Patent Publication Number: US-6705119-B2

Title: Control for an I. S. machine

Description:
The present invention relates to an I.S. (individual section) machine and more specifically to a control for such a machine. 
    
    
     BACKGROUND OF THE INVENTION 
     The first I.S. machine was patented in U.S. Pat. Nos. 1,843,159, dated Feb. 2, 1932, and 1,911,119, dated May 23, 1933. An I.S. (individual section) machine has a plurality of identical sections. Each section has a frame on which are mounted a number of section mechanisms including blank side and blow side mold open and close mechanisms, an invert and neck ring mechanism, a baffle mechanism, a blowhead mechanism, a plunger mechanism and a takeout mechanism. Associated with these mechanisms is process air used for cooling, for example. Each of the section mechanisms and the process air have to be operated at a selected time in the section cycle. 
     In the original I.S. machine, devices (valves which operated the mechanisms and the process air, for example) had to be mechanically turned on and off each cycle and the timing process was controlled by a 360° timing drum which was a cylindrical drum having a number of annular grooves, one for each valve, each supporting “on” and “off” dogs for tripping a corresponding switch associated with a particular valve. The rotation of this mechanical timing drum through 360° has always been equated to the completion of one control cycle of the machine or section and accordingly men skilled in this art have always analyzed machine performance in a wrapped cycle, i.e., one that repeatedly cycles from 0° to 360°. When electronic timing replaced the mechanical timing drum, devices were turned on and off by an electronic sequencer which replicated the wrapped 360° control cycle of the mechanical timing drum. An encoder defined the angular location of the electronic sequencer, and electronic switches were turned on and off at the same angles as was the case with a mechanical timing drum. A very significant development that greatly enhanced the power of the electronic sequencer was the concept of thermodynamic modes (U.S. Pat. No. 3,877,915) wherein groups of these electronic switches were linked so that they could be simultaneously adjusted. These machine controllers allow the user to electronically adjust the on/off schedule (angle) for the various valves which operate the section mechanisms. This conventional approach does not allow an operator to directly command the machine to achieve desired forming durations (e.g. blank contact time, reheat time). It also does not prevent the user from setting invalid or even potentially damaging sequences in which the mechanisms collide. Only with considerable experience, and process insight can an operator effectively adjust the machine timing with the conventional approach and since skill levels vary greatly, the productivity of the machine can vary substantially. 
     OBJECT OF THE INVENTION 
     It is an object of the present invention to provide an improved control system for a glass forming machine which will simplify machine operation and facilitate the tuning of the machine for higher productivity. 
     Other objects and advantages of the present invention will become apparent from the following portion of this specification and from the accompanying drawings which illustrate a presently preferred embodiment incorporating the principles of the invention. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic illustration of one section of an I.S. machine which can have one or more of such sections, 
     FIG. 2 is the first part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 3 is the second part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 4 is the third part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 5 is the fourth part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 6 is the fifth part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 7 is the sixth part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 8 is the seventh part of the Network Constraint Diagram for the Blow and Blow process; 
     FIG. 9 is the eighth part of the Network Constraint Diagram for the Blow and Blow process. 
     FIG. 10 is network model for branch incidence matrix; 
     FIG. 11 is an event timing chart for a 360° electronic sequencer which controls a section of an I.S. machine; 
     FIGS. 12A and 12B are a network diagram for use to unwrap a wrapped cycle; 
     FIG. 13 is a block diagram illustrating the creation of a computerized model of a mathematical representation of a network constraint diagram unwrapped from a wrapped cycle; 
     FIG. 14 is a block diagram illustrating the portion of the computerized model which converts wrapped event angles into unwrapped event times; 
     FIG. 15 is a logic diagram illustrating the operation of a control using the computerized model to analyze an unwrapped schedule re constraint violations such as sequence, collision, or duration violation; 
     FIG. 16 is a logic diagram illustrating the operation of a control using the computerized model to analyze an unwrapped schedule to define the duration of the thermal forming processes; 
     FIG. 17 is a logic diagram illustrating the operation of a control using the computerized model to analyze an unwrapped schedule re the optimization of the schedule; 
     FIG. 18 is a logic diagram illustrating the operation of a control using the computerized model to define the event angles for a feasible schedule with thermal forming process duration “N” inputs; 
     FIG. 19 is a logic diagram illustrating the operation of a control using the computerized model to optimize an unwrapped schedule; 
     FIG. 20 is a logic diagram illustrating the operation of a control using the computerized model to identify, when a schedule is determined to be feasible, any active constraint that restricts further improvement; and 
     FIG. 21 is a logic diagram illustrating the operation of a control using the computerized model to minimize the wear on the displaceable mechanisms. 
    
