Method and apparatus to optimize cut lengths of material

A method and apparatus is disclosed for substantially minimizing or eliminating scrap lenghts of material from being cut from an initial undeterminable length of material. A material is fed toward a cutting device. The material is cut in accordance with an initial cut length schedule that is a value between acceptable customer's limits. The trailing end of the material is monitored and the length of the remaining material is determined. The cut length schedule is then adjusted within the limits in accordance with the determined remaining length to bring the last length of material within the permissible limits or if not possible, then to substantially minimize the scrap cut length of material which falls outside of the limits.

DESCRIPTION 
1. Technical Field 
The present invention relates to a method and apparatus for optimizing cut 
lengths of material and is particularly directed to a method and apparatus 
for eliminating a scrap length end piece, whose length falls outside of 
predetermined length tolerances, by bringing the length of such piece 
within predetermined limits or for substantially minimizing the length of 
a tail-end scrap piece of material. 
2. Background Art 
There are many processes in industry which require the cutting of 
individual lengths of material from a supply of indeterminable length. In 
the steel industry, it is common practice to supply a coil of flat steel 
which is utilized to make pipe or tubing. The flat steel is continuously 
fed into a forming mill where it it shaped into a tube or a pipe and is 
electric-resistance-welded to complete the pipe or tube formation. The 
weld seam is then annealed along the weld seam. 
At a subsequent station, the moving pipe or tube is cut to lengths in 
accordance with a customer's order. Typically, such an order will specify 
a length of pipe or tube with plus and minus tolerances. Standard practice 
has been to cut the pipe or tube into equal lengths equivalent to the 
upper limit specified by the customer. The last remaining piece of pipe or 
tube is referred to as the tail end piece. As the coil runs out, it is 
common for the length of the tail-end piece to be less than the 
permissible lower limit specified in the customer's order but still be of 
substantial length. Such a tail end piece would typically be scrapped and 
is therefore referred to as tail-end scrap. Since the coils of steel vary 
in length and the length of pipe or tube ordered by customers varies, the 
resultant length of the tail-end piece will vary. The quantity of tail-end 
scraps remaining at the end of a working shift can add up to a substantial 
amount of wasted material. 
There have been proposed solutions that attempt to reduce the amount of 
scrap material remaining after cutting customer's orders from an 
intermediate amount of material. One solution proposed when extremely wide 
tolerances are acceptable to a customer calls for making a correction cut 
of material. If the last piece is greater than the allowable limit, one 
cut at a length between the acceptable limits is made during a cutting 
sequence which brings the length of the last cut piece within the wide 
tolerance. This particular approach, however, does not work in all cases, 
particularly when a customer's tolerances are fairly small with respect to 
the overall permissible length. 
Another proposed solution attempts to reduce the amount of scrap material 
by knowing or storing in a memory all the lengths of customer's orders. 
The length of material to be cut must be initially measured prior to any 
cutting. The material is then cut into a number of pieces that would best 
fit an order stored in memory and have the least scrap for that particular 
measured length. Such an approach is not practical when the material used 
is on a coil which can be many hundreds of feet long or even bars or 
billets that are of substantial length that is not conveniently measured. 
DISCLOSURE OF THE INVENTION 
The invention provides a new and improved method and apparatus that is 
particularly adapted for use in optimizing the cutting operation of steel 
tube or pipe to substantially reduce or eliminate tail-end scrap length of 
material at the end of the material supply. The new method and apparatus 
in accordance with the present invention is designed to adjust the 
schedule of cuts to be made after the tail-end of the material is sensed, 
taking into consideration parameters that may vary during the process. 
Such adjustments to the schedule optimizes the cutting operation and 
substantially reduces or eliminates tail-end scrap lengths of material; 
i.e., lengths not falling within customer-specified limits. 
