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The “trim loss problem” (TLP) is one of the most challenging problems in context of optimization research. It aims at determining the optimal cutting pattern of a number of items of various lengths from a stock of standard size material to meet the customers’ demands that the wastage due to trim loss is minimized. The resulting mathematical model is highly nonconvex in nature accompanied with several constraints with added restrictions of binary variables. This prevents the application of conventional optimization methods. In this paper we use synergetic differential evolution (SDE) for the solution of this type of problems. Four hypothetical but relevant cases of trim loss problem arising in paper industry are taken for the experiment. The experimental results compared with those of the other techniques show the competence of the SDE to solve the problem.
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Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume , Article ID , pages
http://dx.doi.org/.//
Research Article
Trim Loss Optimization by an Improved Differential Evolution
Musrrat Ali,1Chang Wook Ahn,1and Millie Pant2
1Department of Computer Engineering, Sungkyunkwan University, Suwon 440746, Republic of Korea
2DepartmentofAppliedScienceandEngineering,IITRoorkee,Roorkee247667,India
Correspondence should be addressed to Chang Wook Ahn; cwan@skku.edu
Received  April ; Accepted  June 
Academic Editor: Alexander P. Seyranian
Copyright ©  Musrrat Ali et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
e “trim loss problem” (TLP) is one of the most challenging problems in context of optimization research. It aims at determining
the optimal cutting pattern of a number of items of various lengths from a stock of standard size material to meet the customers’
demands that the wastage due to trim loss is minimized. e resulting mathematical model is highly nonconvex in nature
accompanied with several constraints with added restrictions of binary variables. is prevents the application of conventional
optimization methods. In this paper we use synergetic dierential evolution (SDE) for the solution of this type of problems. Four
hypothetical but relevant cases of trim loss problem arising in paper industry are taken for the experiment. e experimental results
compared with those of the other techniques show the competence of the SDE to solve the problem.
1. Introduction
Paper industry forms an important part of the world econ-
omy. It is a “round the clock” processindustryconsistingof
dierent processes like inventory management, logistics, and
so forth. e nal aim, however, of a paper industry like any
other industry is the satisfaction of customers. e paper rolls
produced in an industry are to be cut as per the customers
demands which vary from one customer to another. is
leadstoaninevitablelossofpaperknownastrimloss
problem (TLP). It is the goal of every paper industry to
eciently satisfy the customers while minimizing the wastage
due to trim loss. e industries have to maintain an ecient
productionplanwhichiseconomicalaswellassatisfactory
to the customers. A systematic representation of the supply
chain is shown in Figure  (adapted from []).
Considering the practicality and importance of the TLP,
it has been given considerable attention by the researchers for
developing its model and for recommending various methods
to solve it eciently. e trim loss problem comes under
the group of cutting and packing problems. Hence it is also
known as cutting stock problem (CSP). In the present study,
weshalluseTLPandCSPalternatively.
e rst general classication for cutting and packing
problems is introduced in Dyckho []. Dyckho developed
a special classication for cutting and packing problems in
which he called it a typology. An improved typology was
developed by W¨
ascher et al. [], which is partially based on
Dyckho ’s original one, but it adopts new categorization cri-
teria. A typology, as dened by W¨
ascher et al. [], is “a system-
atic organization of objects into homogeneous categories on
the basis of a given set of characterizing criteria.” According
to the typology dened by W¨
ascher et al. [], the problem we
aregoingtosolveinthisstudycanbecategorizedasatwo-
dimensional “single stock size cutting stock problem.”
From the mathematical formulation point of view, many
articles are available in which the TLP has been studied
with dierent goals such as minimizing trim loss [],
minimizing the production costs [,], minimizing the
number of patterns [], and minimizing the total length and
overproduction [].
ItwasobservedthatTLPcanbemodelledasaglobal
optimization problem having a complex formulation (math-
ematical formulation of TLP is discussed in the next section),
and therefore ecient techniques are required for nding its
solution.Fundamentally,solutiontechniquesforCSPexitcan
becategorizedintothreegroups.
(i) Algorithmic methods: these methods though guaran-
teetheoptimalsolutionareusuallyavoidedbecauseof
their high computational complexity.
Mathematical Problems in Engineering
Chip and pulp mills
Paper mills
Converting mills
Paper rolls mills
Paper sheets
Newsprint Printing and
writing Tissue
Paper board
Container
board
Board
boxes
Distribution network
Sales network
Cutting stock problem/trim loss problem
F : e pulp and paper supply chain.
