Abstract:
Methods and apparatus are provided for selecting a maintenance plan such that the cost of the maintenance plan is the lowest or near the lowest. The method comprises receiving a set of maintenance actions, wherein one of the repair actions is likely to repair the failure mode. The set of maintenance actions is sequenced in the increasing order of their waiting times. Each maintenance action has an associated cost equal to a waiting time cost, an execution time cost and a material cost, wherein the waiting time of each maintenance action is the time required to requisition and receive material required to perform the maintenance action. The method also constructs a maintenance plan comprising a first requisition and a second requisition by assigning each of the sequence of maintenance actions to one of the first or the second requisition.

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0001]    This invention was made with Government support under Contract W56 HZV-05-C-0724 awarded by the United States Army. The Government has certain rights in this invention. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention generally relates to the logistics and repair of complex systems. Specifically, it relates to computerized systems in the field of repair parts requisition and maintenance optimization. 
       BACKGROUND 
       [0003]    Man has yet to invent a useful machine or a vehicle that can function throughout its designed useful life without some kind of maintenance or repair being performed. In fact, the lack of reasonable routine maintenance or repair will shorten the useful life of any asset, particularly for complex systems such as aircraft and manufacturing processes. 
         [0004]    When a useful asset suffers a casualty in the field, there are isolation tests that may be applied to isolate and to disambiguate the failure mode (“FM”), and then to narrow repair options down to a finite group of corrective actions (“CA”). Or otherwise, to establish that the group of CAs will not fix the FM. Each isolation test and its subsequent repair procedure have an estimated execution time necessary to complete the test or procedure. 
         [0005]    Further, each isolation test and repair requires a certain set of tools which may or may not be immediately available. If parts and tools are not available, they will have to be requested from another location such as an intermediate maintenance facility/repair depot. The depot can always order tools and parts that are not in stock and ship them to the field location when they become available. Typically, the depot will inform the field location of the estimated time to deliver parts and tools for each isolation action (“I”) and each repair action (“R”) so that the field maintenance facility gets a chance to incorporate this information into a maintenance plan, calculate a maintenance requirement and place an official request for tools and parts with the depot. However, the requisition time requires an additional wait time and a time value cost for that wait time. 
         [0006]    The traditional way of handling ambiguous failures in the field has been to order parts and tools for carrying out all the possible isolation tests at one time. Once the failure mode is isolated, the parts and tools needed for the identified corrective action are then ordered. This would be an optimum solution when all the possible repair actions are expensive (high parts cost, high execution time, and long wait times) compared to the time and costs of executing all the associated isolation tests. As such, the traditional maintenance philosophy required the field maintenance facility to place at most two requisitions, a primary requisition for all of the isolation tools and a secondary requisition for specific repair parts and tools as determined by the isolation procedures. 
         [0007]    Further, when one isolation tool requisition is made for all probable isolation procedures. This means that when parts with a short wait-time are mixed with parts that have a long wait-time in the same order, the longer wait time parts will delay the parts that could be available earlier. There may also be a return penalty for parts that are ordered and not used. As such, the maintenance action sequence planning problem and the parts order scheduling problem are mixed and cannot be solved independently as in a traditional setting. For example, if it is assumed that all parts and tools are always on hand, there will be no need for any parts requisition and one could determine the optimal sequence of maintenance actions by applying a cost function to expected repair procedures. However, the same sequence of maintenance actions may not be optimal if different parts have different wait-times. 
