Abstract:
A method and system are provided for operating a data processing system including a data base computer system and a resource allocation computer for control of resource allocation in a manufacturing plant with a manufacturing line comprising a plurality of stages with manufacturing machines, the resource allocation computer including data storage means. The method includes several steps including: deriving data from the data storage means and computing the targets for each of the stages; obtaining machine capacity data from the data storage means and employing the machine capacity data for allocating machine capacity proportionally and adjusting targets; adding limits to stages of penetration and adjusting targets; determining residual capacity and allocating the residual capacity of the manufacturing machines; and checking the convergence of targets and machine allocation until convergence is achieved.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to resource allocation methods and apparatus for controlling manufacturing plants, and more particularly to semiconductor manufacturing control systems. 
     2. Description of Related Art 
     Disadvantages of a current target generation systems are as follows: 
     1. The number of part-types has been limited with more than 20% of the lots not being directly covered. 
     2. There has been no differentiation of lot priorities and there has been inaccurate machine allocation and estimation of stage penetration. 
     3. There has been a manufacturing process with a homogeneous production rate in terms of WPH (Wafers Per Hour) of the machines over different stages. 
     4. Accuracy of target generation has been low. 
     SUMMARY OF THE INVENTION 
     In accordance with this invention a process is provided in which a manufacturing line is operated with the features defined by the functions or steps as follows: 
     1. Compute the target for each stage; 
     2. Allocate machine capacity proportionally and adjusting targets (according to lot priority); 
     3. Add limits to stages of penetration and adjust targets; 
     4. Allocate the residual capacity; 
     5. Check the convergence of targets and machine allocation until convergence is achieved. 
     The problem solved by this invention is as follows: 
     1. The accuracy of the total target is more than 95%. 
     2. The system is easily maintained. 
     In accordance with this invention, a method is provided for operating a data processing system includes a data base computer system and a resource allocation computer for control of resource allocation in a manufacturing plant with a manufacturing line comprising a plurality of stages with manufacturing machines, the resource allocation computer including data storage means. The method includes several steps including: deriving data from the data storage means and computing the targets for each of the stages; obtaining machine capacity data from the data storage means and employing the machine capacity data for allocating machine capacity proportionally and adjusting targets; adding limits to stages of penetration and adjusting targets; determining residual capacity and allocating the residual capacity of the manufacturing machines; and checking the convergence of targets and machine allocation until convergence is achieved. Preferably the method includes the steps of determining the push target, and determining the pull target. Preferably the process includes the step of determining the upper bound target; the step of determining capacity constraint and machine allocation; the step of limiting stages of penetration and the step of iterating TG&amp;MA --  PR method; and the step of executing SOPEA. 
     In accordance with another aspect of this invention a data processing system includes a data base computer system and a resource allocation computer for control of resource allocation in a manufacturing plant with a manufacturing line comprising a plurality of stages with manufacturing machines, the resource allocation computer including data storage means, the system includes as follows: means for deriving data from the data storage means and computing the targets for each of the stages; means for obtaining machine capacity data from the data storage means and employing the machine capacity data for allocating machine capacity proportionally and adjusting targets; means for adding limits to stages of penetration and adjusting targets; means for determining residual capacity and allocating the residual capacity of the manufacturing machines; and means for checking the convergence of targets and machine allocation until convergence is achieved. 
     Preferably, means are provided for determining the push target, means for determining the pull target, means for determining the upper bound target; means for determining capacity constraint and machine allocation; means for limiting stages of penetration; means for iterating TG&amp;MA --  PR system, and means for executing SOPEA. 
     A computer implemented method for control of resource allocation on a manufacturing line comprises a plurality of stages of manufacturing machines. The method including the steps as follows: determining the targets for each of the stages; obtaining machine capacity data and employing the machine capacity data for allocating machine capacity proportionally and adjusting the targets; adding limits to stages of penetration and adjusting the targets; determining residual capacity and allocating the residual capacity of the manufacturing machines; and checking the convergence of targets and machine allocation until convergence is achieved. 
