjusting feature , which automatically adjusts process input that is out of process specification standards to within specification standards . other system responses include the sending of automated reports to the process engineer , or in more serious cases the sounding of an alarm or even cessation of process execution altogether . for example , it is understood that in certain process control situations , certain vector input combinations will represent input combinations for which the current process can be carried out , however it has been determined from previous history of the given process that the a given input combination is known to yield output which is out of process specification standards , or that the a given input combination contains one or more inputs outside of process specification standards , or that this specific combination of inputs is unacceptable for reasons related to the given process . an embodiment of the present invention allows the process engineer to program the system carrying out the given process so that if input combinations or output that are considered unacceptable for either of these reasons are received during process execution , the machinery and / or mechanisms carrying out the process automatically correct the input to within process specification standards . for more serious cases of this nature , this embodiment of the present invention allows the process engineer to program the system carrying out the given process to automatically sound an alarm instead of or in addition to automatic correction , or to even automatically halt process execution altogether ; or to report an unacceptable input combination or output to the process engineer , or in more serious cases an alarm is sounded or process execution is halted altogether . likewise , in certain process control situations , examination of data in the vector look - up table shows that certain combinations of boundary value ranges for input constants and / or variables which were assumed to yield output that is out of process specification standards do in fact yield output that is within process specification standards . or , certain combinations of boundary value ranges for input constants and / or variables where one or more of the boundary value ranges are considered out of the specification standard for that input do in fact yield output that is within process specification standards . for example , referring again to input constant 10 a . 1 and input variables 10 a . 2 and 10 a . 3 , assume that for 10 a . 1 the boundary value range a 1 is considered out of process specification standards , that for 10 a . 2 the boundary value range b 2 is considered out of process specification standards , and that for 10 a . 3 the boundary value range a 3 is considered out of process specification standards . however , after applying boundary values to the input data and sorting the input data into data vectors according to the embodiments of the present invention , the resulting output is found to be within process specification standards . despite this output , which is within specification standards , such a situation still warrants attention , as the given input combination is still considered to be out of process specification standards . in such a case , an embodiment of the present invention allows the process engineer to program the system carrying out the given process to report input combinations that are out of process specification standards and yet yield output within process specification standards . the input combination can then be analyzed to determine whether the combination constitutes a new and valid set of input or whether the combination constitutes an invalid set of input despite its output yield within process specification standards . the ability of embodiments of the present invention to determine input combinations of this nature with resulting output within process specification standards is a unique feature of the present invention that is unknown in standard methods of process control . data that is categorized as input constants , output constants , input variables , or monitored parameters can be either empirical or derived . empirical data is defined as data obtained from actual process runs . derived data is data obtained from conjecturing or mathematical calculations . an embodiment of the present invention uses either empirical or derived data , or a combination of the two data types , for the purposes of process control optimization . a specific type of derived data that is used by an embodiment of the present invention is data derived from design of experiment ( doe ). doe data is obtained by first choosing a limited sample population representing the spectrum of possible values for input constants and input variables at a given stage in a process , running the process with all possible combinations of this limited sample population , and monitoring the output at the given stage for each possible combination . based on the results of these process runs , a mathematical formula is derived that allows for interpolating or extrapolating process output for any possible combination of input constant values and input variable values , and with this formula output data is derived for the entire range of possible input constant values and input variable values . boundary values are assigned to this data , and with the boundary value ranges the data is arranged into data vectors . these data vectors are entered into the vector look - up table in exactly the same manner as data vectors for empirical data . however , because there is no actual sample population for the derived data , no standard deviation or average value can be calculated . instead , the output value received from the doe formula is entered in the vector look - up table as the average output value by the appropriate vector , and a default standard deviation value is likewise entered in the vector look - up table by the appropriate vector . the preferred embodiment of the present invention is implemented in a run - to - run control mode . in this context , a run - to - run control mode relates to a process control system that determines , based on previous runs of the given process , the optimal setting of process input variable value ( s ) at a given process tool or module in order to obtain or proximate one or more output target values at that tool or module . process control optimization in the run - to - run control mode is implemented at the given tool or module by two methods , known as the feed forward and feed back closed control loops . when implemented in the run - to - run control mode , the preferred embodiment of the present invention accesses data from previous process runs and the current process run ; and determines optimal settings for input variables on - line ( i . e . while the process is occurring ), allowing automated optimization of output for the current process run without significant delays and without halting operation of the given process . fig1 ( generally referenced as ) 7000 illustrates a vector look - up table for an output constant “ o 1 ” 7000 at a given process stage . the table is composed of columns for data vectors 7001 and columns for the output constant data for o 1 7002 . the entries for the vectors in the look - up table are input constant 10 a . 