(ii) Heuristic methods: these methods usually generate a
faster and an acceptable solution but may not nd the
exact optimal solution. A drawback of these methods
is their domain dependency that causes the limited
application of apparently similar problems [].
(iii) Metaheuristic methods: in metaheuristic methods,
the solution process is oen guided by some lower
level heuristic. ese methods usually have an ability
ofnotbeingstuckinlocaloptimathatmighthappen
with traditional heuristic techniques.
A series of articles are available in the literature using
heuristic and metaheuristic methods. Some of these are linear
programming approximations for the reel cutting stock [].
However, it was observed that linear approximation was not a
verypragmaticapproachforsolvingsuchacomplexproblem.
erefore, eorts were made to solve the non linear models
heuristically [,,].
Considering the fact that the decision variables of a TLP
are of integer type, therefore mixed integer linear program-
ming (MILP) has also been applied for solving cutting stock
problems. A review work on dierent formulations of CSP
and the dierent techniques to solve these is given in [].
Instances of application of heuristic techniques for solving
TLP can be found in [,,].
Taking into account the complexity of TLP, metaheuristic
techniques are probably a more pragmatic approach for
solving such problems. Dierent metaheuristics that have
been used include tabu search [], simulated annealing [],
PSO [], genetic algorithm [], hybrid genetic algorithm
[].
Going into the details of modelling of the TLP is out of the
scopeofthepaper.Here,wearemainlyfocusingonshowing
the suitability of the newly proposed SDE [] algorithm for
dealing with such problems. Synergetic dierential evolution
(SDE), an improved version of dierential evolution (DE)
[,]wasappliedtosolvenumericalbenchmarktest
functions, where the results clearly indicated the competence
of the algorithm. For the present study, SDE has been suitably
modied, to deal with the integer/binary restrictions of TLP.
e motivation behind the application of SDE for solving a
TLP is that a very small improvement in a given arrangement
inacuttingcouldsaveaconsiderablylargeamountofmoney.
e remainder of the paper is structured as follows:
Section  gives the brief introduction of SDE.
Section  presents the formulation of the trim loss problem.
Section  states the implementation of SDE for solving trim
loss problem. Finally, Section  provides the summary of the
paper.
2. Synergetic Differential Evolution (SDE)
is section briey describes SDE, an improved version of
classical DE. SDE uses the concepts of opposition-based
learning and random localization and has a one population
set structure. e working of SDE can be understood with the
help of the following steps.
Population Initialization.Constructtwopopulations1and
2of size NP each. Here, 1consists of random solutions
between the lower and upper bounds min and max,
respectively, and 2consists of opposite solutions obtained
by using the opposition-based learning []. Now the initial
population is constructed by taking the NP best solutions
taken from {1U2}.
Mutation. For each individual, the mutation operation is per-
formed by randomly selecting three solutions {𝑟1,𝑟2,𝑟3}
from the population corresponding to target solution 𝑖.
However, unlike DE, SDE holds a tournament between the
selected three individuals, and the region around the best
individual of the tournament is explored:
𝑖=𝑡𝑏 +×𝑟2 −𝑟3,()
Mathematical Problems in Engineering
where 1,2,3{1,...,NP}are randomly selected such that
1 =2 =3 =,𝑡𝑏 is the tournament best individual, and
is the control parameter such that ∈[0,1].
Crossover. Crossover operator of synergetic dierential evo-
lutionisthesameasthatofclassicalDE.isoperationis
performed depending on the crossover probability Cr [0,1]
between the perturbed individual 𝑖=(V1,𝑖,...,V𝑛,𝑖)gener-
ated in mutation step and the target individual 𝑖=(
1,𝑖,
...,𝑛,𝑖)to obtain the trial individual, 𝑖=(1,𝑖,...,𝑛,𝑖),as
follows:
𝑗,𝑖 =V𝑗,𝑖 if rand𝑗Cr ∨=,
𝑗,𝑖 otherwise,()
where =1,...,and {1,...,}is a random parameter’s
index, chosen once for each .
Selection. e selection scheme of SDE is the same as that of
classical DE, but the method of updating the population is
dierent from DE. Aer generation of the new individual,
evaluate the objective function and compare it to its corre-
sponding target individual by the following equation:
󸀠
𝑖=𝑖if 𝑖≤𝑖,
𝑖otherwise.()
If new individual is better than target individual, then it
replaces target individual in the current generation. is is in
contrast to classical DE, where the better one of the two is
added to an auxiliary population to take part in reproduction
in the next generation. erefore, SDE maintains only one
population and the individuals are dynamically updated. e
newly found individual entered in the population may take
part in the reproduction process.