     Preferably the computer implemented method includes the steps of determining the push target, and determining the pull target. Preferably the process includes the step of determining the upper bound target; the step of determining capacity constraint and machine allocation; the step of limiting stages of penetration and the step of iterating TG&amp;MA --  PR method; and the step of executing SOPEA. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other aspects and advantages of this invention are explained and described below with reference to the accompanying drawings, in which: 
     FIG. 1 shows the queuing mode for several stages in a semiconductor production line from the wafer start position to the end where the wafer out position is reached. 
     FIG. 2 shows the queuing mode for a two-stage case. 
     FIG. 3 shows another queuing mode. 
     FIG. 4 shows a computer system with a database computer and a target computer in accordance with this invention. 
     FIG. 5 shows an example of a one stage case for stage j which receives wafers wi from the preceding stage i. 
     FIG. 6 shows an example of a two stage case for stages 1 and 2 for several cases. 
     FIG. 7 shows a steps performed by computer of FIG. 4. 
     FIG. 8 shows steps performed by the computer of FIG. 4. 
     FIG. 9 shows a method and means for calculating the total processing time T TPr  of wafers at any stage. 
     FIG. 10 shows how to determine the value of t ij . 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     To be certain that terms used herein are defined to be interpreted with minimal confusion, a glossary is listed next. 
     GLOSSARY 
     Down-stream: Stages closer to the end of the production line. 
     Penetrate: Pass through a stage. 
     PULL: Pull wafers to the greatest degree possible from the previous stages in the fabrication line. 
     PUSH: Push wafers to the greatest degree possible to the next stage in the fabrication line. 
     SOPEA: Stages Of PEnetrAtion&#34; 
     SOPEA --  PR: Stage Of Penetration Estimation Method with PRiority effect 
     TG&amp;MA --  PR: Target Generation and Machine Allocation with PRiority effect 
     Upstream: Stages closer to the beginning of the production line 
     WIP: Work in Process 
     A system known as a daily Target Generation and Machine Allocation with Priority System (TG&amp;MA --  PR) provides guidelines to technicians on the shop floor for real-time dispatching of wafers to machines for processing. Targets are generated by technology, stage and priority using the PUSH and PULL procedures to maximize production movement and machine utilization and to minimize line WIP. Then, due to the limitations of machine capacity, machine allocation is considered as a factor. The target is finally modified by the concept of limiting Stages of Penetration (SOPEA). An iterative process continues until the targets converge. 
     The manufacturing process for integrated circuit (IC) wafer fabrication is sophisticated. The key factors in competitive IC manufacturing with high performance are usually characterized as: 
     1. High line yield; 
     2. High labor productivity; 
     3. High percentage of on time delivery; 
     4. Short production cycle time; 
     5. High level of equipment automation. 
     In a typical production-to-order type factory; short production cycle time is of primary importance. A daily &#34;Target Generation and Machine Allocation with Priority&#34; (TG&amp;MA --  PR) method and system reduces production cycle time. The TG&amp;MA --  PR method and system fulfill the objectives as follows: 
     1. effective machine utilization, 
     2. line balancing, 
     3. maximizing production move, 
     4. reducing WIP levels. 
     The TG&amp;MA --  PR method consists of two parts, one is the baseline part that computes the target and machine allocation for each production stage while the other estimates the number of fabrication stages that each batch of wafers may finish within a day. 
     Daily Target Generation 
     Consider a queuing model of the re-entrant production line in FIG. 1, where an ellipse corresponds to a server for a stage and a square box corresponds to a buffer. Although the production line of FIG. 1 appears to be a flow line, machines are shared by various stages. For example, stages 1 and j can share one machine group and stages 2 and J can share another. 
     FIG. 1 shows the queuing mode for several stages in a semiconductor production line wip1 wip2 wipi . . . wipJ from the wafer start position at stage wip1 to the end at state wipJ where the wafer out position is reached. 
     At station wip1, a set of buffers B1a, B1b, and B1c are shown leading to the server S1. The output wafers r1 from stage wip1 pass to stage wip2. 
     At station wip2, a set of buffers B2a, B2b, and B2c are shown leading to the server S2. The output wafers r2 from stage wip2 pass to stage wipi. 
     At station wipi, a set of buffers Bia, Bib, and Bic are shown leading to the server Si. The output wafers ri from stage wipi pass to stage wipJ. 
     At station wipJ, a set of buffers BJa, BJb, and BJc are shown leading to the server SJ. The output wafers rJ from stage wipJ pass out to the end of the production line. 
     To formalize the description of quantities on the production line, the following variables are defined: 
     Notations 
     Inputs 
     I: total number of part types; 
     J: total number of stages in global sequence; 
     wipj: the WIP level of stage j at the beginning of the day; 
     wipij: the WIP level of type-i parts at stage j at the beginning of the day, 
     std --  wipj: the standard WIP level at stage j; 
     A: a J*I, stage-part type mapping matrix, which its element aji=1 if type-i process goes through stage j and aji=0, otherwise; 
     The term aji is for a type &#34;i&#34; process. The stages of the process are as follows: 
     
