Abstract:
A tracking unit includes a register for storing initial sequences and an accumulator for effecting a statistical treatment of the initial sequences, and ensures the statistical treatment relates to no more than a number of best sequences based on an evaluating algorithm. A generating unit employs the statistical treatment to present a new sequence. A measuring unit evaluates the new sequence according to the algorithm and cooperates with the tracking unit to effect providing the new sequence to the tracking unit when the evaluating indicates the new sequence is appropriate for the storing and statistical treatment. The measuring unit provides the new sequence to a regulating unit with an indication of efficacy of the new sequence. The regulating unit employs the indication to store a best-sequence-yet-received and responds to a criterion to order the generating unit to present a new sequence or to present the best-sequence-yet-received at an output locus.

Description:
TECHNICAL FIELD 
     The present disclosure may be directed to systems and methods for solving sequencing problems. 
     BACKGROUND 
     Sequencing problems may be common across multiple industries. By way of example and not by way of limitation, a stock cutting problem may be common in the composite aircraft industry. A stock cutting problem may be posed as how one may arrange individual pattern shapes on a material bed for cutting in a manner minimizing the amount of wasted material. By way of further example and not by way of limitation, a resource constrained project scheduling problem may be common in aircraft manufacturing and overhaul operations. A resource constrained project scheduling problem may be posed as how one may order operations to be performed in a manner to maximize utilization of resources and minimize overall duration. 
     Sequencing problems are generally categorized, within the theory of computation, as NP-Complete, meaning that sequencing problems are members of a large family of especially difficult problems where the time required to find a solution grows exponentially with the size of the problem. Without use of specialized methods, the time required to find good solutions to sequencing problems is proportional to N! The symbol N! may be known as “n-factorial”. N!=n*(n−1)*(n−2) . . . *2*1) where N may be the number of items to be sequenced. 
     By way of example and not by way of limitation, potential uses of this method may include
         use to find optimized solutions to conventional sequencing problems in design, manufacturing, maintenance, and transportation, where the objective is to optimize the sequence of operations, actions, events, or other occurrences ordered in time, including traveling salesman problem, line balancing problems, scheduling manufacturing and maintenance, data transmissions in a communications network, and similar problems.   use to find optimized solutions to conventional sequencing problems in design, manufacturing, maintenance, and transportation, where the objective is to optimize the placement of objects, components, or other physical objects ordered in space, including the stock cutting problem, pattern layout problem, circuit board design, communications network design, and similar problems.   use to optimize the results of a decision process by optimizing the sequence of discrete decisions, including use of this method to optimize the sequence of decisions within classical “branch-and-bound” decision trees, commonly used within Operations Research.   use to optimize the results of a physical process by optimizing the sequence of discrete actions, including synthesis of novel organic compounds that arise through sequences of heating, cooling, hydration, dehydration, exposure to ultraviolet light, exposure to electrical discharge, and other physical conditions.   use to find optimized solutions to any problem within the family of NP-Complete problems, by rendering said problems as sequencing problems, using this method to find an optimized solution, including NP-complete problems described in Computer Science literature.   use to optimize the sequence of atoms within a molecular structure, including the sequence of atoms within DNA or RNA, the sequence of atoms within proteins, the sequence of atoms within polymers, or the sequence of layers within a crystal lattice.       

