International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 8- August 2013
Minimization of wastage of material for a flat
production using different heuristics
Bindu Neeharika.V1, Y.L.Vidya2, Kakati.S3
1,2,3
Assistant professor,ANITS,Visakhapatnam,India
Abstract— This paper discusses some of the basic formulation
issues and solution procedures for solving one- and twodimensional cutting stock problems. Linear programming,
sequential heuristic and hybrid solution procedures are
described. For two-dimensional cutting stock problems with
rectangular shapes, we also propose an approach for solving
large problems with limits on the number of times an ordered
size may appear in a pattern.
order for specified quantities of smaller lengths. This is
achieved by generating optimal cutting patterns for the cutting
of the stock lengths. Stock lengths have cost associated with
them. The greater the quantity of stock lengths used in filling
an order, the greater the cost of the order to the customer.
This, essentially, means reducing the quantity of stock lengths
used. It arises from many applications in industry.
.Keywords— Cutting stock, Trim loss, Linear programming,
The Cutting Stock Problem is essentially an integer
programming (IP) problem. However, integer programming
problems are known to be NP-hard and therefore, the Cutting
Stock Problem is formulated as a linear program by relaxing
the integer constraint. This makes Cutting Stock Problems
amenable to linear programming methods of solution. The
resultant LP optimum is rounded to get the IP optimum. The
columns of the basis matrix represent all the cutting patterns
that can be produced from the available stock length. The
number of cutting patterns can be very large and this makes
explicit enumeration of all feasible cutting patterns
impractical. Therefore an initial set of feasible patterns is
generated and used as the basis for the simplex method to
solve for the dual variables.
Two-dimensional knapsack.,
I.INTRODUCTION
Economic resources are scarce and have costs
associated. The scarcity and cost associated with these
resources generally impose certain constraints in their
utilization. The effective management of these constraints
aimedat minimizing the overall cost of input resources and the
maximization of corresponding profit is the subject of
optimization. Practical optimization is the art and science of
allocating scarce resources to the best possible effect
(Amponsah, 2006). Optimization techniques, a branch of
mathematical programming, has enjoyed enormous appeal
after World War II, both in the academia and in practice.
Subsequently, this interest inspired numerous researches that
sought to identify, analyze and substantiate new techniques
for improving industrial and business processes. Currently,
Optimization techniques have become an indispensable tool
for industrial applications including resource allocation,
scheduling, decision-making, etc. Optimization techniques
have various branches and one such branch is linear
programming.
The term “programming” in linear programming
does not assume programming as used in the field of computer
science to denote software development. Instead, it focuses on
mathematical modeling and the requirement of a finite number
of iterations to solve the model. This project focuses on a
special type of linear programming called the Cutting Stock
Problem.
II.GENERAL OVERVIEW OF CUTTING STOCK
PROBLEM
The aim of Cutting Stock Problem is to minimize the
total cost of stock length of given cost that is cut to fill an
ISSN: 2231-5381
III.CLASSIFICATION OF CUTTING STOCK PROBLEM
The Cutting Stock problems can be classified by the
dimensions of the cutting object. This can be one-, two- or
three-dimensional problems.
A. One Dimensional Cutting Stock Method
The one-dimensional cutting stock problem is to obtain a
given set of ordered lengths (patterns) from stock lengths. The
objective is typically to minimize the total cost of stock
materials used (material input). A cutting pattern describes
how many items of each type are cut from a stock material.
The one-dimensional cutting stock problem is defined by the
following data. Let the one dimensional model is as follows:
where is the number of times pattern j is used, is the cost of
stock material used for cutting pattern j, is the number of in
pattern j and is the quantity of ordered.
To be a valid cutting pattern, a pattern must satisfy
where is the length of the kth stock material used to cut the
pattern.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 8- August 2013
The huge number of patterns is not available explicitly for
practical problems. Usually, necessary patterns are generated
during a solution process, hence the term column generation.