         ______________________________________Stage     aij______________________________________j = 1     ai1 = 1j = 2     ai2 = 0j = 3     ai3 = 1j = 4     ai4 = 0j = 5     ai5 = 0j = 6     ai6 = 1j = 7     a17 = 0. .       . .. .       . .. .       . .j = n     ain = 1,  where the stage n is an integer               greater than zero and less than               infinity. The product with process               i will go through stages 1, 3,               6, . . . n because aij = 1 (and will               not go through stages 2, 3, 5, 7,               etc. because aij = 0) If we add up               all of these aij values, it indi-               cates the total number of stages               that the process i will go through.______________________________________ 
    
     M: the total number of machine groups; 
     Nm: number of available machines in group m of the day; 
     Cm: capacity of a machine in group m in term of wafers per day; 
     mj: index of the machine group required by stage j,j=1, . . . J; 
     r o  : wafers started at the beginning of the day; 
     ri o  : type-i wafers started at the beginning of the day, 
     d J  : demanded of output wafers on the day, 
     d iJ  : demanded of output wafers on the day; 
     Intermediate variables 
     i: part type index, i 
     J: stage index, j=1, . . . J; 
     m: the machine group index, m=1, . . . ,M): flow --  inj: number of wafers flowing to stage j from its up-stream stages, 
     flow--inij: number of type-i wafers flowing to stage j from its upstream stages, 
     flow --  out j  : number of wafers leaving stage j to its down-stream stages, 
     flow --  outij: number of type-i wafers leaving stage j to its down-stream stages. 
     Decision variables 
     rj: the target moves of stage j and also the flow-in to stage j+1 during the day; 
     rij: the target move of type-i parts at stage j and also the type-i flow-in to stage j+1 during the day; 
     maj: number of machines allocated to process stage j for the day; 
     maij: number of machines allocated to process type-i parts at stage j for the day, 
     The short term production control function of daily target generation and machine allocation (TG&amp;MA) determines the target move, rij, and the allocation of equipment capacity, maij, for every part type and every stage of the reentrant production line. Given the wafer start (ri o , &#34;∀&#34;, where &#34;∀&#34; means &#34;All&#34; i) of each day, the TG&amp;MA function aims the multiple operation objectives as follows: 
     1. Maximize total moves, Σrj , while meeting daily wafer output targets set by weekly or monthly schedule, i.e., riJ≧di; 
     2. Minimize wafer-in-process (WIP), Σxj, and fabrication cycle times, while maintaining the standard WIP level, Wj, at stage j for &#34;-&#34; j; 
     3. Maximize machine utilization, especially for the bottleneck machines, and 
     4. Balance the production line, i. e., ideally, rj=rj+1 for j=, . . . ,J-1 
     Objective 1 requires that the TG&amp;MA --  PR system provides a capacity for tracking the targets set by its higher level in the production control hierarchy. 
     The achievement of objective 2 or 3 requires that the TG&amp;MA --  PR system sets good targets to its lower level of production control on-line dispatching. 
     As to objective 4, line balancing is a complex issues that requires a good integration of the TG&amp;MA --  PR system with the wafer start schedule and the determination of standard. 
     Definition of TG&amp;MA --  PR Method 
     The basic TG&amp;MA --  PR method can be stated in the following steps: 
     Step 1. Prioritize Lots 
     High Priority Lot: priority &#34;1&#34; in database management system 
     Low Priority Lot: priority &#34;2&#34;, &#34;3&#34;, &#34;4&#34;, &#34;5&#34; in database management system 
     Step 2. Calculate Upper --  Bound Target by technology, stage and priority 
     Step 3. Apply TG&amp;MA --  PR to high priority lots 
     Step 4. Apply TG&amp;MA --  PR to low priority lots using residual machine capacity 
     Step 5. Apply SOPEA --  PR to Compute Stages of penetration Flow-In 
     Step 6. Iterate from step 2 to step 5 
     Step 1. Prioritize Lots 
     According to special orders from customers, some high priority lots must be expedited through the manufacturing production line very rapidly. Such high priority lots are assigned a priority of &#34;1&#34; in the control computer system to make assure that they are always handled immediately. Since high priority lots comprises only about 5% of the total number of production lots, the target system must distinguish between high priority lots and low priority lots. Accordingly, the first step of the TG&amp;MA --  PR system is to prioritize the lots so high priority lots have a priority &#34;1&#34; in the database management system whereas, on the other hand, low priority lots have various priority levels of &#34;2&#34;, &#34;3&#34;, &#34;4&#34; and &#34;5&#34;. 
     Step 2. Calculate Upper Bound Target 
     Maintaining the WIP distribution of the fab at the &#34;standard&#34; WIP distribution is one of the objectives of TG&amp;MA --  PR . This is a fundamental priority of the TG&amp;MA --  PR system. An upper bound target for each stage is first determined by a PUSH and PULL procedure. 
     On the one hand, the PUSH procedure pushes the wafers at a stage except those needed for standard WIP to down-stream stages (closer to the end of the production line) so that the WIP of each stage is minimized and the throughput at each stage is maximized in a heuristic sense. 
     On the other hand, the PULL procedure pulls the production flows from up-stream stages, attempting to maintain the &#34;standard&#34; WIP level and to prevent machine starvation for each stage so that production demands can be met and machine utilization can be maximized. 
     Allocate standard-WIP of stage to each technology 
     The standard-WIP of each stage is allocated to each technology in order to calculate the Push Target and Pull Target of each technology. ##EQU1## 
     The Standard WIP of high priority lots is set as zero to avoid queuing in the production line. 
     Determining Push-target 
     The push-target determination procedure tries to &#34;PUSH&#34; as many wafers through a stage as possible while maintaining the &#34;standard&#34; WIP at the stage. The PUSH procedure starts from the first stage and computes the push-target for each stage as follows: ##EQU2## Determining Pull-target 
     This step implements the PULL concept. It determines the amount of wafers that a stage should supply to the next down-stream stage based on both the output demand and the standard WIP of that down-stream stage. The calculation procedure therefore starts from the final stage, i.e., stage J and computes the pull-target of each stage as follows: 
     