In the preferred embodiment disclosed, the method of the present invention 
comprises the steps of feeding the material to be cut toward a cutter, 
establishing an initial schedule of cut lengths for the material between 
set limits, cutting parts of the material in accordance with the 
established cut lengths, determining a remaining length of material and 
adjusting the cut length schedule within the limits, commensurate with the 
determined remaining length, to substantially minimize or eliminate a 
tail-end length of material outside of the limits. The process further 
controls the length of any tail-end so if possible it is at least long 
enough to be cut from the last usable piece, and not so long that it 
cannot be automatically handled, considering the structured limitations of 
the equipment being used. 
An apparatus of the present invention includes a material cutter, a feeder 
for moving the material toward the cutter, a detector for sensing a 
tail-end of material being moved and cut, and a controller operatively 
coupled to the cutter and the detector, to cut the material in accordance 
with an initial schedule of equal determined lengths within predetermined 
limits, to determine the length of material between the cutter and the 
tail-end of the material and to then adjust the cut length schedule to 
provide cuts that produce lengths of material within the limits to 
substantially minimize or eliminate a tail-end length of material being 
outside of the limits. 
In the preferred embodiment, the material is fed toward a flying cutting 
carriage. Initial cut length limits are established and the material is 
cut in accordance with the initial cut schedule to lengths equal to a set 
upper, but not maximum, limit. The material being fed toward the flying 
cutting carriage is monitored to detect the tail-end of the material by 
using a tail-end detector. The length of material from the blades of the 
flying cutting carriage to the tail-end of the material is measured. A cut 
length adjustment is calculated within the upper and lower limits. The 
schedule is then adjusted in accordance with the calculated cut length 
adjustment to substantially minimize or eliminate the scrap length. 
Another feature of the present invention is the monitoring of the actual 
length of the material cut and then recalculating further cut schedules 
within said limits in accordance with the actual cut lengths. This 
provision accounts for system errors or material slippage during the 
entire operation. Such occurrences are likely because the line length from 
the feed station to the cutting station is quite long, and may typically 
be approximately 500 feet. Because the upper and lower length limits used 
for the schedule are conservative, i.e., are not the maximum and minimum 
specified by the customer, such system errors or slippage do not typically 
resut in scrap. 
Another feature of the present invention is the provision that if the 
tail-end scrap length cannot be eliminated, the last cut is made to leave 
the last length of material within other limits to satisfy system 
constraints, such as handling capabilities. 
Other features and advantages and a fuller understanding of the invention 
will be had from the following detailed description and the accompanying 
drawings.

BEST MODE FOR CARRYING OUT THE INVENTION 
Referring to the drawings, an optimization system 10 is shown for 
substantially reducing or minimizing the amount of scrap material cut from 
a supply of material of indeterminate length or for completely eliminating 
a scrap length of material. The preferred system is shown in connection 
with apparatus for forming steel pipe from flat coiled steel. As shown in 
FIG. 1, flat steel 12 is supplied in the form of a coil 14. The coil is 
unrolled using an uncoiler machine, not shown, and the flat steel is fed 
as a continuous strip into the system along a straight workpath in the 
direction of the arrow 16. The steel is fed lengthwise into a forming mill 
20 where it is continuously formed into a cylindrical pipe or tube shape. 
A welder and localized annealer 22 welds the seam of the pipe and locally 
anneals the seam. The welded pipe 24 procedes along the workpath toward a 
flying cutter carriage 26 that is controlled by a controller 30 to cut the 
pipe into desired lengths. The system 10 controls the cutting of the pipe 
24 so that the resultant lengths remain within limits, e.g., those 
specified by a customer's order, without leaving a scrap piece of pipe 
having a length outside of the customer's limits, or so that the length of 
any scrap piece is substantially minimized. 
An uncoiler tail-out sensor 34, such as a proximity switch, is located 
adjacent the coil of flat steel 14 along the workpath and is operatively 
connected to the controller 30. The uncoiler tail-out sensor 34 monitors 
the steel 12 and provides signals to the controller 30 to indicate that 
the steel 12 is or is not still present in the workpath at the location of 
the sensor. When the tail-end of the steel 12 leaves the roll 14, this 
occurrence of so-called "tail-out" is signalled. 