However, the number of different patterns in a solution cannot
be greater than the number of stock lengths and is usually
comparable with the number of piece types.
Width
B. Two Dimensional Cutting Stock Method
Two important procedures was utilized here in finding
out the Optimal cutting patterns & number of mother coils
against each pattern to meet all the required number of slit
coils in order to minimize the trim loss.
Another important variant of the cutting stock problem is the
two-dimensional cutting stock problem. This variant can be
divided into regular (rectangular, circular) and irregular
shapes (Farley, 1988). Rectangular shapes can be obtained
through guillotine or non-guillotine, oriented or non-oriented
cutting. An oriented cutting means that the lengths of
rectangles are aligned parallel to length of the stock sheet.
Slit
Coil
req.
(Ri)
0
5
0
0
64
0
64
5
Finding optimal cutting patterns:
The optimal cutting patterns are determined using the
algorithm for pattern generation and searching procedure are
as follows.
Generating feasible cutting patterns:
Mother coil width of 1250mm is considered while
A two-dimensional cutting stock problem can be defined as
determining the cutting patterns. Appling pattern generating
follows:
A set of rectangular stock sheets of different types is algorithm in generating feasible cutting patterns.
The coefficients (no. of cuts) against each individual
available. For each type of sheet we know its length and width.
From these sheets we have to cut smaller rectangular pieces of slit widths can be determined b using the following equation.
Aij = Smallestlnt [W - ∑i=1 (AiWi)/Wi)
length and width, in order to satisfy a given demand for pieces
of each type. The objective is to minimize the total area of Step 1: Descending the order of requirement widths (Wi)
stock sheets required. A sequence of cuts of a sheet into Step 2: Elements of first row (j=1)
Finding no. of cuts from the individual pattern
rectangular pieces is a cutting pattern.
A11 = Smallestlnt[(1250-0/121.5] = 10
A21 = Smallestlnt[(1250-10*121.5)/100.4] = 0
IV.CALCULATIONS
A31 = Smallestlnt[(1250-10*121.5*100.4)/89.8] = 0
A41 = Smallestlnt[(1250-10*121.5-0*100.4-0*89.8)/85]
A. One Dimensional Cutting Stock Method
=0
The amount of steel should be indented should
A51 = Smallestlnt[(1250-10*121.5-0*100.4-0*89.8include the possible steel loss during the Slitting operation. 0*85)/80]=0
The Customer orders will be summarized at actual and at the
A61=Smallestlnt[(1250-10*121.5-0*100.4-0*89.8item level (against different OD) and based on which the 0*85number of slit coils required will be determined. The number
0*80)/74]=0
of slit coils required for the month of April 11 is as follows:
A71=Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*85-0*800*74)/70]=0
A81=Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*85-0*80-0*
740*70/60.1]=0
A91 = Smallestlnt [(1250-10*121.5-0*100.4-0*89.80*85-0*800*74-0*700*60.1)/49.8]=0
Step 3: Determining the cutting loss of pattern 1 using eqn.(6)
from section 3.32.
TABLE-I
Cut loss of pattern 1 using eqn.
Tj W - ∑ i=1 (AijWi)
T1=1250Outer
(10*121.5+0*100.4+0*89.8+0*85+0*74+0*70+0*60.1+0*49.
38.1
31.76 28.6 26.99 25.4 23.42 22.22 19.05 15.9
Dia.
8)
(OD)
= 35mm
121.5 100.4 89.8
85
80
74
70
60.1
49.8
Step 4: set the level index 1 from the last row, where Aij>0
Slit
Coil
(row index)= 9.8=1
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 8- August 2013
i.e. level-1, 11>0 and others in that row are equal to zero.
Step 5: Since the current of level-1(A11) is greater than zero, a
new column j=j+1=2 is introduced with the following points.
a)
Value of (A11) will be decrement by 1
b)
Values above (A11) remain same and
c)
Values below (A11) will be determined using eqn.