         ______________________________________ Pull-target.sub.J = d.sub.J, and Pull-target.sub.j = max (0, flow.sub.-- out.sub.j+i +std.sub.-- wip.sub.j+i - wip.sub.j+i);    for j = J - 1, . . . , 1;  where flow.sub.-- out.sub.J = d.sub.Jflow.sub.-- out.sub.j = Target.sub.j.sup.(n-1) ; for j = j, . . . , 2the upper index (n - 1) means the previous iteration______________________________________ 
    
     Determining Upper-bound Target 
     To set an aggressive target for each stage, the upper-bound target of a stage without considering capacity constraint is set by taking the maximum between the Push-target and Pull-target of the stage, i.e., 
     UB --  target j  =max (Push-target j  ; Pull-target j ) 
     Step 3. Apply TG&amp;MA --  PR to High Priority Lots 
     3.1 Compute upper bound of the target by stage by technology case: 
     
         ______________________________________PUSH Target        PULL Target                        Target______________________________________stage1  30             35        30stage2  --             --        --stage3  40             20        55stage4  60             55______________________________________ 
    
     3.2 Compute the Total Target by stage case: 
     There are three technologies in FAB, (i.e. technology A,B,C) 
     
         ______________________________________   Tech. A         Tech. B    Tech. C Total   Target         Target     Target  Target______________________________________stage1    30      10         20    60stage2    145     --         55    200stage3    --      60         40    100stage4    50      65         --    115stage5    55      --         25    80______________________________________ 
    
     3.3 Machine Allocation case: 
     stage 1, 2, 3 use machine XX 
     number of machine XX=4 
     
         ______________________________________Standard          Max Machine                        TotalOut               Limit      Target______________________________________stage1  70            3          60stage2  90            2          200stage3  80            4          100______________________________________ 
    
     3.3.1 Change the target unit to the machine unit 
     The capacity of same machine would differ when involved in a different production process according to a realistic situation. Therefore, machine allocation should based on the ratio of machine unit instead of wafer quantity. In the following example, first transfer the target to machine unit. Ideally, stage1 needs 0.86 machine units stage2 needs 2.22 units and stage3 needs 1.25 units. 
     
         ______________________________________   Total Standard   Target         out          Machine______________________________________stage1    60      70            60/70 = 0.66stage2    200     90           200/90 = 2.22stage3    100     80           100/80 = 1.25                          Total = 4.33______________________________________ 
    
     3.3.2 Proportion the machine allocation 
     Since the number of machines is limited, it is necessary to allocate the actual machines according to ideal machine proportions. 
     
         ______________________________________          Machine______________________________________stage1           4.0 * 0.86/4.33 = 0.80stage2           4.0 * 2.22/4.33 = 2.05stage3           4.0 * 1.25/4.33 = 1.15______________________________________ 
    
     3.3.3 Maximum machine limit 
     Due to some factors, the same kinds of machines may not appropriate for every stage. Therefore, the maximum number of available machine should be considered. The machine allocation is the minimum of the maximum machine limit and the machine allocated in 3.3.2 are as follows: 
     
         ______________________________________Max Machine          Machine______________________________________stage1  3                min(0.80, 3) = 0.80stage2  2                min(2.05, 2) = 2.00stage3  4                min(1.15, 4) = 1.15                    Total = 3.95______________________________________ 
    
     3.3.4 Allocate residual machine capacity 
     Because of the restriction of the maximum number of available machines, there is a residual machine capacity which needs to be allocated. In (3.33) four (4) machines are available-but only 3.95 machines are allocated. Therefore, 0.05 machines will be residual for further allocation. In this case, stage1 needs 0.86 units but is allocated only 0.8 units. So, there is a shortage of 0.06 units. Stage3 is 0.1 units short and stage 2 has a sufficient capacity. According to this ratio of shortages, the residual machine capacity can be allocated as follows: 
     
         ______________________________________          Residual Machine______________________________________stage1           0.05 * 0.06/0.16 = 0.019stage2           --stage3           0.05 * 0.10/0.16 = 0.031______________________________________ 
    
     3.3.5 Compute total machine allocation 
     We add up the numbers from (3.3.4) and (3.3.5) and the sum comprises the actual machine allocation. 
     