A pulse tachometer 38 is positioned along the workpath just upstream of the 
forming mill 20 and generates pulses with a measuring wheel as the steel 
12 passes along the workpath, the number of pulses being a function of the 
linear movement of the steel. The controller 30 includes a counter 40 
(FIG. 2) that counts the pulses from the pulse tachometer 38. The count of 
the counter can therefore be converted to the number of linear feet that 
the steel has traveled. 
A material position tail-out sensor 37 is located along the workpath just 
upstream of the tachometer 38. The material position tail-out sensor 37 
indicates to the controller 30 that steel is or is not still present at 
that location along the workpath. 
The flying cutter carriage 26 includes rotating cutter blades 44 that cut 
the pipe transversely. A cut length measuring wheel 46 is carried by the 
flying cutter carriage and is operatively connected to the controller 30. 
The cut length measuring wheel 46 includes a tachometer 48 (FIG. 2), which 
generates pulses to the controller 30 as the pipe 24 travels past its 
location. The number of pulses generated by the tachometer 48 is a 
function of the relative linear motion between the pipe 24 and the flying 
cutter carriage 26. A cut-off tail-out sensor 50 is positioned upstream 
along the workpath from the flying cutter carriage and is operatively 
coupled to the controller 30 and provides an indication that the pipe 24 
is or is not present at that location along the workpath. 
The operation of a flying cutter carriage is known in the art and will not 
be described herein in detail. Essentially, the flying cutter carriage is 
adapted to travel with the pipe 24 down the workpath as the rotating 
cutter blades 44 cut the pipe. Since the pipe can not be instantaneously 
cut, the flying cutter carriage must travel with and at the same speed as 
the pipe until the rotating cutter blades 44 have completed their cut. 
Once the cut has been completed the flying cutter carriage returns to an 
initial position defined as the home position. 
A carriage home limit switch 52 is operatively coupled to the controller 30 
and indicates to the controller 30 when the flying cutter carriage 26 is 
in its home position. A carriage position limit switch 54 is positioned 
along the workpath downstream with respect to the limit switch 52 and is 
operatively connected to the controller 30. The limit switch indicates to 
the controller 30 when the flying cutter carriage 26 has reached that 
downstream position, which occurs during a cutting operation. 
The controller 30 and its operation will be better understood with 
reference to FIG. 2. The controller 30 includes an optimization computer 
60, a material position sensor 62 and a cut length actuator/monitor 64. 
The pulse tachometer 38 of the material position sensor 36 generates pulses 
that are counted by a counter 40, which is part of the material position 
sensor 62. The material position sensor 62 is operatively connected to the 
optimization computer 60 with the counter 40 being controlled by the 
computer through control logic. The pipe position data is received by the 
computer 60 from the counter 40. The computer 60 converts the count 
indicia from the counter 40 into linear feet of travel that the steel 
moved along the workpath. 
The carriage sensors 52, 54 and the three tail-out sensors 34, 37 and 50 
are all stationary sensors and are all operatively connected to the 
optimization computer 60. Once the steel 12 has tailed-out of the coil 14 
and passes the uncoiler tank-out sensor 34, the computer 60 resets the 
counter 40 to zero. The optimization computer 60 then can determine where 
the end of the steel is at any given time with respect to any fixed point 
along the workpath, such as from the position of the sensor 34. 