    
     BRIEF DESCRIPTION OF THE PREFERRED EMBODIMENT 
     An I.S. machine includes a plurality (usually  6 ,  8 ,  10 , or  12 ) of sections  10 . Each section has a blank station including a mold opening and closing mechanism  12  having opposed mold supports  14  which carry blank mold halves. When these mold supports are closed by a suitable displacement mechanism  16  which can displace the mold support between open (illustrated) and closed positions and which is displaced by a motor  18  such as a servo motor, discrete gobs of molten glass can be delivered to the closed blank mold. The open top of the blank mold will then be closed by a baffle of a baffle support  22  which is displaceable between remote and advanced positions by a motor (such as a servo)  24 . If the section is operating in the press and blow mode, the plunger of a plunger mechanism  26  is advanced vertically upwardly into the gob to form the parison. Cooling air will be supplied to the plunger via a valve V 1 . If the section is operating in the blow and blow mode, the finish is formed by applying settle blow air through a valve V 2  in the baffle mechanism  22 , and the parison is formed with the application of counterblow air to the plunger via a valve V 3 , while vacuum is applied to the baffle through a valve V 4 . 
     After the parison is formed, the baffle support is retracted, the mold supports are retracted and a pair of neck ring holder arms  30  which are rotatively supported by an invert mechanism  31 , will be rotated 180° by a servomotor drive  32 . The blank station also includes a mold opening and closing mechanism  12  having opposed mold supports  14  which carry blow mold halves. These mold supports are displaced between open and closed positions by a suitable displacement mechanism  16  which is displaced by a motor  18  such as a servo motor. With the parison located at the blow station, the mold supports are closed, the neck ring arms are opened to release the parison (each arm is displaceable by a pneumatic cylinder (not shown) which is operated with a suitable valve V 5 ), the invert mechanism returns the neck ring arms to the blank side (the arms close prior to arrival) and a blow head support  34  which is displaceable between a retracted position and an advanced position where a supported blowhead closes the blow mold, is displaced to the advanced position by a suitable motor such as a servo  36  to blow the parison into the bottle. This final blow is controlled by a valve V 6 . 
     When the bottle is formed, the blowhead is retracted, blank molds are opened and a takeout mechanism  38  which is driven by a suitable motor  39 , such as a servo motor, is displaced to pick up the formed bottle and carry it to a location above a deadplate  40  where it is cooled while suspended and then deposited onto the deadplate. In addition to the movement of mechanisms and devices, process air to mechanisms, moveable or stationary, may also be controlled. When the blow molds are closed, mold cooling air is turned on to cool the formed bottle. 
     Each section is controlled by a computer  42  which operates under the control of a 360 degree timing drum (programmable sequencer) which defines a finite number of angular increments around the drum at which mechanisms, etc., can be turned on and off each 360 degree rotation. The control knows the time it takes for rotating 360 degrees and this time can be fixed or defined as the duration between once per cycle pulses such as pulses originating from the feeder of the I.S. machine. Each valve is cycled (turned on and off) and each mechanism is cycled within the time of one machine cycle by an electronic timing drum (programmable sequencer) which is part of the computer  42 . 
     In accordance with the present invention a tool is defined first by making an unwrapped cycle constraint diagram for an actual I.S. machine configuration and then making a mathematical representation of the unwrapped cycle constraint diagram that is capable of automated formulation and solution. “Unwrapped” means the I.S. is a process cycle beginning with the formation of a gob of molten glass by severing the gob from a runner of molten glass and ending with the removal of a formed bottle from the blow station. This process cycle takes more than one 360° machine cycle of the timing drum to complete (normally 2 times 360° machine cycles). 
     FIGS. 2-9 show a possible Network Constraint Diagram for a representative blow and blow process for making glass bottles in an I.S. machine. The cycle starts with the shear cut represented by time node z 1  (“z” and “n” denote a time node). Gob Delivery/M 13  (a block containing an “M” represents an activity that will move between start and end positions with the direction of motion being indicated with arrows) begins at z 1  and ends with n 177 /e 26 /n 6  (a vertically oriented equal sign labeled “e” connecting two nodes indicates that the two connected nodes occur at the same time). Gob Delivery/M 13  motion is subdivided into two submotions: 1. Gob In Collision Zone With Baffle/m 2  (a block containing a “m” represents a submotion) which starts at z 1 /e 1 /n 3  and ends at n 4 . Gob Traverses Blank Mold/m 3  which starts at n 4 /e 2 /n 5  and ends at n 6 . 
     Node z 1  (shear cut) also has another branch Total Process/d 13  which starts at z 1 /e 79 /n 175  and ends at n 176 /e 78 /n 84  (FIG.  9 ). Derived branches are identified with ellipses containing “D” and represent thermal process durations that are defined as a function of the machine events. 
     FIG. 2 also shows that Plunger To Loading Position/MP 1  (“P” means prior cycle) must be completed at n 13 . Node n 13  is the time when the motion Plunger To Loading Position/M 1  was completed at n 15  during the prior cycle. This is indicated by cycle time branch (FIG. 6) that connects n 13  to n 15 . The plunger includes an independently moveable thimble and at the end of Plunger To Loading Position/M 1 , both the thimble and the plunger are up. Node n 177 , the end of Gob Delivery/M 13  must be some time (s 2 ) (“s” along side a pair of closely adjacent direction arrows represents some time (a sequence constraint) that will pass between connected nodes) after n 13   
     FIG. 2 also shows node n 20  which is the time when Baffle Off/MP 15  was completed in the previous cycle t 2 . This is indicated by cycle time branch t 2  that is connected to node n 22  (FIG. 4) which is the time when Baffle Off/M 15  is completed in the subsequent cycle. Node n 20  is connected to n 1  which starts Baffle On/M 14  some time (s 22 ) after n 20 , i.e., Baffle On/M 14  cannot start until Baffle Off/ 15  is completed. Motion branch Baffle On/M 14  ends at node n 93 . Baffle motion is broken down into submotions Baffle Moves To Interference With Gob/m 4  which starts at n 1 /e 27 /n 7  and ends at n 8  and Baffle On Completion/m 5  which starts at n 8 /e 3 /n 9  and ends at n 10 /e 28 /n 93 . Also shown is collision branch Baffle Collides With Gob/c 1  (collision branches are represented by a squiggly line which is identified by “c”) connecting node n 4  to n 8 . This means that the gob must be at n 4  before or no later than the baffle reaches n 8  in order to be sure that no collision will occur. 
     FIG. 2 also shows node n 40  which is the time Blank Molds Closed/MP 9  the last cycle (n 40  is connected to node n 55  (FIG. 6) which is the end of Blank Molds Closed/M 9  the present cycle with t 1  indicating a cycle difference). Blank Molds Closed/MP 9  was complete at n 40  which is some time (s 21 ) before the start of Gob Traverses Blank/m 3  at n 5 . 
     When the gob is fully delivered in the blanks n 177 /e 24 /n 26  Blank Contact/d 1  (FIG. 3) begins and continues until n 25 /e 25 /n 28  when the Blanks Open/M 5 . Prior to Blank Contact/d 1  at time n 5 /e 63 /n 183  (the time when Gob Traverses Blank/m 3  begins) a vacuum valve is opened starting the process branch Vacuum Assist/p 13  (process branches are identified with ellipses containing “P”). Vacuum Assist/p 13  will continue until n 182  when the vacuum valve will be closed. This means that at the same time that the gob is traversing the blank, vacuum is being applied through the neck ring (before completion of the plunger moving to the loading position) to help draw the gob into the neck area of the blank and into the neck ring. 
     At n 12  which is some time (s 5 ) after the gob is delivered (n 177 ) and some time (s 3 ) after the baffle is on (n 10 ), a compressed air valve is opened to start Settle Blow/p 1  which ends at node n 11 /e 73 /n 21 /e 68 /n 155  with the closing of the compressed air valve. When Settle Blow/p 1  ends Settle Blow Vent/p 10  begins and ends at n 19  and Neck Ring Contact/d 8  begins and ends at n 154 /e 69 /n 113 / with Neck Rings Opening/m 21  (FIG.  5 ). This means that at the completion of settle blow the gob is in contact with the neck ring and heat is being removed from the gob until the neck rings are opened. Baffle To Down/M 2  (FIG. 2) begins at n 69 , which is some time (s 1 ) after n 11  and ends at n 35  (closes top of blank for counter blow). At n 172  (FIG. 3) which is some time s 10  after n 177  when the gob is fully loaded into the blank molds and another time s 11  after Blank Cooling/pP 7  ended at n 173  during the last cycle (t 11 ), Blank Cooling/p 7  begins with the opening of a valve and continues until n 171  when the valve is closed. 
     At n 156  (FIG.  3 ), which is time s 40  after Vacuum Assist/p 13  ends at n 182  and which is time s 7  after n 19  when Settle Blow Vent/p 10  is over, the Plunger Is Displaced To Counter Blow Position/M 3  (the thimble is retracted out of the glass), a process that ends at n 70  and at the same time (n 156 /e 70 /n 158 ) the glass in the area of the finish, which is in complete contact with the molds, will reheat (Corkage Reheat/d 9 ) until n 157 /e 71 /n 160  which is some time (s 39 ) after n 70  and which is some time (s 36 ) after n 35  (the end of Baffle To Down/M 2 . At n 160 , Counter Blow/p 11  starts with the opening of a valve and continues until time n 159 /e 80 /n 181 / when a valve opening a vent in the baffle is opened to allow the process Counter Blow Vent/p 12  to start. This process ends at n 180 . At time n 148 , which is some time (s 38 ) after n 159 , the Plunger (is displaced) To Invert Position/M 4  whereat both the thimble and the plunger are down (this takes until n 147 ). 
     At n 149 /e 66 /n 151 , which is some time (s 37 ) following the end of Counter Blow Vent/p 12  at n 180 , the following events begin simultaneously: 1. Parison Bottom Reheat/d 7  which lasts until n 150 /e 65 /n 28  and 2. Baffle Off/M 15  (FIG.  4 )which lasts until n 22 /e 30 /n 33 . Baffle Off/M 15  can be divided into two submotions; the first being Baffle Off Clears Interference With Invert/m 11  which begins at n 149 /e 29 /n 32  (FIG. 3) and ends at n 31 /e 7 /n 34 , and the second being Baffle Off Completion (past interference)/m 12  which starts at n 34  and ends at n 33 . At n 28  (FIG.  3 ), which is some time (s 8 ) after n 149 , the following events simultaneously occur: 1. Blanks Open (blank molds are opened)/M 5  which ends at n 27  (FIG. 4) leaving the bottom of the parison on the bottom plate of the blank molds; 2. Reheat (parison)/d 4  begins at time n 28 /e 15 /n 29  (FIG. 4) and continues until n 61 /e 16 /n 30  (FIG. 6) (some time (s 15 ) after Blow Head On/M 18  movement is completed at n 101 ) when Final Blow/p 2  (FIG. 7) begins, ending at n 63 ; and Inverted Reheat/d 3  which begins at n 28 /e 8 /n 38  (FIG. 3) and continues until n 37 /e 9 /n 39  (FIG. 5) which corresponds to the completion of Invert/M 6  which began at n 24 . At n 36  (FIG.  5 ), some time (s 4 ) following n 37 , reheat will continue with the parison inverted (Parison Invert Recovery/p 4 ) until n 17 . The invert motion is subdivided into a number of submotions. At the beginning of invert displacement (n 24 /e 53 /n 153 ) (FIG.  4 ), there is the submotion Invert To Baffle Interference/m 40  which ends at time n 152 /e 67 /n 125 . The next submotion is Invert Baffle Interference To Invert Blowhead Interference/m 32  which ends at time n 24 /e 52 /n 127 . The next submotion is Invert To Takeout Interference 1  From Blowhead Interference/m 3  which ends at n 126 /e 60 /n 140  when Invert (moves) To Takeout Interference 2 /m 33  begins ending at n 139 /e 61 /n 142 . The next submotion is Invert (moves) To Takeout Interference 3 /m 38  which begins at n 142  and ends at n 141 /e 54 /n 129 . Finally Invert Completion/m 35  (FIG. 5) occurs beginning at N 129  and ending at n 128 /e 55 /n 39 . 