1 ( generally referenced as ) 7003 , input variable 10 a . 2 ( generally referenced as ) 7004 , and input variable 10 a . 3 ( generally referenced as ) 7005 . the types of output constant data recorded in this vector look - up table are average 7006 , standard deviation 7007 , and population 7008 . output constant data is recorded in the look - up table for vectors ( a 1 , a 2 , c 3 ) 7009 , ( b 1 , a 2 , d 3 ) 7010 , ( c 1 , a 2 , a 3 ) 7011 , ( c 1 , a 2 , b 3 ) 7012 , ( c 1 , a 2 , c 3 ) 7013 , ( c 1 , a 2 , d 3 ) 7014 , and ( c 1 , b 2 , c 3 ) 7015 . in the context of the present invention , the feed forward closed control loop relates to optimal correction of input variable values at a given stage in a process determined by the values of that stage &# 39 ; s input constants for the current process run and an output target value or target value range . the first step in implementing the feed forward closed control loop is to choose an optimal vector from the vector look - up table . this optimal vector is chosen by first selecting vectors from the vector look - up table that have boundary value ranges for input constants that include the values of the current run &# 39 ; s input constants . then , from among those selected vectors , the vector with the output constant average value closest to the output target value or to the midpoint of the output target value range is chosen as the optimal vector . for example , assuming the input constants and variables from fig1 a and their boundary values , in fig1 a vector look - up table is presented for output constant o 1 with vectors ( a 1 , a 2 , c 3 ), ( b 1 , a 2 , d 3 ), ( c 1 , a 2 , a 3 ), ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , c 3 ), ( c 1 , a 2 , d 3 ), and ( c 1 , b 2 , c 3 ). in the current process run , the value for input constant 10 a . 1 is 25 . 03 mm , which is in boundary value range c 1 . it is clearly seen that vectors ( c 1 , a 2 , a 3 ), ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , c 3 ), ( c 1 , a 2 , d 3 ), and ( c 1 , b 2 , c 3 ) satisfy the first criterion for choosing a vector in the first step of the feed forward closed control loop , because the value of input constant 10 a . 1 in the current process run is in boundary value range c 1 . next , it must be determined which of these five vectors has the closest average value for output constant o 1 to the output target value . the target value for o 1 is 10 . 00 mm . for vector ( c 1 , a 2 , a 3 ), ō 1 = 10 . 02 mm ; for vector ( c 1 , a 2 , b 3 ), ō 1 = 10 . 04 mm ; for vector ( c 1 , a 2 , c 3 ), ō 1 = 9 . 96 mm ; for vector ( c 1 , a 2 , d 3 ), ō 1 = 9 . 98 ; and for vector ( c 1 , b 2 , c 3 ), ō 1 = 10 . 01 mm . it is clearly seen that vector ( c 1 , b 2 , c 3 ) satisfies the second criterion for choosing the optimal vector in the first step of the feed forward closed control loop , because its average value for o 1 is closest to the output target value . in addition to the vector &# 39 ; s output constant average value , the output constant standard deviation and population number are also considered when choosing the an optimal vector . for example , a vector with an average value that is only 0 . 01 mm from the output target value but with a large standard deviation and low population number may be less preferable than a vector with an average value that is 0 . 02 mm from the output target value but with a low standard deviation and high population number . in such a case , the vector with the lower standard deviation but average value farther from the output target value might be chosen as the optimal vector , and by applying an offset factor to this vector , it is possible to obtain an average output value for this vector closer to the target value . those versed in the art will appreciate that rather than examining a vector &# 39 ; s average and standard deviation , the easiest and most facile way of selecting an optimal vector is to choose the available vector with the highest process capability ratio ( c pk ). however , for the purposes of illustration , the examples presented here refer only to calculation and use of standard deviation , average , and population number of output constant data . it should also be appreciated that there will be situations where choosing an optimal vector also depends on recently used process recipes . for example , it is possible that the vector with the highest c pk contains input variable boundary value ranges that significantly differ from recently used input variables values , and that due to system constraints using this vector will cause delays in reconfiguring a given process . in such a situation , a vector with a lower c pk but with an input combination close to or matching recently used input variables values proves to be the optimal vector since implementation of that vector avoid delays in reconfiguring the given process . the ability to choose an optimal vector based on both average value and standard deviation c pk is an example demonstrates of using of robust optimization for purposes of process control , where robust optimization is defined as the ability to reduce data variance while also attempting to obtain or come as close as possible to approximating a target value . current methods of standard process control are unable to utilize robust optimization , and the ability to utilize robust optimization with embodiments of the present invention is one of the present invention &# 39 ; s unique features that other methods of process control do not offer . another criterion for choosing the optimal vector is input constraints . that is , there may be certain input variable boundary value ranges that are considered unacceptable and will therefore disqualify certain vectors which otherwise meet the optimal vector criteria . for example , assume that an input constraint exists on input variable 10 a . 3 such that boundary value range a 3 is considered unacceptable . this would therefore disqualify vector ( c 1 , a 2 , a 3 ). reasons for input constraints can be cost or efficiency related , or can be related to the subjective considerations of the given process . ( c 1 , b 2 , c 3 ) is found to be the optimal vector . however , the average value ō 1 for vector ( c 1 , b 2 , c 3 ) is 10 . 