If the flying cutter carriage happened to return to its home position, 
tripping the limit switch 52, or reached the limit switch position 54 at 
the exact instant the end of the steel passed the uncoiler tail-out sensor 
34, the length of steel remaining between the cutter blades 44 and the end 
of the steel would be known since the distance between the uncoiler 
tail-out sensor 34 and the cutting blades in the flying cutter carriage at 
either the home position or the position at limit switch 54 can be 
measured and is recorded in the computer. If the flying cutter carriage is 
not in the home position or at the switch 54 when the tail-end of the 
steel passes the uncoiler tail-out sensor 34, which is the typical case, 
the counter 40 will have the indicia of steel length that has traveled 
passed the sensor 34 when the carriage gets to one of the positions, 
because the counter was reset to zero by the computer 60 when the steel 
passed the uncoiler tail-out sensor 34. the remaining length is then 
determined when the flying cutter carriage contacts the limit switch 54. 
The length of steel is equal to the distance of the cutter blades when the 
carriage contact the switch 54 to the uncoiler tail-out sensor, minus the 
length the steel has traveled along the workpath, which was recorded by 
the counter 40. If the last few feet of pipe are scrap because of an open 
weld, such last few feet are accounted for in the calculations. 
The cut length actuator 64 includes an entry means by which an operator can 
set an upper length limit and a lower length limit for permissible cuts of 
the pipe 24. Typically the upper and lower limits are set conservatively 
with respect to a customer's order. For example, if an order was for pipe 
between 43 and 52 feet long (minimum and maximum limits), 45 and 50 feet 
would be the set lower and upper limits. This usually eliminates scrap 
that would otherwise result from small cutting errors due to system 
problems, such as slippage. 
The entry means takes the form of thumbwheel switches in the preferred 
embodiment and are indicated in FIG. 2 as a lower limit block 70 and an 
upper limit block 72. The lower limit 70 and the upper limit 72 are 
monitored by the optimization computer 60. The computer sends control 
information to a carriage controller 80 within the cut length actuator 64, 
which in turn controls the flying cutter carriage 26. 
A keyboard printer 74 is provided so that an operator can enter and receive 
data into and out from the optimization computer 60. The actual limits of 
the customer are entered into the optimization computer 60 through the 
keyboard printer 74 and define a maximum upper and minimum lower limit. 
The tachometer 48 of the cut length measuring wheel 46 is operatively 
connected to the controller 30 through the carriage controller 80 of the 
cut length actuator 64. The computer 60, through control logic, instructs 
the carriage controller 80 to cut pipe to a specific length. 
For comparison, prior art practices can be appreciated by referring to FIG. 
6. In accordance with prior art techniques, the control length actuator 
would cut the pipe 24 to the lengths selected as the upper limit. If 50 
foot lengths were indicated, every cut of pipe 24 would be 50 foot 
lengths. Assuming that the steel coil is 732 feet long, and the lower 
limit entered is 45 as in the above example, this would yield a tail-end 
piece of 32 feet, which is less than the lower limit. FIG. 6 shows the 
last eight cuts in accordance with a prior art practice. Such a tail-end 
piece would typically be scrapped. 
In accordance with the present invention, the pipe 24 is cut to lengths 
equal to the upper limit until the position of the tail-end of the steel 
12 is sensed, after which the computer 60 determines the length of pipe 
remaining to be cut and readjusts the cut lengths within the set limits 
such that the length of the tail-end piece will come within the limits, or 
if scrap length cannot be eliminated, to substantially minimize the length 
that does remain. 
Referring to FIG. 3, the present invention will be better understood. The 
first step 100 of the flow diagram is to establish cut length limits in 
accordance with a customer's order. For example, if a customer's order for 
pipe is between 43 and 52 feet, lower and upper length limits allowing 
some error, such as 45 and 50 feet would be set. The next step 102 is to 
cut the material in accordance with a default schedule. A default schedule 
is a schedule of cut lengths in which all the lengths are equal and is a 
value within the established cut length limits. Typically, the default 
schedule would be the upper limit and in the example would be 50 feet. 