     A number of collision branches are identified. Plunger Collides With Invert/c 2  (FIG. 3) when the plunger M 4  is not displaced to the invert position before the invert moves (time n 147  vs. time n 24 ). Blanks Collide With Invert/c 3  (FIG. 4) when the blanks M 5  are not displaced to the open position before the invert moves (time n 27  vs. time n 24 ). A number of other collisions are shown: Baffle Collides With Invert/c 4  when the baffle m 11  reaches a selected point before n 24  and Baffle Collides With Invert/c 18  when the baffle m 11  reaches its fully off position before n 152  when the invert has reached the outer limits of its interference zone with the baffle. By dividing the interference zone into more than one zones, the mechanism can get an earlier start. The blow head and the invert will collide c 12  if Blow Head Up MP 19  has not occurred (the last cycle t 4 ) before the invert has reached the end of the Invert Baffle Interference To Invert Blowhead Interference (time n 23  vs. time n 124 ). 
     Also shown is the motion of the takeout: Takeout Through Interference 1 /mp 13  (FIG. 4) which ends at n 143  (the last cycle/t 7 ); Takeout Through lnterference 2 /mp 24  which ends at n 144  (the last cycle/t 8 ); and Takeout Through Interferemce 3 /mp 36  (FIG. 5) which ends at n 145  (the last cycle/t 9 ). A number of collisions are identified: Takeout Collides With Invert/c 13  (FIG. 4) if the invert reaches Interference 1  before the takeout does (n 143  vs. n 126 ). Takeout Collides With Invert/c 17  if the invert reaches Interference 2  before the takeout does (n 144  vs. n 139 ). Takeout Collides With Invert/c 16  (FIG. 5) if the invert reaches Interference 3  before the takeout does (n 141  vs. n 145 ). At n 179  (FIG. 4) which is some time (s 34 ) after n 28 , Neck Ring Cooling/p 9  begins with the opening of a valve and continues until n 178 , which is some time s 35  before n 24  when the invert/M 6  begins to move. 
     The blow molds, which were open Mp 24  (FIG. 4) at time n 14  during the last cycle t 10 , begin to close at time n 98 /e 56 /n 146  which is some time (s 17 ) after n 14 . Closing has a number of submotions: Molds Close to Ware Width/m 39  (FIG. 5) which starts at n 146  and ends at n 109 /e 62 /n 85 ; Molds Close To Parison Width/m 16  which starts at n 85  and ends at n 62 /e 32 /n 42 ; Molds Close to Receive Position/m 14  which starts at n 42  and ends at n 41 /e 10 /n 44 ; and Molds Close Shut/m 15  which starts at n 44  and ends at n 43 /e 31 /n 97  (FIG.  6 ). Takeout Clears Ware From Mold/Mp 30  (FIG. 4) must have operated in the prior cycle t 3  before Molds Close To Ware Width/m 39  to avoid a collision of the takeout with the molds c 10  (time n 89  vs. time n 109 . Furthermore, Parison Invert Recovery/p 4  should be complete before the molds close (time n 17  vs. n 62 /e 32 /n 42 ) or a collision Parison Collides With Mold/c 5  will occur. 
     The neck rings open to release the parison at the blow head (Neck Rings Open/M 8 ) (FIG.  5 ). This motion which occurs from n 46  to n 45 /e 44 /n 112  is divided into two parts: Neck Ring Open Delay/m 18  which starts at the same time n 46 /e 45 /n 111  and ends at n 110 /e 43 /n 113  (some time (s 26 ) after n 41 —the end of Molds Close To Receive Position/m 14  and some time (s 25 ) before Blow Molds Close/M 16  is complete at n 97 ) (FIG. 6) when the second part (Neck Rings Opening/m 21 ) starts. This second part ends at n 112 . In the event that Neck Rings Closing/M 7  (FIG. 6) occurs (n 49 ) before Revert To Neck Ring/Blank Interference/m 19  (n 51 ), the collision Neck Rings Collide With Blank Mold/c 6  will occur. At n 100  (FIG.  5 ), which is some time (s 13 ) after the opening of the neck rings (M 8 ) at n 45 / the invert is displaced back to its original position (Revert/M 17 ). Revert is complete at n 99 /e 34 /n 53 . Revert has three submotions 1. starting at n 100 /e 33 /n 48  there is Revert Clears Interference With Blow Head/m 17  which ends at n 47 /e 12 /n 52 , 2. following Revert Clears Interference With Blow Head, there is Revert To Neck Ring/ Blank Interference/m 19  which ends at n 51 /e 13 /n 54  when 3. Revert Completion/ m 20  operates ending at n 53 /e 34 /n 99 . At n 50  which is some time (s 14 ) after n 100 , Neck Rings Closing/M 7  operates until n 49 . If the neck rings are not closed before revert reaches its initial location of interference with the blank mold (time n 49  vs. n 51 ), the collision Neck Rings Collide With Blank Mold/c 6  will occur. 
     At time n 102  (FIG. 5) which is some time (s 23 ) after n 23 , the movement Blow Head On/M 18  (FIG. 6) takes place finishing at n 101 /e 36 /n 59 . This is a two stage displacement beginning with Blow Head To Interference With Revert/m 22  which begins at n 102 /e 35 /n 58  and ends at n 57 . In the event that Revert Clears Interference With Blow Head does not occur before Blow Head To Interference With Revert, Revert Collides With Blow Head/c 8  will occur (n 57  vs. n 47 ). The last portion of the blow head displacement is Blow head On Completion/m 23  which begins at n 57 /e 14 /n 60  and ends at n 59 . 
     At n 56  Blanks Close/M 9  (FIG. 6) begins and continues until n 55 . If the completion of Revert/M 17  at n 99  does not preceed the start of Blanks Close/M 9  at n 56  the Revert Collides With Blank Molds/c 7  collision will occur. At n 16  which is some time (s 6 ) after n 99 , the Plunger To Loading Position/M 1  displacement takes place ending at n 15 . 
     n 30 /e 17 /n 66  (FIG. 7) is the beginning of Mold Contact/d 5  (FIG. 8) which ends at n 65 /e 18 /n 68  and Final Blow/p 2  which ends at n 63 . n 30 /e 11 /n 165  is also the end of Vacuum Blow Lead/d 12  which begins at n 166 /e 77 /n 168 . Also beginning at n 168  is Vacuum Blow/p 5  which ends at n 167  which is some time (s 29 ) before n 68 /e 18 /n 65  (FIG. 8) which is the end of Mold Contact/d 5 . Both Vacuum Blow Lead/d 12  and Vacuum Blow/p 5  begin at n 168 /e 77 /n 166  (FIG. 6) which is some time (s 9 ) after n 97 . At n 91  which is some time (s 27 ) following the end of Blow Molds Close/M 16  at n 97 , Blow Mold Cooling/p 3  (FIG. 8) begins continuing until n 90  which is some time (s 30 ) before the end (n 65 /e 18 /n 68 ) of Mold Contact/d 5 . Additionally Blow Mold Precooling/d 11  (FIG. 6) begins at the same time n 91 /e 74 /n 162  and continues until n 161 /e 75 /n 30 /e 16 /n 61  which is also the end of Reheat/d 4 . Finish Cooling/p 6  (FIG. 7) begins at n 170  which is some time (s 31 ) after the end of the end of Blow Head On/M 18  at n 101  and ends at n 169 . 
     At n 104  (FIG. 7) which is some time (s 32 ) after the end of Finish Cooling/p 6  at n 169 , Blow Head Up/M 19  begins ending at n 103 /e 38 /n 73 . This movement can be broken down into a number of submotions: 1. Blow Head Up To End Final Blow/m 41  which begins at n 104 /e 76 /n 164  and ends at n 163  which is some time (s 20 ) ahead of n 63  (the end of Final Blow/p 2 ), 2. Blow Head Clears Interference 1  With Takeout/m 25  which begins at n 163 /e 37 /n 72  and ends at n 71 , 3. Blow Head Up Clears Interference 2  With Takeout/m 7  which begins at n 71 /e 21 /n 95  and ends at n 92 , 4. Blow Head Up Clears Interference 3  With Takeout/m 8  which begins at n 92 /e 5 /n 96  and ends at n 94  (FIG.  8 ), and 5. Blow Head Up Completion/m 26  which begins at n 94 /e 6 /n 74  and ends at n 73 . 
     Tongs Open/MP 12  (FIG. 6) is complete at n 86  (of the prior cycle t 5 ) and some time (s 28 ) thereafter, at n 119 , Kickback (takeout ready position)/M 22  begins and ends at n 118 . At n 106 , which is some time (s 24 ) after n 118 , Takeout In/M 20  begins ending at n 105 . Takeout movement has a number of sub movements: 1. Takeout In To Interference 1  With Blow Head/m 27  which begins at n 106 /e 39 /n 76  and ends at n 75 , 2. Takeout In To Interference 2  With Blowhead/m 9  which begins at n 75 /e 22 /n 117  and ends at n 116 , 3. Takeout To Interference 3  With Blowhead/m 10  which begins at n 16 /e 19 /n 132  and ends at n 131  and 4. Takeout In Completion/m 28  which begins at n 131 /e 20 /n 78  and ends at n 77 /e 40 /n 105  (FIG.  8 ). A number of collisions are identified: 1. Blow Head Collides With Takeout/c 9  which will occur if n 75  occurs before n 71 , 2. Blow Head Collides With Takeout/c 14  if n 116  occurs before n 92 , and 3. Blow Head Collides With Takeout/c 15  (FIG. 8) if n 131  occurs before n 94 . At n 80  which is some time (s 18 ) after n 105  (the end of Takeout In/M 20 ) Tongs Close/M 11  finishing at n 79 /e 51 /n 120 . Beginning at n 68  and ending at n 67 /e 50 /n 122  Blank Molds Open/M 10 . This motion has a number of submotions: 1. Molds Open To Release Point/ m 29  which begins at n 68 /e 49 /n 121  and ends at n 120 /e 4 /n 64 , 2. Molds Open To Clear Ware/m 6  which begins at n 64  and ends at n 130 /e 48 /n 123 , and 3. Molds Open Completion/m 31  which begins at n 123  and ends at n 122 /e 50 /n 67 . At n 108 , which is some time (s 19 ) after n 79 , the end of Tongs Close/M 11 , Takeout Out/M 21  takes place ending at n 107  (FIG.  9 ). This motion also has a number of sub motions: 1. Takeout Out Through Interference 1 /m 13  which begins at n 108 /e 41 /n 138  and ends at n 133 , 2. Takeout Clears Ware From Mold/m 30  (FIG. 9) which begins at n 133 /e 57 /n 82  and ends at n 81 , 3. Takeout Out Through Interference 2 /m 24  which begins at n 81 /e 23 /n 135  and ends at n 13 , 4. Takeout Out Through Interference 3 /m 36  which begins at n 13 /e 58 /n 137  and ands at n 136 , and 5. Takeout Out Completion/m 37  which starts at n 136 /e 59 /n 88  and ends at n 87 /e 42 /n 107 . The collision Molds Collide With Takeout/c 11  will occur if n 82  occurs before n 130 . 
     Finally at the end of Takeout Out/M 21  (n 107 /e 46 /n 115 )Hanging Dead Plate Cooling/d 6  takes place until n 114 . At n 174  which is some time (s 12 ) after n 107 , Dead Plate High/p 8  takes place lasting until n 18 . Some time (s 33 ) thereafter, at n 84 /e 78 /n 176 /e 47 /n 114  Total Process/d 13  ends, Hanging Dead Plate Cooling/d 6  ends and Tongs Open/M 112  ending at n 83 . 
     While, for illustrative purposes, one specific blow and blow machine configuration has been described, it should be understood that there are a number of operating configurations which machine users use including blow and blow and press and blow and for each users have developed many unique processes that would vary slightly one from another. A person skilled in this art, with an understanding of the illustrated configuration, should be able to define a constraint diagram for his actual configuration. 
     The next step is to convert this network constraint diagram into a representation that is ideal for automated formulation and solution of the schedule synthesis and analysis problems by a computer. A matrix algebraic representation of the network constraint model is utilized in the preferred embodiment but other forms of mathematical representations can be used. The Branch Incidence Matrix, F may be formed as follows: 
     1. Number the branches in the network constraint diagram (NCD) from 1 to M b  where, M b  is the total number of network branches. The ordering of the assigned branch numbers is arbitrary. 
     2. Number the nodes in the NCD from 1 to N n  where N n  is the total number of network nodes. The ordering of the assigned node numbers is arbitrary. 
     3. Form the first row of an M b  row by N n  column matrix F by entering a value of 1 (positive one) in the column corresponding to the source node for the first branch, a value of −1 (negative one) in the column corresponding to the destination node for the first branch, and zeroes in all the other columns. 
     4. Create the second through M b  row of F by repeating the procedure described in Step 3 for the second, third on up to the M b  branch in the network. 
     The result will be a matrix, F with M b  row by N n  columns which is almost entirely filled with zeros, except for one entry of 1 and one entry of −1 in each row. 
     To provide a concrete example, the NCD for a simple network model is shown in FIG.  10 . The network has M b = 7 branches, and N n= 6 nodes. The Branch Incidence Matrix, F for this network will thus have 7 rows and 6 columns. For this model, utilizing the branch and node numbers indicated in FIG. 3, F will then be given by:              F   =     [         1         -   1         0       0       0       0           0       1       0       0         -   1         0           1       0         -   1         0       0       0           0       0       0       0       1         -   1             0       0       1         -   1         0       0           0       0       0       1       0         -   1             0       1       0         -   1         0       0         ]             Equation                 1                         
     Each branch, i, in the network constraint model represents a pair of relationship of the form: 
     