01 , and is not exactly equal to the output target value of 10 . 00 mm . therefore , the second step in implementing the feed forward closed control loop is to find the next best optimal vector that most closely fits the criteria established for finding the optimal vector . if the output constant average value of the optimal vector is greater than the target value , then the output constant average value of the next best optimal vector must be less than the output target value . alternatively , if the output constant average value of the optimal vector is less than the output target value , then the output constant average value of the next best optimal vector must be greater than the output target value . after choosing the next best optimal second vector , a special smoothing algorithm is applied to the two vectors which interpolates the two vectors by means of piece - wise linearization , or linear regression , or the like . the result of this algorithm is a recipe with optimal values for input variables at the given process stage that will yield the output target value . for example , ō 1 for vector ( c 1 , b 2 , c 3 ) is 10 . 01 mm , which is a hundredth of a millimeter greater than the output target value of 10 . 00 . the vector with the closest average value for o 1 that is less than the output target value must be found . we saw previously that for vector ( c 1 , a 2 , d 3 ), ō 1 = 9 . 98 ; and for vector ( c 1 , a 2 , c 3 ), ō 1 = 9 . 96 mm . the average output value of 9 . 98 mm is closer to the output target value of 10 . 00 mm than 9 . 96 mm . therefore , vector ( c 1 , a 2 , d 3 ) is chosen as the next best optimal vector . the smoothing algorithm is then applied to vectors ( c 1 , b 2 , c 3 ) and ( c 1 , a 2 , d 3 ), and returns a resulting recipe for input variables 10 a . 2 and 10 a . 3 in order to receive the output target value of 10 . 00 mm for output constant o 1 . part of this smoothing algorithm involves giving greater priority to the optimal vector according to the proximity of the optimal vector &# 39 ; s output constant average value to the output target value . the smoothing algorithm returns interpolated recipe values for input variables that are more likely to be within the boundary value ranges of the input variables in the optimal vector than in the boundary value ranges of those input variables in the next best optimal vector . for the current example , this means that the recipe value for input variable 10 a . 2 is more likely to be in the b 2 range than in the a 2 range , and that the recipe value for input variable 10 a . 3 is more likely to be in the c 3 range than in the d 3 range . in the feed back closed control loop , optimal correction of input variable values at a given stage in a process is achieved through data analysis of the values of output constants at that stage from previous runs of the process . for example , in the present invention , if an output target value for an output constant is known and actual values of that output constant are known from previous process runs , the actual output values can be used to calculate an average offset of that output constant for the current process run . with this average offset , a new output target value is obtained , and with this new target value embodiments of the present invention automatically implement the feed forward closed control loop . an embodiment of the present invention allows for optimization of multiple outputs in the feed forward closed control loop . in such a case , there are two or more output constants at a given process stage , with output target value ranges for each output constant . current methods of standard process control are substantially unable to optimize multiple outputs in the integrated fashion that the embodiment of the present invention for optimizing multiple outputs does . in this context , the present invention provides a unique and revolutionary approach to process control optimization of multiple outputs . fig1 ( generally referenced as ) 8000 illustrates a vector look - up tables for multiple outputs constants o 1 and o 2 at a given process stage 8000 . the table is composed of columns for data vectors 8001 , columns for the output constant data for o 1 8002 , and columns for the output constant data for o 2 8003 . the entries for the vectors in the look - up table are input constant 10 a . 1 ( generally referenced as ) 8004 , input variable 10 a . 2 ( generally referenced as ) 8005 , and input variable 10 a . 3 ( generally referenced as ) 8006 . the types of output constant data recorded for output constant o 1 are average 8007 , standard deviation 8008 , and population 8009 . the types of output constant data recorded for output constant o 2 are average 8010 , standard deviation 8011 , and population 8012 . output constant data is recorded in the look - up table for vectors ( a 1 , a 2 , c 3 ) 8013 , ( b 1 , a 2 , d 3 ) 8014 , ( c 1 , a 2 , a 3 ) 8015 , ( c 1 , a 2 , b 3 ) 8016 , ( c 1 , a 2 , c 3 ) 8017 , ( c 1 , a 2 , d 3 ) 8018 , and ( c 1 , b 2 , c 3 ) 8019 . the first table is for output constant “ o 1 ” 8001 and the second table is for output constant “ o 2 ” 8002 . the embodiment of the present invention for optimizing multiple outputs carries out optimization of multiple outputs in two stages . the first stage is to try is implemented by first trying to find one or more common vectors for the multiple outputs at the given process stage . a common vector is defined as a vector with average values for the multiple output constants such that each constant &# 39 ; s average value in the vector is within the target value range for that constant . for example , fig1 presents a vector look - up table for multiple outputs based on the vector look - up table for output constant o 1 from fig1 . in fig1 , in addition to all the output data for output constant o 1 that was in the vector look - up table in fig1 , there is output data for an output constant o 2 , and a vector look - up table for an output constant referred to as o 2 . o 1 and o 2 are both output constants at one given stage in a process and the inputs to this process stage are input constant 10 a . 1 and input variables 10 a . 2 and 10 a . 3 . as in the previous example , assume that the value for the input constant 10 a . 1 for the current process run is in the c 1 boundary value range , and that there is an input constraint on input constant 10 a . 3 such that such that boundary values range a 3 is considered unacceptable . therefore , the possible vectors to choose from are ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , c 3 ), ( c 1 , a 2 , d 3 ), and ( c 1 , b 2 , c 3 ). previously , an output target value of 10 . 