     While algorithms may have been developed for some particular sequencing problems, general purpose methods for solution of sequencing problems remain elusive. Genetic Algorithms have proven useful in many computational problems. While Genetic Algorithms may be useful for general sequencing problems, they may have limited value for large problems, problems that have large-scale structure, and other circumstances. 
     There is a need for a system and method for optimizing a sequential arrangement of items that may be substantially generally applied to a wide variety and range of sequencing problems. 
     SUMMARY 
     In one aspect, a system is provided for effecting optimization of a sequential arrangement of items with respect to a predetermined evaluating algorithm. The system includes a computer processor, and a computer-readable storage device having encoded thereon computer-readable instructions that are executable by the computer processor to perform functions that provide an initiating unit, a tracking unit coupled with the initiating unit, a generating unit coupled with the tracking unit, a measuring unit coupled with the generating unit and with the tracking unit, a regulating unit coupled with the measuring unit and the generating unit, and an output locus coupled with the regulating unit. The initiating unit provides a predetermined number of initial sequences to the tracking unit. The tracking unit includes a register for storing the initial sequences and an accumulator for effecting a statistical treatment of the initial sequences. The register and the accumulator are each coupled to an adjusting unit. The register, the accumulator, and the adjusting unit cooperate to ensure the statistical treatment relates to no more than a predetermined number of best sequences based on the predetermined evaluating algorithm. The generating unit employs the statistical treatment to present a new sequence. The measuring unit evaluates the new sequence according to the predetermined evaluating algorithm. The measuring unit cooperates with the tracking unit to effect providing the new sequence to the tracking unit when the evaluating indicates the new sequence is appropriate for the storing and statistical treatment. The measuring unit provides the new sequence to the regulating unit with an indication of efficacy of the new sequence. The regulating unit employs the indication to store at least a best-sequence-yet-received. The regulating unit responds to at least one predetermined criterion to order the generating unit to present a new sequence or to present the best-sequence-yet-received at the output locus. 
     In another aspect, a method is provided for effecting optimization of a sequential arrangement of items with respect to a predetermined evaluating algorithm. The method includes providing an initiating unit, providing a tracking unit configured for coupling with the initiating unit, providing a generating unit configured for coupling with the tracking unit, providing a measuring unit configured for coupling with the generating unit and with the tracking unit, providing a regulating unit configured for coupling with the measuring unit and the generating unit, and providing an output locus configured for coupling with the regulating unit. The initiating unit is operated to provide a predetermined number of initial sequences to the tracking unit. The tracking unit is operated to store and effect a statistical treatment of the initial sequences. The tracking unit includes a register for storing the initial sequences and an accumulator for effecting the statistical treatment. The register and the accumulator are each coupled to an adjusting unit. The register, the accumulator, and the adjusting unit cooperate to ensure the statistical treatment relates to no more than a predetermined number of best sequences based on the predetermined evaluating algorithm. The generating unit is operated to employ the statistical treatment to present a new sequence. The measuring unit is operated to evaluate the new sequence according to the predetermined evaluating algorithm. The measuring unit is operated to cooperate with the tracking unit to effect providing the new sequence to the tracking unit when the evaluating indicates the new sequence is appropriate for the storing and statistical treatment. The measuring unit is operated to provide the new sequence to the regulating unit with an indication of efficacy of the new sequence. The regulating unit is operated to employ the indication to store at least a best sequence-yet-received. The regulating unit is operated to respond to at least one predetermined criterion to order the generating unit to present a new sequence or to present the best-sequence-yet-received at the output locus. 
     In yet another aspect, a system is provided for effecting optimization of a sequential arrangement of items with respect to a predetermined evaluating algorithm. The system includes a computer processor, and a computer-readable storage device having encoded thereon computer-readable instructions that are executable by the computer processor to perform functions that provide an initiating unit, a tracking unit coupled with the initiating unit, a generating unit coupled with the tracking unit, a measuring unit coupled with the generating unit and with the tracking unit, a regulating unit coupled with the measuring unit and the generating unit, and an output locus coupled with the regulating unit. The initiating unit provides a predetermined number of initial sequences to the tracking unit. The tracking unit stores and effects a statistical treatment of the initial sequences. The generating unit employs the statistical treatment to present a new sequence. The measuring unit evaluates the new sequence according to the predetermined evaluating algorithm. The measuring unit cooperates with the tracking unit to effect providing the new sequence to the tracking unit when the evaluating indicates the new sequence is appropriate for the storing and statistical treatment. The measuring unit provides the new sequence to the regulating unit with an indication of efficacy of the new sequence. The regulating unit employs the indication to store at least a best-sequence-yet-received. The regulating unit responds to at least one predetermined criterion to order the generating unit to present a new sequence or to present the best-sequence-yet-received at the output locus. The evaluating indicates the new sequence is appropriate for the storing and statistical treatment when the new sequence is different than sequences previously stored in the tracking unit and the new sequence is better than sequences previously stored in the tracking unit according to the predetermined evaluating algorithm. 