5
Since A11 is the first element in the first pattern
A12 = A10-1=9
A22 = Smallestlnt[(1250-9*121.5)/100.4] = 1
A32 = Smallestlnt[(1250-9*121.5-1*100.4)/89.8]=0
A42 = Smallestlnt[(1250-9*121.5-1*100.4-0*89.8)/85]=0
A52
=
Smallestlnt[(1250-9*121.5-1*100.4-0*89.80*85)/80]=0
A62 = Smallestlnt[(1250-9*121.5-1*100.4-0*89.8-0*850*80)/74]=0
A72=Smallestlnt[(1250-9*121.5-1*100.4-0*89.8-0*850*800*74)70]=0
A82=Smallestlnt[(1250-9*121.5-1*100.4-0*89.8-0*850*800*740*70)/60.1]=0
A92=Smallestlnt[(1250-9*121.5-1*100.4-0*89.8-0*850*800*74-0*700*60.1)/49.8]=1
Step 6 : Cut loss of pattern 2,
0*700*0.1)/49.8)=0
Step 8: The cut loss for the pattern 3.
T3=1250(9*121.5+0*100.4+1*89.8+0*85+0*74+0*70+1*60.1+0*49.8)
=606mm
Searching of Optimal cutting Patterns:
The Haussler heuristic procedure will be applied here in
finding out the optimal patterns that keep the trim loss to
minimum level.
The first descriptor is an estimate of the number of
production rolls needed to satisfy the remaining requirements.
Ist search:
1st Descriptor (∑I RiQi)W
=(0*121.5+5*100.4+0*89.8+0*85+64*80+0*74+64*70+2*64
.1+16*48.9)
/1250
= 8.815 Mother coils required to satisfy slit coils requirement
The second descriptor is the average number of rolls to be
obtained from each production roll. i.e.
2nd Descriptor (∑i Ri/[∑i RjWi)/W))
= (0+5+0+0+64+0+64+2+16)/8.815
= 17.1295 avg. no. of rolls from each Mother coil
These descriptors of the remaining requirements are used
to set an aspiration level for the next pattern to enter the
solution.
The characteristics used are
T2=1250a) Trim Loss:
(9*121.5+1*100.4+0*89.8+0*85+0*74+0*70+0*60.1+1*49.8)
Maximum allowable trim loss MAXTL
= 6.3mm
W-∑i AijWi <=MAXTL (Max. allowable trim loss)
Step 7: Similarly with the step 4
This will reduce the number of feasible patterns over
Row index = 9-7=2
which search will be carried out. We assumed that MAXTL
A22>0
should be less than 25 and its all optimal cut patterns are
For a new column j=j+1=3
found out from these patterns.
A23=A22-1=0
W-∑i AijWi <=25
A13=A12=9
Minimum allowable Trim loss MINTL (allowing the
The remaining elements in the pattern are determined by slitting m/c for coil holding purpose)
eqn. (5).
W-∑i AijWi > = MINTL (Min. allowable trim loss)
A33 =Smallestlnt[(1250-9*121.5-0*100.4)/89.8]=1
This is the constraint from the production of slit coils. A
A43 =Smallestlnt[(1250-9*121.5-0*100.4-1*89.8)/85]=0
minimum trim loss should be allowed while doing slitting
A53
=Smallestlnt[(1250-9*121.5-0*100.4-1*89.8- operation for holding purposes and also to square an
0*85)/80)=0
unevenness in the edges. With the physical set up in External
A63 =Smallestlnt[(1250-9*121.5-0*100.4-1*89.8-0*85- Processing agency responsible for the slitting operation, a
0*80)/74)=0
minimum of 5mm is allowed/taken on either side of the
A73=Smallestlnt[(1250-9*121.5-0*100.4-1*89.8-0*85mother coil.
0*80-0*
W-∑i AijWi >=10
This will further reduces the cut patterns generated.