00 mm was assumed for output constant o 1 . a target value range of 9 . 97 - 10 . 03 mm is now assumed . assuming the same output constant average values for vectors ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , c 3 ), ( c 1 , a 2 , d 3 ), and ( c 1 , b 2 , c 3 ) that were mentioned previously , it is clearly seen that vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) fall within the target value range , since for vector ( c 1 , a 2 , d 3 ), ō 1 = 9 . 98 ; and for vector ( c 1 , b 2 , c 3 ), ō 1 = 10 . 01 mm . although vector ( c 1 , b 2 , c 3 ) has an average value closer to the midpoint of the target value range , for the purposes of optimizing multiple outputs with the common vector embodiment of the present invention , all vectors with average values within the target value range are considered . assume that the output constant o 2 measures the density of the item produced by inputs 10 a . 1 , 10 a . 2 , and 10 a . 3 at the given process , and that output constant o 2 has an output target value range of 999 . 50 - 1000 . 50 grams / cm 3 . for vector ( c 1 , a 2 , c 3 ), ō 2 = 999 . 87 grams / cm 3 ; for vector ( c 1 , a 2 , d 3 ), ō 2 = 1000 . 22 grams / cm 3 ; and for vector ( c 1 , b 2 , c 3 ), ō 2 = 999 . 46 grams / cm 3 . it is clearly seen that vectors ( c 1 , a 2 , c 3 ) and ( c 1 , a 2 , d 3 ) fall within the range 999 . 50 - 1000 . 50 grams / cm 3 for output constant o 2 , and that vector ( c 1 , b 2 , c 3 ) does not . output constant o 1 was found to have vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) in its output target value range , and output constant o 2 was found to have vectors ( c 1 , a 2 , c 3 ) and ( c 1 , a 2 , d 3 ) in its output target value range . therefore , it is clearly seen that these two output constants share the common vector ( c 1 , a 2 , d 3 ) in their respective target value ranges . ( c 1 , a 2 , d 3 ) is the optimal vector for output constants o 1 and o 2 , and the recipe for optimizing these multiple outputs is determined by taking the midpoints of the boundary value ranges of input variables 10 a . 2 and 10 a . 3 , which in this case are the midpoints of boundary value ranges a 2 and d 3 . although this example illustrates implementation of the embodiment of the present invention for optimization of multiple outputs on only two outputs , the methods presented here can be applied to any number of outputs . it is important to note that the optimal vector for multiple outputs would not necessarily be the optimal vector for each output constant if each output constant is analyzed on its own without regard to the other output constants . for example , output constant o 1 was found to have vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) in its output target value range , with vector ( c 1 , b 2 , c 3 ) having the closest average value to the midpoint of the target value range . if o 1 were analyzed on its own , ( c 1 , b 2 , c 3 ) would be its optimal vector . however , vector ( c 1 , b 2 , c 3 ) does not fall within the target value range for output constant o 2 , whereas vector ( c 1 , a 2 , d 3 ) does . likewise , in regards to output constant o 2 , vector ( c 1 , a 2 , d 3 ) would not be its optimal vector if o 2 is analyzed on its own , since vector ( c 1 , a 2 , c 3 ) has the closest average value to the midpoint of the target value range . however , vector ( c 1 , a 2 , c 3 ) does not fall within the target value range for output constant o 1 , whereas vector ( c 1 , a 2 , d 3 ) does . therefore , since vector ( c 1 , a 2 , d 3 ) is the only vector to fall within the target value ranges for output constants o 1 and o 2 , ( c 1 , a 2 , d 3 ) becomes the optimal vector when both output constants are analyzed together . this situation can be understood in terms of output constraints . that is , each output presents a constraint on other outputs , such that a vector which would be considered an optimal vector for a given output when that output is analyzed independent of other outputs cannot be used because it is outside of the output target value range ( s ) of the other output ( s ); and a common vector which may not be the optimal vector for some or all of the given output constants is selected because it is within the output target value ranges for all outputs and best optimizes multiple outputs when all the various output constraints are taken into consideration . when applying the embodiment of the present invention for optimization of multiple outputs , it is also possible that more than one common vector is shared by the multiple outputs . for example , in the previous example , an output target value range of 999 . 50 - 1000 . 50 grams / cm 3 was assumed for output constant o 2 , with vectors ( c 1 , a 2 , c 3 ) and ( c 1 , a 2 , d 3 ) falling within that target value range . assume now that the target value range is expanded to 999 . 45 - 1000 . 50 grams / cm 3 . the vector ( c 1 , b 2 , c 3 ), with an average value of 999 . 46 grams / cm 3 for o 2 , now falls within the target value range o 2 . further , in the previous example the output constant o 1 had an output target value range of 9 . 97 - 10 . 03 mm and the vector ( c 1 , b 2 , c 3 ), with an average value of 10 . 01 mm for o 1 , also fell within the target value range for o 1 . therefore , there are now two vectors , ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ), which fall in the output target value ranges for output constants o 1 and o 2 . if from amongst the two or more common vectors there is one vector with average values for all of the output constants closer to the midpoint of each output target value range for the respective output constants than any of the other vectors , then that vector is clearly the optimal vector for optimizing the given multiple outputs . however , if from among the two or more common vectors there is no one vector with average values for all of the output constants closer to the midpoint of each output target value range for the respective output constants than any of the other vectors , then there is a need to determine which of the two or more common vectors will best optimize the multiple outputs . the present example illustrates this latter possibility . it was determined that both vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) fall within the output target value ranges for output constants o 1 and o 2 . regarding output constant o 1 , ( c 1 , b 2 , c 3 ) with ō 1 = 10 . 01 mm is clearly preferable over vector ( c 1 , a 2 , d 3 ) with ō 2 = 9 . 98 mm . however , regarding output constant o 2 , ( c 1 , a 2 , d 3 ) with ō 2 = 1000 . 22 grams / cm 3 is clearly preferable over ( c 1 , b 2 , c 3 ) with ō 2 = 999 . 