       
           t   destination,i   −t   source,i ≦δ max,i   Equation 2 
       
     
     
       
           t   destination,i   −t   source,i ≧δ min,i   Equation 3 
       
     
     Where: 
     t  destination,i =time assigned to the destination node of the i th  branch 
     t  source,i =time assigned to the source node of the i th  branch 
     δ max,i =maximum allowable branch duration for the i th  branch 
     δ min,i =minimum allowable branch duration for the i th  branch 
     Define the vector t of nodal times, where the j th  element of t is the time assigned to the j th  network node. Denoting the i th  row of the branch incidence matrix F by F i , Equation 2 and Equation 3 can be rewritten as follows: 
     
       
         −F i t≦δ max,i    Equation 4 
       
     
     
       
         −F i t≧δ min,i    Equation 5 
       
     
     This results from the fact that that the matrix multiplication of the i th  row of the constraint matrix, F i , with the nodal time vector, t, selects only the source and destination node times, because all other entries in the row are zero. In accordance with conventional practice a value of positive one is assigned to the element corresponding to a source node and a value of negative one is assigned to the destination node. 
     Since Equation 3 and Equation 4 hold for each branch in the network the Fundamental Matrix Constraint Equations can be written as follows: 
     
       
         −Ft≦δ max    Equation 6 
       
     
     
       
         −Ft≧δ min    Equation 7 
       
     
     For branches that have no upper bound on their duration, δ max,i , is set to positive infinity. Similarly for branches which have no lower bound set on their duration, δ min,i  is set to negative infinity. For branches which must exactly match a target value, both the upper and lower bound are set equal to the target value, δ target,i . 
     The fundamental matrix constraint equations (Equation 6 and Equation 7) are augmented to account for three types of additional constraints. These three additional types of constraints are: 
     1. The branch duration for all cycle branches must be equal. This is required to enforce a uniform cycle period on the whole system. 
     2. The branch duration for each submotion branch must remain a constant fraction of the branch duration for its corresponding main motion branch. 
     3. The absolute event time for one node in the network must be set to a desired reference value (typically zero). 
     These requirements can be expressed in terms of the previously defined branch incidence matrix, F as follows: 
     Each cycle branch must have a duration equal to the cycle period, T, and therefore, regardless of the particular value of the cycle period all N t  cycle branches must have the same branch duration. Denote the branch numbers corresponding to the cycle branches by the set {i 1 , i 2 , . . . i N     t   }. The duration of the k th  cycle branch can then be expressed as: 
     
       
         −F i     k   t=δ i     k      Equation 8 
       
     
     Where F i     k    represents the i k  row of the Branch Incidence Matrix, F. 
     The uniform branch duration constraint is then enforced by setting each of the cycle branch durations to be equal to the duration of the first cycle branch as follows:                  [           -     F     i   2                   -     F     i   3                 ⋮             -     F     i     N   t                 ]        t     =       [           -     F     i   1                   -     F     i   1                 ⋮             -     F     i   1               ]        t             Equation                 9                         
     Which can be rearranged to obtain:                  [             -     F     i   2         +     F     i   1                     -     F     i   3         +     F     i   1                 ⋮               -     F     i     N   t           +     F     i   1               ]        t     =     [         0           0           ⋮           0         ]             Equation                 10                         
     Defining the matrix A t  to represent the left hand side of Equation 10 so that:                A   t     =     [             -     F     i   2         +     F     i   1                     -     F     i   3         +     F     i   1                 ⋮               -     F     i     N   t           +     F     i   1               ]             Equation                 11                         
     Equation 10 can be rewritten more compactly as: 
     
       
         A t t=0   Equation 12 
       
     
     If the branch duration for a main motion branch is varied, then the submotion branches (if any) associated with this branch should be proportionally resealed. 
     To represent this set of auxiliary constraints some notation must first be defined. Denote the branch numbers corresponding to the main motion branches (including only branches that have associated submotion branches) by the set {M 1 , M 2 , . . . M Nm } where Nm is the total number of main motion branches that have associated submotion branches. Denote the submotion branches associated with the k th  main motion branch by the set {m k1 , m k2 , . . . m kNk }, where N k  is the total number of submotion branches associated with the k th  main motion branch. Each submotion branch duration represents a fixed fraction of the associated main branch duration. Let α kj  denote this fixed fraction for the j th  submotion branch associated with the k th  main motion branch. 
     The required set of constraints associated with the kth main motion branch can now be represented by the equation:                  [             F     m   k1       -       α   k1          F     M   k                       F     m   k2       -       α   k2          F     M   k                   ⋮               F     m     k                   N   k           -       α     k                 N            F     M   k                 ]        t     =     [         0           0           ⋮           0         ]             Equation                 13                         
     Defining the matrix A M     k    to represent the left hand side of Equation 13 so that:                A     M   k       =     [             F     m   k1       -       α   k1          F     M   k                       F     m   k2       -       α   k2          F     M   k                   ⋮               F     m     k                   N   k           -       α     k                 N            F     M   k                 ]             Equation                 14                         
     we can rewrite Equation 13 more compactly as: 
     