74)/70]=0
The total trim loss associated with the problem:
A83=Smallestlnt[(1250-9*121.5-0*100.4-1*89.8-0*85= (W*Mother coils assigned-trim loss associated with the
0*80-0*74optimal patterns
W*Mother coils assigned
0*70)/60.1)=1
= ((1250*9-(T1+T2+T3+T4+T5)) *100
A93 = Smallestlnt[(1250-9*121.5-0*100.4-1*89.8-0*851250*9
0*80-0*74= 2.05%
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 8- August 2013
The periodic inventory model considered the Lead time
period of five weeks and a Review period of four weeks is
developed for minimizing the indenting quantity.
B. Two Dimensional Cutting Stock Method
The amount of steel should be indented should
include the possible steel loss during the Slitting operation.
The Customer orders will be summarized at actuals and at the
item level (against different OD) and based on which the
number of slit coils required will be determined. The number
of slit coils required for the month of April 11 is as follows:
Two important procedures was utilized here in
finding out the Optimal cutting patterns & number of mother
coils against each pattern to meet all the required number of
slit coils in order to minimize the trim loss.
Finding optimal cutting patterns:
The optimal cutting patterns are determined using the
algorithm for pattern generation and searching procedure are
as follows.
Generating feasible cutting patterns:
Mother coil width of 1250mm and thickness 40 mm
are considered while determining the cutting patterns. Appling
pattern generating algorithm in generating feasible cutting
patterns.
The coefficients (no. of cuts) against each individual
slit widths can be determined by using the following equation.
Aij = Smallestlnt [W - ∑i=1 (AiWi)/Wi)
Step 1: Descending the order of requirement widths (Wi)
Step 2: Elements of first row (j=1)
Finding no. of cuts from the individual pattern
A11 = Smallestlnt[(1250-0/121.5] = 10
A21 = Smallestlnt[(1250-10*121.5)/100.4] = 0
A31 = Smallestlnt[(1250-10*121.5*100.4)/89.8] = 0
A41 = Smallestlnt[(1250-10*121.5-0*100.4-0*89.8)/85]
=0
A51 = Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*85)/80]=0
A61=Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*850*80)/74]=0
A71=Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*85-0*800*74)/70]=0
A81=Smallestlnt[(1250-10*121.5-0*100.4-0*89.80*85-0*80-0*
740*70/60.1]=0
A91 = Smallestlnt [(1250-10*121.5-0*100.4-0*89.80*85-0*800*74-0*700*60.1)/49.8]=0
Step 3: Determining the cutting loss along the width using
following
ISSN: 2231-5381
Cut loss of pattern 1 using eqn.
(Tj W - ∑i=1 (AijWi))*t
T1=(1250(10*121.5+0*100.4+0*89.8+0*85+0*74+0*70+0*6
0.1+0*49.8)) *40
= 1400mm2.
Step 4: The coefficients (no. of cuts) against each individual
slit thickness can be determined by using the following
equation.
Aij = Smallestlnt [t - ∑i=1 (Aiti)/t i)
Step 5: Descending the order of requirement thickness (ti)
Step 6: Elements of first row (j=1)
Finding no. of cuts from the individual pattern
B11 = Smallestlnt[(40-0/3.66] = 10
B21 = Smallestlnt[(40-10*3.66)/2.89] = 1
B31 = Smallestlnt[(40-10*3.66-1*2.89)/2.54] = 0
B41 = Smallestlnt[(40-10*3.66-1*2.89-0*2.54)/2.39] =
0
B51
=
Smallestlnt[(40-10*3.66-1*2.89-0*2.540*2.39)/2.24]=0
B61=Smallestlnt[(40-10*3.66-1*2.89-0*2.54-0*2.39
0*2.24/2.05]=0
B71=Smallestlnt[(40-10*3.66-1*2.89-0*2.54-0*2.390*2.240*2.05)/1.93]=0
B81=Smallestlnt[(40-10*3.66-1*2.89-0*2.54-0*2.390*2.24-0*
2.050*1.93/1.58]=0
B91 = Smallestlnt [(40-10*3.66-1*2.89-0*2.54-0*2.390*2.24-0*0*2.05-0*1.930*1.58)/0.9]=0
Step 7 : Determining the cutting loss along the thickness using
following
Cut loss of pattern 1 using eqn.