         [0008]    The traditional way is not always the best in practice. Some corrective actions could be done without performing an isolation procedure if they are inexpensive, even if the probability of success is small. For example, if there is a 1% probability that a failure may be caused by a $0.25 faulty light bulb or a 99% probability the failure is caused by a $100,000 line replaceable unit (“LRU”), then one would opt to replace the bulb without conducting the associated isolation procedure since the probability weighted cost is deminimus. The small probability cost that the bulb is the cause of the FM makes it cost effective not to do the isolation test for the bulb. Regardless of whether or not the casualty was actually fixed by the replacement bulb, a maintenance technician would have isolated the problem very quickly as to whether or not the bulb was the problem. The technician may have also repaired the problem at the same time. In this example, the objective was purely to reduce the repair cost. There could be more to the objective than cost such as time cost. There may be different times required to execute each repair or to order the parts for them. If all those are taken into account, a different action could be optimum to execute up front. 
         [0009]    Finding an optimal solution to this problem involves knowledge of the future choices. The only way to know all possible future choices is to do an exhaustive search of all possible permutations of maintenance actions and parts ordering schedules against a cost function. The time required for executing the exhaustive search increases exponentially with the number of actions in the probable group of corrective actions and makes it impractical for field execution. Such a computationally intensive process may take days to provide a solution for a casualty that may involve only a handful of probable maintenance actions. 
         [0010]    Accordingly, it is desirable to have a maintenance ordering and scheduling system that quickly determines the lowest cost maintenance plan for a given failure condition with multiple probable failure modes even if such a determination is sub-optimal but yet close to optimal. In addition, it is desirable to obtain a near to optimal maintenance plan that accomplishes the corrective action in no more than two supply requisitions. Furthermore, other desirable features and characteristics of the present invention will become apparent from the subsequent detailed description of the invention and the appended claims, taken in conjunction with the accompanying drawings and this background of the invention. 
       BRIEF SUMMARY 
       [0011]    A method is provided for minimizing the cost of a maintenance plan for correcting a failure mode in a complex machine. The method comprises receiving a plurality of maintenance actions related to the failure mode, wherein one of the maintenance actions of the plurality is likely to repair the failure mode. Each maintenance action has an associated cost equal to a waiting time cost, an execution time cost and a material cost, wherein the waiting time of each maintenance action is the time required to requisition and receive material required to perform the maintenance action. The method also comprises constructing a plurality of different maintenance plans each comprising a sequence of maintenance actions by assigning zero or more of the maintenance actions of the sequence with the longest of the waiting times to the second requisition, each maintenance plan includes a first requisition and a second requisition. The method further comprises calculating an associated cost of each maintenance plan and determining which of the associated cost maintenance plans is lowest. 
         [0012]    A method is provided for determining a lowest total cost maintenance plan. The method comprises receiving a sequence of maintenance actions in an order of a waiting time for each maintenance action, wherein one of the maintenance actions is likely to repair the failure mode. Each maintenance action has an associated cost equal to a waiting time cost, an execution time cost and a material cost, wherein the waiting time of each maintenance action is the time required to requisition and receive material required to perform the maintenance action. The method also constructs a maintenance plan comprising a first requisition and a second requisition by assigning each of the sequence of maintenance actions to one of the first and second requisition. 
         [0013]    A computer readable medium is provided for. The computer readable medium includes instructions that when executed by a computing device receive a sequence of maintenance actions in an order of a waiting time for each maintenance action. Each one of the maintenance repair actions is likely to repair the failure mode and has an associated cost equal to a waiting time cost, an execution time cost and a material cost. The waiting time of each maintenance action is the time required to requisition and to receive material required to perform the maintenance action. A plurality of different maintenance plans are constructed from the sequence of maintenance actions, each maintenance plan includes a first requisition and a second requisition which are established by assigning the maintenance actions of the sequence with the longest waiting times to the second requisition. The associated cost of each maintenance plan is calculated and the lowest associated cost maintenance plan is determined. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]    The present invention will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements. 
           [0015]      FIG. 1  is a functional block diagram of system described herein. 
           [0016]      FIG. 2  is a simplified flow chart illustrating the exemplary embodiments disclosed herein for creating a heuristic method for optimizing a maintenance plan. 
       