     In yet another aspect, a method is provided for effecting optimization of a sequential arrangement of items with respect to a predetermined evaluating algorithm. The method includes providing an initiating unit, providing a tracking unit configured for coupling with the initiating unit, providing a generating unit configured for coupling with the tracking unit, providing a measuring unit configured for coupling with the generating unit and with the tracking unit, providing a regulating unit configured for coupling with the measuring unit and the generating unit, and providing an output locus configured for coupling with the regulating unit. The initiating unit is operated to provide a predetermined number of initial sequences to the tracking unit. The tracking unit is operated to store and effect a statistical treatment of the initial sequences. The generating unit is operated to employ the statistical treatment to present a new sequence. The measuring unit is operated to evaluate the new sequence according to the predetermined evaluating algorithm. The measuring unit is operated to cooperate with the tracking unit to effect providing the new sequence to the tracking unit when the evaluating indicates the new sequence is appropriate for the storing and statistical treatment when the new sequence is different than sequences previously stored in the tracking unit and the new sequence is better than sequences previously stored in the tracking unit according to the predetermined evaluating algorithm. The measuring unit is operated to provide the new sequence to the regulating unit with an indication of efficacy of the new sequence. The regulating unit is operated to employ the indication to store at least a best sequence-yet-received. The regulating unit is operated to respond to at least one predetermined criterion to order the generating unit to present a new sequence or to present the best-sequence-yet-received at the output locus. 
     It is, therefore, a feature of the present disclosure to present a system and method for optimizing a sequential arrangement of items that may be substantially generally applied to a wide variety and range of sequencing problems. 
     Further features of the present disclosure will be apparent from the following specification and claims when considered in connection with the accompanying drawings, in which like elements are labeled using like reference numerals in the various figures, illustrating the preferred embodiments of the disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of a system for optimizing a sequential arrangement of items. 
         FIG. 2  is a schematic diagram of a representation of a first exemplary sequence stored according to the teachings of the present disclosure. 
         FIG. 3  is a schematic diagram of a representation of a second exemplary sequence stored according to the teachings of the present disclosure. 
         FIG. 4  is a schematic diagram of a representation of a third exemplary sequence stored according to the teachings of the present disclosure. 
         FIG. 5  is a schematic diagram of a representation of a fourth exemplary sequence stored according to the teachings of the present disclosure. 
         FIG. 6  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 2 and 3 . 
         FIG. 7  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 4 and 6 . 
         FIG. 8  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 5 and 7 . 
         FIG. 9  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after subtracting the representation illustrated in  FIG. 2  from the representation illustrated in  FIG. 8 . 
         FIG. 10  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure as illustrated in  FIG. 8 , with row and column sums presented. 
         FIG. 11  is a schematic diagram of a representation of column probabilities associated with data presented in  FIG. 10 . 
         FIG. 12  is a schematic diagram of a representation of row probabilities associated with data presented in  FIG. 10 . 
         FIG. 13  is a schematic diagram of a representation of cell probabilities associated with data presented in  FIG. 10 . 
         FIG. 14  is a flow diagram illustrating a method for optimizing a sequential arrangement of items. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a schematic diagram of a system for optimizing a sequential arrangement of items. In  FIG. 1 , a system  10  for optimization of a sequential arrangement of items with respect to a predetermined evaluating algorithm may include an initiating unit  12 , a tracking unit  14  coupled with initiating unit  12 , a generating unit  16  coupled with tracking unit  14 , a measuring unit  18  coupled with generating unit  16  and coupled with tracking unit  14 , a regulating unit  20  coupled with measuring unit  18  and generating unit  16  and an output locus  22  coupled with regulating unit  20 . 
     Tracking unit  14  may include a store or register  30  coupled with initiating unit  12 , an adjusting unit  32  coupled with register  30  and an accumulator  34  coupled with adjusting unit  32  and generating unit  16 . Adjusting unit  32  may include an adder  36  coupled with register  30  and accumulator  34 , and may include a subtractor  38  coupled with register  30  and accumulator  34 . 
     Generating unit  16  may include a generator  40  coupled with accumulator  34 , and may include an improver  42  coupled with generator  40  and measuring unit  18 . 
     Measuring unit  18  may include an evaluator  44  coupled with improver  42 , and may include a selector  46  coupled with evaluator  44 , register  30  and regulating unit  20 . Regulating unit  20  may be coupled with generator  40 . 
     In operation, initiating unit  12  may begin an optimizing process by providing a predetermined number of initial sequences to tracking unit  14 , specifically delivering the initial sequences to register  30 . Tracking unit  14  may store the initial sequences in register  30 . Register  30  may employ adder  36  to provide the initial sequences to accumulator  34 . Tracking unit  14  may effect a statistical treatment of the initial sequences, specifically employing accumulator  34 . 
     Generating unit  16  may employ the statistical treatment provided by accumulator  34  to generate an initial new sequence, specifically employing generator  40 . Improver  42  may improve the initial new sequence generated by generator  40  to present a new sequence to measuring unit  18 . Improver  42  may be eliminated, if desired, so that an initial new sequence presented by generator  40  is presented to measuring unit  18  as a new sequence. 
     Measuring unit  18  may evaluate the new sequence according to the predetermined evaluating algorithm (not shown in  FIG. 1 ) specifically employing evaluator  44 . Evaluator  44 , selector  46  and register  30  may cooperate to ascertain whether the new sequence is appropriate for storage in register  30  and statistical treatment by accumulator  34 . A new sequence may be appropriate for storage in register  30  and statistical treatment by accumulator  34  when a new sequence is different from sequences stored in register  30  and is better (as measured by the predetermined evaluating algorithm) than sequences stored in register  30 . 