46 grams / cm 3 . with the present invention , optimization of multiple outputs in such a situation is done by means of a special weighting algorithm that is part of the embodiment for optimization of multiple outputs . when using this weighting algorithm , the process engineer assigns weightings to ( i . e . prioritizes ) the multiple outputs . the weighting assigned to each given output is then applied to the most preferable vector for that given output constant from amongst the common vectors , and the weighting algorithm derives a recipe for optimization of the multiple outputs based on each output &# 39 ; s preferable vector and its corresponding weighting . in most cases , the recipe values for input variables will be within the input variable boundary value ranges of the vector with the greatest weight . the assignment of these weightings can also be understood in terms of output constraints . that is , the more weight ( or greater priority ) that is assigned to one output will place greater constraints on the optimization of the other outputs . in our current example with common vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ), assume that output constant o 2 receives a greater weighting than o 1 . therefore , the greater weighting is applied to vector ( c 1 , a 2 , d 3 ), which is o 2 &# 39 ; s more preferable vector from amongst the two common vectors . o 1 receives the lesser weighting , and this lesser weighting is applied to vector ( c 1 , b 2 , c 3 ), which is o 1 &# 39 ; s more preferable vector from amongst the two common vectors . the two vectors have input variables 10 a . 2 and 10 a . 3 . since vector ( c 1 , a 2 , d 3 ) receives the higher priority , the recipe value for 10 a . 2 will be in the a 2 boundary value range and the recipe value for 10 a . 3 will be in the d 3 boundary value range . the extent to which the vector ( c 1 , a 2 , d 3 ) received a greater weighting will determine to what extent the recipe values for input variables are closer to the midpoint of the boundary value ranges of the input variables . in our current example , if ( c 1 , a 2 , d 3 ) received a much greater weight than ( c 1 , b 2 , c 3 ), then the recipe values for 10 a . 2 will be close to the midpoint of boundary value range a 2 ; whereas if ( c 1 , a 2 , d 3 ) received only a slightly greater weighting than ( c 1 , b 2 , c 3 ), then the recipe value for 10 a . 2 will still be within boundary value range a 2 but will be closer to the end of range a 2 that borders boundary value range b 2 . in certain situations , it is not always possible to find one or more common vectors within the output target value ranges of the multiple outputs . optimization of any one given output at a given stage in a process presents a constraint on optimizing all other outputs constants at that given process stage , and as the number of output constants and constraints increase , so does the difficulty in finding one common vector within the output target value ranges for all the multiple outputs . in the above example , common vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) were found within the two output target value ranges . if another third output constant were taken into consideration , it is possible that vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ) would not fall within the target value range of that third output constant , and no common vector would be found that falls within the output target value ranges for all three output constants . in such a situation , the weighting algorithm is also be applied to optimize multiple outputs . for example , assume that in addition to the output constants o 1 and o 2 from the previous example , there is a third output constant o 3 at the given process stage and that the vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ), which were determined to fall within the output target value ranges of o 1 and o 2 , do not fall within the output target value range of o 3 . further , assume that o 3 has an optimal vector ( c 1 , a 2 , b 3 ), and that vector ( c 1 , a 2 , b 3 ) does not fall within either of the output target value ranges for output constants o 1 and o 2 . therefore , there is no common vector that falls within the output target value ranges for the three output constants . the weighting algorithm is applied to this situation in the same manner as it was applied in the previous example with output constants o 1 and o 2 and common vectors ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ). now , weightings must be assigned to the three vectors ( c 1 , a 2 , b 3 ), ( c 1 , a 2 , d 3 ) and ( c 1 , b 2 , c 3 ). assume now that the optimal vector for o 3 , ( c 1 , a 2 , b 3 ), receives the greatest weight of the three vectors , and that vector ( c 1 , a 2 , d 3 ) receives a greater weighting than vector ( c 1 , b 2 , c 3 ) as in the previous example . the recipe value for the input variable 10 a . b will now very likely be close to the midpoint of boundary value a 2 since vector ( c 1 , a 2 , a 3 ) ( with the greatest weighting ) and vector ( c 1 , a 2 , d 3 ) ( with the second greatest weighting ) both have boundary value range a 2 for input variable 10 a . 2 . the recipe value for input variable 10 a . 3 in this case will greatly depend on the degree to which vector ( c 1 , a 2 , b 3 ) is weighted over vector ( c 1 , a 2 , d 3 ) and the degree to which vector ( c 1 , a 2 , d 3 ) is weighted over vector ( c 1 , b 2 , c 3 ). if vector ( c 1 , a 2 , b 3 ) has a much greater weight than vector ( c 1 , a 2 , d 3 ), then the recipe value is likely to be somewhere in the b 3 range ; whereas if the weighting for ( c 1 , a 2 , b 3 ) is only slightly greater than the weighting for vector ( c 1 , a 2 , d 3 ) and the weighting for ( c 1 , a 2 , d 3 ) is much greater than the weighting for vector ( c 1 , b 2 , c 3 ), the recipe value for 10 a . 