         ______________________________________          Machine Allocation______________________________________stage1           0.80 + 0.019 = 0.519stage2           2.00 + 0.000 = 2.000stage3           1.15 + 0.031 = 1.181______________________________________ 
    
     3.3.6 Compute target 
     Finally the machine units are transferred into actual wafer quantities. 
     
         ______________________________________           Target______________________________________stage1            0.819 * 70 = 57.33stage2            2.000 * 90 = 180.00stage3            1.181 * 80 = 94.48______________________________________ 
    
     The machine allocation of high priority lots can be completed by following steps (3.1), (3.2) and (3.3) . 
     Step 4. Apply TG&amp;MA --  PR to Low Priority Lots 
     After the machine allocation of high priority lots, the machine capacity is checked. If there are residual machines, the capacity is allocated to low priority lots by following the same steps. 
     Step 5. Apply SOPEA --  PR 
     In the above procedures for calculating targets, the time delay of a wafer which is moving from one stage to another stage due to queuing and processing times is not accounted for. As a result, a wafer may be moved throughout the production line within a day, which is obviously not realistic. 
     Actually, a wafer may need about thirty-five days to move throughout a production line if account is taken of the waiting time, process time and hold time. 
     In order to prevent this from happening, it is required to compute the number of stages through which the initial WIP of each stage can be moved during one day. Add that sum as a constraint to the calculation of flow-in WIPs for each stage. 
     Details of estimating the number of stages that WIP can penetrate (pass through) in a single day &#34;Stages Of PEnetrAtion&#34; (SOPEA) is described in the following section. Given the SOPEA of each-stage in the line, it is possible to identify the up-stream stages whose WIP may flow into each stagej within the current day. WIPs of stagej and its up-stream stages are then summed up as an estimate of WIP penetration limit for the stage which the daily target of stagej should not exceed. Thus, all of the targets (target j  ; j=1, . . . J) are then modified. 
     In this section, steps referred to as SOPEA are used to estimate how many stages that wafers at each stage can penetrate within one day. The maximum number of incoming WIPs for each stage is then determined, and finally the targets are modified individually. The number of stages of penetration is a function of part type, processing time, WIP distribution, machine capacity and dispatching. In the fab, wafers of different product types (even the same product type) at the same stage can be anticipated to get different processing priority, and can be anticipated to be randomly queued in the buffer waiting for processing. 
     To reduce the complexity of the problem to a practical level and to focus on the key factors about a one day long penetration estimation, the assumptions are made, as follows: 
     1. The wafers dispatching rule is that the first work in is the work which is served first or that a &#34;First In First Out&#34; (FIFO) system is used. 
     2. Neglect consideration of the transfer time which is the time between WIP track-in machine and being processed. 
     3. The unit being processed being considered is a single piece, but in practice the wafers processed unit is a lot of pieces. Some machines process wafers by batch which may be seven pieces or twenty-four pieces or even more. 
     Wafers of different types at a stage or at another stage may compete for the same type of equipment to be processed, as different stages may use the same type of equipment in processing. The result is that the SOPEA of wafers of a given type at a stage could be greater or less due to the quantity of machine capacity allocated to process the given type at the stage. In addition, this means that the Total Processing Time (T TP ) of wafers at any stage is defined by the basic equation of the SOPEA method which is as follows: ##EQU3## 
     To describe the development of SOPEA method and system, certain notation is defined. Since it considers one product type at a time in the method of SOPEA, for simply and easily capturing the ideas, the type index &#34;i&#34; was omitted from the latter discussion. Thus it is necessary to redefine some notation to prevent confusion with those defined in previous section and further define extra notation in the follows: 
     wj: WIP level for type-i wafers at stage j; 
     Cj: machine capacity in terms of number of machines allocated to process type-i wafers at stage j during a day; 
     τj: processing time (in hours) for type-i wafers at stage j; 
     tjk: time needed for all the wafers for type-i at stage j to be completed at stage k, k&gt;j. In other words, &#34;tjk&#34; is the time to move from stage j to stage k. 
     Method development 
     First, calculate the time needed for all wafers at one stage to be completed at the stage immediately down-stream (called the &#34;two-stage condition&#34;) until all of those wafers are done for wafers at every stage. Then the resulting data of two-stage condition is used to calculate another time interval needed for wafers to be processed through three stages (called the &#34;three-stage condition&#34;) . Finally these results can be converted to a general condition (n-stage condition) and the time needed to complete wafers from one stage to its n-th down-stream stage is calculated. Then an estimate is easily made as to how many stages that wafers at one stage can penetrate. A determination is made as to the maximum incoming WIPs of each stage during one day. 
     (I.) Two-stage case 
     FIG. 2 illustrates the queuing mode for the two-stage case with a stage W1 and a stage W2. 
     At station W1, a set of buffers B1a, B1b, and B1c are shown leading to the server τ1 with machine capacity C 1 . The output wafers from stage W1 pass to stage W2. 
     At station wip2, a set of buffers B2a, B2b, and B2c are shown leading to the server τ2 with machine capacity C 2 . The output wafers from stage W2 pass to the next stage (which is not relevant to the two stage case). 
     Consider the following two cases: 
     (1) If w 2  τ2/C 2  &gt;w 1  τ 1  /C 1 , it means by the time when w 2  wafers are completed at stage 2, all w 1  wafers have arrived at the buffer of stage 2. Thus, 
     