     Register  30  may store all appropriate sequences, but accumulator  34  may be limited to treating a predetermined number of best sequences. Thus, register  30  may cooperate with accumulator  34  and adder  32  to add better sequences (i.e., sequences evaluated as better than sequences currently treated by accumulator  32 ) to accumulator  34  for treatment. Once the predetermined number of best sequences is treated by accumulator  34 , whenever register  30  seeks to add a new best sequence to accumulator  34  for treatment, register  30  and subtractor  38  may cooperate to remove a “least best” sequence (i.e., a sequence evaluated as not as good as all other sequences currently treated by accumulator  34 ) from accumulator  34  treatment. That is, sequences may be added to accumulator  34  from register  30  via adder  36  until accumulator  34  is dealing with a predetermined number of sequences. After accumulator  34  may be dealing with the predetermined capacity of sequences, if register  30  adds a sequence to accumulator  34  for treatment, a sequence may be withdrawn from accumulator  34  using subtractor  38 . 
     Selector  46  may provide the new sequence to regulating unit  20  (regardless of whether it is deemed appropriate for storing and effecting statistical treatment by tracking unit  14 ). It may be preferred that an indication of quality of the new sequence accompany the new sequence to regulating unit  20  so that regulating unit  20  may identify and store at least a best-sequence-yet-received. Regulating unit  20  may employ at least one predetermined criterion to either: (1) order generator  40  to generate another initial new sequence, or (2) present the best-sequence-yet-received to output locus  22 , thus terminating the optimization operation. 
     If generator  40  is ordered to generate another initial new sequence, system  10  may continue the optimization operation using improver  42 , evaluator  44 , selector  46 , tracking unit  14  and regulating unit  20  as described until occurrence of at least one predetermined criterion. The predetermined criterion may be embodied in, by way of example and not by way of limitation, a predetermined number of new sequences being evaluated by evaluator  44  or may be one or more other criteria. 
       FIG. 2  is a schematic diagram of a representation of a first exemplary sequence stored according to the teachings of the present disclosure. In  FIG. 2 , a grid or table  50  is arranged with elements of a sequence, such as numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, may be arrayed in columns  52  and in rows  54 . Individual cells may be identified by a (row, column) designation so that, for example, a cell  56  in row  9 , column  5  may be identified as cell (9, 5) or cell [9][5]. Grid  50  may represent a statistical record of all occurrences of one item before another item in sequences which accumulator  34  ( FIG. 1 ) may currently treat. As each sequence is created (e.g., by generator  40 ;  FIG. 1 ) and processed for storing in register  30  and treatment by accumulator  34 , accumulator  34  may present in grid  50  a count in each cell (i,j) of how many times an item [i] may appear preceding an item [j] among the sequences currently treated by accumulator  34 . 
     Data reflected in grid  50  may be managed by adder  36 , subtractor  38 , and accumulator  34 . If a sequence may be added to accumulator  34  then “1” may be added for each instance of an element preceding another. If a sequence may be subtracted or removed from accumulator  34  then “1” may be subtracted for each instance of an element preceding another. That is, “1&#39;s” that may have been added to accumulator  34  when the sequence was recorded may be subtracted from accumulator  34  when the sequence may be removed or subtracted. Register  30 , adder,  36  and subtractor  38  may cooperate so that there may be no circumstance in which a sequence may be subtracted without first having been added in some prior step or iteration. Accumulator  34  may only contain zero or positive numbers. 
     By way of simplistic example and not by way of limitation, when grid  50  may be populated using the above routine given the Sequence #1 of the following four sequences 
     Sequence #1: 1, 3, 5, 7, 6, 4, 2, 8, 0, 9 
     Sequence #2: 0, 1, 7, 9, 8, 3, 4, 5, 6, 2 
     Sequence #3: 2, 1, 7, 8, 9, 6, 5, 4, 3, 0 
     Sequence #4: 8, 4, 5, 2, 6, 9, 0, 1, 7, 3 
     grid  50  may appear as illustrated in  FIG. 2 . 
     To illustrate, in Sequence #1 (1, 3, 5, 7, 6, 4, 2, 8, 0, 9): 
     “1” precedes 3, 5, 7, 6, 4, 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (1, 3), (1, 5), (1, 7), (1, 6), (1, 4), (1, 2), (1, 8), (1, 0) and (1, 9). 
     “3” precedes 5, 7, 6, 4, 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (3, 5), (3, 7), (3, 6), (3, 4), (3, 2), (3, 8), (3, 0) and (3, 9). 
     “5” precedes 7, 6, 4, 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (5, 7), (5, 6), (5, 4), (5, 2), (5, 8), (5, 0) and (5, 9). 
     “7” precedes 6, 4, 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (7, 6), (7, 4), (7, 2), (7, 8), (7, 0) and (7, 9). 
     “6” precedes 4, 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (6, 4), (6, 2), (6, 8), (6, 0) and (6, 9). 
     “4” precedes 2, 8, 0 and 9. Therefore, “1” may be added to each of cell (4, 2), (4, 8), (4, 0) and (4, 9). 
     “2” precedes 8, 0 and 9. Therefore, “1” may be added to each of cell (2, 8), (2, 0) and (2, 9). 
     “8” precedes 0 and 9. Therefore, “1” may be added to each of cell (8, 0) and (8, 9). 
     “0” precedes 9. Therefore, “1” may be entered in cell (0, 9). 
     “9” precedes nothing, so no entry may be made in the row of grid  50  associated with “9”. 
     If accumulator  34  may not have reached a predefined capacity, then the representation of Sequence #1 illustrated in  FIG. 2  may be added to accumulator  34 , preferably in a matrix format substantially similar with grid  50 . If accumulator  34  has reached its predefined capacity, the representation of Sequence #1 illustrated in  FIG. 2  may be added to accumulator  34  and, as an additional step, the worst sequence represented by entries in into accumulator  34  (as measured by a predetermined criterion) may be removed or subtracted from accumulator  34 . The process of subtraction may be described further in connection with  FIG. 9 . 
       FIG. 3  is a schematic diagram of a representation of a second exemplary sequence stored according to the teachings of the present disclosure. In  FIG. 3 , when a grid  60  may be populated using the above routine given the Sequence #2 of the following four sequences 
     Sequence #1: 1, 3, 5, 7, 6, 4, 2, 8, 0, 9 