       
         A M     k   t=0   Equation 15 
       
     
     Further, defining the matrix A m  as:                A   m     =     [           A     M   1                 A     M   2                 A     M   3               ⋮             A     M     N                 M               ]             Equation                 16                         
     the complete set of auxiliary submotion constraints can be expressed by the equation: 
     
       
         A m t=0   Equation 17 
       
     
     One reference node in the network is selected, and the absolute time at which this event is to occur is set to zero. Denoting the node number of the reference node as k, this constraint can be expressed as: 
     
       
         A z t=0  Equation 18 
       
     
     where the kth element of the row vector A z  has a value of 1 and all other elements are zero. 
     Finally, the Augmented Constraint Matrix A is defined by:              A   =     [           -     F   r                 A   t               A   m               A   z           ]             Equation                 19                         
     where F r  is the reduced branch incidence matrix formed by eliminating all of the now redundant rows in F. Specifically, the rows corresponding to all of the submotion branches, and all but the first cyclic branch are removed from F to form F r . The length N b +N t +N M +1 vectors b min  and b max  are defined by:                b   min     =     [           δ   min             0         ]             Equation                 20                 b   max     =     [           δ   max             0         ]             Equation                 21                         
     The complete set of network constraints can then be expressed by combining Equation 6, Equation 7, Equation 12, Equation 17, Equation 20 and Equation 21 into the single set of augmented constraint equations: 
     
       
         At≦b max    Equation 22 
       
     
     
       
         At≧b min    Equation 23 
       
     
     In the right hand side of Equation 22 and Equation 23 the final N t +N M +1 elements of the vectors b max  and b min  respectively, are always identically zero. The objective is to find a set of event times (schedule) which satisfy all the required network constraints. In general, there will be more than one, in fact, infinitely many, schedules that will fit the network constraints. A constrained optimization based methodology is therefore employed to select the most desirable schedule from the many available possibilities. The general approach can be tailored to fit a variety of practical problems by making judicious choices of the optimization criterion. Some informally stated examples of optimization criteria that are of practical interest include: 
     1. Minimizing the cycle period with specified thermal process durations 
     2. Maximizing particular thermal process durations, for example reheat, within a fixed cycle period. 
     3. Minimizing wear and tear by slowing down mechanisms as much as possible with a fixed cycle period and specified set of thermal process durations. 
     Optimal schedules utilizing such criteria are readily obtained using the new methodology which has been developed. 
     In terms of the matrix algebraic model representation, described previously, the general problem to be solved is to find a length N n  vector t of nodal times which satisfies: 
     
       
         minimize f(t)   Equation 24 
       
     
     subject to the constraints: 
     
       
         At≦b max   
       
     
     
       
         At≧b min   
       
     
     The scalar function f, referred to as the objective function, specifies the criterion for distinguishing the most desirable of the many possible solutions to the problem. This is known as a Constrained Optimization Problem (as opposed to an Unconstrained Optimization Problem) because we are seeking an optimal solution but are limiting the set of possible solutions to those that satisfy a specified set of constraints. In this case, the constraints are expressed as a set of linear inequalities. 
     A wide variety of practical criteria can be expressed in terms of a quadratic objective function of the form (actually, the constant term, f o , is not strictly required since it has no effect on the location of the system&#39;s minima and maxima. It is only retained here because it later allows the value of the objective function to be given a more obvious interpretation as the distance of the actual branch durations from the desired target values. 
     
       
           f ( t )={fraction ( 1 / 2 )} t   T   Ht+Ct+f   0    Equation 25 
       
     
     As will be detailed subsequently, the essential machine scheduling problems can in fact be expressed using a quadratic objective function of the form given in Equation 25. 
     An optimization problem that has this combination of a quadratic objective function, and linear constraints is known as a Quadratic Programming Problem. A large variety of fast and reliable numerical algorithms exist for solving Quadratic Programming Problems. In some practical cases (for example minimizing the cycle period) the optimization criteria can be expressed using a linear objective function of the form of given: 
     
       
           f ( t )= Ct+f   0    Equation 26 
       
     
     This combination of a linear objective function with linear constraints is known as a Linear Programming Problem. Linear Programming problems, in many cases, can be solved with less computational effort and therefore even faster than Quadratic Programming Problems, but the Quadratic Programming Solver used to save linear and quadratic objective functions is the most economical solution. 
     The basic idea of the GTSSM (General Target Schedule Synthesis Methodology) is to assign a target value for the duration of each branch in the network. These target values represent the ideal set of values the user would like to attain for all of the branch durations. Because of the many network constraints that must also be satisfied, it may not in fact be possible to obtain all of the target branch duration values. The GTSSM therefore finds a schedule that matches the target values as closely as possible. 
     The GTSSM achieves it ability to provide a single approach to a variety of problems through the use of four major features: 
     1. Quadratic Objective Function—A quadratic objective function makes mathematically precise the notion of a schedule being as close to the target value as possible. 
     2. Hard Limits—Hard upper and lower limits can be imposed on the allowable durations for each network branch. 
     3. Locking—Durations of specified branches can be locked so that they are achieved exactly in the resulting schedule. 
     4. QP Solver—Use of a robust QP (Quadratic Programming) numerical solver. 
     Each of the above features will now be described in further detail. 
     The intuitive notion of a schedule being close to the target value must be made mathematically precise in order to implement an automated numerical solution. For this purpose define the objective function, f(t), as follows:                f        (   t   )       =       ∑     i   =   1       N   b              (       w   i          (         δ   i          (   t   )       -     δ     t   i         )       )     2               Equation                 27                         
     where: 
     w i =a constant which weights the importance of the deviation between the target and actual duration for the i th  network branch 
     δ i (t)=duration of the i th  network branch as a function of t, the length N n  vector of nodal event times (schedule) 
     δ t =target duration for the i th  network branch 
     N b =total number of network branches 
     Thus, the distance, from the target is expressed as the weighted, sum of the squared deviations between the target and actual branch durations. It is noted that for the two or three-dimensional case (N b =2 or N b =3) and w i =1, Equation 27 expresses the familiar Euclidean Distance Formula. 
     Noting that the branch duration for the i th  branch duration can be expressed in terms of the i th  row of the branch incidence matrix as: 
     
       
         δ i =−F i t, 
       
     
     Equation 27 can be expressed in terms of the previously defined Matrix Algebraic System Model as: 
     
       
           f ( t )=( W ( Ft +δ)) T ( W ( Ft +δ))   Equation 28 
       
     
     where: 
     W=weighting matrix 
     δ=vector of target branch durations 
     F=branch incidence matrix 
     T=length N n  vector of nodal event times (schedule) superscript T=matrix transpose 
     Following routine algebraic manipulation Equation 28 can be rewritten as: 
     
       
           f ( t )= t   T   F   T   W   T   WFt+ 2δ t   W   T   WFt+δ   t   T   W   T   W   T   Wδ   t    Equation 29 
       
     
     Equation 29 can then be expressed in the standard form given in Equation 25 for a quadratic objective function: 
     