3 is likely to be somewhere in the c 3 range . the schematic diagram in fig1 ( generally referenced as ) 9000 illustrates in a schematic diagram a stage in the chemical / mechanical polishing process of a silicon wafers at “ tool “ 1 ” 9001 . the inputs constants at “ tool - 1 ” are : thickness - in 9002 , and pad life 9003 , input variables at “ tool - 1 ” are time 9004 , pressure 9005 , and speed 9006 ; output constants at “ tool - 1 ” are uniformity 9007 and final - thickness 9008 . the following example illustrates how embodiments of the present invention are applied in the chemical / mechanical polishing ( cmp ) of silicon wafers . a stage in the cmp process is depicted in fig1 , with the following inputs : and two output constants final - thickness and uniformity . it is clearly seen in fig1 that these two output constants are dependent on the 5 inputs . embodiments of the present invention are applied in the apc run - to - run algorithm mode is applied to this process and calculates the feed back and feed forward corrections closed control loops are implemented for process run n as follows : 1 . calculate the offset average for the output constant final - thickness for run n for the last m previous runs , where m is any number from 1 to n − 1 , as follows : actual ( n − j ) is the actual measured value of output constant final - thickness at run ( n − j ), model pred ( n − j ) is the model predicted value of output constant final - thickness at run ( n − j ), which is the average value for output constant final - thickness taken from the vector look - up table for process run ( n − j ). σ n − j is the standard deviation of output constant final - thickness at run ( n − j ), taken from the vector look - up table for process run ( n − j ). p is a correction coefficient used to correct discernible deviation trends . 2 . with final - thickness ( δ n ), calculate the adjusted output target value range adj . target n for output constant final - thickness according to the following formula : where target n is the original output target value range for output constant final - thickness . in the event that the adjusted target value range goes above the ucl ( upper control limit ) or below the lcl ( lower control limit ) for output constant final - thickness , the system will cease operation . 3 . repeat steps 1 and 2 for the output constant uniformity , calculating the offset average for uniformity at run n with the formula from step 1 that was used to calculate the average offset for final - thickness , and using the offset average for uniformity at run n to determine the adjusted output target value range for uniformity . 4 . with the adjusted output target value ranges for uniformity and final - thickness and the values of input constants thickness - in and pad life for the current process run , refer to the vector look - up table for uniformity and final - thickness to find vectors within the respective adjusted output target value ranges . 5 . with the two groups of vectors obtained in step 4 , look for common vectors in the two groups . i . one common vector is found — use this vector to determine the process recipe . the recipe will be determined by taking the midpoints of the boundary value ranges in the vector for input variables pressure , time , and speed . ii . more than one common vector is found — from the common vectors found , determine the preferable vector for uniformity and the preferable vector for final - thickness . assign weightings to uniformity and final - thickness and apply the weighting algorithm to their preferable vectors to obtain the process recipe for input variables pressure . time , and speed . iii . no common vector is found — determine the optimal vector for uniformity and the optimal vector for final - thickness . assign weightings to uniformity and final - thickness and apply the weighting algorithm to their optimal vectors to obtain the process recipe for input variables pressure , time , and speed . in the above example , the determination of optimal values for input variables is based on both output values from previous runs , and the values of input constants for the current process run . that is , the calculation of the average offset for uniformity and final - thickness and their new output target value ranges for run n is a case of using output values from previous runs to determine optimal values of input variables for run n ; and the selection of optimal vectors from the vector look - up table is a case of using the values of input constants for the current process run to determine the optimal values of input variables . simply stated , the feed back and feed forward closed control loops are implemented together . the preferred embodiment of the present invention implements both control loops together . however , it should be noted that both control loops can be implemented independent of the other . for example , in an embodiment of the present invention , input constant data for the current process run is used to determine the optimal values for input variables without known output data , thus operating only in the feed forward closed control loop and not the feed back closed control loop . likewise , in another embodiment of the present invention , output value data is used to determine the optimal values for input variables without using input constant data , thus operating only in the feed back closed control loop and not the feed forward closed control loop . an embodiment of the present invention allows for updating the data stored in the vector look - up table . every time a process run is executed , the data for that run is processed according to the various methods that have been described up to this point , and with that data , the vector look - up table is eventually updated . the ability for the look - up table to be updated is of particular significance when data in the look - up table has been generated through design of experiment ( doe ). because the data from doe is almost entirely derived data , output data derived from doe is inherently less accurate than output data obtained from recording the results of actual process runs . therefore , the ability to update a look - up table containing doe data with data from actual process runs is essential . the embodiment of the present invention for updating the look - up table with recent empirical data is designed so that the process engineer can determine with what frequency the look - up table will automatically be updated . in many process control situations , it is neither efficient nor desirable to update the look - up table after every process run or even after every few process runs , and it is necessary to allow the process engineer to set the frequency for updating the look - up table based on the specific requirements of and constraints on the given process . this frequency can sometimes be even periods of weeks or months . other embodiments of the present invention allow for analysis of output data in the look - up table and correction of model predicted output values . according to one of these embodiments , the actual values of output constants recorded during recent process runs are compared with the model predicted values for that output constant according to the vector look - up table . recent process runs are defined as process runs whose data has not been used to update the vector look - up table . the differences , hereinafter residuals , between the actual values and the model predicted values of the output constant are calculated , and the residuals for each vector are examined . if , for all the residual values calculated , a common residual value is observed ; then this common residual value is used to correct the model predicted output values . fig1 ( generally referenced as ) 10 , 000 illustrates a vector look - up table 10 , 001 for output constant o 1 at a given process stage , an actual values column 10 , 002 which is not part of the vector look - up table , and a residual values column 10 , 003 which is not part of the vector look - up table . the vector look - up table 10 , 001 is composed of columns for data vectors 10 , 004 and columns for the output constant data for o 1 10 , 005 . the entries for the vectors in the look - up table are input constant 10 a . 