     Sequence #2: 0, 1, 7, 9, 8, 3, 4, 5, 6, 2 
     Sequence #3: 2, 1, 7, 8, 9, 6, 5, 4, 3, 0 
     Sequence #4: 8, 4, 5, 2, 6, 9, 0, 1, 7, 3 
     grid  60  may appear as illustrated in  FIG. 3 . 
     To illustrate, in Sequence #2 (0, 1, 7, 9, 8, 3, 4, 5, 6, 2): 
     “0” precedes 1, 7, 9, 8, 3, 4, 5, 6 and 2. Therefore, “1” may be entered in cells (0, 1), (0, 7), (0, 9), (0, 8), (0, 3), (0, 4), (0, 5), (0, 6) and (0, 2). 
     “1” precedes 7, 9, 8, 3, 4, 5, 6 and 2. Therefore, “1” may be added to each of cell (1, 7), (1, 9), (1, 8), (1, 3), (1, 4), (1, 5), (1, 6) and (1, 2). 
     “7” precedes 9, 8, 3, 4, 5, 6 and 2. Therefore, “1” may be added to each of cell (7, 9), (7, 8), (7, 3), (7, 4), (7, 5), (7, 6) and (7, 2). 
     “9” precedes 8, 3, 4, 5, 6 and 2. Therefore, “1” may be added to each of cell (9, 8), (9, 3), (9, 4), (9, 5), (9, 6) and (9, 2). 
     “8” precedes 3, 4, 5, 6 and 2. Therefore, “1” may be added to each of cell (8, 3), (8, 4), (8, 5), (8, 6) and (8, 2). 
     “3” precedes 4, 5, 6 and 2. Therefore, “1” may be added to each of cell (3, 4), (3, 5), (3, 6) and (3, 2). 
     “4” precedes 5, 6 and 2. Therefore, “1” may be added to each of cell (4, 5), (4, 6) and (4, 2). 
     “5” precedes 6 and 2. Therefore, “1” may be added to each of cell (5, 6) and (5, 2). 
     “6” precedes 2. Therefore, “1” may be added to cell (6, 2). 
     “2” precedes nothing, so no entry may be made in the row of grid  60  associated with “2”. 
     If accumulator  34  may not have reached a predefined capacity, then the representation of Sequence #2 illustrated in  FIG. 3  may be added to accumulator  34 , preferably in a matrix format substantially similar with grid  60  (see  FIG. 6 ). If accumulator  34  has reached its predefined capacity, the representation of Sequence #2 illustrated in  FIG. 3  may be added to accumulator  34  and, as an additional step, the worst sequence represented by entries in into accumulator  34  (as measured by a predetermined criterion) may be removed or subtracted from accumulator  34 . The process of subtraction may be described further in connection with  FIG. 9 . 
       FIG. 4  is a schematic diagram of a representation of a third exemplary sequence stored according to the teachings of the present disclosure. In  FIG. 4 , when a grid  62  may be populated using the above routine given the Sequence #3 of the following four sequences 
     Sequence #1: 1, 3, 5, 7, 6, 4, 2, 8, 0, 9 
     Sequence #2: 0, 1, 7, 9, 8, 3, 4, 5, 6, 2 
     Sequence #3: 2, 1, 7, 8, 9, 6, 5, 4, 3, 0 
     Sequence #4: 8, 4, 5, 2, 6, 9, 0, 1, 7, 3 
     grid  62  may appear as illustrated in  FIG. 4 . 
     To illustrate, in Sequence #3 (2, 1, 7, 8, 9, 6, 5, 4, 3, 0): 
     “2” precedes 1, 7, 8, 9, 6, 5, 4, 3 and 0. Therefore, “1” may be entered in cells (2, 1), (2, 7), (2, 8), (2, 9), (2, 6), (2, 5), (2, 4), (2, 3) and (2, 0). 
     “1” precedes 7, 8, 9, 6, 5, 4, 3 and 0. Therefore, “1” may be added to each of cell (1, 7), (1, 8), (1, 9), (1, 6), (1, 5), (1, 4), (1, 3) and (1, 0). 
     “7” precedes 8, 9, 6, 5, 4, 3 and 0. Therefore, “1” may be added to each of cell (7, 8), (7, 9), (7, 6), (7, 5), (7, 4), (7, 3) and (7, 0). 
     “8” precedes 9, 6, 5, 4, 3 and 0. Therefore, “1” may be added to each of cell (8, 9), (8, 6), (8, 5), (8, 4), (8, 3) and (8, 0). 
     “9” precedes 6, 5, 4, 3 and 0. Therefore, “1” may be added to each of cell (9, 6), (9, 5), (9, 4), (9, 3) and (9, 0). 
     “6” precedes 5, 4, 3 and 0. Therefore, “1” may be added to each of cell (6, 5), (6, 4), (6, 3) and (6, 0). 
     “5” precedes 4, 3 and 0. Therefore, “1” may be added to each of cell (5, 4), (5, 3) and (5, 0). 
     “4” precedes 3 and 0. Therefore, “1” may be added to each of cell (4, 3) and (4, 0). 
     “3” precedes 0. Therefore, “1” may be added to cell (3, 0). 
     “0” precedes nothing, so no entry may be made in the row of grid  62  associated with “0”. 
     If accumulator  34  may not have reached a predefined capacity, then the representation of Sequence #3 illustrated in  FIG. 4  may be added to accumulator  34 , preferably in a matrix format substantially similar with grid  62  (see  FIG. 6 ). If accumulator  34  has reached its predefined capacity, the representation of Sequence #3 illustrated in  FIG. 4  may be added to accumulator  34  and, as an additional step, the worst sequence represented by entries in into accumulator  34  (as measured by a predetermined criterion) may be removed or subtracted from accumulator  34 . The process of subtraction may be described further in connection with  FIG. 9 . 
       FIG. 5  is a schematic diagram of a representation of a fourth exemplary sequence stored according to the teachings of the present disclosure. In  FIG. 5 , when a grid  64  may be populated using the above routine given the Sequence #4 of the following four sequences 
     Sequence #1: 1, 3, 5, 7, 6, 4, 2, 8, 0, 9 
     Sequence #2: 0, 1, 7, 9, 8, 3, 4, 5, 6, 2 
     Sequence #3: 2, 1, 7, 8, 9, 6, 5, 4, 3, 0 
     Sequence #4: 8, 4, 5, 2, 6, 9, 0, 1, 7, 3 
     grid  64  may appear as illustrated in  FIG. 5 . 
     To illustrate, in Sequence #4 (8, 4, 5, 2, 6, 9, 0, 1, 7, 3): 
     “8” precedes 4, 5, 2, 6, 9, 0, 1, 7 and 3. Therefore, “1” may be entered in cells (8, 4), (8, 5), (8, 2), (8, 6), (8, 9), (8, 0), (8, 1), (8, 7) and (8, 3). 
     “4” precedes 5, 2, 6, 9, 0, 1, 7 and 3. Therefore, “1” may be added to each of cell (4, 5), (4, 2), (4, 6), (4, 9), (4, 0), (4, 1), (4, 7) and (4, 3). 
     “5” precedes 2, 6, 9, 0, 1, 7 and 3. Therefore, “1” may be added to each of cell (5, 2), (5, 6), (5, 9), (5, 0), (5, 1), (5, 7) and (5, 3). 
     “2” precedes 6, 9, 0, 1, 7 and 3. Therefore, “1” may be added to each of cell (2, 6), (2, 9), (2, 0), (2, 1), (2, 7) and (2, 3). 
     “6” precedes 9, 0, 1, 7 and 3. Therefore, “1” may be added to each of cell (6, 9), (6, 0), (6, 1), (6, 7) and (6, 3). 
     “9” precedes 0, 1, 7 and 3. Therefore, “1” may be added to each of cell (9, 0), (9, 1), (9, 7) and (9, 3). 
     “0” precedes 1, 7 and 3. Therefore, “1” may be added to each of cell (0, 1), (0, 7) and (0, 3). 
     “1” precedes 7 and 3. Therefore, “1” may be added to each of cell (1, 7) and (1, 3). 
     “7” precedes 3. Therefore, “1” may be added to cell (7, 3). 
     “3” precedes nothing, so no entry may be made in the row of grid  64  associated with “3”. 
     If accumulator  34  may not have reached a predefined capacity, then the representation of Sequence #4 illustrated in  FIG. 5  may be added to accumulator  34 , preferably in a matrix format substantially similar with grid  64  (see  FIG. 6 ). If accumulator  34  has reached its predefined capacity, the representation of Sequence #4 illustrated in  FIG. 5  may be added to accumulator  34  and, as an additional step, the worst sequence represented by entries in into accumulator  34  (as measured by a predetermined criterion) may be removed or subtracted from accumulator  34 . The process of subtraction may be described further in connection with  FIG. 9 . 