The uncoiler tail-out sensor 34 is monitored by the controller in step 104 
of the flow chart. If tail-out of the steel has not occurred, the material 
is continued to be cut in accordance with the default schedule. Once 
tail-out has occurred, which means that the tail-end of the steel 12 has 
passed the tail-out sensor 34, the computer determines the remaining 
length of material in step 106 by calculation. To make this calculation, 
the computer first resets the counter 40 to zero at tail-out and monitors 
when the carriage 26 contacts the limit switch 54. The length of material 
will be equal to the distance between the rotating cutting blades 44 and 
the uncoiler tail-out sensor 34 when the flying cutter carriage 26 first 
contacts the limit switch 54 minus the distance the steel has traveled as 
indicated by the counter 40. 
The computer 60 then establishes a new schedule of cut lengths for the 
remaining length of steel to ensure that the length of the tail-end piece 
is within the lower and upper limits or if the length can not be brought 
within such limits, it substantially minimizes the length of a scrap piece 
whose length is less than the lower limit of the customer's order. This is 
done in step 108 of the flow diagram. 
Referring now to FIG. 4, the program proceeds from step 108 (FIG. 3) 
through "A" where it is determined in step 120 whether the length of the 
tail-end piece of material would be greater than the lower set limit 70. 
If it is, the remaining cut schedule follows the default schedule (step 
122) established in step 102 of the flow diagram in FIG. 3 and the program 
returns to FIG. 3 through "B". If the tail-end piece is not greater than 
the lower limit, a determination is made in step 124 whether it is 
possible to eliminate the scrap length of material at the end. If the 
computer makes an analysis that it is possible to eliminate a scrap 
length, in step 126 the cut length schedule is adjusted to an optimized 
schedule and the program returns to FIG. 3 through "B". The program then 
proceeds to step 140 (FIG. 3). 
If it is determined that the length of the tail-end cannot be brought 
within the limits, a scrap routine 141 is established. 
There are several algorithms to eliminate scrap length that can be used in 
step 126. One such algorithm divides the remaining length of material by 
the upper limit to determine the length of the remaining piece and the 
number of cuts that can be made at the upper limit. The difference between 
the set lower and upper limits times the number of possible cuts is equal 
to the maximum length that can be added to the last piece. In the above 
example, assume that 482 feet remain after tail-out. 50 divided into 482 
is 9 with a remainder of 32. The limit difference is 5 (50 minus 45). A 
total length of 15 feet can be added to the last length by making 3 cuts 
at 45 feet. The three pieces prior to the last piece are cut at 45 feet 
and the remaining piece is 47 feet (32 plus 15). This is shown in FIG. 7. 
One particular scrap routine 141 can be used to minimize the scrap length 
piece of material if it is determined in step 124 that the scrap length 
cannot be eliminated. Assume an example in which the initial default cut 
schedule is at the lower limit. If it is determined that a scrap length 
cannot be eliminated, a cut schedule adjustment would be made to the upper 
limit which would minimize the remaining scrap length. If the default cut 
schedule was at some other value between the set limits and if it was 
determined by the computer that the last length could not be brought 
within the limits, a similar approach can be used to add material to the 
lengths remaining to be cut to keep them within the set limits and 
minimize the remaining scrap length. 
Step 140 of the flow diagram is to make the next cut in accordance with the 
new established schedule of cuts. The computer monitors the cut length 
measuring wheel to determine, in step 142, whether the new cut length 
schedule is being met. For example, if the new cut length schedule 
determined that the next piece should be 45 feet long and because of a 
slip in the machine or system, it actually is cut to 46 feet in length, a 
readjustment could be necessary. Because the actual cut length does not 
fall outside the maximum and minimum limits, for the length, the piece is 
not scrap. In step 142 a determination is made of whether the program can 
or cannot still be met based on an accumulation of such deviations from 
the initial schedule, necessitating a return to step 106 if it cannot. If 
the program returns to step 106, the remaining length of material is 
recalculated, correcting for the known error. The cut schedule is then 
readjusted in step 108. If the schedule is being met, the computer in step 
144 monitors the end of the material through the cut-off tail-out sensor 
50. If the end of the material has not been reached, the program returns 
to step 140 to make the next cut in accordance with the established 
schedule. Each cut length is monitored to assure that the intended length 
has in fact been cut in accordance with the schedule and if it has not 
been, the computer, if necessary, will readjust the schedule accordingly 
to substantially reduce or eliminate a scrap length piece falling outside 
of a customer's limits. Once the end of the material has been reached, the 
program ends in step 116. 