       
           f ( t )=½ t   T   Ht+Ct+f   0    Equation 30 
       
     
     where: 
     H=2F T W T WF 
     C=2δ t W T WF 
     f 0 =δ T W T Wδ t    
     Some flexibility is available in the definition of the elements of the diagonal weighting matrix. The simplest alternative is to set each of the branch weights w i  to a value of 1(one) so that W becomes the identity matrix. This gives equal weight to the absolute error (deviation) between the desired and target duration values for all of the network branches. Although in some cases the absolute error approach may be appropriate, it is more commonly the case that we are concerned with the relative error, in which the error for each branch is normalized by its typical duration. Thus, with the relative error approach a deviation of 1 millisecond for a branch whose typical duration is 10 milliseconds is considered to be of the same significance as a deviation of 1 second for a branch whose typical duration is 10 seconds. For the relative error approach we thus define the weighting matrix W by:              W   =     [           1   /     (       δ     high   1       -     δ     low   1         )           0       0       0           0         1   /     (       δ     high   2       -     δ     low   2         )           0       0           0       0                   0           0       0       0         1   /     (       δ     high     N   b         -     δ     low     N   b           )             ]             Equation                 31                         
     where: 
     δ high1 =The high scale value for the i th  network branch 
     δ low1 =The low scale value for the i th  network branch 
     It is often useful to have the ability to limit the allowable ranges of particular branch durations. Example situations for requiring this capability might include mechanisms that have a lower bound on their motion duration, and process steps that have lower and/or upper bounds on their duration. These bounds are set in the GTSSM by assigning appropriate values to the elements in the b min  and b max  vectors forming the right hand sides of the matrix constraint relationships given by Equation 24. 
     In some cases it is desirable to specify that particular branch durations are to be exactly equal to the target values. This will be referred to as locking the target value. For example, in some cases it is necessary to lock the duration of the cycle branches because the cycle period of upstream equipment such as the feeder can not be readily adjusted. This capability is implemented in the GTSSM by setting the value of the appropriate elements of the upper and lower bounds (vectors b min  and b max  forming the right hand sides of the matrix constraint relationships given by Equation 24) both equal to the target value. The matrix H should be positive definite). To avoid complications with this numerical problem, small weights can be assigned even to branches whose duration&#39;s are not of interest. or a solver can be used that specifically treats the case where H is only positive semidefinite. 
     Based upon either prior experience or specific tests, the desired durations of all of thermal forming process steps (reheat, final blow, etc.) may be known and the bottle maker may not wish these values to vary. With the cycle period branch duration unlocked, all of thermal-forming related branch durations locked, and mechanism motion branches duration locked at the value corresponding to the fastest possible mechanism travel durations, the target duration for the cycle period could be set to zero (indicating that it is desired to have it as short as possible). The QP solver will then find the schedule with the shortest possible cycle period consistent with all of the network constraints (These constraints include the locked thermal process durations and mechanism motion durations along with the requirement for collision avoidance, proper sequencing, etc.). 
     It is quite possible that a particular schedule may achieve the required cycle period and desired set of thermal formal process durations but that it requires moving some mechanisms faster than is strictly necessary to achieve these objectives. It may be desirable instead to operate the mechanisms only as fast as is absolutely necessary to achieve the other desired objectives. This would reduce the average and peak current to the servomotors (and associated motor heating) and perhaps otherwise reduce general wear and tear on the system. To do this, the cycle period and other thermal process branch durations would be locked at their desired values. All of the motion branch durations would be unlocked, and their target values set to a relatively large value. The QP solver would then have the freedom to speed up the mechanism if required to meet the constraints on cycle period and thermal process durations, but otherwise it would increase the motion durations as much as possible. 
     When the desired target values can not be exactly achieved, the user can be provided with some indication of which bounds must be relaxed in order to more closely achieve the desired objective. This can be done by examining the Lagrange Multiplier Values computed at the location of the optimum schedule. The Lagrange Multipliers can be given the interpretation of being the partial derivatives of the objective function with respect to the elements in the b min  and b max  vectors forming the right hand sides of the matrix constraint relationships given by Equation 24. Thus non-zero values for a particular Lagrange Multiplier indicates that the objective function would be either increased or decreased (depending on the algebraic sign of the Lagrange Multiplier) by changing the value of the associated element of the b min  and b max  vectors. Such constraints are said to be active. Other constraints whose Lagrange Multiplier values are zero are said to be inactive. By appropriately displaying to the user the active constraints ranked by the relative magnitude of their Lagrange Multiplier values the user would be informed as to which bounds are imposing the greatest limitation on achieving the desired results. Further, the sign of the Lagrange multiplier could be used to determine, and subsequently display to the user, whether the target value (in the case of a locked branch) should be increased or decreased to further improve the ability to achieve the target values of the unlocked branches. Most constrained optimization algorithms provide for the capability to compute Lagrange Multiplier values (or already compute them as part of their normal operation), so this additional information could be utilized to provide further guidance to the user if desired. 
     If the user overly constrains the system there may be no feasible solution to the QP problem that has been posed. In such a case it is important to recognize that the problem is infeasible, and to relax the bounds enough to allow a feasible solution to occur. QP solvers typically are able to recognize that there is no feasible solution and return an appropriate flag. This flag can be used by the software that implements the GTSSM to trigger a prompt to the user to relax any constraints as much as possible. 
     The MAR (Matrix Algebraic Representation) also allows a proposed schedule to be analyzed in order to discover any potentially damaging or undesirable constraint violations. This capability provides a mechanism for performing intelligent input qualification on user requested changes to event times which goes well beyond conventional high and low bound checking. 
     The basic purpose of the schedule analysis methodology is to provide the capability to check a proposed schedule for constraint violations and then to report any violations that might be found. The method also allows violations to be reported in a way that allows a user to understand the consequences of the violation and, to the extent possible, indicates a remedy. 
     The actual checking of the constraint violations is computationally quite simple involving only a single matrix multiplication and subtraction. To fully obtain the desired functionality there are some additional considerations that must also be taken into account. The additional complexity arises primarily from the fact that the can only schedules nodal (event) times for a subset of the nodes in the overall system model. This subset of nodes is referred to as the set of independent nodes. The nodal times for the remaining, dependent nodes, is then automatically calculated from the independent nodal times and known, fixed branch durations. 
     The overall methodology then consists of the following components: 
     1. Solving for Dependent Nodal Times 
     2. Detecting Constraint Violations 
     3. Diagnosing and Categorizing Violations 
     Dependent Nodal Times can be solved for in terms of the previosly defined constraint set by utilizing the following procedure. 
     1. Form the subset of equality constraints as: 
     
       
         A eq t=b eq    Equation 32 
       
     
     By retaining only those rows of A and b (as defined in Equations 19 and 20 respectively) for which the upper and lower bounds are equal. Note that the upper and lower bounds for branches with known, fixed, durations will both be set to this known fixed value. The upper and lower bounds of these fixed duration branches will therefore be equal and the rows of A corresponding to these branches along with the auxilliary constraints will thus be retained in A e  Typically the branches with known fixed values will be the Motion, Cyclic and Simultaneity branches. In order to have a well posed problem, the row dimension of A eq  must be greater than or equal to the number of dependent nodal times. It is required that a sufficient number of branches be assigned fixed values such that this condition is met. 
     2. By reordering the columns of A eq , form the partitioned incidence matrix A p , in which the first N I columns of A p  correspond to the independent nodal times. Form the partitioned nodal time vector t p  by sorting the columns of t to correspond to the new column order in F p , Equation 32 can then be rewritten as:                  [       A     P   I            A     P   D         ]          [       t     p   I         t     p   D         ]       =     b   eq             Equation                 33                         
     3. Rearrange Equation 33 to form the set of linear equations 
     
       
           A   p     D     t   p     D   −( b   eq   −A   p     I     t   p     I   )=0   Equation 34 
       
     
     5. Assign current values to the independent nodal event times and elements of b eq  corresponding to fixed branch duration and solve the overdetermined system of Equations 34, for t p     D   . This can be done using standard numerical methods available for solving overdetermined systems of linear equations, e.g. a Linear Least Squares Solver. For a consistent set of fixed branch durations and properly constructed Network Constraint Diagram an exact solution to this overdetermined problem may be obtained. That is, a vector of dependent nodal times t p     D    can be found that satisfies Equation 34 without any error. If an exact solution can not be found then the user should be notified accordingly so that the situation can be remedied. It is noted that the zero reference node should be included with the independent event times and be consistent with the definition provided in Equation 18. 
     6. The elements of of t p     D    and t p     I    are resorted into their original order corresponding to the rows of Equations 22 and 23 to form a vector of times, t proposed  consituting the proposed schedule. 
     Once the dependent times have been calculated and a proposed schedule is available, actually detecting constraint violations is relatively straightforward. Let the proposed schedule be given by the vector of nodal times, t proposed . From Equation 22 and Equation 23 the conditions to be checked are then given by the set of inequalities: 
     
       
           At   proposed   −b   max ≦0   Equation 35 
       
     
     