       FIG. 6  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 2 and 3 . In  FIG. 6 , a cell-by-cell addition of cells in  FIGS. 2 and 3  (i.e., Sequences #1 and #2) may yield the cell-by-cell sum-array or grid  66  illustrated in  FIG. 6 . Entries in cells of grid  66  may indicate a number of occurrences of a particular numeral appearing before another particular number in Sequences #1 and #2. 
     Thus, by way of example, cell (6,2) contains a “2” entry, indicating that the number 6 may appear before the number 2 in both of Sequences #1 and #2. Cell (2,5) contains a “0” entry, indicating that the number 2 may not appear before the number 5 in both of Sequences #1 and #2. Cell (5,0) contains a “1” entry, indicating that the number 5 may appear before the number 0 in one of Sequences #1 and #2. The annotation scheme may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective cell (i, j) will not be provided here. 
       FIG. 7  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 4 and 6 . In  FIG. 7 , a cell-by-cell addition of cells in  FIGS. 2 ,  3  and  4  (i.e., Sequences #1, #2 and #3) may yield the cell-by-cell sum-array or grid  68  illustrated in  FIG. 7 . Entries in cells of grid  68  may indicate a number of occurrences of a particular numeral appearing before another particular number in Sequences #1, #2 and #3. 
     Thus, by way of example, cell (6,2) contains a “2” entry, indicating that the number 6 may appear before the number 2 in two of the Sequences #1, #2 and #3. Cell (2,5) contains a “1” entry, indicating that the number 2 may appear before the number 5 in one of Sequences #1, #2 and #3. Cell (5,0) contains a “2” entry, indicating that the number 5 may appear before the number 0 in two of Sequences #1, #2 and #3. The annotation scheme may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective cell (i, j) will not be provided here. 
     In the cell-by-cell sum-array or grid  68  presented in  FIG. 7 , numerals contained in respective cells (i,j) may indicate the number of occurrences of ordering relationships experienced among the sequences represented in the array. In  FIG. 7 , cells (i,j) containing a “0” may show ordering relationships that did not occur in Sequences #1, #2 and #3. Cells (i,j) containing a “1” may show ordering relationships that occurred only one time in Sequences #1, #2 and #3. Cells (i,j) containing a “2” may show ordering relationships that occurred only two times in Sequences #1, #2 and #3. Cells (i,j) containing a “3” may show ordering relationships that occurred three times in Sequences #1, #2 and #3. 
     In  FIG. 7 , the sum of “opposite” cells—that is, the sum of cell (i,j) and cell (j,i)—equals n, where n equals the number of sequences that have been employed to construct the cell-by-cell sum-array or grid  50 . In the case of  FIG. 7 , n=3. This symmetry between opposite cells may permit desired results from statistically weighted sorts described in connection with  FIGS. 11-13 . 
       FIG. 8  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after adding representations illustrated in  FIGS. 5 and 7 . In  FIG. 8 , a cell-by-cell addition of cells in  FIGS. 2 ,  3 ,  4  and  5  (i.e., Sequences #1, #2, #3 and #4) may yield the cell-by-cell sum-array or grid  70  illustrated in  FIG. 8 . Entries in cells of grid  70  may indicate a number of occurrences of a particular numeral appearing before another particular number in Sequences #1, #2, #3 and #4. 
     Thus, by way of example, cell (6,2) contains a “2” entry, indicating that the number 6 may appear before the number 2 in two of the Sequences #1, #2, #3 and #4. Cell (2,5) contains a “1” entry, indicating that the number 2 may appear before the number 5 in one of Sequences #1, #2, #3 and #4. Cell (5,0) contains a “3” entry, indicating that the number 5 may appear before the number 0 in three of Sequences #1, #2, #3 and #4. The annotation scheme may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective cell (i, j) will not be provided here. 
       FIG. 9  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure after subtracting the representation illustrated in  FIG. 3  from the representation illustrated in  FIG. 8 . By way of example and not by way of limitation, let it be assumed that according to a predetermined criterion Sequence #2 is of lesser value than Sequences #1, #3 and #4. Let it be further assumed, for purposes of illustration, that the capacity of accumulator  34  ( FIG. 1 ) is three sequences. A result of such a situation may be to require subtracting the least desirable sequence (i.e., Sequence #2) from the cell-by-cell sum-array or grid  70  illustrated in  FIG. 8 . The result of such a subtraction may be viewed in  FIG. 9 . 
     In  FIG. 9 , the “1” entries associated with Sequence #2 (presented in  FIG. 3 ) have been subtracted from the array illustrated in  FIG. 8  in a cell-by-cell fashion to present a cell-by-cell sum-array or grid  72  representing cell-by-cell addition of cells in  FIGS. 2 ,  3 ,  4  and  5 , less cell entries presented in  FIG. 3  (i.e., Sequences #1, #3 and #4). 
     Register  30 , adder  36  and subtractor  38  ( FIG. 1 ) may cooperate to assure a sequence may not be subtracted without having been added in a prior step. Accumulator  34  may preferably contain only zeros or positive numbers. 
       FIG. 10  is a schematic diagram of a representation of data stored in the accumulator of the present disclosure as illustrated in  FIG. 8 , with row and column sums presented. In  FIG. 10 , sums of entries in each row and in each column of the cell-by-cell sum-array or grid  70  presented in  FIG. 8  are presented adjacent to respective rows and columns in a grid  74 . The row sums and column sums may be useful in carrying out statistical probability calculations described in connection with  FIGS. 11-13 . 
     Accumulator  34  ( FIG. 1 ) may employ grid  74  to effect statistical treatment of sequences. Generator  40  ( FIG. 1 ) may cooperate with accumulator  34  using grid  74  to create an initial new sequence that is statistically similar to sequences treated by accumulator  34 . By way of example and not by way of limitation, Generator  40  may perform a statistical analysis by beginning with a random sequence. The random sequence may then modified by a series of three steps that perform a column-based sort, a row-based sort, and a cell-based sort to produce a final generated sequence. 
     Accumulator  34  and generator  40  may cooperate to extract statistics from columns of grid  74  to create a new sequence that may roughly measure the number of successors for each item in a solution sequence. 
     The initial random sequence may then be sorted into reverse order, using the successor data from the column-extraction to weight probability of exchanging two items. Probability of exchanging two items [i] and [j] may be expressed as:
 