1 ( generally referenced as ) 10 , 006 , input variable 10 a . 2 ( generally referenced as ) 10 , 007 , and input variable 10 a . 3 ( generally referenced as ) 10 , 008 . the types of output constant data recorded in this vector look - up table are average 10 , 009 , standard deviation 10 , 010 , and population 10 , 011 . output constant data is recorded in the look - up table for vectors ( a 1 , a 2 , c 3 ) 10 , 012 , ( b 1 , a 2 , d 3 ) 10 , 013 , ( c 1 , a 2 , a 3 ) 10 , 014 , ( c 1 , a 2 , b 3 ) 10 , 015 , ( c 1 , a 2 , c 3 ) 10 , 016 , ( c 1 , a 2 , d 3 ) 10 , 017 , and ( c 1 , b 2 , c 3 ) 10 , 018 . the actual values column 10 , 002 contains values act o1 ( a 1 , a 2 , c 3 ) 10 , 019 , act o1 ( b 1 , a 2 , d 3 ) 10 , 020 , act o1 ( c 1 , a 2 , a 3 ) 10 , 021 , act o1 ( c 1 , a 2 , b 3 ) 10 , 022 , act o1 ( c 1 , a 2 , c 3 ) 10 , 023 , act o1 ( c 1 , a 2 , d 3 ) 10 , 024 , and act o1 ( c 1 , b 2 , c 3 ) 10 , 025 . the residual values column 10 , 003 contains values δ o1 ( a 1 , a 2 , c 3 ) 10 , 026 , δ o1 ( b 1 , a 2 , d 3 ) 10 , 027 , δ o1 ( c 1 , a 2 , a 3 ) 10 , 028 , δ o1 ( c 1 , a 2 , b 3 ) 10 , 029 , δ o1 ( c 1 , a 2 , c 3 ) 10 , 030 , δ o1 ( c 1 , a 2 , d 3 ) 10 , 031 , δ o1 ( c 1 , b 2 , c 3 ) 10 , 032 . the actual values 10 , 002 are compared with the predicted values according to the vector look - up table for output constant “ o 1 ” 10 , 001 and the calculated difference between them is the residual value 10 , 003 . for example , referring to fig1 , depicted is the vector look - up table for output constant o 1 at a given process stage which has been seen previously in fig1 and 12 , with the noticeable difference that the population numbers for each vector are much greater now than they were in fig1 and 12 . assume that the process is now executed another 70 times with varying input combinations that correspond to the vectors in the look - up table depicted in fig1 . the values for o 1 for these 70 process runs are recorded but the look - up table is not updated with these values . the column to the right of the look - up table contains the average of the actual values of o 1 received during the 70 recent process runs for each vector in the look - up table , and the column on the far right contains the residual values of o 1 for each vector in the look - up table . that is , assume that during the 70 recent process runs , the process was executed 10 times for each vector in the look - up table . therefore the value act o1 ( a 1 , a 2 , c 3 ) is the average of the 10 values received for o 1 the when the process was executed with input combinations corresponding to the boundary value ranges a 1 , a 2 , and c 3 ; the value act o1 ( b 1 , a 2 , d 3 ) is the average of the 10 values received for o 1 when the process was executed with input combinations corresponding to the boundary value ranges b 1 , a 2 , and d 3 , and likewise preferably for all the values listed in the actual values column . the column on the far right contains the residual values of o 1 for each vector in the look - up table . that is the value δo 1 ( a 1 , a 2 , c 3 ) in the first row of the residual values column is the difference between ō 1 of vector ( a 1 , a 2 , c 3 ) and act o1 ( a 1 , a 2 , c 3 ); δ o1 ( b 1 , a 2 , d 3 ) is the difference between ō 1 of vector ( b 1 , a 2 , d 3 ) and act o1 ( b 1 , a 2 , d 3 ); etc . assume that for vector ( a 1 , a 2 , c 3 ), ō 1 = 9 . 99 mm ; for vector ( b 1 , a 2 , d 3 ), ō 1 = 10 . 00 mm ; for vector for vector ( c 1 , a 2 , a 3 ), ō 1 = 10 . 02 mm ; for vector ( c 1 , a 2 , b 3 ), ō 1 = 10 . 04 mm ; for vector ( c 1 , a 2 , c 3 ), ō 1 = 9 . 96 mm ; for vector ( c 1 , a 2 , d 3 ), ō 1 = 9 . 98 ; and for vector ( c 1 , b 2 , c 3 ), ō 1 = 10 . 01 mm . further , assume that act o1 ( a 1 , a 2 , c 3 )= 9 . 97 mm , act o1 ( b 1 , a 2 , d 3 )= 9 . 98 mm , act o1 ( c 1 , a 2 , a 3 )= 10 . 00 mm , act o1 ( c 1 , a 2 , b 3 )= 10 . 02 mm , act o1 ( c 1 , a 2 , c 3 )= 9 . 94 mm , act o1 ( c 1 , a 2 , d 3 )= 9 . 96 mm , and act o1 ( c 1 , b 2 , c 3 )= 9 . 99 mm . it is clearly seen that there is a common residual value for each vector in the look - up table of 0 . 02 mm . because there is a common residual value , the model predicted output values for o 1 are decreased by 0 . 02 mm , and these adjusted predicted values will be used for the purposes of process control optimization at the given process stage in the future . the example here presents a simple case of detecting an exact common residual value for all vectors in the look - up table . however , in practice , a significant range for residual values will likely be determined such that even if residual values are not exactly equal , if residual values fall within the significant range , then a common residual value is considered to have been detected and the model predicted output values will be corrected by the average value of those residual values . another embodiment of the present invention analyzes respective residuals and their corresponding model predicted values for each vector in the output constant &# 39 ; s vector look - up table to determine whether a functional relationship exists between the output constant &# 39 ; s model predicted values and the corresponding residual value . if a functional relationship is determined , then that functional relationship is used to correct the model predicted output values for all vectors of the given output constant . it is also possible that residual values will be analyzed and that no common residual value is detected , nor is any functional relationship determined between model predicted output values and their corresponding residual values . this can also be true even when applying a significant range , and not an exact value , to detect common residuals . in such a situation , if it is determined that the residual values are significant enough to warrant correction of model predicted output values , then the vector look - up table must be disregarded and a new vector look - up is calculated according to the embodiments of the present