       
           At   proposed   −b   min ≧0   Equation 36 
       
     
     if the inequalities given in either Equation 35 or Equation 36 are not completely satisfied then the proposed schedule violates at least one constraint. 
     Each row in Equation 35 and Equation 36 represents a particular system constraint. Explanatory text and a severity level can accordingly be assigned to each row in these equations. A proposed schedule would then be tested by evaluating Equation 35 or Equation 36. The row numbers of any rows that did not satisfy the required inequality would then provide an index for recalling and displaying corresponding error message text. The severity level could be used to sort multiple constraint violations in order of severity, and could also key for an appropriate color code, or other attribute (flashing) on the graphical user interface. 
     This assignment of text and severity level can be done in an automatic manner. To understand how such an automatic assignment can be performed, recall that the rows in Equation 35 or Equation 36 are derived from network branches. The violation implied by each branch type can therefore be an attribute that is assigned to the particular branch type and then further specified for the particular branch. For example for a collision branch we could automatically define the violation text to read “collision occurs between invert and baffle” this event could also be assigned a severity level for example a number between 1 and 10 with 10 being the most severe. The corresponding row in Equation 35 or Equation 36 would then inherit these descriptions from the branches they descend from. Alternatively, once a Network Constraint Diagram was fully defined for a particular forming process, individual messages could be manually entered or the automatically generated default set could be edited and the resulting data stored in a table for each of the finite number of constraint violations which might occur. While this manual approach could perhaps allow some enhancement to the readability of the messages, it could also be error prone, and would have to be updated if any changes were made to the Network Constraint Diagram. The automatic approach is thus preferred. 
     In a state of the art control, the operation of one of these mechanisms/processes is controlled by turning the mechanism, etc., “on” and “off” at selected angles within a 360° cycle. The turning “on” of a mechanism is an event and the turning “off” of a mechanism is an event. FIG. 12 illustrates a conventional list of timed events with their angular “on” and “off” times for an I.S. machine. This list is available from the machine control. 
     The unwrapped schedule can be converted to a corresponding wrapped schedule using the known cycle period and calculating event angles modulo 360 degrees (event angle=modulo 360(unwrapped event time/cycle period)×360. To go from a wrapped schedule to an unwrapped schedule the original network constraint diagram is augmented with a new set of directed branches called unwrapping branches. The subgraph formed from the unwrapping branches along with the motion and sequence branches and any nodes that are incident on these branches will be referred to as the cycle unwrapping graph. An example of a Cycle Unwrapping Graph is illustrated in FIG. 12 which shows a press and blow cycle. The CUG (Cycle Unwrapping Graph) is created so as to have the following properties. 
     Property 1. The CUG is a connected graph 
     Property 2. The nodes of the CUG are exactly the set of all the source and destination nodes for all of the motion and process branches in the NCD (Network Constraint Diagram). This means that every “on” and “off” angle of the timing drum (sequencer) is represented on the graph. 
     Property 3. Every branch in the CUG is part of a cycle (path from an event to the next periodic repetition of that event). For example, the lowermost line of FIG. 11 proceeds: M 120  (Tongs Open), M 110  (Tongs Close), M 210  (Takeout Out) and M 120 . Similarly, the next line up proceeds M 210  (Takeout Out),M 220  (Kickback), M 200  (Takeout In), and M 210 . The next line up proceeds, M 190  (Blow Head Up), M 180  Blow Head Down), p 2  (Final Blow), and M 190 . The next line up proceeds MP 1000  (Blow Molds Open), M 240  (Blow Molds Closed), M 1000 . The next line proceeds MP 100  (Plunger To Loading Position), M 230  (Pressing), M 80  (Neck Rings Open), M 70  (Neck Rings Closing) and M 100 . The next line proceeds MP 90  (Blanks Close), M 230 , M 40  (Plunger To Invert Position), M 60  (Invert), M 70  (Revert), M 90  and the first line from the left proceeds MP 150  (Baffle Up), M 140  (Baffle Down), M 230 , M 150  (Baffle Up). 
     Property 4. The branches incident on the nodes of the CUG either fan in or fan out but not both. That is, if there is more than one branch directed towards a given node then there is only one branch leaving that node (fan in). If there is more than one branch leaving a given node then there is exactly one node entering it (fan out) 
     The above Properties imply that the CUG also has the following additional properties: 
     Property 5. Any sequence of three nodes that can be traversed by two interconnecting branches following in the branch direction, will be part of at least one common cycle. So each node in the CUG is between two other events in a cyclic sequence. 
     Property 6. Since each branch in the CUG is part of a cycle it must be less than one period long. 
     The problem is then solved in a series of steps, which include checks that the input data is properly ordered to provide a valid solution. 
     1. Form the branch incidence matrix for the CUG. 
     2. Divide the nodes of the CUG into two sets: The independent nodes whose values are given in the input set of wrapped event angles, and the remaining dependent nodes whose event angles are as yet unknown. For a well posed problem all of the dependent nodes must be connected to an independent node by a branch whose duration is known. 
     3. Assign the known input event angles to the independent nodes in the CUG to which they correspond. 
     4. Determine the event angles for the dependent event angles using:                Θ   i     =     mod        (       (       Θ   j     ±         d     i   -   j         T   cycle       *   360       )     ,   360     )               Equation                 37                         
     Where: 
     Θ i  is the event angle to be calculated for ith dependent node 
     Θ i  is the dependent node connected to node i through a branch with known temporal duration d i−j . 
     The algebraic sign in Equation 37 is chosen as positive when the dependent node is downstream from the independent node and negative otherwise. 
     5. Assign the event angle for any periodic repetition nodes to equal the value of the node which it replicates (the node to which it is connected to by a cyclic branch in the NCD). 
     6. Check that all event angles are in the correct cyclic order. This is done by checking that the event angle assigned to each node is between the value of any pairing of its immediate upstream and downstream adjacent nodes. 
     7. Find the angular branch durations for all of the branches in the CUG using: 
     
       
         δ=mod(− F Θ,360)   Equation 38 
       
     
     Where: 
     F is the branch incidence matrix for the CUG 
     Θ is the vector of nodal event angles in the CUG 
     δ is the vector of angular branch durations in the CUG 
     8. Convert δ, the vector of angular branch durations to a vector d of temporal duration using: 
     
       
           d =δ/360* T    Equation 39 
       
     
     where T is the cycle period. 
     9. Solve for the unwrapped nodal event times by solving, using standard numerical methods, the possibly overdetermined system: 
     