Another scrap routine 141' of the present invention is shown in FIG. 5. 
This scrap routine is used when parameters of a particular facility 
necessitate that the last piece must be at least a minimum length so that 
the system line can support the last remaining length of material so it 
will not fall behind the carriage when cut and necessitate that the 
material does not exceed another length so that it can be automatically 
removed from the line. 
The scrap routine, shown in FIG. 5, is therefore used only when system 
constraints make it more important to satisfy system criteria rather than 
minimize scrap length. One example is where pipe is desired to be cut 
between 45 and 50 feet and system contraints requires the last piece to be 
between 25 and 72 feet. For example, in a particular handling system 
embodying this invention, if the last length to be cut is shorter than 25 
feet, the flying cutter cannot make the cut, and if the last piece will be 
longer than 72 feet, special handling is required to remove it from the 
workpath. The optimization computer, in a case where the scrap length 
cannot be eliminated, will minimize the length of the last piece down to a 
length of 25 feet. If it is determined that the last piece is shorter than 
25 feet, the last cut will not be made and adjustments in the cutting 
schedule will be made to ensure that the last length is shorter than 72 
feet. Such a long last length can be subsequently cut off-line to the 
necessary limits manually. 
In step 152 a determination is made whether or not the last piece can be 
cut. If it can be cut, the default cut schedule is carried out with the 
program continuing on as shown in FIG. 3 until the last cut has been made. 
In step 154, the tail-end piece is then scrapped. 
If it is determined in step 152 that the last piece cannot be cut because 
it would leave a piece whose length is below the system minimum, it is 
determined in step 156 whether adding the scrap piece to the last piece 
would exceed a permissible system handling length. If the remaining length 
or scrap piece can be added to the last piece without exceeding the 
permissible length, then the final cut of the tail-end piece is not made 
in step 157. If it is determined in step 156 that adding the scrap piece 
to the last piece would exceed a permissible handling length, a 
determination is made in step 158 of whether or not cut length adjustments 
can be made in the schedule that would permit addition of the scrap piece 
to the last length of pipe to bring it within the handling limits. If so, 
the cut schedule is again adjusted in step 160 and the program returns to 
step 140 in FIG. 3, except that the last piece is not cut in step 157. 
In the event it is determined that the cut schedule cannot be adjusted in 
step 158 to bring the last length of pipe plus the scrap length within the 
upper handling limits, step 162 determines whether, if cut lengths are 
increased to the maximum upper limit of a customer's order, which data was 
entered into the computer 60 by the keyboard, it will bring the last 
length within system limits. If the answer is yes, the adjustment is made 
in step 164 to make such a cut schedule. The program then returns again to 
the basic program in step 140 with the new upper limit. If the answer is 
no in step 162, step 166 reestablishes the original default schedule and 
the last cut is not made. The material would then have to be handled 
specially by personnel. 
The invention heretofore has been described assuming that the length of 
material exceeds the line length. It is typical for the line length, that 
is the distance between the uncoiler tail-out sensor 34 and the flying 
cutter carriage 26, to be approximately 500 feet. Thus the description has 
assumed that the length of steel on the roll is greater than 500 feet. 
Typically the distance between the pipe position tracking wheel of the 
tachometer 38 and the uncoiler tail-out sensor 34 would be half the line 
distance or approximately 250 feet in the arrangement described. If the 
material length is greater than the length between the pipe position 
tracking wheel of the tachometer 38 and the uncoiler tail-out sensor 34, 
that is, between 250 and 500 feet, measurement of the remaining length of 
material is done in a similar manner as heretofore described. In general 
operation, when the material is first formed into a pipe, the front end is 
not initially welded. The welding typically begins 5 to 10 feet past the 
beginning end of the material. In such a case when the material length is 
between 250 and 500 feet, the measurement is not completed until the front 
end open weld section is cropped. This is done to avoid confusion with 
material that may be in the flying cutter carriage from a previous 
material feed. 