         t.sub.12 =w.sub.1 τ.sub.1 /C.sub.1 +w.sub.2 τ.sub.2 /C.sub.2 
    
     (2) If w 2  τ 2  /C 2  ≦w 1  τ 1  /C 1  there can be two additional subcases as follows: 
     (a) When the last piece of wafer is finished at stage 1, all other wafers (w 2  +(w 1  -1)) have been completed at stage 2, i.e., 
     
         if w.sub.1 τ.sub.1 /C.sub.1 &gt;((w.sub.1 -1)+w.sub.2)τ.sub.2 /C.sub.2 then t.sub.12 =w.sub.1 τ.sub.1 /C.sub.1+τ2 /C.sub.2 
    
     (b) If w 2  τ 2  /C 2  ≦w 1  τ 1  /C 1  ≦((w 1  -1)+w 2 )τ 2  /C 2 , then there are w 1  -(w 1  τ 1  /C 1  -w 2  τ 2  /C 2 )/(τ 2  /C 2 ) pieces remaining in the buffer of stage 2. So, 
     
         t.sub.12 =w.sub.1 τ.sub.1 /C.sub.1 +w.sub.1 -(w.sub.1 τ.sub.1 /C.sub.1 -w.sub.2 τ.sub.2 /C.sub.2)/(τ.sub.2 /C.sub.2))) (τ.sub.2 /C.sub.2)=(w.sub.1 +w.sub.2)(τ.sub.2 /C.sub.2) 
    
     
         t.sub.12 =(w.sub.1 +w.sub.2)/(τ.sub.2 /C.sub.2)+(w.sub.2 τ.sub.2 -τ.sub.1) 
    
     Note that there cannot be any vacant time for the stage 2 machine in this subcase; otherwise it goes returns to subcase (a). 
     In summary, we have the following 
     
         if w1τ1/C1&gt;((w1-1)+w.sub.2)τ2/C.sub.2, then t12=w1τ1/C1+τ2/C.sub.2 
    
     
         if w1τ1/C1≦((w1-1)+w2)τ2/C.sub.2, then t12=(w1+w.sub.2)(τ2/C.sub.2 
    
     II) General case 
     Assume that ti(j-1) and t(i+1)j have been obtained, we can conclude the method to calculate the time need for wafers to be processed from stage i to stage j as follows: 
     
         if t.sub.i(j-1) &gt;t.sub.(i+1)j +(w.sub.i -1)τ.sub.j /C.sub.j ; t.sub.ij =t.sub.i(j-1) +τ.sub.j /C.sub.j ; 
    
     
         if ti(j-1)≦t.sub.(i+1)j +(w.sub.i -1)τ.sub.j /C.sub.j ; t.sub.ij =t.sub.(i+i)j +w.sub.i τ.sub.j /C.sub.j ; 
    