       
         −F r t=δ  Equation 40 
       
     
     Where F r  is the branch incidence matrix of the CUG with the column corresponding to the zero reference node deleted. (The choice of zero reference node is arbitrary but should be consistent with that of the NCD.) Although the above system is overdetermined the least squares solution will in fact have zero error because δ is in the column space of F r . This should be verified to identify any computational problems. 
     10. The independent nodes in the NCD are assigned values using the corresponding unwrapped event times that are determined from equation 4. The dependent nodes in the NCD can then be determined as previously described. 
     FIG. 13 is a block diagram illustrating the making of the analytical tool (Tool). The first thing is to Define A Network Constraint Diagram For A Bottle Forming Process In An I.S. Machine  60  (an unwrapped cycle following the formation of the gob, its delivery to the blank station, the transfer of a parison from the blank station to the blow station and the removal of a formed bottle from the blow station). Then Translate The Network Constraint Diagram Into A Data Table 61 which is a compilation of the essential data in the constraint diagram and includes lists of all branches and for each, identifies its name, type, start node number, end node number, etc.. This can be done manually by examining a drawing of the constraint diagram or a computer can do this from a computerized drawing of the constraint diagram. The next step is to Translate The Data Table Into A Mathematical Representation  62  which is then transformed into a Computerized Model  64  for evaluation. In the illustrated embodiment the computerized model is based on matrix mathematics but other mathematical approaches could be used. Depending on the nature of the bottle and the process used to make the bottle (press and blow, blow and blow, for example), different data tables may be required. As shown, a number of data tables (Data Table “N”) may be stored in memory and input as desired. 
     FIG. 14 illustrates the portion of the Computerized Model  64  which will Unwrap 360° Machine Cycle Event Angles Into Bottle Forming Process Event Times  66  (Unwrapping). It receives, as inputs, either as downloaded data from a machine control  42  (FIG. 1) or the like, or via manual entry at a console, terminal or the like, the Event Angles, the Machine Cycle Time (the cycle time for a 360° timing drum for an I.S. machine) and the Motion Durations (the “M” (capital M) durations for the displaceable mechanisms) and outputs Event Times in the bottle forming process. It should be noted that while the event angles and machine cycle time are normally available data from an existing job file, motion durations would have to be defined for the job. 
     FIG. 15 illustrates the use of this Computerized Model  64  to Analyze An Unwrapped Schedule Re Constraints  68  (Constraint Violation Avoidance). With Event Times, Machine Cycle Time, Motion Durations, Submotion Durations (the “m” (small m) motion durations for the displaceable mechanisms), the Collision Branch Lower Limits, the Sequence Branch Lower Limits, and the Thermal Forming Process “N” Limits as inputs, the computerized model can determine whether There Will Be A Constraint Violation?  70 . While the word input means that the inputted information is made available, it might be available form different sources. For example, the event angles and machine cycle time could be available from an existing job file, whereas the rest of the inputs would be inputted at the time when the data table is inputted into Translate The Data Table Into A Mathematical Representation  62 . 
     Whenever an input can have a range of values that may be selected by the operator, such input will include the upper and lower limits of that input, and a choice as to whether the setting is to be locked at a specific value or unlocked to permit its location somewhere within the limits. Nominally the lower limits for the Collision and Sequence Branches can be set at zero or at a selected margin of error and this can be locked out from the operator or the operator may be given access to these inputs so that the operator can define any desired lower limits. One constraint violation would be a schedule that would result in something happening in the wrong sequence. Another would be a schedule that would result in a collision. Either of these constraint violations could be determined without Thermal Forming Process Duration “N” Limits. With this additional input(s), the unwrapped schedule could be evaluated to determine whether one or more of the Thermal Forming Process Durations will be either too short or too long and thereby violate one or more of the thermal forming process constraints. These inputs and ouputs, as well as inputs and outputs in latter discussed embodiments, could be available for viewing on any suitable screen. 
     If either inquiry obtains an affirmative answer, the control will Operate Alarm and/or Reject Inputs  74  and Output Constraint Violation(s)  76 . If neither inquiry is answered in the affirmative the control can Output The Calculated Margins  78  to give the operator some idea as to how tight the schedule is and then Wrap The Event Times Into Event Angles And Print The Event Angles and New Machine Cycle Time  79 . “Print” is intended to mean the presentation of data in either operator readable form as an output presented on a screen or a document or machine readable form so that the machine control can automatically operate on the data such as by resetting the machine with the new event angle and machine cycle time. 
     In one mode, an I.S. machine may be running and the operator may want to change one or more of the event angles in the 360° timing drum. A particular job is being run and basic data for that job (the durations and limits) has already been inputted into the control. This data along with the machine cycle time can be downloaded from the machine control. The Event Angles including any proposed Event Angle change can be downloaded to the unwrapper  66  so that Event Times can be defined. In another mode, an operator may have a record (Event Angles and Machine Cycle Time) of a job that was run previously and want to evaluate some changes before he starts the job. 
     In a conventional I.S. machine which has a number of mechanisms that are operated via pneumatic cylinders, Motion Durations and Submotion Durations may have to be empirically defined, as with high-speed cameras. Where interferences involve actuators which are displaced pursuant to motion profiles, the submotion zones can either be empirically defined or they may be mathematically determined. 
     FIG. 16 illustrates the use of this computerized model to monitor the thermal forming process durations (Thermal Forming Process Durations). With Event Times, Motion Durations, Submotion Durations and Machine Cycle Time known or as inputs, the Computerized Model  64  will Analyze An Unwrapped Schedule Re Thermal Forming Process Durations  80 . and then the Computerized Model  64  will Output The Thermal Forming Process Durations  82 . The operator can, accordingly, at any time, see the Thermal Forming Process Durations and based on his experience, make changes to the 360° Event Angles and Machine Cycle Time. With the additional input of Thermal Forming Process Duration “N” Limits, the computer model can also Output Thermal Forming Process Duration “N” Margins  81  so that the operator can see where the time of any process is relative to its allowable time window. 
     FIG. 17 illustrates the use of the computerized model to define, for an existing machine set up, the optimized cycle time (Optimized Cycle Time) and the optimized Event Angles for that schedule. With Motion Durations, Submotion Durations, Collision Branch Lower Limits, Sequence Branch Lower Limits, Event Times, Machine Cycle Time, and Optimized Machine Cycle Time/Target/Lock Status known or as inputs to the Optimize Unwrapped Schedule For Minimum Cycle Time  82 , the Computerized Model  64  will determine whether There Is A Feasible Schedule?  83 . If not the model will Reject The Inputs  85 . The Machine Cycle Time and the Event Times may be supplied from the unwrapper and the Optimized Machine Cycle Time may be inputted by the operator. The Event Times and Machine Cycle Time are only required to determine the thermal forming durations so that these valves could be locked before doing the optimization. Equivalent inputs would be the Thermal Forming Durations. The operator can set the Optimized Machine Cycle Time Target to zero with an unlocked status and the Computerized Model will try to optimize the proposed schedule at the lowest possible cycle time. In the event the operator decides that rather than reduce the machine cycle time from the current Machine Cycle Time to the fastest Machine Cycle Time, he would prefer to reduce the cycle time to some machine cycle time therebetween. He can set the Optimized Machine Cycle Time Target at a time intermediate the Machine Cycle Time and the fastest machine cycle time with a locked status. If there is a feasible schedule, the model will Wrap Optimized Event Times Into Event Angles  84  and Print The Event Angles And The New Machine Cycle Time  86  for the schedule cycle so that it will be available for input into the machine controller portion of the control. 
     FIG. 18 illustrates the use of the Computerized Model  64  to tune an operating I.S. machine in response to operator inputs defining one or more of the Thermal Forming Process Durations (Thermal Forming Process Duration “N”, and associated Target, Limits and Lock Status). With Machine Cycle Time and Event Times (or Thermal Forming Process Durations), as inputs, and with Motion Durations, Submotion Durations, Collision Branch Lower Limits, Sequence Lower Limits, also as inputs, the Optimize Unwrapped Schedule  88  portion of the Computerized Model  64  will determine whether There is A Feasible Schedule?  90 . As shown, there is an additional input: Thermal Forming Process Duration “N”, which includes the Target (time), Limits and Lock Status. 
     The operator may, for example, decide that a defect is occurring because there isn&#39;t enough “reheat” time and input a proposed new reheat time. The operator could also input more than one new Thermal Forming Process Durations N 1 , N 2 , . . . , during an off line evaluation of the process. In either of these modes the Event Angles for the entire schedule would be available and these could all be inputted by the operator or downloaded from the control for the machine. 
     If no schedule is feasible, the Computerized Model will Reject The Inputs  92 . If a schedule is feasible, the Computerized Control will Output The Thermal Forming Process Durations  89 . Such an output might, for example, be a printout for each duration, of the target duration, an indication of whether or not its target duration was locked, and the actual duration located in a window extending between the high and low limits for the duration. Should there be a solution the Wrap Optimized Event Times Into Event Angles  84  portion of the Computerized Model converts the Event Times to Event Angles and proceeds to Print Event Angles And New Machine Cycle Time  94 . 
     FIG. 19 illustrates the use of the computerized model for complete schedule optimization (Schedule Optimization). Machine Cycle Time, Event Times, Motion Durations, Submotion Durations, Thermal Forming Process Durations, Collision Branch Durations and Sequence Branch Durations which represent target values, are possible inputs to Optimize The Unwrapped Schedule  96 . In addition, a number of limits are also inputs: 1. Min/Max Motion Duration “N”, 2. Min/Max Thermal Forming Process Duration “N”, 3. Min/Max Collision Branch “N”, and 4. Min/Max Sequence Branch “N”. The Min/Max Motion Duration “N” relates to servomotor driven displacements which can be selectively varied. Given these inputs, The Optimize Unwrapped Schedule finds and optimized schedule if a feasible schedule exists. In the event that the query There Is A Feasible Schedule ?  98  is answered in the negative, the operator will be advised to Loosen Limits  100  so that the operator will try to find a solution by modifying the limits. In the event that the query There Is A Feasible Schedule?  98  is answered in the affinnative, the control may Set Collision/Sequence Branches To Max, Lock All Other Durations and Again Optimize The Unwrapped Schedule  101 . This will maximize these branches to further reduce the rate of collision of missequencing. The computer model will then Wrap Event Times Into Event Angles  102 , Print The Event Angles And The New Machine Cycle Time  104  and Output Optimized Durations VS. Limits  106 . The operator accordingly has the ability to manipulate the unwrapped schedule to the fullest. He can start with an existing job file which traditionally would have the cycle time, event angles, and the servo-motion branch durations and work to define an optimized schedule. Alternatively, he could enter the Thermal Forming Process Durations and convert them to the Event Times (a screen, not shown could display all of this information to facilitate his analysis). 
     The Computerized Model can, if There Is A Feasible Schedule  107  (FIG.  20 ), determine if There Is An Active Constraint(s) That Restricts Further Improvement?  108  and will Output The Active Constraint(s) (including the direction to move for improvement)  110 . For example, the computerized model may show that the constraint that is preventing optimization is blow mold cooling time. This then enables the operator to address this specific problem to increase the flow of coolant through or at the molds. If there is no solution the operator is advised to Loosen Limits  100 . 
     FIG. 21 illustrates the use of this technology to optimize the wear on a mechanism operated by a servomotor (Wear Optimization). Here the Computerized Model  64  is used to optimize an unwrapped schedule and when There Is A Solution  107 , the next step is for the computer model to Optimize The Unwrapped Schedule Locking All Variables Except Servo Motion Durations And Setting Target Servo Motion Durations At Large Value. 112. The next step is for the computer model to print The Optimized Duration For Servo Motor “N:” and to Deliver The Optimized Duration For Servo Motor “N” To Servo Motor “N” Controller  114 , which will then Route Duration Of Servo Motor “N” From Servo “N” Controller To Servo “N” Amplifier Drive Card  116  which will then Change To Optimized Duration In Amplifier Digital Signal Processor  118 . The Amplifier Digital Signal Processor could, for example, scale a normalized motion profile for the mechanism to be driven to accommodate any duration of motion. In this environment, an ideal motor to adjust in this fashion is a servomotor which has a normalized motion profile that could be scaled from a minimum duration to a maximum duration. While the preferred embodiment of profiled activator is a servo motor, other electronic motors, such as a stepping motor, could be used. 
     The disclosed control can be used with a glass forming machine either directly as a part of the machine control or indirectly as a control of a machine which is virtually operated for evaluation purposes.