invention previously described here . the description of the embodiments of the present invention , up to this point , has focused on implementation of these embodiments at the process stage , or tool , level . it should be recalled that the tool level is the highest level of resolution in process control , such that process control optimization focuses on optimization of output at only one given stage in a process and no more . however , the embodiments of the present invention described here are also applied to process control optimization on the module level , where a module is defined as two or more contiguous tools in a process . in the simplest case , a module consists of two contiguous tools . in such a case , there is some output from the first tool that is input to the second tool . process control optimization with embodiments of the present invention in such a case focuses on optimizing output at the second tool in the module by means of the input variables and constants of the first tool in the module . the schematic diagram in fig1 ( generally referenced as ) 11 , 000 illustrates a process module of two tools , a schematic diagram for apc on a module level 11 , 000 . “ tool “ 1 ” 11 , 001 and “ tool - 2 ” 11 , 002 . “ tool - 1 ” has input constants thickness - in 11 , 003 and pad life - 111 , 004 ; has inputs variables : “ time - 1 ” 11 , 005 , “ pressure - 1 ” 11 , 006 , and “ speed - 1 ” 11 , 007 , and input constants ( whose values are used in feed forward 11 , 001 to adjust the aforementioned input variable values ): “ pad life 1 ” 11 , 003 , and thickness - in 11 , 004 ; and output constant thickness - out 11 , 008 . “ tool - 2 ” has input constants thickness - out 11 , 008 ( which is also an output constant from “ tool - 1 ”) and pad life - 2 11 , 009 ; input variables time - 2 11 , 010 , pressure - 2 11 , 011 , and speed - 2 11 , 012 ; and output constants uniformity 11 , 013 and final - thickness 11 , 014 . for example , fig1 presents a process module of two tools , labeled tool - 1 and tool - 2 . tool - 1 has input variables time - 1 , pressure - 1 , and speed - 1 , input constants pad life - 1 and thickness - in , and output constant thickness - out , which is also an input constant to tool - 2 . tool - 2 has input variables time - 2 , pressure - 2 , and speed - 2 , input constants pad life - 2 and thickness - out ( the output constant from tool - 1 ), and two output constants uniformity and final - thickness . embodiments of the present invention can be applied to a process module such as the one depicted in fig1 for optimization of the output constants uniformity and final - thickness at tool - 2 by means of the input constants and variables at tool - 1 . assume that a recipe of input values at tool - 2 has been determined for the optimization of the multiple outputs constants uniformity and final - thickness . this recipe contains a specific value or value range for the input constant thickness - out at tool - 2 . because thickness - out is also an output constant for tool - 1 , embodiments of the present invention are applied to tool - 1 to determine a recipe for the output target value or target value range of thickness - out , where the output target value or target value range for thickness - out is the input constant value or value range of thickness - out in the recipe for optimizing multiple outputs uniformity and final - thickness . that is , the recipe value of thickness - out when optimizing multiple outputs constants final - thickness and uniformity serves as the output target value for the recipe at tool - 1 . the example illustrated in fig1 presents a simple case of applying embodiments of the present invention on the module level . however , it should be noted that embodiments of the present invention are applied on the module level where the module is any number or configuration of contiguous or non - contiguous tools . for example , assume a module of five sequential tools . if each tool in the module has one or more output constants that are inputs to the next tool in module , then a recipe for obtaining one or more output target values or target value ranges at the fifth tool in the module will serve as the basis for determining the recipe of input variables at the first tool in the module . the embodiments of the present invention are applied at each tool in the module starting from the last tool and working back to the first tool . at each tool , starting from the second to last tool in the module , target values or target value ranges for the output constants in the recipe at that tool will be the recipe values or value ranges that those output constants have as input constants in the recipe for the immediately following tool in the module , until a recipe is received for input values at the process engineer the first tool in the module . fig1 ( generally referenced as ) 12 , 000 illustrates a schematic diagram of a module of four tools in a non - sequential configuration . “ tool - 1 a ” 12 , 001 , “ tool - 1 b ” 12 , 002 , and “ tool - 1 c ” 12 , 003 are parallel tools which run in either a simultaneous or alternate manner and send output to “ tool - 2 ” 12 , 004 . “ tool - 1 a ” has input constants 12 , 005 , input variables 12 , 006 , and output constants 12 , 007 which are also input constants to “ tool - 2 ”. “ tool - 1 b ” has input constants 12 , 008 , input variables 12 , 009 , and output constants 12 , 010 which are also input constants to “ tool - 2 ”. “ tool - 1 c ” has input constants 12 , 011 , input variables 12 , 012 , and output constants 12 , 013 which are also input constants to “ tool - 2 ”. “ tool - 2 ” has input constants 12 , 007 , 12 , 010 , and 12 , 013 , and the output constant final output 12 , 014 . likewise , embodiments of the present invention can be applied on non - sequential module configurations such as the module depicted in fig1 . fig1 depicts a module of four tools , where tools 1 a , 1 b , 1 c run parallel to each other in either a simultaneous or alternate manner and send output to tool - 2 . the output constants of tools 1 a , 1 b , and 1 c are input constants to tool - 2 . assume that an output target value is known for the output constant final output at tool - 2 . a recipe for receiving the target value of final output is obtained using embodiments of the present invention . the values of the input constants in the recipe for final output at tool - 2 then serve as the output constant target values for the recipes at tools 1 a , 1 b , and 1 c . on the basis of the recipe at tool - 2 , this embodiment of the present invention determines the output target values at each of the parallel tools and then determines the recipe for input variables and constants at each parallel tool .