    
    
     DETAILED DESCRIPTION 
       [0017]    The following detailed description is merely exemplary in nature and is not intended to limit the invention or the application and uses of the invention. Furthermore, there is no intention to be bound by any theory presented in the preceding background or the following detailed description. Nor is there an intention to be bound by a particular data source. 
         [0018]    A heuristic alternative method, for optimizing the costs of Correction Action (“CA”) is presented herein below. When a malfunction occurs in a complex machine, the root cause may stem from one of several sources. Some root causes may be expected and some may not be readily apparent. A group of several potential root causes of a malfunction may be known as an ambiguity group or an ambiguity set. The corrective action (“CA”) for the failure mode (“FM”) is probably included in the ambiguity set but there is general uncertainty as to which one. 
         [0019]    When using a traditional global search method to create a maintenance plan, a computing device executes a software application resident on a computer readable medium that searches one of more databases for each and every possible cause of the failure regardless of probability of occurrence. This creates an exhaustive ambiguity group. The software application then compiles a list of all of the repair and isolation procedures for each of the possible causes. 
         [0020]    The global search method examines all of the possible combinations and permutations of the ambiguity set&#39;s direct costs, repair wait times and probabilities of repair success to determine the absolute optimum maintenance plan with which to affect the repair. 
         [0021]    However, the time and computational power required for such an endeavor makes a global search system cumbersome and relatively expensive. 
         [0022]    The system and method described herein reduces costs by eliminating computational overhead by making several assumptions in order to produce an optimal or a nearly optimal maintenance plan in a faster and less expensive manner The subject matter described herein will be disclosed by way of non-limiting examples where an equipment casualty may occur due to a particular failure mode(s). 
         [0023]      FIG. 1  is a functional block diagram of a system including the subject matter disclosed herein. In an embodiment, the system  10  may be a stand alone system and communicate with a computerized diagnostics system  50  via a computer interface  30 . The system  10  may be comprised of hardware, software, firmware or a combination thereof. In other embodiments, the system  10  may be a module within the computerized diagnostics system  50  or vice versa. The system  10  may also be implemented in another device or distributed over or within a network  60 . 
         [0024]    The system  10  includes a processor  15  in operable communication with both of the computer interface  30  and a memory storage device  20 . Processor  15  may be any suitable type of processor that currently exists or that may exist in the future. As non-limiting examples, the processor  15  may be any suitable stand alone processor, a programmable logic device, a general purpose processor, a special purpose processor, or a co-processor. Processor  15  is also a non-limiting example of a computer readable medium. 
         [0025]    The memory device  20  stores thereon a set of instructions that when executed by the processor  15  causes the system  10  to process a specific ambiguity set of failure modes and related corrective actions received from the computerized diagnostics system  50 . The defined corrective actions may, in some embodiments be sequenced in order of an expected requisition time for each corrective action. However, one of ordinary skill in the art will recognize that the corrective actions may be sequenced using a different index. 
         [0026]    The memory device  20  may be any suitable type of volatile or non-volatile memory devices. Exemplary, non-limiting memory devices may include flash memory, random access memory, read only memory, programmable read only memory, electronic erasable read only memory, programmable logic devices, magnetic disks, optical disks and any suitable memory devices that currently exists or is developed in the future. The foregoing memory devices are also non-limiting examples of computer readable media. 
         [0027]    In one exemplary embodiment, a failure mode may have four likely causes (X, Y, Z and C) that are included in an ambiguity group shown in Table 1. In the context of the system  10 , an ambiguity group is a set of known probable causes of the failure mode. Any of the probable causes may have caused the failure and each cause will have a probability of being the actual cause. In some ambiguity groups, it is possible for the actual cause not to be included in the group, but may be appended by other means to an ambiguity group for the purposes described herein. 
         [0028]    Whether or not the probabilities of each failure mode of the ambiguity group cover every possible cause such that their summed probabilities total 100%, the probabilities of each failure mode making up the ambiguity group are normalized such that the sum of the probabilities in the ambiguity group will total 100%. It will be appreciated by those skilled in the art that some failure modes that are improbable may not be included in the ambiguity group for the sake of efficiency and cost reduction. An exemplary ambiguity group may look like the example in Table 1 below. 
         [0000]                                            TABLE 1                       Failure Mode   X   Y   Z   C                           Probability   10%   33%   33%   24%           Isolation Procedure   I x     I y     I z     I c             Repair Procedure   R x     R y     R z     R c                          
The selection, probabilities and ranking of the various repairs (R) and isolation tests (I) in Table 1 may be compiled by the computerized diagnostics system  50 . Exemplary, non-limiting computer based diagnostics systems that disambiguate a reported casualty into its probable causes and then ranks the isolation and repair actions related to the cause according to their wait times have been developed, and are described more fully in U.S. Pat. No. 6,748,304 to Felke et al., which is incorporated herein by reference in their entirety.
 
         [0029]    As an output, the computerized diagnostics system  50  may produce a prioritized list of repair procedures intermingled with isolation procedures in order of ascending wait times for parts. For example the computerized diagnostics system  50  may generate the hypothetical prioritized list: R x , I x , I y , I z , I c , R c , R y , R z . Where parts for repair X may be on hand and parts for repair Z may take weeks or months to be delivered. 
         [0030]    This particular example illustrates that, based on the wait times for parts and tools, repair X (R x ) should be executed first followed by isolation and repair procedures I x , I y , I z , I c , R c , R y , and R z . Executing the actions in the order provided by the computerized diagnostics system  50  may minimize the overall wait time where all parts and tools are ordered simultaneously. If the performance of R x  was unsuccessful, isolation procedures I y , I z  and I c  would then be performed in order, followed by repair R c , R y , or R z  as may be identified by isolation action I y , I z  and I c . The computerized diagnostics system  50  may also determine the overall cost of ordering and performing the entire list of corrective actions. 
         [0031]    It will be appreciated by one of ordinary skill in the art that in this simplified example that the performance of repair R x  is suggested to be accomplished before the isolation procedure I x  for cause X. A successful repair R x  will nullify any need for conducting I x  and any other repair and/or isolation procedures subsequent to completing R x . If R x  was unsuccessful, the need to conduct I x  would still be obviated because the associated repair R x  failed to clear the casualty. Therefore, the technician already knows that cause X is not the problem. However, in other exemplary embodiments, if I x  is capable of detecting multiple FMs then I x  should be conducted after repair R x  in regard to the additional FMs. 
         [0032]    The system  10 , as described herein, would apply a wait cost minimization feature to the output of computerized diagnostics system  50 . This is accomplished by displacing specific repair and isolation actions that have the longest wait times from the list provided by the computerized diagnostic system  50  into a secondary parts requisition. The system  10  thereby creates a first requisition and a second requisition, where the second requisition is placed if needed after the primary requisition has arrived and the corrective actions have been executed. The system  10  implements an iterative process. Both repair and isolation actions are removed one-by-one from the first requisition into the second requisition until a minimum cost maintenance plan is reached. The “minimum cost” comprises a materials cost and a time cost. The time cost further comprises a wait time cost (which equals the longest wait time in the requisition) and an execution time cost. In other exemplary embodiments, the minimum cost function could include other or additional variables of concern, such as personnel costs. 
         [0033]    In another exemplary embodiment, the methodology implemented by the system  10  is disclosed in further detail in regard to  FIG. 2  using the data in Table 2. Although this relatively simple example is presented for the sake of brevity and clarity, the method below is applicable to more complex cases such as that illustrated in Table 1. It will be appreciated by those of ordinary skill in the art that the assumptions and the processes disclosed below may be altered without straying from the intended scope of this disclosure. Different failure mode assumptions may be made and logical processes may be combined, split apart, rearranged and substituted with similar processes and still fall within the intended scope of the subject matter disclosed herein. 
         [0034]    In one exemplary embodiment, the basic assumptions of the method include:
       1) The likelihood of each FM occurring in an ambiguity set is normalized to be the probabilities of each FM occurring.   2) There is only one cause in the ambiguity set so the probabilities of each FM sum to total 100%.   3) Parts and tools needed for a maintenance action will be requested from a depot using no more than two requisitions and are available at the depot or on hand. If they are not available, the estimated time of when they will be made available is known.       
 
         [0038]    Table 2 below is a simple two FM data chart where the cost function of the overall maintenance action will be assumed to equal the dollar cost of tools and parts plus the time value costs of any wait time and the time to repair. 
         [0000]    
       
         
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Failure Mode 
                 1 
                 2 
               
               
                   
                   
               
             
             
               
                   
                 Probability of Occurrence 
                 80% 
                 20% 
               
               
                   
                 Isolation Cost (IC) 
                  0 $ 
                 20 $ 
               
               
                   
                 Repair Cost (RC) 
                 40 $ 
                 10 $ 
               
               
                   
                 Execution Time Isolation (EI) 
                 10 $ 
                 10 $ 
               
               
                   
                 Execution Time Repair (ER) 
                 20 $ 
                 10 $ 
               
               
                   
                 Waiting Time Isolation (WI) 
                 11 $ 
                  1 $ 
               
               
                   
                 Waiting Time Repair (WR) 
                  0 $ 
                 10 $ 
               
               
                   
                   
               
             
          
         
       
     
         [0039]      FIG. 2  is a flow chart of the method using the exemplary two-failure-mode ambiguity data of Table 2. Having been presented with the pertinent casualty data, a computerized diagnostics system  50  determines that there are two (n=2) likely failure modes (1 and 2) in the ambiguity group, produces a list of four (2n) repair and isolation actions and determines the cost data for each repair (RC 1  and RC 2 ) and each isolation (IC 1  and IC 2 ) at process  105 . 
         [0040]    At process  110 , the computerized diagnostics system sequences the four possible actions in an increasing order of logistics wait time cost, for example, isolation wait time cost (WI) and repair wait time cost (WR). As such, the corrective actions are grouped together where each group of equal wait times is arranged in order of ascending wait time costs (WI and WR). According to Table 2 the list would could be R 1  (0), I 2  (1), R 2  (10), I 1  (11). 
         [0041]    It should be noted that if the waiting time cost for I 1  and I 2  were $10 and $0, respectively instead of 11 hours and 1 hour, the list may have any of the orders (R 1 , I 2 , R 2 , I 1 ), (R 1 , I 2 , I 1 , R 2 ), (I 2 , R 1 , R 2 , I 1 ) or (I 2 , R 1 , I 1 , R 2 ). This is so because the wait time cost of R1 and I2 would both be zero and the wait times of I 1  and R 2  would both be 10. Given this flexibility, under these assumptions, the identical wait time actions in this order may be sequenced as per the minimum cost sequence determined by the process in  125 . 
         [0042]    At process  115 , all possible first requisition combinations are determined In the non-limiting example, there are four possible primary requisitions for tools and parts given their wait time order and the superset of the resulting secondary requisitions. The actual secondary requisition will be determined via optimization after the result information for the execution of actions in the first requisition is available. Note that multiple possibilities exist for the second requisition for a given first requisition. For example, for case 4 in Table 3, {I2, R2} could be a second requisition or {R2} alone could be the second requisition or the second requisition could be a null set if the execution of R1 in the first requisition fixed the problem. 
         [0000]                                TABLE 3                           Superset of           First Order   Second Orders                           1) R 1 , I 2 , R 2 , I 1     —           2) R 1 , I 2 , R 2     I 1             3) R 1 ,I 2     R 2 ,I 1             4) R 1     I 2 , R 2 , I 1                          
It will be appreciated by those skilled in the art that adding more procedures requires more computing time as the complexity of the plan grows exponentially
 
         [0043]    At process  120 , a cost function is defined by the computerized diagnostics system  50  for evaluating the cost of each set of requisitions. Although other cost elements may be included as the situation so requires, a basic cost function for one known requisition will generally have the form: 
         [0000]      Requisition Cost=Σ tool and part costs+C {maximum waiting time value for any part in the requisition+weighted average execution time value of the actions related to the requisition}
 
         [0044]    Where, the value of parameter C determines relative importance of cost and time to the user. For this exemplary embodiment, C may equal 0.5. The total cost of a maintenance plan is the sum of the costs for the two requisitions in that maintenance plan. For a given first requisition, different corresponding second requisitions may be possible depending on the outcomes of the actions in the first requisition. These corresponding second requisitions form a set of possible corresponding second requisitions; each corresponding second requisition in the set will have a known probability. The total cost of the second requisition will be probability weighted costs of all these possible second requisitions in the superset. 
         [0045]    At process  125 , the maintenance actions in the primary requisition (R 1 , I 2 , R 2 , I 1 ) are sequenced by the computerized diagnostic system  50  to minimize the cost function assuming that all parts and tools are instantaneously available, the lowest cost sequence for the chosen primary requisition (R 1 , I 2 , R 2 , I 1 ) is (I 1 -R 1 -R 2 ). Since I 2  is not needed in this chosen execution sequence of the first requisition, the first requisition effectively reduces to (I 1 , R 1 , R 2 ) from (R 1 , I 2 , R 2 , I 1 ). The execution cost for the first requisition (I 1 , R 1 , R 2 ) based on Table 2 is: 
         [0000]    
       