P[Exch[i]&amp;[j])˜{Σcol[j]}÷{Σcol[i]+Σcol[j]}  [1]
 
     That is, the probability of exchanging item [i] with item [j] may be proportional to the sum of elements in column [j] divided by a quantity which is the sum of elements in a column containing element [i] plus the sum of elements in a column containing element [j] This operation may be referred to as a column-based sort. 
       FIG. 11  is a schematic diagram of a representation of column probabilities associated with data presented in  FIG. 10 . In  FIG. 11 , a column-based sort may be carried out as described below.  FIG. 11  may be regarded as a column-column array or grid  76 . That is “columns” in  FIG. 11  and “rows” in  FIG. 11  each represent columns in expression [1]. Said another way, the column sum for column [i] may be read at the end of a horizontal “row” in  FIG. 11 , and the column sum for a column [j] may be read at the bottom of a vertical “column” in  FIG. 11 . 
     By way of example and not by way of limitation, to evaluate probability of exchanging a “3” with a “5” in the sequences represented in  FIG. 10 , one may employ expression [1] as follows:
 
P[Exch[i]&amp;[j])˜{Σcol[j]}÷{Σcol[i]+Σcol[j]}  [1]
 
P[Exch[3]&amp;[5])˜{Σcol[5]}÷{Σcol[3]+Σcol[5]}
 
P[Exch[3]&amp;[5])˜{10}÷{18+10}
 
P[Exch[3]&amp;[5])˜{10}÷{28}
 
P[Exch[3]&amp;[5])˜0.4
 
     This result may be found in grid  76  at the matrix location (3, 5). 
     The annotation scheme of  FIG. 11  may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective column-based sort represented in  FIG. 11  will not be provided here. 
     Using column [j] as the numerator and using (column [i]+column [j]) as the denominator in expression [1] may have the effect of sorting items in the random sequence into approximately the same order as the columns if sorted into descending order by their respective column sums. 
     Statistics may be extracted from rows of grid  76  that may roughly measure the number of predecessors for each item in a solution sequence. The sorted sequence of the column-based sort may be sorted into order, using the predecessor data from the column-extraction above to weigh probability of exchanging two items. Probability of exchanging two items [i] and [j] may be expressed as:
 
P[Exch[i]&amp;[j])˜{Σrow[i]}÷{Σrow[i]+Σrow[j]}  [2]
 
     That is, the probability of exchanging item [i] with item [j] may be proportional to the sum of elements in row [i] divided by a quantity which is the sum of elements in a row containing element [i] plus the sum of elements in a row containing element [j]. This operation may be referred to as a row-based sort. 
       FIG. 12  is a schematic diagram of a representation of row probabilities associated with data presented in  FIG. 10 . In  FIG. 12 , a row-based sort may be carried out as described below.  FIG. 12  may be regarded as a row-row array or grid  78 . That is “columns” in  FIG. 12  and “rows” in  FIG. 12  each represent rows in expression [2]. Said another way, the row sum for row [i] may be read at the end of a horizontal “row” in FIG.  12 , and the row sum for a row [j] may be read at the bottom of a vertical “column” in  FIG. 12 . 
     By way of example and not by way of limitation, to evaluate probability of exchanging a “3” with a “5” in the sequences represented in  FIG. 10 , one may employ expression [2] as follows:
 
P[Exch[i]&amp;[j])˜{Σrow[i]}÷{Σrow[i]+Σrow[j]}  [2]
 