If the material length is less than the distance between the pipe position 
tracking wheel of the tachometer 38 and the uncoiler tail-out sensor 34, 
that is, less than 250 feet in the arrangement described, tail-out of the 
uncoiler would occur before the beginning end reaches the pipe position 
monitor tracking wheel of the tachometer 38. In such a circumstance, the 
counter 40 is reset and tracking begins when the beginning end of the 
steel reaches the pipe position tracking wheel. When the tail-end reaches 
the pipe position tracking wheel of the tachometer 38 the material length 
is indicated on the counter 40 minus any initial crop cut that is made. 
A manual override 180 is provided as part of the cut length actuator 64. 
This permits the operator to in essence disable the optimization computer 
60 from controlling the cut length actuator 64. It is contemplated that, 
under such conditions, the cut length actuator would then cut all pieces 
to the upper limit as entered in block 72. 
To summarize, the preferred method for reducing end scrap in the successive 
cutting of the moving pipe 24 to produce pieces of desired length within 
predetermined lower and higher limits, and where the differences between 
the limits is small with respect to the lengths being cut, e.g., no 
greater than one-half the desired length, includes the steps of (a) 
cutting pieces within said limits from the moving length, (b) sensing a 
trailing end of the length of material at a known distance from where the 
material is cut, the distance being at least three times the minimum 
desired length, and (c) after the trailing end is sensed, controlling when 
the cuts are made as follows: (i) calculate the number of pieces of a 
length equal to the higher limit that can be cut and the length of any 
remaining material forming a last-piece, (ii) if the last-piece is longer 
than said lower limit, cut the remaining material into pieces of a length 
equal to the higher limit, (iii) if the last-piece is shorter than said 
lower limit, calculate whether one or more of the pieces to be cut can be 
reduced in length from the higher limit but to a length no shorter than 
the lower limit so said last-piece will be of a length within said limits 
and if so cut said one or more pieces to said reduced length, (iv) if 
under the calculation of subparagraph (iii) hereof the last-piece will not 
fall within said limits, determine if said last-piece is longer than a 
minimum length that can be cut from the preceding piece and if so cut said 
one or more of the pieces remaining to be cut to a length equal to the 
higher limit, and scrap the last-piece, (v) if under the determination of 
subparagraph (iv) said last-piece is shorter than said minimum length, 
determine if the sum of the length of the last-piece and said higher limit 
is less than a maximum length that can be accommodated automatically due 
to handling constraints and if so do not sever said last-piece, (vi) if 
said sum is greater than said maximum length, calculate whether cutting 
said one or more pieces to a length equal to the higher limit will reduce 
the length of said last-piece to less than said maximum length and if so 
cut said one or more pieces to the higher limit and do not sever said 
last-piece, (vii) if the calculation of subparagraph (vi) will not reduce 
the length of said last-piece to less than said maximum length, calculate 
whether making the said one or more pieces of a shorter length than the 
higher limit but at least as large as the lower limit will result in said 
last piece being between said minimum and maximum lengths and if so cut 
said one or more pieces to said shorter length, and (viii) if said 
last-piece is not shorter than said maximum length under the calculation 
of subparagraph (viii), make said one or more pieces of a length equal to 
the higher limit, do not sever said last-piece, and remove said last-piece 
including an attached length equal to the higher limit from handling and 
cutting apparatus nonautomatically. 
Other modifications and variations of the invention will be apparent to 
those skilled in the art in view of the foregoing detailed disclosure. 
Therefore, it is to be understood that, within the scope of the appending 
claims, the invention can be practiced otherwise than as specifically 
shown and described.