     FIG. 3 shows the queuing mode for several stages from stage i to stage j as for stage W i , W i+1 , . . . τ j-1 , τ j . 
     At station &#34;i&#34; , a set of buffers Bia, Bib, and Bic are shown leading to the server τi. The machine capacity at stage &#34;i&#34; is Ci. The output wafers from stage &#34;i&#34; pass to stage i+1. 
     At station i+1, a set of buffers Bi+1a, Bi+1b, and Bi+1c are shown leading to the server τi+1. The machine capacity at stage &#34;i+1&#34; is Ci+1. The output wafers from stage i+1 pass to stage j-1. 
     At station j-1 a set of buffers Bj-1a, Bj-1b, and Bj-1c are shown leading to the server τj-1. The machine capacity at stage &#34;j-1&#34; is Cj-1. The output wafers from stage j-1 pass to stage j. 
     At station j, a set of buffers Bia, Bjb, and Bjc are shown leading to the server τj. The machine capacity at stage &#34;j&#34; is Cj. The output wafers from stage j pass out to the next stage of the production line. 
     Following the procedures of this method development, we can use SOPEA to determine all the lead time between upper-stream and down-stream stages beginning from 2-stage condition, finally we can know the time need for wafers to be processed from any stage to its any down-stream stage. By summing up the WIPs at the stage during, and then adjust the targets in order not to exceed this quantity. 
     Iterative Scheme in TG&amp;MA --  PR Method 
     The TG&amp;MA --  PR method is intrinsically an iterative method for generating daily targets and machine allocation. Using the PUSH and PULL procedures to determine the Push --  target and the Pull-target for a stage, the TG&amp;MA --  PR method uses the data of numbers of wafers flowing into and flowing out of a stage (flow --  in and flow --  out) individually. The flow --  in data and flow --  out data of a stage are just the daily targets of its upstream stage and this stage itself was determined in the previous iteration. 
     In the beginning for executing the TG&amp;MA --  PR method, it needs a given initial set of targets used as flow-in data and flow-out data. Then a new set of targets and machine allocation can be determined according to the PUSH and PULL procedures. As this machine allocation is determined, SOPEA can be executed then to estimate wafer penetration limits to modify the targets. These targets (after some tuning) are further used to determine new flow --  in data and new flow --  out data to generate daily target data and machine allocation data for the next iteration. The iteration process continues until the targets converge. 
     FIG. 4 shows a computer system in accordance with this invention including database computer 10 and target computer 14. The database computer 10 contains a database system employed in operation of the system of this invention. The data from database computer 10 is supplied to the central processing unit (CPU) 16 in target computer 14. The target computer 14 includes CPU 16 and memory and direct access storage device (DASD) 20. CPU 16 is connected as follows: by cable 12 to computer 10; by cable 17 to memory 18; and by cable 19 to DASD 20. 
     FIG. 5 shows an example of a one stage case for stage j which receives wafers wi from the preceding stage i with parameters as follows: ##EQU4## 
     Accordingly, 75 minutes is required to process 100 pieces through this stage given the allocation of two machines. As a result, two machines are included in the specifications which shows how the system and method of this invention are tied to physical devices. 
     FIG. 6 shows an example of a two stage case for stages 1 and 2 for several cases with parameters as follows: 
     First consider the case in which as follows: ##EQU5## 
     Thus, if t14=25.4 hours, t13=20.5 hours means the wafer can pass through three stages within one day. 
     The computer 14 in FIG. 4 performs a sequence of steps shown in a flow chart sequence in FIG. 7. 
     In step 30, the computer 14 computes the target for each stage of the production line. 
     Following step 30, in step 32, the computer 14 allocates machine capacity proportionally and adjusts targets (according to lot priority). 
     Following step 32, in step 34, the computer 14 adds limits to stages of penetration and adjusts targets. 
     Following step 34, in step 36, the computer 14 allocates the residual capacity of the manufacturing line. 
     Following step 36, in step 38, the computer 14 checks the convergence of targets and machine allocation until convergence is achieved. 
     Features of the system of this invention are described in the following sections with reference to FIG. 8 where a sequence of steps is performed by the computer system 14 of FIG. 4. 
     1. Determining Push-Target 
     Referring to FIG. 8, in step 40, the push target is determined by computer 14 using the equation as follows: 
     
         ______________________________________Push-Target.sub.j = max(0, wip.sub.j + flow.sub.-- in.sub.j - std.sub.--wip.sub.j); for j = 1, . . . J (1)where:    flow.sub.-- in.sub.1 = r.sub.0 (wafer release) and     flow.sub.-- in.sub.j+1 = Target.sub.j (n - 1); for j = 1, . . J     - 1 and     the superscript index (n - 1) refers to the     previous iteration.______________________________________ 
    
     2. Determining Pull-Target 
     Referring to FIG. 8, following step 40 in step 42, the pull target is determined by computer 14 using the equation as follows: 
     
         ______________________________________Pull-Target.sub.j = max(0, flow.sub.-- out.sub.j+1 + std.sub.-- wip.sub.j+1 - wip.sub.j+1) (2)for j = J - 1, . . 1where:flow.sub.-- out.sub.j = d.sub.j (demand for output wafers for the day)andflow.sub.-- out.sub.j = Target.sub.j (n - 1); for j = J, . . 2 and the superscript index (n - 1) refers to the previous iteration.______________________________________ 
    
     3. Determining Upper-Bound Target 
     Referring to FIG. 8, following step 42 in step 44, the upper bound target is determined by computer 14 using the equation as follows: 
     
         UB.sub.-- target.sub.j =max (Pull-Target.sub.j, Push-Target.sub.j)(3) 
    