Cv = A11W1 * [1250 – B11t1]
= 10*121.5 *[1250 – 10*3.99]
= 1470271 mm2.
Step 8: set the level index 1 from the last row, where Aij>0
(row index)= 9.8=1
i.e. level-1, 11>0 and others in that row are equal to zero.
Step 8: Since the current of level-1(A11) is greater than zero, a
new column j=j+1=2 is introduced with the following points.
d)
Value of (A11) will be decrement by 1
e)
Values above (A11) remain same and
f)
Values below (A11) will be determined using eqn.
Since A11 is the first element in the first pattern
Optimum Solution
There are 14 feasible cutting patterns available to cut
raw material with the dimensions 1250 mm *40 mm into
required rectangular shaped items. The mathematical model is
developed to design generated cutting patterns so that waste
(cut loss) will be minimized and the optimum solution to the
model is given in Table II:
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 8- August 2013
TABLE-II
Journal of Operational Research Society, Vol.39, pp.41-53, 1988.
The total trim loss associated with the problem:
[3] Farley, A., A note on bounding a class of linear programming problems,
including cutting stock problems, Journal of Operations Research, Vol.38,
pp.922-923, 1990a.
[4] Farley, A., The cutting stock problem in the canvas industry, European
Journal of Operational
Research, Vol.44, pp.247-255, 1990b.
[5] Gilmore, P. C. and Gomory, R. E., Multistage cutting-stock problems of
two and more dimensions,
Operation Research, Vol.13, pp.94-120, 1965.
[6] Gilmore, P. C. and Gomory, R. E., A linear programming approach to the
cutting stock problem,
[7] Gilmore, P.C., and Gomory, R.E., "A linear programming
approach to the cutting stock problem-Part II", Operations
Research 11 (1963) 863-888.
= (W*Mother coils assigned-trim loss associated with the
optimal patterns)
W*Mother coils assigned
= ((1250*40*9-(T1+T2+T3))
1250*40*9
= 1.90%
*100
[8] Haessler, R.W., "Single-machine roll trim problems and
solution procedures", Tappi 59 (1976) 145-149.
[9] Haessler, R.W., "A note on some computational modifications
to the Gilmore-Gomory cutting stock algorithm",
Operations Research 28 (1980) 1001-1005.
[10] Pierce, J.F., Some Large Scale Production Scheduling Problems
in the Paper Industry, Prentice-Hall, Englewood Cliffs,
N J, 1964.
[11] Sculli, D., "A stochastic cutting stock procedure: Cutting
rolls of insulating tape", Management Science 27(8) (1981)946-952.
V.CONCLUSION
In this work an algorithm to solve an important
variant of the one-dimensional cutting-stock problem and twodimensional cutting-stock problem encountered in paper and
steel industries, was proposed. This algorithm first identifies
all feasible non dominated combinations through a pattern
generation procedure, to construct the constraints matrix.
Then a relaxation of the problem is performed in order to
obtain a linear formulation, and finally a solution to our
original problem is generated from the solution of the relaxed
problem. The propose algorithm has been tested on a set of
randomly generated problems. The experimental work shows
the following performances of our heuristic: a good quality of
the solution (the difference between the solution and the lower
bound to the optimal solution, is less than 1%) and this
heuristic is capable to solve problems with medium size (not
only small size) because the search tree does not consider
dominated feasible patterns.
ACKNOWLEDGMENT
The authors thank the referees for their constructive comments,
which have contributed to the improvement of this paper.
VI.REFERENCES
[1] Benati, S., An algorithm for a cutting stock problem on a strip, Journal of
the Operational Research
Society, Vol.48, pp.288-294, 1997.
[2] Farley, A., Mathematical programming models for cutting stock problems
in the clothing industry.
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