P[Exch[3]&amp;[5])˜{Σrow[3]}÷{Σrow[3]+Σrow[5]}
 
P[Exch[3]&amp;[5])˜{9}÷{9+17}
 
P[Exch[3]&amp;[5])˜{9}÷{26}
 
P[Exch[3]&amp;[5])˜0.4
 
     This result may be found in grid  78  at the matrix location (3, 5). 
     The annotation scheme of  FIG. 12  may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective column-based sort represented in  FIG. 12  will not be provided here. 
     Using row [i] as the numerator and using (row [i]+row [j]) as the denominator in expression [2] may have the effect of sorting items in the random sequence into approximately the same order as the rows if sorted into ascending order by their respective row sums. 
     The row-based sort may be refined using contents of individual cells of grid  78  to weight probability of exchanging two items. Probability of exchanging two items [i] and [j] may be expressed as:
 
P[Exch[i]&amp;[j])˜{Σcell[i][j]}÷{Σcell[i][j]+Σcell[j][i]}
 
     That is, the probability of exchanging item [i] with item [j] may be proportional to the element in cell [i][j] divided by a quantity which is the sum of elements in cell [i][j] (i.e., a cell located by a row containing element [i] and a column containing element [j]) plus the sum of elements in cell [j][i] (i.e., a cell located by a row containing element [j] and a column containing element [i]). This operation may be referred to as a cell-based sort. 
     The result of the cell-based sort may be presented by generator  40  ( FIG. 1 ) as a new sequence. 
       FIG. 13  is a schematic diagram of a representation of cell probabilities associated with data presented in  FIG. 10 . In  FIG. 13 , a cell-based sort may be carried out as described below. By way of example and not by way of limitation, to evaluate probability of exchanging a “3” with a “5” in the sequences represented in  FIG. 10 , one may employ expression [3] as follows:
 
P[Exch[i]&amp;[j])˜{Σcell[i][j]}÷{Σcell[i][j]+Σcell[j][i]}  [3]
 
P[Exch[3]&amp;[5])˜{Σcell[3][5]}÷{Σcell[3][5]+Σcell[5][3]}
 
P[Exch[3]&amp;[5])˜{1}÷{1+2}
 
P[Exch[3]&amp;[5])˜{1}÷{3}
 
P[Exch[3]&amp;[5])˜0.3
 
     This result may be found in a grid  80  illustrated in  FIG. 12  at the matrix location (3, 5). 
     The annotation scheme of  FIG. 12  may be understood by one skilled in the art of statistical evaluations. In order to avoid prolixity a detailed description of each respective column-based sort represented in  FIG. 12  will not be provided here. 
     Using cell[i][j] as the numerator and using (cell[i][j]+cell[j][i]) as the denominator in expression [3] may have the effect of sorting items in the random sequence into approximately an order in which the probability of an item [i] occurring before an item [j] may be determined by the statistical data contained within accumulator  34  ( FIG. 1 ). 
     The result of this cell-based sort may be an output from generator  40  ( FIG. 1 ). The combination of steps described above may be important. Simple, deterministic sorts may not yield the careful balance between similarity and difference. In practice, a single sort (e.g., sorting by successors, by predecessors or by comparative weighting) may not fully reflect statistical characteristics of sequences represented in accumulator  34 . 
       FIG. 14  is a flow diagram illustrating a method for optimizing a sequential arrangement of items. In  FIG. 14 , a method  100  for optimizing a sequence of items according to an algorithm may begin with generating a predetermined number of initial sequences, as indicated by a block  102 . 
     Method  100  may continue with storing the initial sequences in a data store to establish a set of stored sequences, as indicated by a block  104 . 
     Method  100  may continue with statistically evaluating the stored sequences and storing statistical evaluation of the stored sequences, as indicated by a block  106 . 
     Method  100  may continue with employing the statistical evaluation to generate a new sequence, as indicated by a block  108 . 
     Method  100  may continue with evaluating the new sequence with respect to the stored sequences according to the algorithm, as indicated by a block  110 . 
     Method  100  may continue with posing a query whether the new sequence is acceptable, as indicated by a query block  112 . If the new sequence is acceptable, method  100  may proceed from query block  112  via a YES response line  114  to add the new sequence to the stored sequences, as indicated by a block  116 . 
     Method  100  may continue with posing a query whether the number of statistically evaluated sequences has reached a predetermined number, as indicated by a query block  118 . If the number of statistically evaluated sequences has reached the predetermined number, method  100  may proceed from query block  118  via a YES response line  120  to remove the least desired sequence from statistical evaluation, as indicated by a block  122 . 
     If the number of statistically evaluated sequences has not reached the predetermined number, method  100  may proceed from query block  118  via a NO response line  124 . 
     Method  100  may continue with including the new, acceptable, sequence to the statistically evaluated sequences, as indicated by a block  126 . Method  100  may continue with posing a query whether at least one predetermined condition has been met, as indicated by a query block  130 . 
     If the new sequence is not acceptable, method  100  may proceed from query block  112  via a NO response line  128  to pose the query indicated by query block  130 . 
     If the at least one predetermined condition has not been met, method  100  may continue from query block  130  via a NO response line  132  and method  100  may proceed to a locus  107 . Method  100  may proceed from locus  107  substantially as described above in connection with blocks  108 ,  110 ,  112 ,  116 ,  118 ,  122 ,  126 ,  130 . 
     If the at least one predetermined condition has been met, method  100  may continue from query block  130  via a YES response line  134  to present the best-sequence-yet-received at an output locus  136 . 
     It is to be understood that, while the detailed drawings and specific examples given describe preferred embodiments of the disclosure, they are for the purpose of illustration only, that the system and method of the disclosure are not limited to the precise details and conditions disclosed and that various changes may be made therein without departing from the spirit of the disclosure which is defined by the following claims: