Abstract:
A demand-driven, mixed-model manufacturing line design system utilizes Demand Flow Technology to design an efficient mixed-model manufacturing line. Concurrent manufacture of several products in a continuous flow, mixed-model product line designed utilizing the design system exhibits essentially none of the inefficiencies associated with mixed-model, batch-oriented manufacturing lines. The method according to the present invention includes a product synchronization step for synchronizing product flow between unit processes, a resource balancing step to ensure sufficient resources are provided at each process, thereby reducing the product cycle time at each process to be approximately equal to an operational cycle time calculated for each process, a cell definition step for consolidating constant volume processes, an operational definition step for grouping tasks associated with the consolidated processes within each cell, and an operation balancing step for defining additional cells based on actual product cycle time diversity. A design system according to this invention includes a computer system wherein machine readable code accessible to the computer system enables execution of the above described method.

Description:
BACKGROUND 
     1. Field of the Invention 
     This invention relates to computer systems and methods for process design, and more particularly to a computer system and method for designing a mixed-model manufacturing process. 
     2. Description of the Related Art 
     When designing a manufacturing process, the tasks performed by individual workers and work stations and the physical flows between work stations and materials storage areas must be considered. To make a product, the process design engineer prepares drawings and charts which indicate the various components and subassemblies going into the product as well as the sequence of events that should be followed in combining them. The resulting manufacturing line design is straight forward since a single-model product may be made in a single, static manufacturing line. 
     The design of a mixed-model manufacturing line, on the other hand, must be consistent with a process technology strategy that will permit making all functional variations (i.e., models) in a family of products. The wide variation in product features and functionality necessitates flexibility in the manufacturing line in order to accommodate the various process changes, customer demanded changes and material variations associated with the variety of products in the family of products. Conventional mixed-model manufacturing lines are organized into fairly autonomous departments grouped by either like-equipment or like-skills in order to produce various components of the products. 
     To produce each product, the manufacturing line for each product includes a set of processes. Traditionally, mixed-model process lines use batch processing since only a single product “run” is produced at any given time. The path of individual products through various work stations and departments is specified using “routing sheets,” which contain the operating steps and routing required for each part, and “process flow charts,” which describe the desired sequence of specific tasks (e.g., inspections, movements, and storage operations) for that product. 
     One problem with conventional mixed-model manufacturing lines is that the variety of products produced by the factory are forced to compete for manufacturing resources and equipment. This competition for resources manifests itself in the batch scheduling where only one product is manufactured at a time. This batch scheduling in turn results in long manufacturing latency, or lead time, for other products. Costs associated with this latency include the opportunity cost of customers unwilling or unable to tolerate the long manufacturing lead times, the costs of re-tooling a machine or work station when switching production from one product to another, and the costs of maintaining sufficient work in process (WIP) at each of the processes to permit filling anticipated product orders. 
     Customers who are unwilling or unable to tolerate a long manufacturing lead time will look to other sources to fill their demand. Lost customers affect the factory&#39;s profitability as they represent lost revenue. 
     Retooling increases the product&#39;s manufacturing cost. In that event, either the cost is absorbed by the factory, thereby reducing unit profit, or else the cost is passed on to the customer in the form of higher selling prices. Unless the product is unique and unavailable elsewhere, a higher selling price usually results in lower demand and, consequently, in lower revenues for that product. 
     Work-in-process represents a lost opportunity cost. Material and labor in WIP represents tied-up capital that cannot be recovered until processing has been completed and the finished goods (the products) are shipped to the customer. 
     Manufacturers have developed various manufacturing strategies which seek to mitigate some of the costs associated with mixed-model manufacturing process lines. One strategy, commonly called “just-in-time” (JIT) manufacturing, teaches that just enough work is started to ensure that WIP is minimized. This strategy, however, does not alleviate the batch scheduling inefficiencies of mixed model manufacturing lines, and may result in an excess amount of finished goods inventory. The capital that was invested in WIP is, in JIT processing, invested in finished goods. Other strategies that promise to solve the problems of manufacturing processes, such as “total market quality” (TQM) and the like, suggest changes in the way conventional mixed-model manufacturing lines are managed but do not solve the problems of conventional mixed model manufacturing lines. Accordingly, use of these various strategies has resulted in improvements to the batch-oriented scheduled processing associated with a mixed-model manufacturing line, but the fundamental problems of manufacturing latency, retooling, and unfinished and finished goods inventory remain. 
     Another problem with conventional mixed-model manufacturing line designs is that dissimilarity in the types of processes used to manufacture the various products introduces resource inefficiencies, such as machine and labor inefficiencies, associated with idle process lines. As discussed above, conventional mixed-model manufacturing is a batch process in which a process group or department works on only one particular product at any given time. Work orders are used to schedule and issue materials required to create the components and subassemblies of that product. These materials are batch-processed into the components and subassemblies, and then staged as WIP until the next process is ready to receive them. During the time the batch process is occurring, the WIP in upstream processes must wait until it can be scheduled into the current process, thereby tying-up capital in the upstream WIP. In addition, downstream processes may be idle while the upstream process completes the batch processing. This idle machine time and the labor and overhead required to maintain the upstream idle process represents an undesirable inefficiency and an additional cost to the factory. The accumulation of WIP and the frequency of idle processes is aggravated when the actual process time (the time to process a standard batch size) at each process is different. 
     It is thus apparent that there is a need for a system and method for designing a mixed-model manufacturing process line which eliminates manufacturing latencies, and which embodies manufacturing efficiencies and WIP control without incurring a large finished goods inventory. There is also a need for a system and method for designing a mixed-model manufacturing process line which coordinates resources and processes to eliminate idle processes caused by batch-process delays, and delays due to process time imbalances. 
     SUMMARY OF THE INVENTION 
     In accordance with this invention, a demand-driven, mixed-model manufacturing line design system and method permit design of a mixed-model manufacturing line for concurrent manufacturing of product families. The resulting mixed-model manufacturing line, or Demand Flow™ (DF) line, exhibits essentially none of the inefficiencies associated with the mixed-model, batch-oriented manufacturing lines of the prior art. 
     Embodiments of the invention include a computer and software for implementing a method of designing a demand flow manufacturing line. In most embodiments, the computer includes input devices (e.g., keyboards, pointers, and bar code readers), display devices (e.g., video terminals and printers), internal memory (e.g., RAM and ROM) for storing data and program instructions, a fixed memory (e.g., magnetic hard drives, floppy drives, CD ROM drives) for long term storage of data and programs, and a CPU that interacts with the input devices and the display devices and processes data according to program instructions. 
     Embodiments of the method incorporated in the program instructions include algorithms for product and process definitions and balancing or synchronization of product flow. Each product in the family of products to be manufactured by the manufacturing line is defined in terms of a set of processes that are used to manufacture that product. A process is defined in terms of a set of tasks for making the product and is parametrized in terms of process yield, process path type and other parameters. Processes are characterized in that product fallout (e.g., scrap or routing onto a rework or option path) only occurs during the final task of the process. Processes are members of process paths. Process paths are sequences of processes and can be categorized as one of four types: a main path, a feeder path, an option path, or a rework path. 
     Once each product is defined in terms of a set of processes, families of products that share a particular process are defined. In addition, knowing what the final end-of-line demand-at-capacity will be for each product, a demand-at-capacity for each process can be determined. The demand-at-capacity is the maximum amount of total product that the manufacturing line or process is required to produce. For example, if the manufacturing line is required to produce 100 units of product A and 100 units of product B, the total demand-at-capacity at end-of-line (EOL) is 200 units. If both products are members of a family of products defined in terms of process C, then process C has a demand-at-capacity of 200 units. In some embodiments, a demand-at-capacity for each product at each process is determined. 
     Following the demand-at-capacity calculation, an operational-cycle-time (alternatively referred to as the takt time) is calculated. The operational-cycle-time is defined as the maximum amount of time in which a unit of product or a component for a unit of product must be produced by a process in order to ensure that the demand-at-capacity for that product at that process is satisfied. 
     The takt time for each process can be balanced, resulting in a continuous mixed-model process flow, by adjusting the operating hours for the process (i.e., by placing additional resources in the process), relocating work, or redesigning product definition or process sequences so that the takt time for each process is approximately a constant. When the balancing is completed, the manufacturing line to accommodate the demand-at-capacity for each product in the family of products manufactured includes sequences of processes that produce to the takt time (i.e., all of the processes take approximately the same time to complete). The continuous mixed-model process flow defined in this fashion, therefore, eliminates WIP accumulation and idle processes. Additionally, the concurrent manufacture of mixed models (i.e., products) essentially eliminates long manufacturing lead times associated with traditional batch-processing manufacturing lines. 
     In some embodiments, further balancing of the manufacturing line is incorporated. Each of the products can additionally be defined in terms of the work content performed at each process that defines the product. The work content is defined by a sequence of events that in turn is defined as a series of tasks. Some embodiments include a standardized sequence of events to facilitate definition of processes and work contents. Each task in a sequence of events can be characterized in terms of work content times for machines and labor and an attribute for indicating whether the work content adds value to the product or does not add value to the product. For example, machine set-up and transport tasks do not add value to the product. Manufacturing line improvements that eliminate or reduce tasks that do not add value may then be incorporated. 
     In another embodiment, a demand weighted average cycle time is computed for each process. The demand weighted average cycle time can be computed from the demand-at-capacity for each product at each process and the actual time for each product to pass through each process. The demand weighted average cycle time for each process is compared with the takt time. If the demand weighted average cycle time for a process significantly exceeds the takt time for that process, the number of resources used to perform that process can be increased until the demand weighted average cycle time is acceptably comparable with the takt time. 
     In accordance with another embodiment of the methods of this invention, each process can be redefined as at least a portion of a cell. A cell is defined as a grouping of dissimilar resources (for example, people and machines) in a logical and sequential manner to facilitate the flow of a product or a family of products. Different products that flow through the cell may require different times. A cell is usually characterized as a series of processes with similar work contents (i.e. taking similar times). Processes having product fallout (i.e., less than 100% yield) are typically delegated to the last process in a cell or isolated in an independent cell, however these processes may also occur within cells. In one embodiment, each process defines a cell. In another embodiment, a cell is defined by adding sequential processes to the cell under the direction of a user of the design system. In yet another embodiment, the design system defines cells by adding sequential processes to the cell in accordance with some requirement such as that the cell does not exceed a maximum work volume. Once the cells are defmed, operations within the cell can be defined. Operations can be defined according to machine dominated tasks if the work content is dominated by machine work, according to labor dominated tasks if the work content is dominated by labor, or according to general mixed tasks. An actual cycle time for each operation in each cell can be calculated and compared with the takt time of the cell, which relates to the takt time of the processes within the cell, to calculate a time index. To balance the operations in the cell, the time index is arranged to fall within a defmed range by modifying the cell. These modifications include reducing the work content time per task, reducing move or setup times, identifying overlap times, adding resources, building up resources in upstream processes to compensate and increasing the operating time of the operation, and redesigning processes or products. 
     Finally, the family of products that is produced by the manufacturing line can be balanced such that the flow of each product through each process requires a process time that does not exceed a time index limit. For each product at each process, a time index can be calculated by comparing the actual time for that product to flow through that process with the takt time for that process. If the time index falls outside of a predetermined range, the product can be removed from the product family and placed within an alternate product family having similar characteristics. The removed products may, then, be dropped from the manufacturing line, or produced in a separate manufacturing line, or may flow through a separate option path that includes the offending process. The resulting manufacturing line operating on the remaining family of products is balanced such that each product requires approximately the same time as every other product to flow through each process. 
     These calculations and redefinitions described above may be iterated multiple times until an optimized manufacturing line is designed. In some embodiments, daily manufacturing line management can be accomplished. Given the actual product demand for a given day, for example, the actual required resources for that day can be determined and unused resources can be idled. For example, a process that utilizes multiple identical machines may, on a given day, require many fewer machines or none of the machines. Additionally, resource utilization in the manufacturing line can be calculated and monitored from the actual demand for products. 
     Embodiments of this invention are described below in relation to the figures. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a conceptual block diagram of a computerized mixed-model manufacturing process design system according to the present invention. 
     FIG. 2 is a flow diagram overview of one embodiment of the mixed-model man acturing design method according to the present invention. 
     FIG. 3 is a detailed flow diagram of one embodiment of the product and process definition steps of the mixed-model manufacturing design method and system according to the present invention as shown in FIG.  2 . 
     FIG. 4 is a detailed flow diagram of one embodiment of the product synchronization step according to the present invention as shown in FIG.  2 . 
     FIG. 5 is a detailed flow diagram of one embodiment of the calculation steps of the method and system according to the present invention preliminary to balancing process resources in a mixed-model manufacturing line design. 
     FIG. 6 is a detailed flow diagram of the resources balancing step according to the present invention as shown in FIG.  2 . 
     FIG. 7A shows a block diagram of a portion of a manufacturing line having a rework path. 
     FIG. 7B shows a block diagram of a portion of a manufacturing line having an option path. 
     FIG. 7C shows a block diagram of a portion of a manufacturing line having a rework path and an option path. 
     FIGS. 8A-8D shows a detailed flow diagram of one embodiment of the cell definition step of the method and system according to the present invention as shown in FIG.  2 . 
     FIG. 9 shows a detailed flow diagram of one embodiment of the operational definition steps of the method and system according to the present invention as shown in FIG.  2 . 
     FIG. 10 shows a detailed flow diagram of one embodiment of the operational balancing steps of the method and system according to the present invention as shown in FIG.  2 . 
     FIG. 11 shows an example manufacturing flow line. 
    
    
     In the figures, elements having the same or similar functions have the same identifying label. 
     DETAILED DESCRIPTION OF THE INVENTION 
     While the invention is described below with reference to certain illustrated embodiments, it is understood that these embodiments are presented by way of example and not by way of limitation. The intent of the following detailed description is not to cover all modifications, alternatives and equivalents as may fall within the spirit and scope of the invention. For example, it is understood by a person of ordinary skill practicing the method according to this invention that a balancing of resources may be performed after cells have been defined rather than before. It is also understood that the calculations are presented by way of example and that a person skilled in the art, in view of the disclosure of this invention, may utilize other algorithms to perform a demand-pull, mixed-model manufacturing line design according to this invention. 
     Definitions 
     The following definitions are listed here as an aid to understanding the description of embodiments of the invention: 
     
       
         
               
               
             
           
               
                   
               
             
             
               
                 actual cycle time 
                 The actual cycle time is the actual amount of time 
               
               
                 (AT) 
                 required to perform a task, process, or sequence of 
               
               
                   
                 processes for a product. The actual cycle time for 
               
               
                   
                 product i at process j, At ij , is the actual amount 
               
               
                   
                 of time required for a unit of product i to flow through 
               
               
                   
                 process j. 
               
               
                 cell 
                 A grouping of dissimilar resources (for example people 
               
               
                   
                 and machines) in a logical and sequential manner to 
               
               
                   
                 facilitate the flow of a product or a family of products. 
               
               
                 demand weighted 
                 The demand weighted average cycle time is the 
               
               
                 average cycle 
                 average actual cycle time weighted by the demand-at- 
               
               
                 time (ATw) 
                 capacity. For example, the demand weighted average 
               
               
                   
                 cycle time at process j, ATw j , is the sum over all 
               
               
                   
                 products of the demand-at-capacity for each product at 
               
               
                   
                 process j multiplied by the actual cycle time for each 
               
               
                   
                 product at process j divided by the total demand-at- 
               
               
                   
                 capacity at process j. 
               
               
                 demand-at- 
                 The demand-at-capacity, Dc, refers to the maximum 
               
               
                 capacity Dc 
                 demand for a product. Demand-at-capacity may also 
               
               
                   
                 refer to the total demand for all products. For example, 
               
               
                   
                 the demand-at-capacity for product i at process j, 
               
               
                   
                 Dc ij , refers to the maximum number of product units 
               
               
                   
                 required to produce product i that process j is expected 
               
               
                   
                 to produce. 
               
               
                 downstream 
                 Refers to a process in the same process path or 
               
               
                 process 
                 sequence that occurs closer to the end-of-line. 
               
               
                 end-of-line 
                 The end-of-line is the final point on the manufacturing 
               
               
                 (EOL) 
                 line. The product is completed at this point on the 
               
               
                   
                 manufacturing line and is in the form of finished 
               
               
                   
                 product or finished goods. 
               
               
                 family of 
                 A family of products is a grouping of products based 
               
               
                 products 
                 on a common feature. Two primary examples of 
               
               
                   
                 families of products include all of the products that are 
               
               
                   
                 manufactured by the manufacturing line and all of the 
               
               
                   
                 products defined in terms of a particular process. 
               
               
                 feeder path 
                 A feeder path manufactures sub-assemblies or 
               
               
                   
                 components of products in the family of products 
               
               
                   
                 manufactured by the manufacturing line. Feeder paths 
               
               
                   
                 run parallel to another production path and terminate 
               
               
                   
                 by insertion of the manufactured sub-assembly or 
               
               
                   
                 component into the process 
               
               
                 main path 
                 A main path is a process path that terminates at the 
               
               
                   
                 end-of-line. This process path is typically the 
               
               
                   
                 principal production path used to manufacture products 
               
               
                   
                 and is shared by all or most of the products in the 
               
               
                   
                 family or products produced by the manufacturing line 
               
               
                 operational- 
                 The operational-cycle-time, or takt time, is the 
               
               
                 cycle-time 
                 maximum amount of time that a process or sequence 
               
               
                 (OPc/t) 
                 of processes can use to produce a unit of product. The 
               
               
                   
                 operational cycle time for a particular process, 
               
               
                   
                 (OPc/t j , is the maximum amount of time that process j 
               
               
                   
                 can use to produce a unit of product and meet the 
               
               
                   
                 demand-at-capacity requirements at EOL. 
               
               
                 option path 
                 An option path is a process path wherein the work 
               
               
                   
                 content imparted to the product results in functional 
               
               
                   
                 or cosmetic changes to the product that are different 
               
               
                   
                 from those imparted to products in alternate option 
               
               
                   
                 paths. Option paths are typically responsible for 
               
               
                   
                 the performance and appearance differences between 
               
               
                   
                 products. Option paths input product from a down- 
               
               
                   
                 stream process in the production path. 
               
               
                 process path 
                 A process path is a sequence of processes. Process 
               
               
                   
                 paths can be categorized into one or four basic 
               
               
                   
                 types: main paths, option paths, feeder paths or 
               
               
                   
                 rework paths. 
               
               
                 process yield 
                 The process yield refers to the ratio of the units 
               
               
                   
                 of product that exit a product to the units of product 
               
               
                   
                 the enter the process. A process having less than a 
               
               
                   
                 100% yield results in a scrap output or output to a 
               
               
                   
                 rework path. 
               
               
                 product 
                 A product refers to the manufactured item. The 
               
               
                   
                 product can be defined in terms of the sequence of 
               
               
                   
                 processes required to produce the product. 
               
               
                   
                 Additionally, a quantity of product at a process 
               
               
                   
                 refers to the material in the process used to 
               
               
                   
                 produce the product. One unit of product is the 
               
               
                   
                 amount of material that exists in one final product. 
               
               
                 product fallout 
                 Refers to the amount of product that leaves the 
               
               
                   
                 process path, usually as scrap, or by entering a 
               
               
                   
                 rework path. 
               
               
                 Product/Process 
                 A matrix indicating which processes are included in 
               
               
                 map 
                 the definition of each product 
               
               
                 product/task work 
                 A matrix indicating the work content imparted to 
               
               
                 content map 
                 each product by tasks in a process. 
               
               
                 resources 
                 Resources refers to individual machines or quantities 
               
               
                   
                 of labor. The amount of resources, for example, can 
               
               
                   
                 be increased for a particular task by increasing 
               
               
                   
                 the number of machines or people or by increasing 
               
               
                   
                 the number of hours that the machines and people 
               
               
                   
                 work. 
               
               
                 rework path 
                 A rework path inputs rejected (i.e., out-of- 
               
               
                   
                 specification) quantities of product from a process, 
               
               
                   
                 remedies the fault in the quantities of product 
               
               
                   
                 and re-inserts the quanitites of product into an 
               
               
                   
                 upstream process in the process path. In some cases, 
               
               
                   
                 the rework path inserts quantities of product at 
               
               
                   
                 the point in the process path where rejected 
               
               
                   
                 quantities of product were diverted into the rework 
               
               
                   
                 path. The point of insertion of the recovered 
               
               
                   
                 quantities of production into the manufacturing 
               
               
                   
                 line depends on the effective cumulative work content 
               
               
                   
                 of the quantities of product and will typically 
               
               
                   
                 be at point in the process path where first-run parts 
               
               
                   
                 have the same cumulative work content. The effect 
               
               
                   
                 of the rework path is to increase the effective 
               
               
                   
                 yield of the process from which the rework part 
               
               
                   
                 originated. 
               
               
                 sequence of 
                 The sequence of events refers to a sequential 
               
               
                 events (SOE) 
                 sequence of tasks. In some embodiments, standardized 
               
               
                   
                 sequences of events (SSOE) are defined. 
               
               
                 takt time 
                 See operational cycle time 
               
               
                 task 
                 A task is a small unit of work. A task is usually 
               
               
                   
                 defined in terms of work content and the resources 
               
               
                   
                 required to impart the work content to the product. 
               
               
                 time index 
                 A time index is a comparsion of a time, usually an 
               
               
                   
                 actual cycle time for a process, an operation, or a 
               
               
                   
                 cell, with a takt time for that process, operation, 
               
               
                   
                 or cell. The comparison is typically a ratio of the 
               
               
                   
                 two quantities. 
               
               
                 upstream 
                 Refers to a process in the same process path or 
               
               
                 process 
                 sequence as the current process but occurs towards 
               
               
                   
                 the beginning of the manufacturing line. 
               
               
                 work content 
                 Work content refers to the amount of working time 
               
               
                 or work 
                 that hs been invested in a quantity of product. A 
               
               
                 content time 
                 process or process path can impart work content to 
               
               
                   
                 a product. Work content includes machine and labor 
               
               
                   
                 work times, machine and labor setup times and move 
               
               
                   
                 times. Work content from a particulr process or task 
               
               
                   
                 may be characterized by whether the work adds value 
               
               
                   
                 to the final product or does not add value to the 
               
               
                   
                 final product. Work content may also be petitioned 
               
               
                   
                 into machine work and labor work. 
               
               
                 Work in Process 
                 Product units that exist in the manufacturing line. 
               
               
                 (WIP) 
                 WIP can include subassemblies and components of the 
               
               
                   
                 finished product. 
               
               
                 work volume 
                 The amount of product that flows through a process, 
               
               
                 (throughput) 
                 task, or process path in order to produce the demand 
               
               
                   
                 at capacity amount of product at the end-of-line. 
               
               
                   
               
             
          
         
       
     
     Description 
     FIG. 1 shows a computerized mixed-model flow manufacturing design system  1 . Design system  1  includes at least one processor  2 , at least one input structure such as a keyboard  3 , a mouse  4 , or other input device such as a bar code scanner, at least one visual display such as a monitor or printer  5  and program and data storage means  6  such as a holographic, magnetic, or optical storage unit and main memory  7  including volatile and non-volatile program and data storage devices. Main memory  7  may include RAM, DRAM, SRAM, or other computer readable medium. Machine readable code for controlling design system  1  according to the present invention is embodied on the program and data storage means  6  and read into the main memory  7  at the time of execution. Main memory  7  includes an operating system code  8 , computer code  9  for effectuating the mixed-model manufacturing design method of the present invention, and at least one additional memory space  10  for storing programs and data. At least one bus  11  (busses  11   a ,  11   b ,  11   c ,  11   d  and  11   e  are shown in FIG. 1) provides electronic communication between the elements of design system  1 . 
     Computer code  9  on computer main memory  7 , or alternately on storage unit  6 , includes machine readable code segments of a computer program according to the present invention. Computer code  9  directs computerized mixed-model flow manufacturing design system  1  according to embodiments of the present invention. 
     FIG. 2 shows a block diagram of one embodiment of the method for designing a mixed-model manufacturing line. The method is implemented in software that is executed in design system  1  of FIG.  1 . 
     In product definitions  15  the definition of a product in terms of the processes required for manufacturing that product is inputted. In process definition  50 , each process in the manufacturing line is defmed and characterized. Each product in the family of products that is to be manufactured by the manufacturing line is defined and stored as data. Typically, user input is required for product definition  15  and process definition  50 , and more than one product and process will be defined. Processes are defmed in terms of process parameters that are inputted and stored for use in subsequent calculations. Process parameters include: 
     process yield (e.g., the amount of product that flows to a subsequent process with respect to the amount of product that entered the present process); 
     process path type attributes (e.g., main path, feeder path, option path, or rework path); 
     process identification indicating the position of the process in a sequence of processes; 
     if the process is a member of an option path, the amount of product that is routed to the option path and the identification of the process to which the option path directs its outputted amount of product; and 
     if the process is a member of a rework path, the identification of the process to which the rework path directs its outputted amount of product. 
     Additionally, the tasks required for each process and the individual work times for each task by product are inputted into the system and stored. 
     The result of the product synchronization step  100  is to define families of products that share a common process, calculate the demand-at-capacity for each product at each process, and calculate the work content time for each product at each task in each process. 
     In resource balancing  200 , an operational cycle time defining the maximum cycle time for each product unit at each process is calculated and compared to a weighted average actual cycle time for that process. Resources at each process are balanced (for example, by adding additional machines or people) so that the actual weighted cycle time is approximately equal to the operational cycle time. 
     In cell definition  300 , cells are defined. A cell is a logical grouping of dissimilar resources in a logical and sequential manner to facilitate the flow of quantities of product through the manufacturing line. Organizing machines and people into cells reduces or eliminates queues, reduces setup times, and reduces throughput times in a mixed-model manufacturing flow line. In some embodiments, each cell may be defined to include a single process. In other embodiments, a cell may be defined to include multiple processes. In other embodiments, a cell may group additional manufacturing tasks associated with other processes so long as those additional tasks are logically, and conveniently, grouped with the manufacturing tasks already within that cell. In some embodiments the additional tasks represent the tasks associated with contiguous processes, thereby grouping dissimilar machines. 
     In one embodiment, an automatic cell definition is accomplished by product model balancing. Since the resources for each process were based on a demand weighted average product cycle time, individual product cycle times at each cell may be significantly greater than or less than the operational cycle time. Cell definition  300  can balance each cell against the family of products and provides for the creation of additional cells or process paths for those products having cycle times significantly greater or less than the calculated operational cycle time of the processes comprising the original cell. 
     The tasks identified with each of the processes comprising the cell are regrouped into operations at operation definition  400 . Operations are defined such that the total time for the group of tasks at each operation is within a predetermined range around the previously calculated operational cycle time. Conveniently, if the processes were previously balanced (i.e., the processes are defined such that actual time to complete each process is approximately the takt time) prior to forming cells, the actual cycle time for the group of tasks within each operation will tend to be within the predetermined range around the operation cycle time. 
     Design output  500  outputs the results of mixed-model manufacturing design system  1 . The results are then implemented at the manufacturing factory. The components of the method shown in FIG. 2 are implemented in the mixed-model manufacturing design system  1 . Each of the steps in FIG. 2 is further explained below. 
     1. Product Definitions  15  and Process Definitions  50   
     FIG. 3 shows a block diagram of one method according to an embodiment of the present invention for performing define process  50  and define product  15 . The product models to be manufactured on the proposed manufacturing line are defmed at define product  15  in terms of the processes required for their manufacture. These processes are inputted for each product model at input processes  16 . The process definition for each product model includes assigning a sequence of events to each process in Assign SSOE  51 . The specific tasks included in the sequence of events (SOE) are, in this embodiment, selected from a standardized sequence of events (SSOE) list. Alternatively, the sequence of events, and the tasks that constitute the SOE, can be defined at this step. The work content for each task identified is estimated and input into the system at define work content step  52 . The work content includes machine and labor work content times, machine and labor setup and move times, and a flag indicating whether the task is “value-add” or “non-value add” work content. Additional information inputted for each task includes whether the machine and labor work content time within the task overlap, and whether the machine and/or labor work content times overlap with the machine and/or labor work content times of other tasks included in the SOE for that process. 
     Global process variables are obtained in steps  53  and  54 . The global process variables include, but are not limited to, process and path I.D.s (locating the process in a manufacturing sequence by production path and location in the production path), a process yield and a path type. In the event that the process yield inputted at step  53  is less than 100 percent, the percentage of product directed towards a rework path and scrap are also inputted at input process yield  53 . The process type is inputted at input process type  54 . Process types in this embodiment are determined by whether a process is a member of a main path, feeder path, rework path, or option path. Examples of the process paths are graphically shown in FIGS. 7A through 7C. 
     A main path is a process path that terminates at an end-of-line (EOL) where the finished product leaves the manufacturing line. For example, in FIG. 7A, process P1, P2 and P3 are main path processes. 
     A feeder-path inserts components or subassemblies into another production path. For example, in FIG. 7C, process FP1 and FP2 are feeder path processes. 
     A rework path removes a quantity of component from a production path and reinserts a re-worked quantity of component into the production path at an insertion point that precedes or is the same as the process where the quantity of component was removed. For example, in FIG. 7C process RP1 is a rework path process that accepts inputted quantities of product outputted from feeder path process FP2 and reinserts reworked quantities of product at feeder path process FP1. 
     An option path is used to produce products not produced in the main path or, in some manufacturing lines, to produce individual variations (such as differing product features) between products. For example, in FIG. 7B, one option path includes option path process OP1 and a second option path includes option path process OP2. 
     In order to fully characterize feeder path processes, additional information is inputted, including: the quantity of sub-component that needs to be produced by the feeder path to manufacture one unit of finished product and an index indicating the process in the production path into which the feeder path directs the quantity of product that it outputs. The demand-at-capacity for each feeder path process is multiplied by the quantity of the component produced by the feeder path that is consumed per unit of product produced by the main-line path. 
     A rework path process must direct product in an upstream direction in the production path from which it originated. It may return to its originating process, to just prior to its originating task, to any prior process upstream of the originating process, or to its point of origin. If a product falls out of a main path and completes the remaining work to finish the product in a production path parallel to the main process path, then this parallel path is an option path. 
     In FIG. 3, optional process  55  identifies processes that belong to an option path. Option paths may occur in the main line path, in rework paths, or in feeder paths and run parallel to the respective production process. Unlike rework paths, option paths re-enter the respective production path at a process downstream from the option path&#39;s origination process. Alternately, option paths can insert into other option paths, or can go to the end-of-line (EOL) directly. Option paths may include rework paths to any process within the option path. If an option process is identified at option process  55 , input block  56  inputs additional process information, including the percentage of the originating production path product that is shunted, or diverted, to that option process, and the production path process position index receiving the output of the option path. Rework process steps are recognized at step  57  and, for rework processes, input block  58  inputs additional information including the rework yield and the process position index of the process receiving the output of the rework production path. 
     If decision block  59  identifies additional processes in the product definition, the foregoing steps  50  through  58  are repeated for the next process associated with the product model. If, at decision block  59 , it is determined that no additional processes have been identified for that product, decision block  60  determines if there are further products to define. If there are, then steps  15  through  59  are repeated for the next product model. If at decision block step  60  no more product models are to be defmed, the product/process definition step is completed and the method of this invention continues to director  61  and the mixed-model flow manufacturing design system  1  (FIG. 1) proceeds to product synchronization  100  (FIG.  2 ). 
     2. Product Synchronization  100   
     According to the present invention, mixed-model product flow is synchronized between processes and within each process in accordance with product synchronization  100 . Product synchronization  100  comprises: 
     1) taking product and process definitions from the databases created in define product  15  and define process  50  and calculating the demand-at-capacity for each product at each process; 
     2) calculating 
     a demand weighted average cycle time for each process, and 
     an operational cycle time for each process; and 
     3) adjusting the amount of resources at each process such that the demand weighted average product cycle time is within a predefined range around the operational cycle time. 
     Product synchronization  100  is described in FIGS. 4 and 5 and includes the steps of assembling process design input matrices, such as a product/process map and a product task map from the input data provided according to the product definitions and process definitions described in FIG.  3 . The process shown in FIG. 3 links with the process shown in FIG.  4  through director  61  (shown in both figures). 
     The process definition input matrices are calculated at step  62 . The process definition matrices include a product vs. process map and a product/task work content matrix. Process  63  prepares a product/process map (alternatively referred to as a product/process matrix). Process  63  begins by initializing the process index j to 1 at step  65 . All product models including process j in their process definition are identified in step  66  as a family of products. Query  68  determines if all processes have been analyzed. If additional processes remain, j is incremented in step  67  and process  63  returns to step  66 . If all processes have been identified, a product/process matrix is generated by process  63 . Process  63  stores the product/process matrix in a computerized mixed-model flow manufacturing design system  1  (see FIG. 1) in storage step  69 . Some embodiments of the invention utilize other groupings of products to define a family of products, including products that include similar sets of processes in their definitions or other operator controlled definitions. 
     By way of example, a product/process map generated according to process step  63  is illustrated in Table I. Mixed-model flow manufacturing design system  1  may display such a product/process matrix on monitor or printer  5  of design system  1  (See FIG.  1 ). 
     
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE I 
               
             
             
               
                   
                   
               
               
                   
                 PROCESSES 
               
             
          
           
               
                 MODEL NO. 
                 P1 
                 P2 
                 P3 
                 P4 
                 P5 
                 P6 
                 P7 
                 P8 
                 P10 
               
               
                   
               
               
                 Model #1 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                   
                 X 
                 X 
               
               
                 Model #2 
                   
                 X 
                 X 
                 X 
                 X 
                   
                 X 
                 X 
                   
               
               
                 Model #3 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 Model #4 
                 X 
                 X 
                 X 
                 X 
                 X 
                   
                   
                 X 
                 X 
               
               
                 Model #5 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                   
                 X 
                   
               
               
                 Model #6 
                   
                   
                   
                   
                 X 
                 X 
                   
                 X 
                   
               
               
                 Model #7 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                   
               
             
          
         
       
     
     In Table I, there are seven (7) products, labeled model #1 through model #7, defined in terms of ten (10) processes labeled P1 through P10. The “X” in each cell signifies that the corresponding process is included in the definition of the corresponding product model. Model #6, for example, includes processes P5, P6 and P8. The product-process map clearly indicates the family of products for each process. Models 1, 3, 4, 5 and 7, for example, are members of the family of products using process P1. Alternatively, the X&#39;s in Table I may be replaced by process yield, process type, or the value of any parameter or parameters that have been inputted for that process with respect to that product. 
     The set of process design input matrices may also include a product/task work content map. Preparation of the product/task work content map is accomplished in step  64 . The product number i and process number j are each initialized to 1 at initialization step  71 . In one embodiment, product number i refers to the product number in a family of products all having process j in common. The index j is a process number identifying a particular process in the set of processes defined across all product models. Other systems of indexing may be used to access and identify products within families of products. 
     The product/process map generated in process  63  is read at input  72  in order to identify the family of products associated with process j. In step  73 , the tasks for each process j, in product i and their associated work content times (both machine and labor) and attribute data, inputted during steps  51  and  52  (see FIG.  3 ), are retrieved from memory  10  or from storage  6  (see FIG.  1 ). The machine work content times for all of the tasks defining process j in product i are summed during step  74 . Labor work content times for process j in product i are summed during step  75 . Overlap attributes for each task are identified and overlapping work content time is deleted in step  76 , eliminating double counting of time, unless two independent resources are co-involved with performance of a task. Overlap work content times refers to work content time from two sources that occur simultaneously, such as the labor time and machine time in a task involving a manually operated machine. Work content time from two separate machines, even if the two machines are operating on the same quantity of product simultaneously, however, is included. Static quantities (work content independent of product volume or throughput), such as set-up times and move times, are included in the summations. 
     Once the work content summations and overlap adjustments for process j are completed, step  77  checks to determine if all processes have been analyzed. If additional processes remain for product i, then process number j is incremented to refer to the next process, and the work content for the next process in the manufacture of product i is analyzed according to steps  73  through  76 . If at step  77  no more processes remain for product i, the actual cycle times for product i at process j are calculated at step  82 . The actual cycle time, At ij , is the sum of the machine work content and the labor work content for product i at process j less the overlap work content for product i at process j. A check for additional product models is made at step  79 . If not all products have been analyzed, at step  80  the product index variable, i, is incremented and the process index, j, is re-initialized to 1, representing the first process in the sequence of processes required for the manufacture of product i+1. Steps  73  through  80  are repeated until all products and their processes have been analyzed. The resulting data and actual cycle times are stored in memory  10  (FIG.  1 ). 
     Off-page connector  70  continues product synchronization  100  (FIG. 2) onto connector  70  on FIG.  5 . FIG. 5 shows the process of calculating the parameter values necessary to synchronize product flow in the manufacturing design method according to this invention. 
     The end-of-line (EOL) demand-at-capacity for each product, Dc i , is inputted and stored at step  83 . Alternately, these parameters may be inputted along with information regarding products and processes at an initial data entry step. The planned hours per shift at each process, H j , and number of shifts per day at each process, S j , are inputted and stored at steps  84  and  85 , respectively. These parameters may be inputted from a central data base stored on storage  6  of mixed-model manufacturing process design system  1  (see FIG. 1) or may be inputted from a user. Initialization step  86  sets product counter i and process counter j both to 1. 
     The demand-at-capacity, Dc ij , for product i at process j is determined at step  87 . Various algorithms may be used to calculate Dc ij . In some embodiments of the present invention, a derivative method is used whereby a starting value, x, is chosen for process 1 of each product i and the EOL yield is calculated, process-by-process, based on the yield at each process. The starting value x represents the number of units of product i that need to enter the manufacturing line in order that Dc i  units of product are produced at EOL. 
     Rework paths in the process definition result in adjusting x until the desired overall demand-at-capacity Dc i  for product i is obtained. In implementation, the computer system assigns a quantity x 1  to a first process and calculates an output quantity for the first process, x 1  being x adjusted for the yield of the first process. At least a portion of the output quantity x 1  is delivered as the input quantity to an adjacent downstream process (i.e., the next process in the path). The amount of product x that is carried forward to the next process is based on the amount of product x that is redirected toward rework and option process paths. The quantity x is dependent on the yield and scrap factors associated with the process. 
     The foregoing procedure is repeated through the entire production process path (including rework, option, and feeder production paths) until a final yield x F  is calculated at the EOL. The final yield x F  is used to calculate a multiplier which will adjust the final yield x F  to be one. For example, if 87.5% of initial product x arrives at the EOL, then the multiplier is 1/0.875=1.14286. This multiplier is then applied to the starting process to determined the associated absolute amount of product throughput, output and scrap for each process. 
     In general, the calculation of demand-at-capacity for product i at process j, Dc ij , in step  87  of FIG. 5 follows the following procedure, for each path: 
     1) consider a quantity of product x inputted to the product path (one unit of product is that amount of material present in one unit of final product); 
     2) follow the production path sequentially and, for each process in the production path 
     a) determine the amount of product inputted to the process, including inputs of product from rework paths re-entering the production path at that point (i.e., an insertion point) and, if the current process is an optional process, the amount of product that is diverted into the optional path from the production path, and 
     b) calculate the amount of material that is outputted from the process considering the amount of material directed into a rework path or a scrap path; and 
     3) determine the initial quantity of material by: 
     a) if the product path terminates at the EOL, normalizing the output of the last process to 1 so that 1 unit of final product is produced, or 
     b) if the product path is a feeder path, normalizing the output of the last process in the feeder path to be the amount of product present in the production path at the point where the feeder path terminates, taking into account the number of components per produced product that is required at the termination point of the feeder path. 
     In the calculations, the amount of product outputted from a rework path is included in the input of the process following the rework path insertion point in the production path. Feeder paths are balanced so that the number of components that arrive at the insertion point of the production path matches the requirement for those components at that insertion point. From the above-described calculations, the demand-at-capacity at each process j for each product i is calculated from the total demand-at-capacity for each product i that is previously inputted and stored in the mixed-model system  1  (see FIG. 1) during step  83  (see FIG.  5 ). 
     The above-described method for calculating demand-at-capacity at each process is exemplified in the following discussion with reference to FIGS. 7A-7C. FIG. 7A is a block diagram showing a main process path comprising processes P1, P2, and P3, a rework path including rework process RP1, and a scrap output. The parameter designation “xxx/yy/zz” indicates the percent yield, the rework percentage (the percentage amount of product directed towards the rework path) and the scrap percentage (the percentage amount of product directed towards scrap), respectively, for that process. In the following discussion, a reference to the amount of “product” at a particular process refers to the amount of materials (or parts) present at that process that is required to produce a product. 
     The derivative process for calculating Dc ij  for the manufacturing process line configuration of FIG. 7A includes: 
     1) considering a quantity of product x inputted to process P1; 
     2) calculating the amount of product outputted from process P1, which in this example is x because the yield of process P1 is 100%; 
     3) determining the amount of product inputted to process P2, which in this example is the amount of product outputted from process P1 (x) plus the amount of product y outputted from rework process RP1; 
     4) determining the amount of product inputted to the rework process RP1, which in this example is 0.05 (x+y); 
     5) determining the amount of product outputted from the rework process RP1, y, and solving for y in order to obtain the actual amount of product inputted to process P2, which in this example is 0.05 (x+y) because the yield of the rework process RP1 is 100%, therefore y=0.05 (x+y) and y=0.05/0.95x=0.0526x; 
     6) determining the amount of product inputted to process P3, which in this example is 90% of (1.0526)x, 0.947x, because the amount of product that is inputted to process P2 is (1.0526)x and the yield is 90%, with 5% going to rework process RP1 and 5% going to scrap; 
     7) determining the amount of product at EOL in terms of x, which in this example is 0.947x because the yield of process P3 is 100%; and 
     8) determining x such that the amount of product at EOL is normalized to 1, which in this example implies that x=1/0.947=1.0556. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE II 
               
               
                   
               
               
                 PROCESS/ 
                 PATH 
                   
                 EXT- 
                 IN- 
                 IN/ 
               
               
                 ID 
                 TYPE 
                 YIELD 
                 RACT&#39;N 
                 SERT‘N 
                 OUT 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                 1X/1X 
               
               
                 P2/2 
                 Main 
                  90/5/5 
                 N/A 
                 N/A 
                 1.0526X/ 
               
               
                   
                   
                   
                   
                   
                 0.947X 
               
               
                 P3/3 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                 0.947X/ 
               
               
                   
                   
                   
                   
                   
                 0.947X 
               
               
                 RP1/1 
                 Rework 
                 100/0/0 
                 P2 Out 
                 P2 In 
                 0.0526X/ 
               
               
                   
                   
                   
                   
                   
                 0.0526X 
               
               
                 Scrap 
                 N/A 
                 N/A 
                 P2 Out 
                 N/A 
                 0.0526X 
               
               
                 EOL 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 0.947X 
               
               
                   
               
             
          
         
       
     
     Therefore, in the example of FIG. 7A, 1.0556 units of product must be inputted to process P1 in order for 1.0 unit of product to reach EOL. Table II shows the input parametric values and the calculated values of x for the manufacturing line of FIG. 7A, including the extraction and insertion points (the points at which material exits the production path or enters the production path, respectively) of the product streams. 
     FIG. 7B is a block diagram showing an example with a main path including processes P1 and P2, and two option paths including processes OP1 and OP2, respectively, all having 100 percent yields. As shown, the amount of product outputted at process P1 is split 50% to OP1, and 50% to OP2. The derivative method for calculating Dc ij  for the manufacturing process line of FIG. 7B includes: 
     1) considering a quantity of product x inputted to process P1; 
     2) calculating the amount of product that is outputted by process P1, which in this example is x because the yield of process P1 is 100%; 
     3) determining the input to optional process OP1, which in this example is 0.5x because the parameter list for optional process OP1 indicates that 50% of the yield of process P1 is directed into optional process OP1; 
     4) calculating the amount of product that is outputted by process OP1, which in this example is 0.5x because the yield of optional process OP1 is 100%; 
     5) determining the amount of product inputted to optional process OP2, which in this example is 0.5x because the parameter list for optional process OP2 indicates that 50% of the yield of process P1 is directed into optional process OP2; 
     6) calculating the amount of product that is outputted by optional process OP2, which in this example is 0.5x because the yield of optional process OP2 is 100%; 
     7) determining the amount of product that is inputted to process P2, which in this example is x, the sum of the amount of product from optional processes OP1 and OP2; 
     8) calculating the amount of product that is outputted from process P2 to the EOL, which in this case is x because the yield of process P2 is 100%; and 
     9) calculating x so that the amount of product at EOL is normalized to 1, which in this example results in x being 1. 
     Table III shows the input parametric values and the calculated values of x for the manufacturing line of FIG. 7B, including the extraction and insertion points of the option product streams. From the above calculations, as illustrated in Table III, the multiplier factor is 1, i.e. for each unit inserted at process P1, one unit is produced at the EOL. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE III 
               
               
                   
               
               
                 PROCESS/ 
                 PATH 
                   
                 EXT- 
                 IN- 
                 IN/ 
               
               
                 ID 
                 TYPE 
                 YIELD 
                 RACT&#39;N 
                 SERT‘N 
                 OUT 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                 1X/1X 
               
               
                 P2/2 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                 1X/1X 
               
               
                 OP1/1 
                 Option 
                 100/0/0 
                 P1 Out 
                 P2 In 
                 0.5X/0.5X 
               
               
                 OP2/1 
                 Option 
                 100/0/0 
                 P1 Out 
                 P2 In 
                 0.5X/0.5X 
               
               
                 EOL 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 1X 
               
               
                   
               
             
          
         
       
     
     FIG. 7C is a block diagram showing an example of a main process path comprising processes P1-P5, one option path including process OP1, one feeder path including process FP1 and FP2, and a rework process path originating from feeder path process FP2 and including process RP1. The derivative method for calculating Dc ij  for the manufacturing process line configuration of FIG. 7C includes: 
     1) considering an amount of product x inputted into process P1; 
     2) calculating the amount of product that is outputted by process P1, which in this example is x because the yield of process P1 is 100%; 
     3) determining the amount of product that is inputted into process P2, which in this example is 0.8x because the parameter list of optional process OP1 shows that 20% of the output of process P1 is directed into option path process OP1; 
     4) calculating the amount of product outputted from process P2, which in this example is 0.8x because the yield of process P2 is 100%; 
     5) determining the amount of product inputted to process P3, which in this example is 0.8x; 
     6) calculating the amount of product that is outputted by process P3, which in this case is 0.6x because the yield of process P3 is 75% with 25% of the product going to scrap; 
     7) determining the amount of product inputted to option path process OP1, which in this example is 0.2x because the parameter list for optional process OP1 shows that 20% of the product outputted by process P1 is diverted into optional process OP1; 
     8) calculating the amount of product that is outputted by optional process OP1, which in this case is 0.18x because optional process OP1 has a yield of 90% with 10% being directed towards scrap; 
     9) determining the amount of product inputted to process P4, which is the sum of the outputs of processes P3 and OP1 and, in this example, is 0.78x. 
     10) calculating the amount of product that is output by process P4, which in this example is 0.78x because process P4 has a yield of 100%; 
     11) determining the amount of product inputted to process P5, which in this example is 0.78x; 
     12) determining the amount of product outputted to EOL, which in this example is 0.468x because the yield of process P5 is 60% with 40% going to scrap; 
     13) calculating a multiplier x by normalizing the amount of product outputted to the EOL to the value 1, which in this case is 1/0.468=2.14; 
     14) considering an amount of component (a component being the product produced in a feeder path) y inputted to the beginning of the feeder path that includes feeder process FP1 and FP2 and rework path RP1; 
     15) determining the amount of component that is inputted into feeder process FP1, which in this case is y+z where z is the amount of component that is outputted by rework process RP1; 
     16) calculating the amount of component that is outputted by feeder process FP1, which in this case is 0.75(y+z) because feeder FP1 has a yield of 75% with 25% going to scrap; 
     17) determining the amount of component that is inputted to feeder process FP2, which in this case is 0.75(y+z); 
     18) calculating the amount of component that is outputted from process FP2, which in this case is 0.675(y+z) because process FP2 has a yield of 90% with 10% of the product being routed to the rework process; 
     19) determining the amount of component inputted to rework process RP1, which is 0.075(y+z); 
     20) calculating the amount of component that is outputted by rework process RP1, which in this example is 0.075(y+z) because the yield of process RP1 is 100%; 
     21) determining a value for z using that, from step  15 , z represents the output from rework process RP1, z=0.075(y+z) therefore z=0.075y/(1−0.075)=0.081y; 
     22 determining the total amount of component outputted from the feeder process, which from steps  18  and  21  is 0.675(y+z)=0.730y; and 
     23) determining y by balancing the amount of components outputted by the feeder path with the amount of product required in the product path to which the feeder path outputs (i.e., if the feeder path must supply 4 components for each unit of finished goods at the EOL, then y is calculated such that 4 components are produced for each unit of product in the main path at the insertion path), which in this example (with a 1:1 correspondence assumed) is 0.78x=0.73y (see steps  9 ,  13  and  22 ) or y=1.07x=2.29. 
     Table IV shows the input parametric values and the calculated values of x for the manufacturing line of FIG. 7C, including the extraction and insertion points of the feeder path, and the insertion point of the feeder path. From the above calculation, x in this example is 2.78. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE IV 
               
               
                   
               
               
                 PROCESS/ 
                   
                   
                 EXT- 
                 IN- 
                 IN/ 
               
               
                 ID 
                 PATH 
                 YIELD 
                 RACT&#39;N 
                 SERT‘N 
                 OUT 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                   x/x 
               
               
                 P2/2 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                  .8x/ 
               
               
                   
                   
                   
                   
                   
                  .8x 
               
               
                 P3/3 
                 Main 
                 75/0/25 
                 N/A 
                 N/A 
                  .8x/ 
               
               
                   
                   
                   
                   
                   
                  .6x 
               
               
                 P4/4 
                 Main 
                 100/0/0 
                 N/A 
                 N/A 
                  .78x/ 
               
               
                   
                   
                   
                   
                   
                  .78x 
               
               
                 P5/5 
                 Main 
                 60/0/40 
                 N/A 
                 N/A 
                  .78x/ 
               
               
                   
                   
                   
                   
                   
                  .468x 
               
               
                 OP1/1 
                 Option 
                 90/0/10 
                 P1 Out 
                 P4 In 
                  .2x/ 
               
               
                   
                   
                   
                   
                   
                  .18x 
               
               
                 FP1/1 
                 Feeder 
                 75/0/25 
                 N/A 
                 N/A 
                 1.16x/ 
               
               
                   
                   
                   
                   
                   
                  .87x 
               
               
                 FP2/2 
                 Feeder 
                 90/10/0 
                 N/A 
                 P4 In 
                  .87x/ 
               
               
                   
                   
                   
                   
                   
                  .78x 
               
               
                 RP1/1 
                 Rework 
                 100/0/0 
                 FP2 Out 
                 FP2 Out 
                  .078x/ 
               
               
                   
                   
                   
                   
                   
                  .078x 
               
               
                 EOL 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 1X 
               
               
                   
               
             
          
         
       
     
     From the above three examples, one skilled in the art can readily determine a procedure for determining the demand at each process j for each product i. Examples of calculations involving option paths, rework paths, and feeder paths were given. 
     After the demand-at-capacity for product i at process j is calculated in step  87  of FIG. 5, the actual cycle time (also known as the actual process time) for each product at each process is obtained in step  88 . The actual cycle time for each process is either calculated or, alternatively, retrieved from memory as calculated at step  75  of FIG.  4 . As previously described, the actual cycle time for a process is the sum of the previously defined non-overlapping work-content times (either machine or labor associated with the tasks comprising the SOEs at each process. 
     The demand-at-capacity for product i at process j, Dc ij , and the actual cycle time for product i at process j, AT ij , are calculated for all of the products in the product family for a given process j. Query step  90  checks to determine if all products in the product family utilizing process j have been analyzed. If not, then the product index i is incremented by one at step  89  and Dc ij  and AT ij  are calculated for the next product. 
     If no additional products in the product family for that process remain, the total demand at that process, Dc j , is then calculated at step  91 . The total demand at process j is calculated by summing Dc ij  across all products in the product family at process j, i.e., Dc i =Σ j  Dc ij . 
     The operational cycle time, also known as the takt time, for all products at process j is then calculated in step  92  by the following equation: 
     
       
         OPc/t j =[H j *S j ]/(Σ i Dc ij ),  (1) 
       
     
     where: 
     i ranges from 1 to the total number of products in the product family at process j; 
     j is the process identifier (process ID) and ranges from 1 to the total number of processes in the production path; 
     H j  is the number of work hours per shift for process j; 
     S j  is the number shifts per day in process j; and 
     Dc ij  is the demand at capacity of unit/sub-assembly/component associated with model i at process j. 
     For each process, the number of hours per shift and the number of shifts per day are defined in steps  84  and  85  of FIG.  5 . The EOL demand-at-capacity for product i is obtained in step  83 . The demand-at-capacity for product i at process j is obtained in step  87 . 
     A demand weighted average actual cycle time for all products at process j, ATw j , is calculated at step  93 . The demand weighted average cycle time considers the average time to produce products and is given by 
     
       
         ATw j =[Σ(Dc ij *AT ij )]/ΣDc ij ,  (2) 
       
     
     where AT ij  is the actual cycle time (or process time) for product i at process j obtained in step  83  of FIG.  4 . 
     When these parameters are calculated for each product i and process j, the system proceeds through off-page connector  96 , to balance resources  200  (see FIG.  2 ). 
     Table V shows takt time and the demand-at-capacity for a single product having a total EOL demand-at-capacity of 100 units in the manufacturing line shown in FIG.  7 A. The values shown in Table V are based on processes where H j  and S j  are set to 7.3 hrs. and 1, respectively. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE V 
               
               
                   
               
               
                   
                   
                 TAKT 
                   
                 ORIGINAL 
                   
               
               
                 PROCESS/ 
                   
                 TIME 
                 DESIGNED 
                 THROUGH- 
                   
               
               
                 ID 
                 PATH 
                 (min) 
                 OUTPUT 
                 PUT 
                 SCRAP 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                  4.16 
                 105.263 
                 105.263 
                 0 
               
               
                 P2/2 
                 Main 
                  3.96 
                 100 
                 110.53 
                 5.263 
               
               
                 P3/3 
                 Main 
                  4.38 
                 100 
                 100 
                 0 
               
               
                 RP1/1 
                 Rework 
                 83.22 
                 5.263 
                 5.263 
                 0 
               
               
                   
               
             
          
         
       
     
     In Table V, designed output refers to the actual amount of product that is outputted from the process and original throughput refers to the actual amount of product inputted to the process. Values for the designed output and the original throughput are obtained from Table II. The takt time is calculated using the Original Throughput value. For example, the takt time for process P2 given by (7.3 hrs)/110.53=3.96 minutes (see Equation 1). Notably, the takt time for process 2 is lower than the other processes in the example because the lower yield associated with process P2 requires a higher throughput rate. 
     Table VI shows the takt times and flows for a single product in the manufacturing line shown in FIG.  7 B. In accordance with Table III, in order to obtain one unit of product at EOL, one unit of product must be inputted to process P1. Again a 7.3 hour work shift (H=7.3), one work shift per day (S=1) for each process, and an EOL demand-at-capacity of 100 units is assumed. 
     The takt time is calculated using the Original Throughput value. Notably, the takt times for processes OP1 and OP2 are longer because of the smaller demand associated with each option process which requires a lower throughput rate. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE VI 
               
               
                   
               
               
                   
                   
                 TAKT 
                   
                 ORIGINAL 
                   
               
               
                 PROCESS/ 
                   
                 TIME 
                 DESIGNED 
                 THROUGH- 
                   
               
               
                 ID 
                 PATH 
                 (min) 
                 OUTPUT 
                 PUT 
                 SCRAP 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                 4.38 
                 100 
                 100 
                 0 
               
               
                 P2/2 
                 Main 
                 4.38 
                 100 
                 100 
                 0 
               
               
                 OP1/1 
                 Option 
                 8.76 
                  50 
                  50 
                 0 
               
               
                 OP2/1 
                 Option 
                 8.76 
                  50 
                  50 
                 0 
               
               
                   
               
             
          
         
       
     
     Table VII shows the takt times for a single product manufactured by the manufacturing line illustrated in FIG.  7 C. Again, a demand-at-capacity for the product is 100 units, H j =7.3 hrs and S j =1 are considered. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE VII 
               
               
                   
               
               
                   
                   
                 TAKT 
                   
                 ORIGINAL 
                   
               
               
                 PROCESS/ 
                   
                 TIME 
                 DESIGNED 
                 THROUGH- 
                   
               
               
                 ID 
                 PATH 
                 (min) 
                 OUTPUT 
                 PUT 
                 SCRAP 
               
               
                   
               
             
             
               
                 P1/1 
                 Main 
                  4.38 
                 214 
                 214 
                 0 
               
               
                 P2/2 
                 Main 
                  5.48 
                 171.2 
                 171.2 
                 0 
               
               
                 P3/3 
                 Main 
                  5.48 
                 128.4 
                 171.2 
                 42.8 
               
               
                 P4/4 
                 Main 
                  4.38 
                 166.92 
                 166.92 
                 0 
               
               
                 OP1/? 
                 Option 
                   
                 38.52 
                 42.8 
               
               
                 P5/5 
                 Main 
                  4.38 
                 100 
                 166.92 
                 66.92 
               
               
                 FP1/1 
                 Feeder 
                  8.76 
                 186.18 
                 248.24 
                 62.060 
               
               
                 FP2/2 
                 Feeder 
                  8.76 
                 166.92 
                 186.18 
                 0 
               
               
                 RP1 
                 Rework 
                 43.8  
                 16.7 
                 16.7 
               
               
                   
               
             
          
         
       
     
     3. Balancing Resources  200   
     The operational cycle time Opc/t j  (takt time) at a process is calculated by dividing the process time by the summed demand at capacity of each product at each process Dc ij  (Equation 1) and defines the maximum cycle time that can be allowed to produce a product in that process in order that the demand-at-capacity at the end-of-line is provided. Balance resources  200  adjusts the capabilities of each process (i.e., by allocating resources to the process) in order that the operational cycle time (takt time) target is met, thereby synchronizing or balancing the flow of product through all processes. 
     The computerized mixed-model manufacturing design system  1  (see FIG. 1) calculates the number of resources required at each process to manufacture the set of products. In some embodiments, the term “resources” refers to individual machines or quantities of labor required for a process, i.e., operations, machines, pieces or people required to support the manufacture of products in each process. In the present exemplary embodiment, the resources at each process are analyzed and adjusted, if necessary, to balance the demand weighted average cycle time AwT j  with the takttime OPc/t j  at each process j. This resource balancing ensures smooth product flow between adjacent processes. Other embodiments of this invention may balance the resources after the definition process cells have been created and operations within those cells defined. 
     FIG. 6 shows a resource balancing  200  method for balancing resources according to the present invention. Off-page connector  96  indicates the continuation of the method according to the present invention into the resource balancing step  200 . After process index j is initialized to 1, i.e., pointing to the first process, at step  201 , the resource multiplier, N j , for process j is calculated in step  202 . The resource multiplier is given by 
     
       
         N j =ATw j /(OPc/t j )  (3) 
       
     
     where: 
     OPc/t j  is the operational cycle time (or takt time) at process j (see equation 1); and 
     ATw j  is a demand weighted average cycle time at process j (see equation 2). 
     The resources multiplier is the ratio of the weighted average cycle time, ATw j =[Σj (Dc ij *AT ij )]/Σj Dc ij , to the operational cycle time OPc/t j . The weighted average cycle time ATw j  is calculated in step  93  of FIG.  5  and the takt time OPc/t j  is calculated in step  92  of FIG.  5 . In other embodiments of this invention, N j  may be calculated in other ways, including substituting an un-weighted cycle time average, a medium cycle time value, a root mean square (RMS), demand-at-capacity estimate, and the like, for the takt time. 
     The actual cycle times for each product at a process will be dependent on the work performed on that product model. Consequently, the demand weighted average cycle time at a process may deviate significantly from the takt time. Alternatively, adjusting the amount of resources required may occur at either the process level, prior to grouping processes into cells, or at the cell level. A resource multiplier is used to scale the amount of resources required in order that the takt time (OPc/t j ) be nearly the demand weighted average cycle time (ATw j ). 
     Also, parameters other than ATw j  or Dc j  may be used to calculate a resource multiplier N j  with comparable results. Other parameters on which a calculation of a resource multiplier can be based include an un-weighted actual cycle time average, a median actual cycle time value, a root mean square (RMS) actual demand estimate, and the like. 
     If at step  203  N j  is determined to be greater than an upper threshold, UTL, then additional resources must be provided in an amount equal to the original resource amount multiplied by the resource multiplier N j  minus the original resource amount. Typically, UTL is one. However, UTL may be slightly higher than one, for example 1.1 or 1.2, without significantly affecting the flow or product through the process. If N j  is significantly less than 1 (e.g., 0.8 or 0.9), then the process has more resources available than are required to meet the demand-at-capacity. For purposes of balancing resources, concern is with the case when N j  is greater than the threshold indicating that the amount of resources available is unacceptably less than the amount required to meet the demand-at-capacity for that process. When N j =1, there are just enough resources to meet the demand-at-capacity. 
     Where the resources are determined in step  212  to be grouped principally by process equipment, or machines, the original number of machines is multiplied against N j  as shown in step  204 . Where resources are determined at step  213  to be defined primarily in terms of labor, the original non-static labor amount is multiplied by N j  in step  206 . If resources are neither machine nor labor intensive, then both machine and labor resources are multiplied by N j  in step  205  to provide the required amount of resources sufficient to satisfy the demand-at-capacity at process j. 
     When calculating machine resources in some embodiments, the total number of machines required is rounded to the next whole number. Also, since both the total number of machines and the demand-at-capacity at that process are known, the throughput of each machine is easily calculated. 
     The deviation of the demand weighted average cycle time from the takt time may also be compensated for by improving the process, such as reducing non-productive work time (“non-value add” tasks such as set-up and move times) or by reducing actual machine time or labor time. Alternately, the takt time at a process may be modified by increasing the number of hours worked per day (i.e., increasing H, S or both) or by increasing the yield through process improvements or through rework, thereby reducing the demand-at-capacity at that process. 
     Alternately, resources may be balanced after processes common to a product family have been grouped into a cell. In that case, the work times for each of the dissimilar machines and/or for each person performing dissimilar tasks is compared to the takt time as defmed in Equation 1. The number of machines or persons is adjusted so that the cumulative work time for each set of dissimilar machines or persons is within a predetermined range of the takt time for the cell. The cell takt time is calculated as in Equation 1, with the exception that the demand at capacity is calculated for the cell rather than for a process. 
     In some embodiments, multiplier N j  is also interpreted as a machine utilization index. If N j  is approximately equal to one, then machine utilization is high. If N j  is less than about 0.8, then machine utilization is low. Accordingly, the index, N j , permits resource planning and monitoring after the manufacturing line has been designed and built. Using the mixed-model manufacturing process design system  1  (see FIG. 1) at the start of a routine manufacturing day, the demand weighted average cycle time, ATw j , and the daily demand rate, Dr j , are provided as inputs to permit calculation of a flow rate and a daily resource index N j . The calculations are in accordance with Equation 1 through Equation 3, with the exception that EOL demand-at-capacity, Dc, is replaced with that day&#39;s EOL demand rate, Dr. During lower volume production days, when the calculated number of required resources are correspondingly smaller, machine units can be switched off, or taken off-line, in integer increments, thereby bringing the utilization closer to unity. 
     When resources include primarily labor resources, the calculation provides the number of people (or the number of person hours) required for the process. As in the calculation of the number of machines, the number of people also provides a labor utilization index whereby the amount of labor provided may be reduced in accordance with the daily rate of production. 
     Query step  207  determines whether there are additional processes to be resource balanced. If there are additional processes, then the process index j is incremented at step  209  and steps  202 - 207 ,  212  and  213  are repeated. Otherwise, balance resources  200  terminates at off-page connector  210 . 
     4. Cell Definition  300   
     In some embodiments of the invention, processes are combined to produce “cells”. A cell is defined as a convenient grouping of dissimilar resources (people and/or machines) in a logical and sequential manner in order to facilitate the flow of product or products. The cells of the manufacturing line impart work content at constant volume product flow (i.e. no yield loss in the cell, or yield loss only at the output of the cell). In some embodiments, the selection of processes to include in a cell is at the discretion of the manufacturing line designer using the system. In other embodiments, the processes are selected, at least in part, by the system. 
     Although flexible, cells are defined in a fashion to facilitate the flow of quantities of product through the manufacturing line. For example, a cell may be defined to include several machine dominated processes along with a single person who operates all of the machines. Definitions of this fashion maximize the utilization of the various resources of the manufacturing line and reduce the amount of scrap produced. 
     Referring now to FIG. 8A, the procedure resumes from off-page connector  210  (FIG. 6) with the cell definition step at  300 . In some embodiments, the operator of the mixed-model manufacturing design system  1  (FIG. 1) may elect to select the method by which processes are assigned to cells at a step  301 . In some embodiments, the operator may choose not to define the cells, in which case each process is defined as its own cell. In the embodiment illustrated in FIG. 8A, the operator may select one of three methods of cell definition: a manual definition  302 , a simple definition  315 , or an automatic definition  322 . Cell definition  300  exits at connector  306 . 
     FIG. 8B shows an embodiment of manual definition  302 . Initialization step  303  initializes pointers I, J, and K to 1. The pointer I refers to the cell that is currently being defined, pointer J refers to the process that is currently being evaluated, and pointer K refers to the product in the family of products that includes process J. A process is selected in select process  304 . In step  305 , product K is added to the definition of cell I. Adding product K to cell I indicates that process J for producing product K is included in cell I. Initially, the first product (K=1) of the family of products that includes process J is added to cell I. 
     Decision step  306  determines if there are further products in the family of products that includes process J. If there are, K is incremented in step  307 . In step  308 , the operator is queried as to whether to include product K in cell I. If not, then the cell pointer I is incremented in step  309  and product K is included in the new cell I at step  305 . If, at step  308 , the operator elects to include product K in the current cell I, then product K is added to cell I at step  305 . 
     If, in step  306 , it is determined that there are no further products in the family of products that includes process J, then step  310  determines if there are further processes to analyze. If there are further processes, the process counter is incremented in step  311 . In step  312 , the operator is queried as to whether or not to copy the cell definitions for the J−1 process into the definitions for the Jth processes. If the operator elects to copy the processes from the J−1th process, then step  310  again determines whether there are further processes. In some embodiments, the operator may copy any previously determined set of cell definitions for a process. 
     If, in step  312 , the operator elects not to copy a previous cell definition, then the cell pointer I is incremented and the product pointer K is set to 1 in step  313 . After step  313 , manual definition  302  returns to step  304 . 
     If, at step  310 , it is determined that there are no further processes to consider, step  314  repeats the steps that lead to resource balancing before exiting manual definitions  302 . At this point, steps  64  through  210  of FIGS. 4 through 6 are repeated except that the definitions referring to processes now refer to cells. 
     FIG. 8C shows a simple cell assignment  315 . Cell counter I and process counter J are initialized at step  316  and set equal to one. The first process is assigned to the first cell at step  317 . The system then identifies any additional processes at step  318  and increments J at step  319 . In step  320 , the operator indicates whether or not to begin a new cell. If the operator opts not to begin a new cell, simple definition  315  returns to step  317  and process J is added to cell I. 
     Not all processes may appropriately be included in a multi-process cell. For example, the proximity to other processes, and/or the character of the sequential process (e.g., clean room versus non-clean-room, or use of hazardous or incompatible chemical raw materials), or non-constant volume flow may compel isolating a process as a single-process cell. 
     If, in step  320 , the operator elects to begin a new cell, cell pointer I is incremented in step  321  and process J is added to cell I in step  317 . The foregoing steps are repeated until all processes have been assigned to a cell. In some embodiments of the design system, the operator may opt a default to assign each process to a single cell. 
     In still other embodiments of the system of the present invention, processes may be automatically assembled into cells at step  322  (FIG.  8 A). While not shown in FIG. 8A, various criteria may be programmed into the system to determine whether processes may be combined. For example, sequential processes may include a flag in their process definitions indicating that the sequential set of processes are combinable in a single cell. Alternately, process characteristics such as proximity to an adjacent process, whether or not an operator or technician will be shared with an adjacent process, the absence or presence of hazardous chemicals, and the like may be considered by an artificial intelligence module in the system to determine whether adjacent processes are appropriately combinable into a single cell. 
     FIG. 8D shows an embodiment of automatic cell definition  322 . The embodiment of automatic cell definition  322  shown in FIG. 8D is based on product balancing. The takt time for a process is based on a demand weighted average cycle time across all products comprising the family of products at that process. It is possible that the actual cycle time for an individual product model may deviate significantly from the takt time. Those product models having significantly different actual cycle times from the demand weighted average cycle time will cause temporary disruptions in the product flow through that process. 
     Accordingly, these product models are identified in a product balancing step by comparing the “time index” of product i in cell (or process) l, with the takt time for cell (or process) l. The time index is defined as follows: 
     
       
         TI il =AT il /(OPc/t l )  (4) 
       
     
     where TI il  is the time index of product i in process l, AT il  is the actual cycle time of product i in process 1, and Opc/t l  is the operational cycle time or takt time of cell 1. The takt time for cell 1, OPc/t l , is equal to the takt time of the processes comprising that cell. It is clear that a TI il  value of 1.0 is desirable as the actual cycle time for the product is equal to the takt time of the cell, thereby ensuring smooth product flow at that cell. Products having a time index which deviate significantly from 1.0 are identified and alternate operations, cells or option paths can be provided to accommodate them. 
     The embodiment of automatic cell definition  322  shown in FIG. 8D, which is based on product balancing, begins with a definition of the threshold limits that bound the time index at step  323 . A lower threshold value (LTV), typically 0.8, and an upper threshold limit (UTL), typically 1.1, are defined to the system at step  323  by the manufacturing line designer (the operator). These threshold values place limits on how much deviation the actual process time at each operation varies from the operational cycle time for the cell and depend on the takt time OPc/t j ; for the process. Long operational cycle times can tolerate more deviation in actual process time from the takt time. For example, for takt times less than 3.0 minutes, the limits may span the range of 0.9 to 1.1. If the takt time is in the range of 3 to 9 minutes, then the limits may span the range 0.8 and 1.1. If the takt time is greater than 9, but less than 30 minutes, then the limits may span the range 0.7 and 1.2. If the takt time is greater than 30 minutes, then the limits may be in the range between 0.7 and 1.3. Additionally, the operator may initially set the threshold limits high when initially defining cells and refining the limits at a later stage of cell definition. 
     Initialization  324  sets cell counter I, product counter K, and process counter J to one. Process J is selected for consideration in step  325  and product K is selected for consideration in step  326 . Product K is a member of the family-of-products that includes process J In step  327 , the actual cycle time for product K at process J, AT KJ  and the operational cycle time (takt time) time for process J OPc/t J  are retrieved. The time index for product K, TI K , according to Equation 4 is calculated in step  328  and compared against the upper threshold limit and lower threshold value in step  329 . If TI K  falls within the LTV to UTL limits, then product K is added to the definition of cell I (i.e., process J for product K is added to the cell definition) in step  330 . If TI K  falls outside of the LTV to UTL limits, product K is added to the cell definition of cell I+1. 
     Step  332  determines if there are more products in the family of products for process J. If there are more products, the product pointer K is incremented in step  333  and product K is selected for consideration in step  326 . If step  332  determines that there are no more products, then step  334  re-executes a resource balancing for cell I. Step  334  executes steps  64  through  210  of FIGS. 4 through 6 in order to balance the resources in cell I. 
     Step  335  determines if there are any products that were placed in cell I+1. If cell I+1 is empty, step  346  determines if there are more processes to consider. If not, then automatic cell definition  322  exits. If step  346  determines that there are more processes, then product pointer K is reset to 1, process pointer J is incremented and a new process is selected for consideration in step  325 . 
     If it is determined in step  335  that cell I+1 is not empty, then cell pointer I is incremented by 1 and a product counter G is set to 1 in step  336 . The resource balance procedure for cell I (steps  64  through  210  of FIGS. 4 through 6) are executed in step  337 . Product G is selected in step  338  and the actual cycle time ATG and the operational cycle time OPc/t I  for product G and cell I, respectively, are retrieved in step  339 . The time index for product G in cell I, TI G , is calculated in step  340  and compared with the lower threshold value and the upper threshold limit in step  341 . If time index TI G  is within the limits of LTV and UTL, then product G is left in Cell I in step  342 . However, if time index TI G  is not within the limits of LTV and UTL, then product G is added to cell I+1 in step  343 . Step  344  determines if there are more products to consider in cell I. If there are more products to consider, the product counter G is incremented in step  345  and product G is selected in step  338 . If there are no more products in cell I to consider, then step  334  rebalances the resources for cell I. 
     Once cells have been defined, the system exits cell definition at connector  306  and enters the operational definition  400  (FIG. 2) step. 
     5. Operational Definition  400   
     The system enters the operational definition step  400  at off-page connector  306  (FIG.  8 A). An operation is a subset of a cell and is defined as a grouping of tasks taken from the SOEs (see step  51  of FIG. 3) that define the work to be performed on the products by a person or machine. Operations are conveniently defmed in relation to particular types of machine, or to a labor operation. In an anticipated high-volume, machine intensive manufacturing line, operations are advantageously identified with the work content associated with a machine type. For example, in a high volume computer disk manufacturing process, a cell may comprise a surface etch process and a nickel phosphate plating process. It would be convenient to divide the cell into two operations, i.e., a surface etch operation and a plating operation. Typically, as in the case of the foregoing example, the operations are divided along process boundaries although that is not always the case. Alternately, in a low volume labor intensive manufacturing line, it is convenient to divide operations around labor boundaries. 
     The grouping of tasks to produce an operation is constrained by the requirement that each group of tasks, based on their cumulative previously defined work content times, have an actual cycle time that is approximately equal to the takt time, or operational cycle time, for the processes corresponding to the tasks in the group. If the resource balancing was performed at the process level, as described above, and if operational boundaries within the cell correspond to the process boundaries, then the cycle times of each operation will, in fact, be approximately equal to the takt time of the constituent processes. 
     FIG. 9 shows an embodiment of the define operation step  400  (see FIG.  2 ). A cell pointer is initialized to point at cell one in step  415 . If it is indicated to the system at step  401  that the intended process is a high-volume, machine-intensive process, then operations are defined according to machine work content (in step  402 ). If, on the other hand, it is indicated at step  404  to the system that the anticipated manufacturing line is a low-volume, labor intensive line, the operations are defined around labor content times, as at step  405 . After either of steps  402  or  405 , the system proceeds to step  413  and, if there are remaining cells returns to step  401 . 
     Defining operations according to machine type allows tasks that are machine intensive to be grouped. For example, if the work content for the tasks is primarily from machines and labor work content is minimal, then operations defined by machine tasks are desired. Contrarily, if the work content is dominated by labor then grouping by labor tasks is desired. 
     Alternately, when the anticipated process is neither machine nor labor intensive or where divisions according to process boundaries are not logical, and particularly where the individual processes have not been previously resource balanced, the system moves on to initialization step  406 . In some embodiments steps  401 ,  402 ,  404  and  405  may be omitted and the system immediately proceeds to initialization step  406 . Initialization step  406  sets task counter k and operation counter m both to one. A first task is included in a first operation in the cell at step  407 . The cumulative cycle time of the task is compared to the corresponding process takt time to ensure that the cumulative process time of the first operation does not exceed the takt time for the task&#39;s corresponding process. If the takt time is not exceeded, and if additional tasks remain to be allocated as determined at step  410 , then the task counter k is incremented by one at step  409  and the next task is included in the first operation. If, at step  408 , the takt time is exceeded by addition of the last task (or, alternately, exceeded by a predetermined amount or percentage), then the last task added to the operation is removed from that operation at step  411 . The operation counter is then incremented by one at step  412  and the previously removed task is included in the next operation of the cell. Tasks are added to each operation, subject to the constraint that the cycle time for that operation does not exceed the takt time, until all of the tasks for all processes comprising that cell have been allocated to an operation. The last operation may contain tasks whose combined work content time is significantly shorter than the process takt time. 
     If no more tasks are to be allocated, as determined in step  410 , then the system checks whether there are more cells to process in step  413 . If there are more cells, step  414  adjusts the cell pointer to the next cell and the system returns to step  401 . Otherwise, the operation definition step  400  exits at page connector  403 . 
     If a cell is comprised of a single process, or several constant volume processes, and if operational boundaries in the cell are defined along the process boundaries of the processes comprising the cell, then the resource balancing step (see FIG. 6) will ensure that the operational cycle time of each operation within the cell is approximately equal to the takt time of the processes comprising that cell. If the processes comprising the cells were not previously resource balanced, then each operation must now be balanced against the takt time for processes within the cell. The cell takt time is equal to the takt time of any process in the cell. 
     Operations, according to this method, are constructed until all tasks have been assigned. In some embodiments, a cell will comprise a single process involving only a single machine type or single person. In that event, the cell will typically comprise a single operation containing the task associated with that process, as defined by the SOE list for that process. Where the resources have been previously balanced for that process, the operational cycle time will be within predefined tolerance limits. If the resources in that process are not yet balanced, then the number of resources (machines and/or persons) are now balanced such that the demand weighted average cycle time of operations (calculated in the same fashion as was previously described for the demand weighted average cycle time for processes) is approximately equal to the takt time. 
     In other embodiments, multiple dissimilar machine types or multiple persons, each performing a different task, are involved, and the cell will be divided into a plurality of operations, each operation reflecting a single machine type or person. As in the single operation cell, multiple-operation cells are resource balanced either prior to defining the operations (i.e. the individual processes are resource balanced) or else after the operations have been defined so that the demand weighted average cycle time for each operation is within a predefined range around the takt time for the cell. 
     In view of the above discussion, it is clear that the inter-operational movement of quantities of product within the cell is an essentially continuous flow of the mixed-model production, and that the boundaries between operations represent natural demand-pull, or kanban type interfaces where visible inventory controls can be implemented. 
     Operation definition/balancing continues at connector  403  in FIG.  10 . with the operation balancing step  450 . A lower threshold value (LTV), typically 0.8, and an upper threshold limit (UTL), typically 1.1, are defined to the system at step  451  by the manufacturing line designer. These threshold values place limits on how much deviation the actual process time at each operation varies from the operational cycle time for the cell and depend on the takt time OPc/t j  for the process. As was previously discussed in relation to automatic cell definition in define cells  300 , long operational cycle times can tolerate more deviation in actual process time from the takt time. 
     Initialization step  452  sets operation counter m and cell counter  1  both to one, thereby pointing to the first operation in the first cell. A time index is calculated at step  465  for the current operation in the current cell according to the following equation: 
     
       
         OTI ml =AT ml /OPc/t l   (5) 
       
     
     where OT ml  is the time index for operation m in cell l, AT ml  is the actual cycle time of operation m in cell l and is equal to the cumulative work content time of the group of tasks comprising that operation, and OPc/t l  is the operational cycle time, or takt time, of cell 1, which relates to the individual takt time of the processes that comprise that cell. 
     The time index for the operation is compared to the threshold values LTV and UTV in step  453 . If the time index for the operation is within the range bounded by LTV and UTV then that operation is considered to be balanced. In that case, the operation counter m is incremented by one in step  455  and the system determines if there are any remaining operations in the cell at step  454 . If additional operations remain, then steps  465  and  453 - 454  are repeated. If no more operations remain in the current cell, the system determines whether any cells are left to be analyzed at step  467 , and if so, then cell counter l is incremented by one, and the operation counter m is reset to one at step  468  and the system returns to step  465 . If no more cells remain, then the operational balancing step of the method of this invention is complete as indicated by off-page connector  464 . 
     If at step  453  the time index is outside of the range bounded by LTV and UTL, then the designer/operator may perform any of a number of mitigating steps, including those identified in block  456 . These mitigating actions in step  456  include reducing the work content time per task  457 , modifying the production time  458 , reducing moves or step times  459 , building inventory in upstream operations  460 , identifying overlap times  461 , redesign processes or products  462 , or identify additional resources  463 . For example, if the time index exceeds the UTV, then process improvements, or reduction or elimination of move or setup times may be made that will reduce the work content time of the process, thereby reducing the actual cycle time. Overlapping machine times, labor times, or machine/labor times will also reduce the work content time at a process. Additional actions that may be taken to make the actual cycle time approximately equal to the cell takt time include increasing the number of hours worked per day at that operation as at  458 , permitting a small inventory build up at upstream operations as at  460 , and redesigning the product(s) or process to improve the throughput rate as at  462 . 
     If, in step  467 , it is determined that there are no more cells to evaluate, then the results of the manufacturing design is stored and outputted in step  500  (FIG.  2 ). The mixed-model flow manufacturing design system  1  (FIG. 1) stores the design in storage  6  and outputs the results on monitor  5  or on an attached printer or other hard-copy device or both. 
     In accordance with yet another aspect of the method of this invention, actual measurements of product flow permit closed-loop design refinements to the designed manufacturing line. Estimates of actual cycle times per product by process, as well as estimates of work content times by product by process, are replaced by actual measurements to better define the resulting takt time, number of resources needed, and to balance each cell. 
     Some methods according to this invention include daily manufacturing line management. Advantageously, should the daily demand for each product model deviate from the demand at capacity design criteria, the cell and operation definitions do not change. Rather, as demand and product mix fluctuates, the resources required at each cell fluctuates accordingly, thereby permitting the manufacturing line manager to merely use only that portion of the total number of machines or persons in the cell necessary to accommodate the daily demand. 
     Illustrative Numerical Example 
     As a concrete example of a mixed-model manufacturing line design system according to the present invention, consider the manufacture of the nine products listed below: 
     Product 1 is a 2 inch gearbox; 
     Product 2 is a 3 inch gearbox; 
     Product 3 is a 4 inch gearbox; 
     Product 4 is a 6 inch gearbox; 
     Product 5 is a 8 inch gearbox; 
     Product 6 is a 12 inch gearbox; 
     Product 7 is a 18 inch gearbox; 
     Product 8 is a 24 inch gearbox; and 
     Product 9 is a 36 inch gearbox. 
     FIG. 11 shows a schematic drawing of a manufacturing line for producing products 1 through 9 listed above. In FIG. 11, two feeder paths feed assembly  1101 . Feeder path 1  1111  includes a cover assembly  1103 , drill and tap  1104  and deburr  1105  step. Feeder path 2 includes worm shaft  1102 . Assembly  1101 , test  1106 , and pack  1108  comprise a main path terminating at EOL  1109 . Test  1106  initiates a scrap  1110  fallout and a rework path that includes rework  1107 . 
     In step  15  of FIG. 3, descriptions of each of the products are entered into design system  1 . In steps  16  through  59 , each of the products is defined in terms of the following processes: 
     Process 1 is assembly  1101  (main path); 
     Process 2 is worm shaft  1102  (feeder path 2); 
     Process 3 is cover assembly  1103  (feeder path 1); 
     Process 4 is drill and tap  1104  (feeder path 1); 
     Process 5 is deburr  1105  (feeder path 1); 
     Process 6 is test  1106  (main path); 
     Process 7 is test rework  1107  (rework path); 
     Process 8 is pack  1108  (main path). 
     Table VIII shows a product definition as generated in steps  15  through  60  of FIG.  3 . Additionally, the system inputs that from process 6 (test  1106 ), 20% of products 1 through 6 are routed to process 7 (test rework  1107 ) and 5% of products 7 through 9 are routed to scrap  1110 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE VIII 
               
             
             
               
                   
               
               
                 Definition of Products 
               
             
          
           
               
                   
                   
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                   
                   
                   
                   
               
               
                   
                 Proc. 1 
                 Worm 
                 Cover 
                 Drill &amp; 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                 Assem. 
                 shaft 
                 Assem. 
                 Tap 
                 Deburr 
                 Test 
                 Rework 
                 Pack 
               
               
                   
                   
               
             
          
           
               
                 Prod. 1 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (2″) 
               
               
                 Prod. 2 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (3″) 
               
               
                 Prod. 3 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (4″) 
               
               
                 Prod. 4 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (6″) 
               
               
                 Prod. 5 
                 X 
                   
                   
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (8″) 
               
               
                 Prod. 6 
                 X 
                   
                   
                 X 
                 X 
                 X 
                 X 
                 X 
               
               
                 (12″) 
               
               
                 Prod. 7 
                 X 
                 X 
                   
                   
                   
                 X 
                   
                 X 
               
               
                 (18″) 
               
               
                 Prod. 8 
                 X 
                 X 
                   
                   
                   
                 X 
                   
                 X 
               
               
                 (24″) 
               
               
                 Prod. 9 
                 X 
                 X 
                   
                   
                   
                 X 
                   
                 X 
               
               
                 (36″) 
               
               
                   
               
             
          
         
       
     
     From the definition of products given in Table VIII, families of products are identified in step  63  (FIG.  4 ). Families of products are defined as groups of products that share at least one process. In the above example, there are several families of products that can be defined. For example, all of products 1 through 8 are included in the family of products that utilize processes 1, 6 and 8. Products 1-4 and 7 through 9 utilize process 2, Products 1-4 utilize process 3, Products 1 through 6 utilize processes 4, 5 and 7. Another grouping of products can be defined utilizing similar processes. Products 1-4 (GB4) utilize processes 1 through 8. Products 5 and 6 (GB8) utilize processes 1 and 4-8. Products 7-9 utilize process 1, 2, 6 and 8. For purposes of this discussion, the latter definition of families of products is utilized. Note that in design system  1  (FIG.  1 ), the system can utilize a default definition of families of products (searching for all products that utilize each process) or can group products in other ways based on operator input. 
     In step  64  (FIG.  4 ), the tasks for each product i at process j is identified, along with the actual time for completion of each task. Table IX shows a sequence of events for producing product 1 (the 2″ gearbox) at process 1 (assembly). 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE IX 
               
             
             
               
                   
               
               
                 Sequence of Events for Product 1 at Process 1 (times in minutes) 
               
             
          
           
               
                 Sequence 
                   
                 Machine 
                 Labor 
                 TQC/ 
               
               
                 ID 
                 Task 
                 Time 
                 Time 
                 verify 
               
               
                   
               
               
                  10 
                 Retrieve housing 
                 0.0 
                 0.1 
                 Check tapped hole for 
               
               
                   
                   
                   
                   
                 paint; none should be 
               
               
                   
                   
                   
                   
                 present 
               
               
                  20 
                 Place Bottom 
                 0.0 
                 0.1 
                 Be certain it is seated 
               
               
                   
                 Seal 
                   
                   
                 properly 
               
               
                  30 
                 Grease Housing 
                 0.0 
                 0.5 
                 In area where worm- 
               
               
                   
                   
                   
                   
                 shaft will go; fill 
               
               
                   
                   
                   
                   
                 recess to top of 
               
               
                   
                   
                   
                   
                 housing 
               
               
                  40 
                 Insert worm-shaft 
                 0.0 
                 1.0 
                 Into greased housing 
               
               
                   
                 into greased 
                   
                   
                 recess 
               
               
                   
                 housing 
               
               
                  50 
                 place gear 
                 0.0 
                 0.3 
                 Over bottom seat; 
               
               
                   
                   
                   
                   
                 mesh with worm gear 
               
               
                  60 
                 install switch 
                 0.0 
                 2.0 
                 Secure with 2 screws; 
               
               
                   
                   
                   
                   
                 wires to exit bottom 
               
               
                   
                   
                   
                   
                 hole in housing. 
               
               
                  70 
                 place top seal 
                 0.0 
                 0.1 
                 Over gear; be certain 
               
               
                   
                   
                   
                   
                 it is seated properly 
               
               
                  80 
                 install cover 
                 0.0 
                 0.3 
                 Rotate left and right 
               
               
                   
                 assembly 
                   
                   
                 to install; seat 
               
               
                   
                   
                   
                   
                 properly; pull wires 
               
               
                   
                   
                   
                   
                 through hoe in cover; 
               
               
                   
                   
                   
                   
                 secure with 2 screws 
               
               
                  90 
                 dress wires 
                 0.0 
                 0.5 
                 Separate and single- 
               
               
                   
                   
                   
                   
                 knot wires for power 
               
               
                   
                   
                   
                   
                 (white), ground 
               
               
                   
                   
                   
                   
                 (green), and common 
               
               
                 100 
                 place completed 
                 0.0 
                 0.1 
                 (black) 
               
               
                   
                 unit in IPK 
               
               
                   
               
             
          
         
       
     
     From Table XI, Assembly process 1 for producing product 1 (a 2″ gearbox) takes 5 minutes of labor and 0 minutes of machine time. From tables like Table XI, a product/task work map such as that shown in Table XII is produced. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE XI 
               
             
             
               
                   
               
               
                 Product/process map wth sequence-of-event times (M = Machine time; 
               
               
                 L = Labor time; times are in minutes) 
               
             
          
           
               
                   
                 Proc. 1 
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                   
               
             
          
           
               
                 Prod. 1 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 5.0 
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 1.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 2 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 6.0 
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 1.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 3 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 7.0 
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 2.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 4 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 8.0 
                 L: 10.0 
                 L: 8.0 
                 L: 8.2 
                 L: 2.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 5 
                 M: 0.0 
                   
                   
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 9.0 
                   
                   
                 L: 8.2 
                 L: 3.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 6 
                 M: 0.0 
                   
                   
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 10.0 
                   
                   
                 L: 8.2 
                 L: 3.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                 Prod. 7 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                 L: 50.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                 Prod. 8 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                 L: 55.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                 Prod. 9 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                 L: 60.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                   
               
             
          
         
       
     
     From the data in Table XI, actual cycle times for each product at each process can be calculated as in steps  74  through  82 . Table XII lists the actual cycle times, At ij , for each product i at each process j. In Table XII, the overlap times have been deleted as in step  76  of FIG.  4 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE XII 
               
             
             
               
                   
               
               
                 Actual cycle times (in minutes) for each product at each process, AT ij   
               
             
          
           
               
                   
                 Proc. 1 
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                   
               
             
          
           
               
                 Prod. 1 
                 5.0 
                 10.0 
                 5.0 
                 8.2 
                 1.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 2 
                 6.0 
                 10.0 
                 5.0 
                 8.2 
                 1.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 3 
                 7.0 
                 10.0 
                 5.0 
                 8.2 
                 2.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 4 
                 8.0 
                 10.0 
                 8.0 
                 8.2 
                 2.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 5 
                 9.0 
                   
                   
                 8.2 
                 3.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 6 
                 10.0 
                   
                   
                 8.2 
                 3.0 
                 5.0 
                 10.0 
                 2.0 
               
               
                 Prod. 7 
                 50.0 
                 12.0 
                   
                   
                   
                 6.0 
                   
                 3.0 
               
               
                 Prod. 8 
                 55.0 
                 12.0 
                   
                   
                   
                 6.0 
                   
                 3.0 
               
               
                 Prod. 9 
                 60.0 
                 12.0 
                   
                   
                   
                 6.0 
                   
                 3.0 
               
               
                   
               
             
          
         
       
     
     In step  83  (FIG.  5 ), the demand at capacity of each product is defined. The demand at capacity Dc i  is the maximum daily number of units of product that the manufacturing line designed by design system  1  is capable of producing. In the present example, Dc i  is given in Table XIII: 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE XIII 
               
             
             
               
                   
               
               
                 EOL Demand at Capacity for each product 
               
             
          
           
               
                   
                 Product ID 
                 Dc j   
               
               
                   
                   
               
               
                   
                 1 
                 10 
               
               
                   
                 2 
                 20 
               
               
                   
                 3 
                 30 
               
               
                   
                 4 
                 10 
               
               
                   
                 5 
                 15 
               
               
                   
                 6 
                  5 
               
               
                   
                 7 
                  5 
               
               
                   
                 8 
                  3 
               
               
                   
                 9 
                  2 
               
               
                   
                   
               
             
          
         
       
     
     In step  84 , the number of hours for each shift is defmed. Typically, all recurring lost or unavailable times is subtracted from available time. In the present example, there are 8 hours per shift but 0.7 hours are lost for employee breaks, equipment downtime, maintenance timer, recurring meetings or other similar occurrences. Therefore, each of process 1 through 8 has 7.3 hours per shift. Also, in this example, it is assumed that there is one shift for each process per day. Therefore, H j =7.3 hours and S j =1 for all j. 
     In step  87  (FIG.  5 ), the demand at capacity for each product at each process Dc ij  is calculated. The demand at capacity for a product at a process depends on the process yield, rework, option and scrap factors. In the present example, there is one rework process, one instance of scrap and no option process. Consider product 1, the 2″ gearbox. Following the procedures that have previously been outlined for calculating Dc ij , if we start with x units of product 1 at the beginning of the main path, assembly  1101 . There is no scrap or diversion of product into option or rework paths at assembly  1101  and therefore x amount of product leaves assembly  1101 . The amount of product that is input to test  1106  is x+z where z is the output of product from rework  1107 . At the conclusion of test  1106 , 20% of product 1 is diverted to rework  1107 . Therefore, the amount of product input to rework  1107  is 0.2(x+z). The output of rework  1107 , therefore, is also 0.2(x+z) which is z. Therefore, z=0.25x. None of product 1 is diverted to scrap  1110 . The amount of product that is input to pack  1108 , therefore, is 0.8(x+z)=0.8(1.25)x=x. Therefore, the amount of product 1 that is output from pack  1108  to EOL  1109  is x. Normalizing x so that the demand at capacity Dc of product 1 is 10 indicates that x=10. 
     Because product 1 requires one worm shaft to be produced by worm shaft  1102 , then the demand at capacity for product 1 at worm shaft  1102  is also 10. Additionally, feeder path  1111  has no losses or rework paths. As such, the demand at capacity for product 1 at cover assembly  1103 , drill and tap  1104  and deburr  1105  is also 10. 
     Table XIV indicates the demand at capacity for each product at each process. Note that products 7-9 require that worm shaft  1102  produce two worm shafts for each gearbox instead of one and that products 7 through 9 do not utilize rework  1107 , directing 5% of products 7-9 to scrap  1110  instead. Therefore, starting with x units at assembly  1101 , x units of product are input to test  1106 . From test  1106 , 5% of product (0.05x) is directed to scrap  1110  while 95% (0.95x) of product is input to pack  1108  to yield 0.95 x at EOL. Therefore, to yield a demand at capacity for product 7 of 5 units, x=5.26 units of product 7. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE XIV 
               
             
             
               
                   
               
               
                 Demand at Capacity for each product at each process Dc ij . 
               
             
          
           
               
                   
                   
                   
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                   
                   
                   
                   
               
               
                   
                 Dc 
                 Proc. 1 
                 worm 
                 cover 
                 drill &amp; 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                 EOL 
                 Assem. 
                 shaft 
                 assem. 
                 tap 
                 deburr 
                 test 
                 rework 
                 pack 
               
               
                   
                 1109 
                 1101 
                 1102 
                 1103 
                 1104 
                 1105 
                 1106 
                 1107 
                 1108 
               
               
                   
                   
               
             
          
           
               
                 Prod. 1 
                 10 
                 10 
                 10 
                 10 
                 10 
                 10 
                 12 
                 2 
                 10 
               
               
                 (2″) 
               
               
                 Prod. 2 
                 20 
                 20 
                 20 
                 20 
                 20 
                 20 
                 24 
                 4 
                 20 
               
               
                 (3″) 
               
               
                 Prod. 3 
                 30 
                 30 
                 30 
                 30 
                 30 
                 30 
                 36 
                 6 
                 30 
               
               
                 (4″) 
               
               
                 Prod. 4 
                 10 
                 10 
                 10 
                 10 
                 10 
                 10 
                 12 
                 2 
                 10 
               
               
                 (6″) 
               
               
                 Prod. 5 
                 15 
                 15 
                   
                   
                 15 
                 15 
                 18.75 
                 3.75 
                 15 
               
               
                 (8″) 
               
               
                 Prod. 6 
                 5 
                 5 
                   
                   
                 5 
                 5 
                 6.25 
                 1.25 
                 5 
               
               
                 (12″) 
               
               
                 Prod. 7 
                 5 
                 5.26 
                 10.52 
                   
                   
                   
                 5.26 
                   
                 5 
               
               
                 (18″) 
               
               
                 Prod. 8 
                 3 
                 3.16 
                 6.32 
                   
                   
                   
                 3.16 
                   
                 3 
               
               
                 (24″) 
               
               
                 Prod. 9 
                 2 
                 2.1 
                 4.2 
                   
                   
                   
                 2.1 
                   
                 2 
               
               
                 (36″) 
               
               
                 Total 
                 100 
                 100.52 
                 91.04 
                 70 
                 90 
                 90 
                 119.52 
                 19 
                 100 
               
               
                 Dc j   
               
               
                   
               
             
          
         
       
     
     From Table XIV, the operational cycle time (Takt time) OPc/t j  for each process is calculated in step  92  (FIG.  5 ). As has been previously discussed, Opc/t j =(H j *S j )/Dc j  where in this example H j =7.3 hours (the number of hours in one shift of process j) and S j =1 (the number of shifts for process j per day). In that case, Table XV gives the takt times for each of processes 1 through 9. 
     From Table XI and Table XIV, the actual weighted cycle time ATw j  is calculated in step  93  of FIG.  5 . The actual weighted cycle time ATw j  is given by Equation 2. In Table XV, the actual cycle time for each product at each process, segregated into machine time (M) and labor time (L), is given along with the actual weighted cycle time ATw j . For convenience, Table XV also includes the total demand at capacity for each process Dc j . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE XV 
               
             
             
               
                   
               
               
                 Calculation of the takt time for each process and the actual weighted 
               
               
                 cycle time for each process. 
               
             
          
           
               
                   
                   
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                   
                   
                   
                   
               
               
                   
                 Proc. 1 
                 Worm 
                 Cover 
                 Drill &amp; 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                 Assem. 
                 Shaft 
                 Assem. 
                 Tap 
                 Deburr 
                 Test 
                 Rework 
                 Pack 
               
               
                   
                 1101 
                 1102 
                 1103 
                 1104 
                 1105 
                 1108 
                 1107 
                 1109 
               
               
                   
                   
               
             
          
           
               
                 Prod. 1 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 82.0 
                 M: 0.0 
                 M: 60.0 
                 M: 0.0 
                 M: 0.0 
               
               
                 (2″) 
                 L: 50.0 
                 L: 100.0 
                 L: 50.0 
                 L: 82.0 
                 L: 10.0 
                 L: 60.0 
                 L: 20.0 
                 L: 20.0 
               
               
                 Min 
               
               
                 Prod. 2 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 164.0 
                 M: 0.0 
                 M 120.0 
                 M: 0.0 
                 M: 0.0 
               
               
                 (3″) 
                 L: 120.0 
                 L: 200.0 
                 L: 100.0 
                 L: 164.0 
                 L: 20.0 
                 L: 120.0 
                 L: 40.0 
                 L: 40.0 
               
               
                 Min 
               
               
                 Prod. 3 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 246.0 
                 M: 0.0 
                 M: 180.0 
                 M: 0.0 
                 M: 0.0 
               
               
                 (4″) 
                 L: 210.0 
                 L: 300.0 
                 L: 150.0 
                 L: 246.0 
                 L: 60.0 
                 L: 180.0 
                 L: 60.0 
                 L: 60.0 
               
               
                 Min 
               
               
                 Prod. 4 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 82.0 
                 M: 0.0 
                 M: 60.0 
                 M: 0.0 
                 M: 0.0 
               
               
                 (6″) 
                 L: 80.0 
                 L: 100.0 
                 L: 80.0 
                 L: 82.0 
                 L: 20.0 
                 L: 60.0 
                 L: 20.0 
                 L: 20.0 
               
               
                 Min 
               
               
                 Prod. 5 
                 M: 0.0 
                   
                   
                 M: 123.0 
                 M: 0.0 
                 M: 93.75 
                 M: 0.0 
                 M: 0.0 
               
               
                 (8″) 
                 L: 135.0 
                   
                   
                 L: 123.0 
                 L: 45.0 
                 L: 93.75 
                 L: 37.5 
                 L: 30.0 
               
               
                 Min 
               
               
                 Prod. 6 
                 M: 0.0 
                   
                   
                 M: 41.0 
                 M: 0.0 
                 M: 31.25 
                 M: 0.0 
                 M: 0.0 
               
               
                 (12″) 
                 L: 50.0 
                   
                   
                 L: 41.0 
                 L: 15.0 
                 L: 31.25 
                 L: 12.5 
                 L: 10.0 
               
               
                 Min 
               
               
                 Prod. 7 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 27.67 
                   
                 M: 0.0 
               
               
                 (18″) 
                 L: 263.0 
                 L: 126.2 
                   
                   
                   
                 L: 31.67 
                   
                 L: 15.0 
               
               
                 Min 
               
               
                 Prod. 8 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 15.8 
                   
                 M: 0.0 
               
               
                 (24″) 
                 L: 173.8 
                 L: 75.84 
                   
                   
                   
                 L: 18.96 
                   
                 L: 9.0 
               
               
                 Min 
               
               
                 Prod. 9 
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 10.5 
                   
                 M: 0.0 
               
               
                 (36″) 
                 L: 126.0 
                 L: 50.4 
                   
                   
                   
                 L: 12.6 
                   
                 L: 6.0 
               
               
                 Min 
               
               
                 Sum i 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 738.0 
                 M: 0.0 
                 M: 599.0 
                 M: 0.0 
                 M: 0.0 
               
               
                 AT ij   
                 L: 1207.8 
                 L: 952.5 
                 L: 380.0 
                 L: 738.0 
                 L: 170.0 
                 L: 608.2 
                 L: 190.0 
                 L: 210.0 
               
               
                 Total 
                 100.52 
                 91.04 
                 70 
                 90 
                 90 
                 119.52 
                 19 
                 100 
               
               
                 Dc j   
               
               
                 ATw 
                 M: 0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.00 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 12.02 
                 L: 10.46 
                 L: 5.43 
                 L: 8.2 
                 L: 1.89 
                 L: 5.09 
                 L: 10.0 
                 L: 2.10 
               
               
                 Opc/t j   
                 4.36 
                 4.81 
                 6.26 
                 4.87 
                 4.87 
                 3.66 
                 23.05 
                 4.38 
               
               
                 (Min) 
               
               
                 N j   
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 1.68 
                 M: 0.0 
                 M: 1.4 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 2.76 
                 L: 2.16 
                 L: 0.86 
                 L: 1.68 
                 L: 0.39 
                 L: 1.4 
                 L: 0.43 
                 L: 0.48 
               
               
                 OPER. 
                 0/3 
                 0/3 
                 0/1 
                 2/2 
                 0/1 
                 2/2 
                 0/1 
                 0/1 
               
               
                   
               
             
          
         
       
     
     From the compilation of Table XV, the number of resources required at each process can be calculated. The number of resources required is given by the ratio between the actual weighted cycle time ATw j  for that process and the takt time for process j, OPc/t j . The number of resources required is calculated according to step  202  (FIG.  6 ). The number of resources is given by the ratio of the actual weighted cycle time Atw j  for process j and the takt time for process j. This ratio is given in Table XV as N j . Notice that, for these calculations, the labor time and the machine times are calculated separately. The number of operations is also tabulated. The number of operations is the number of machine resources (i.e., the number of machines) and the number of labor resources (i.e., the number of shifts) required to meet the demand at capacity at EOL Dc requirements of the manufacturing line. Note that the number of operations is an integer number, adding only whole shifts or whole machines, and that the number of resources is always greater than the number required. In this example, the value of UTL used in step  203  of FIG. 6 is 1. 
     In step  301  of FIG. 8A, the operator must decide a method of cell definition. The automatic cell definition, step  322 , described here uses time indexing in order to separate processes into cells. Time indexing is a technique that is used to design a mixed model line when significant variation exists in the actual process times various products at particular processes. For example, process 1 (assembly  1101 ) has a variation in time to produce products of between 5 minutes and 60 minutes. If a 5 minute product is followed by a 60 minute product, an imbalance is created in the flow and downstream processes would become idle. 
     The operator may also choose other considerations for forming cells. The original demand at capacity for each product should be considered as well as presentation of material to the line, physical constraints, and general knowledge of the processes involved. If the products that cause variation are low in volume, the operator may choose to produce those products in a separate cell. If products do not share common material, cells may be separated to facilitate material delivery to the manufacturing line. Formation of cells may also be influenced by building requirements, safety requirements, and other factors that are determined by the operator. Following the automatic definition of cells, step  322 , shown in FIG. 8D, it is clear from Table XV that products 1 through 6 at process 1 can be grouped in one cell and that products 7-9 at process 1 can be grouped in a third. Each of processes 2 through 9 for all products are grouped into separate cells as well. To see that this is true, the takt time for process 1 is 4.36 minutes. For products 1 through 6, the time index given by the ratio of the actual time to the takt time ranges from 1.1 to 2.3 whereas the time index for products 7-9 ranges from 11.6 to 14.0. Table XVI shows the refinement of the balance resources step with the new cell definitions. 
     
       
         
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE XVI 
               
               
                   
                   
               
               
                   
                   
                   
                 Cell 3 
                 Cell 4 
                 Cell 5 
                   
                   
                   
                   
               
               
                   
                 Cell 1 
                 Cell 2 
                 Proc. 2 
                 Proc. 3 
                 Proc. 4 
                 Cell 6 
                 Cell 7 
                 Cell 8 
                 Cell 9 
               
               
                   
                 Proc. 1 
                 Proc. 1 
                 worm 
                 cover 
                 drill &amp; 
                 Proc. 5 
                 Proc. 6 
                 Proc. 7 
                 Proc. 8 
               
               
                   
                 assem. 
                 Assem. 
                 shaft 
                 assem. 
                 tap 
                 deburr 
                 test 
                 rework 
                 pack 
               
               
                   
                 1101 
                 1101 
                 1102 
                 1103 
                 1104 
                 1105 
                 1106 
                 1107 
                 1108 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 Prod. 1 
                 M: 0.0 
                   
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 5.0 
                   
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 1.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (10) 
                   
                 (10) 
                 (10) 
                 (10) 
                 (10) 
                 (12) 
                 (2) 
                 (10) 
               
               
                 Prod. 2 
                 M: 0.0 
                   
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 6.0 
                   
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 1.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (20) 
                   
                 (20) 
                 (20) 
                 (20) 
                 (20) 
                 (24) 
                 (4) 
                 (20) 
               
               
                 Prod. 3 
                 M: 0.0: 
                   
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 7.0 
                   
                 L: 10.0 
                 L: 5.0 
                 L: 8.2 
                 L: 2.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (30) 
                   
                 (30) 
                 (30) 
                 (30) 
                 (30) 
                 (36) 
                 (6) 
                 (30) 
               
               
                 Prod. 4 
                 M: 0.0: 
                   
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 8.0 
                   
                 L: 10.0 
                 L: 8.0 
                 L: 8.2 
                 L: 2.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (10) 
                   
                 (10) 
                 (10) 
                 (10) 
                 (10) 
                 (12) 
                 (2) 
                 (10) 
               
               
                 Prod. 5 
                 M: 0.0: 
                   
                   
                   
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 9.0 
                   
                   
                   
                 L: 8.2 
                 L: 3.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (15) 
                   
                   
                   
                 (15) 
                 (15) 
                 (18.75) 
                 (3.75) 
                 (15) 
               
               
                 Prod. 6 
                 M: 0.0: 
                   
                   
                   
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 10.0 
                   
                   
                   
                 L: 8.2 
                 L: 3.0 
                 L: 5.0 
                 L: 10.0 
                 L: 2.0 
               
               
                   
                 (5) 
                   
                   
                   
                 (5) 
                 (5) 
                 (6.25) 
                 (1.25) 
                 (5) 
               
               
                 Prod. 7 
                   
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                   
                 L: 50.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                   
                   
                 (5.26) 
                 (10.52) 
                   
                   
                   
                 (5.26) 
                   
                 (5) 
               
               
                 Prod. 8 
                   
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                   
                 L: 55.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                   
                   
                 (3.16) 
                 (6.32) 
                   
                   
                   
                 (3.16) 
                   
                 (3) 
               
               
                 Prod. 9 
                   
                 M: 0.0 
                 M: 0.0 
                   
                   
                   
                 M: 5.0 
                   
                 M: 0.0 
               
               
                   
                   
                 L: 60.0 
                 L: 12.0 
                   
                   
                   
                 L: 6.0 
                   
                 L: 3.0 
               
               
                   
                   
                 (2.1) 
                 (4.2) 
                   
                   
                   
                 (2.1) 
                   
                 (2) 
               
               
                 Total 
                 90 
                 10.52 
                 91.04 
                 70.0 
                 90.0 
                 90.0 
                 119.52 
                 19 
                 100 
               
               
                 Dc 
               
               
                 Takt 
                 4.87 
                 41.63 
                 4.81 
                 6.26 
                 4.87 
                 4.87 
                 3.67 
                 23.05 
                 4.38 
               
               
                 time 
               
               
                 ATs 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 8.2 
                 M: 0.0 
                 M: 5.0 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 7.17 
                 L: 53.5 
                 L: 10.5 
                 L: 5.43 
                 L: 8.2 
                 L: 1.9 
                 L: 5.1 
                 L: 10.0 
                 L: 0.5 
               
               
                 N 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 0.0 
                 M: 1.7 
                 M: 0.0 
                 M: 1.4 
                 M: 0.0 
                 M: 0.0 
               
               
                   
                 L: 1.5 
                 L: 1.3 
                 L: 2.2 
                 L: 0.9 
                 L: 1.7 
                 L: 0.4 
                 L: 1.4 
                 L: 0.4 
                 L: 0.5 
               
               
                 Res. 
                 0/2 
                 0/2 
                 0/3 
                 0/1 
                 2/2 
                 0/4 
                 2/2 
                 0/1 
                 0/1 
               
               
                   
               
             
          
         
       
     
     In operational balance  400 , steps  401  through  415  of FIG. 9, operations within each cell are defined according to the takt time for the cell. Referring to cells 1 and 2 and the sequence of events shown in Table IX, cells 1 and 2 can be divided into two operations. Operation 1 includes tasks 10 through 50 and operation 2 includes tasks 60-100. 
     In operations balancing  450 , indexing may be applied to each operation within the cell. Additionally, imbalances in timing are recognized and non-value added work is eliminated, work is reallocated, additional resources are assigned, or other actions are taken in order to balance the flow of product through each cell. 
     The above-described examples of this invention are demonstrative only. Those skilled in the art will recognize a plethora of variations based on these examples, all of which are intended to be included within the scope of this description. As such, the invention is limited only by the following claims.