         
           
             
               
                 
                   Cost 
                   = 
                     
                    
                   
                     
                       IC 
                       1 
                     
                     + 
                     
                       
                         ( 
                         
                           P 
                           2 
                         
                         ) 
                       
                       × 
                       
                         RC 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           P 
                           1 
                         
                         ) 
                       
                       × 
                       
                         RC 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     0 
                     + 
                     
                       ( 
                       
                         .8 
                         × 
                         40 
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         .2 
                         × 
                         10 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   $34 
                 
               
             
           
         
       
     
         [0000]    where, 
         [0046]    IC 1 =cost of performing I 1 ; 
         [0047]    P 2  is the probability of R 2  correcting the problem; 
         [0048]    P 1  is the probability of R 1  correcting the problem; 
         [0049]    RC 2  is the cost of performing R 2 ; and 
         [0050]    RC 1  is the cost of performing R 1    
         [0000]    It will be appreciated by one of ordinary skill in the art that the cost for isolation action IC 2  in the sequence are dropped from the equation because I 2  is the last action in the sequence and is essentially not worth performing or evaluating. As a principle, it may be dropped from the calculation. As such, the execution cost for (I 1 -R 1 -R 2 ) is $34. 
         [0051]    At process  130 , the cost of the second requisition is calculated. However, because a second requisition will not be needed for this particular first requisition, the cost of the second requisition is zero. The first plan based on the first requisition is received and recorded by the system  10 . 
         [0052]    The cost of the maintenance plan with first requisition (I 1 , R 1 , R 2 ) and no second requisition is computed using, 
         [0000]      the waiting time=MAX( WI   1   ,WR   1   ,WR   2 )hr= WI   1 =11; 
         [0000]      the total execution time=( EI   1   +P (2) EI   2   +ER   1 )=(10+0.8*(20)+0.2*(10))=$28; therefore, 
         [0000]      the total time value=$28+$11=$39; and 
         [0000]      the total cost of the maintenance plan=($34+0.5×$39)=$53.50=execution cost+time cost.
 
         [0053]    At process  140 , the last maintenance action R 2  in the waiting time sequence is kicked out of the primary first requisition and placed into the superset of the possible corresponding second requisition actions to create a secondary first requisition (I 1 , R 1 ). At process  145 , the system  10  sends the second requisition to the computerized diagnostics system  50  which sequences the maintenance actions in the second requisition corresponding to (I 1 , R 1 ) as the first requisition in order to reduce the cost of the second requisition. Since, at this point there is only a single maintenance action R 2 , the only possible sequence of the second requisition is R 2  if the execution of the maintenance plan for the first requisition failed to fix the problem. The other possibility is a null action set for the second requisition if the problem got fixed by execution of the actions in the secondary first requisition (I 1 , R 1 ). 
         [0054]    At process  150 , the system  10  re-evaluates the cost of the first requisition and the corresponding second requisition based upon the cost function. The first requisition sequence is now I 1 , R 1 . The second requisition sequence is R 2  with a probability of 20% (P 2 ) because the probability of that the casualty being repaired using the primary first requisition with R 1  is 80% (i.e. probability of being the cause) and the secondary requisition will never have to be placed. 
         [0055]    The total execution cost for both the first and the second requisitions based on Table 2 is, then: 
         [0000]    
       
         
           
             
               
                 
                   Cost 
                   = 
                     
                    
                   
                     
                       IC 
                       1 
                     
                     + 
                     
                       
                         ( 
                         
                           P 
                           2 
                         
                         ) 
                       
                       * 
                       
                         RC 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           P 
                           1 
                         