     4. Capacity Constraint and Machine Allocation 
     Referring to FIG. 8, following step 44 in step 46, the capacity constraint and machine allocation are determined by computer 14, as follows: 
     4.1 Apply UB-Target to high priority lots (having a higher priority of &#34;1&#34; in database management system). 
     4.2 Apply UB-Target to low priority lots (having lesser priorities of &#34;2&#34;, &#34;3&#34;, &#34;4&#34;, &#34;5&#34; in a database management system) using the residual machine capacity. 
     Consider a stage j, let the set S be the collection of stages competing for machine group Mj. ##EQU6## 
     The number of machines allocated to stage j in a day is ##EQU7## 
     5. Limiting Stages of Penetration 
     Referring to FIG. 8, following step 46 in step 48, the limits of stages of penetration are determined by computer 14, as follows: 
     Given the stages of penetration of each stage in the line, we can identify the up-stream stages whose Work in Process (WIP) may flow into each stage j within the day. WIPs of stage j and its upstream stages are then summed up as an estimate of the WIP penetration limit for the stage which the daily target of stage j should not exceed. So, all targets (Target j; j=1. . . J) are then modified. 
     At this point reference is made to FIG. 3 above where: 
     W j  : WIP level for type-i at stage j; 
     Cj: machine capacity in terms of number of machines allocated to process type-i wafers at stagej during a day. 
     τ j: processing time (in hours) for type-i at stage j; 
     τ jk: time, needed for all the wafers for type i at stage j to be completed at stage k, k&gt;j. 
     Referring to FIG. 9, a method and means for calculating the total processing time T TPR  of wafers at any stage is shown, as follows: ##EQU8## where T SPr  : stage processing time 
     N MA  : Number of machines allocated 
     Referring to FIG. 9, the stage processing time T SPr  in block 22, WIP in block 26 and number of machines allocated N MA  in block 24 are derived form DASD 20. In block 28, the value T SPr  from block 22 is multiplied by the WIP value in block 26 yields the product WIP×T SPr . 
     In block 30, the value of T TPr  is calculated by dividing the value of WIP×T SPr  in block 28 by the value N MA  from block 24. 
     FIG. 10 shows the method and means for determining the value of t ij . Block 32 determines the value x for equations (7) and (9) below by block 32 which derives data from DASD 20 to calculate x. 
     In equation (7) below, a test is made as shown by decision block 34 where the test is made whether t i (j-1) &gt;x. If so, then the result in block 38 is that t ij  =Q which is the value in equation (8) . 
     In equation (9) below, a test is made as shown by decision block 36 where the test is made whether t i (j-1) ≦x. If so, then the result in block 40 is that t ij  =R which is the value in equation (10) . 
     When the values of t i (j-1) and t.sub.(i+1) have been obtained, a method in accordance with this invention for calculating the time need for wafers to be processed from stage i to stage j is as follows: 
     
         if t.sub.i(j-1) &gt;t.sub.(i+1)j +(w.sub.i -1)τ.sub.j /C.sub.j ;(7) 
    
     
         t.sub.ij =Q=t.sub.i(j-1) +τ.sub.j /C.sub.j ;           (8) 
    
     
         if ti(j-1)≦t.sub.(i+1)j +(w.sub.i -1)τ.sub.j /C.sub.j ;(9) 
    
     
         t.sub.ij =R=t.sub.(i+i)j +w.sub.i τ.sub.j /C.sub.j ;   (10 ) 
    
     By summing up the WIPs at the stages of penetration, an estimate is made of the maximum number of wafers that can flow into each stage during a day from 7:00 AM to 19:00 PM, every day, and then the targets are adjusted to avoid exceeding this maximum quantity. 
     6. Iterate TG&amp;MA --  PR Method 
     Referring to FIG. 8, following step 48 in step 50, the TG&amp;MA-PR is iterated by computer 14, as follows: 
     In the beginning for executing TG&amp;MA --  PR , an initial set of targets as to flow-in data and flow-out data is required to determine new targets and machine allocation data according to the Push &amp; Pull target generation procedures. 
     7. Determine SOPEA 
     Referring to FIG. 8, following step 50 in step 52, the SOPEA is determined by computer 14. As machine allocation is determined, SOPEA can then be executed to estimate wafer penetration limits for modifying the targets. These targets are further used as new flow-in and flow-out to generate daily target and machine allocation for the next iteration. The iteration continues until the targets are converged. 
     While this invention has been described in terms of the above specific embodiment(s), those skilled in the art will recognize that the invention can be practiced with modifications within the spirit and scope of the appended claims, i.e. that changes can be made in form and detail, without departing from the spirit and scope of the invention. Accordingly all such changes come within the purview of the present invention and the invention encompasses the subject matter of the claims which follow.