                         ) 
                       
                       * 
                       
                         RC 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     0 
                     + 
                     
                       ( 
                       
                         .8 
                         * 
                         40 
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         .2 
                         * 
                         10 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   $34 
                 
               
             
           
         
       
     
         [0056]    The total cost of the complete maintenance plan with the secondary first requisition (R 1 , I 1 ) and the corresponding second requisition R2 is now the wait time cost of the first requisition (R 1 , I 1 ) and the probability weighted wait time cost of the second requisition R 2  plus the total execution time costs and the parts and tools. The total cost of the maintenance plan which includes first requisition (I1,R1) and if needed a second requisition R2 is calculated as follows: 
         [0000]      the waiting time=MAX( WI   1   ,WR   1 )+0.2 *WR   2 =$11+0.2*$10=$13; 
         [0000]      the total execution time=( EI   1   +P (2) EI   2   +ER   1 )=(10+0.8*(20)+0.2*(10))=$28; therefore, 
         [0000]      the total time value=$41; and 
         [0000]      the total cost of the maintenance plan=($34+$41/2)=$54.5 
         [0057]    The displacement of R 2  from the primary first requisition to a possible second requisition kept the total plan&#39;s total parts and tools cost the same at $34. However, the additional 20% probability that the maintenance plan would have to wait 10 additional hours for the second requisition to arrive, should the first fail, would drive the probable wait time up to $13 from $11. In this case, kicking out the longest delayed tools and parts to a second requisition actually increased the probable cost of the maintenance plan. 
         [0058]    At decision point  155 , it is determined whether the cost of the second plan is more or less costly than the previous plan. If the second plan is more costly, the method ends and the previous plan is placed in memory at process  156 . If the second plan is less costly, then the method loops to process  140  where the next to last maintenance action in the first requisition sequence is also displaced into the second requisition and the process continues until the total cost of a plan actually rises as compared to the immediately prior plan. 
         [0059]    At decision point  160 , it is determined if a minimum cost plan for each of the sequences in Table 3 have been determined. If not, then the process loops back to process  120  where the next sequence received from the computerized diagnostic system  50  (See, Table 3) is processed as was the first two. If a minimum cost plan for all of the sequences in Table 3 is determined, the process ends and the lowest cost plan in memory is rendered to a user at process  165 . 
         [0060]    Those of skill in the art will appreciate that the various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. Some of the embodiments and implementations are described above in terms of functional and/or logical block components (or modules) and various processing steps. However, it should be appreciated that such block components (or modules) may be realized by any number of hardware, software, and/or firmware components configured to perform the specified functions. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention. For example, an embodiment of a system or a component may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, or the like, which may carry out a variety of functions under the control of one or more microprocessors or other control devices. In addition, those skilled in the art will appreciate that embodiments described herein are merely exemplary implementations 
         [0061]    The various illustrative logical blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. The word “exemplary” is used exclusively herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments. 
         [0062]    The steps of a method or algorithm described in connection with the embodiments disclosed herein may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor such the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC. The ASIC may reside in a user terminal. In the alternative, the processor and the storage medium may reside as discrete components in a user terminal 
         [0063]    In this document, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Numerical ordinals such as “first,” “second,” “third,” etc. simply denote different singles of a plurality and do not imply any order or sequence unless specifically defined by the claim language. The sequence of the text in any of the claims does not imply that process steps must be performed in a temporal or logical order according to such sequence unless it is specifically defined by the language of the claim. The process steps may be interchanged in any order without departing from the scope of the invention as long as such an interchange does not contradict the claim language and is not logically nonsensical. 
         [0064]    Furthermore, depending on the context, words such as “connect” or “coupled to” used in describing a relationship between different elements do not imply that a direct physical connection must be made between these elements. For example, two elements may be connected to each other physically, electronically, logically, or in any other manner, through one or more additional elements. 
         [0065]    While at least one exemplary embodiment has been presented in the foregoing detailed description of the invention, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing an exemplary embodiment of the invention. It being understood that various changes may be made in the function and arrangement of elements described in an exemplary embodiment without departing from the scope of the invention as set forth in the appended claims.