3. Mathematical Formulation
e TLP, in the context of paper, appears when a set of
ordering paper products is to be cut from the large paper
roll, having specied widths. e cutting method is simply a
winding process, where the large paper roll is wound through
the slitter and is cut by a set of knives positioned on the
line (Figure ). e objective is to minimize the trim loss
while satisfying the demand specications. In this paper, the
mathematical formulation of TLP suggested by Adjiman et
al. []andYenetal.[] is taken into consideration. e
problem is dened as follows.
It is assumed that a paper roll of width max is to be cut in
dierent sizes to satisfy the customer’s demands.
e order specications are taken as follows.
(i) 𝑖rolls of order with a width 𝑖are to be produced,
where = 1,...,indicates dierent products. All
products rolls are assumed to be of equal length.
(ii) In order to design the best overall scheme, a maxi-
mum of  = 1,...,; dierent cutting patterns are
Trim loss
Raw paper reels
Product paper reels
F:Aschematicillustrationoftrimlossproblem.
Product 1
Product 2
Product 3
Product 4
Wast e
Bmax
b1
b2
b3
b4
m1=2 m3=1m2=1
Δ
n11 =3 n12 =2
n22 =1
n23 =1
n33 =1
n43 =1
F : e cutting pattern.
assumed, where a pattern is identied by the position
of the knives.
(iii) Integer and binary restrictions: All the variables are
either integer or binary in nature.
(iv) 𝑗is integer variable indicating the number of times
pattern is repeated.
(v) 𝑖𝑗 is integer variable indicating the existence of a
product in a given pattern.
(vi) 𝑗is binary variable to introduce a change in pattern.
If a new pattern is introduced (𝑗>0), then 𝑗is
equal to one.
AsampleofcuttingpatternisshowninFigure .
Objective function: the actual cost of the trim loss is
the total amount of raw materials used, that is, the sum all
repeated patterns multiplied by a cost factor 𝑗, in addition
to the cost of changing knife positions between patterns. Let
the pattern change be weighted by a coecient 𝑗.etrim
loss problem may now be dened as:
Minimize
𝑚𝑗,𝑦𝑗,𝑟𝑖𝑗
𝑃
𝑗=1 𝑗⋅𝑗+𝑗⋅⋅𝑗,()
subject to the following constraints.
Mathematical Problems in Engineering
e number of rolls of each product must be greater than
the customer’s order.
𝑗𝑗𝑖𝑗 ≥𝑖, =1,...,. ()
e width of each pattern must be less than the width of
rawpaperroll,andwidthofcutproductineachpatternmust
exceed a certain minimum :
max −𝑗
𝑖𝑖𝑖𝑗 ≤max 𝑗, =1,...,. ()
is constraint imposes a lower bound on the total number
of patterns made:
𝑃
𝑗=1𝑗max 𝑁
𝑖=1 𝑖
max ,𝑁
𝑖=1 𝑖𝑖
max . ()
ere must be at least one product in a pattern, and the total
number of knives is limited to max:
𝑗𝑁
𝑖=1𝑖𝑗 ≤max 𝑗, =1,...,. ()
ere must be at least one pattern aer a knife change, and
the maximum number of pattern repetitions is limited to 𝑗:
𝑗≤𝑗≤𝑗𝑗, =1,...,. ()
Constraints () introduce an order on and variables to
reduce degeneracy:
𝑘+1 ≤𝑘, =1,...,−1,
𝑘+1 ≤𝑘, =1,...,−1, ()
𝑗{0,1}, =1,...,,
𝑗∈0,𝑗∩, =1,...,,
𝑖𝑗 ∈0,max, =1,...,,=1,...,,
𝑗=1, =1,...,,
𝑗=0.1, =1,...,,
()
where is a set of integers.
e presence of bilinear inequality () makes the problem
nonlinear and nonconvex.
4. Implementation of SDE for
Trim Loss Problem
In [] SDE has been applied for solving problems having
continuous variables. e TLP, however, turns out to be a
MINLP problem (having binary variables as well), therefore
suitable changes are made in SDE to adapt it for dealing with
integer as well as binary variables. is is described in the
following section.
4.1. Handling of Integers and Binary Variables in SDE. In its
standard form SDE can only handle the continuous variables.
To make it handle the integer variables is, however, an easy
task and needs only a couple of simple modications. Integer
values can be used in the evaluation of the objective function
and constraints, while it works internally with continuous
oating points.
According to the literature, getting the integer values for
evaluating the objective function and constraints can be done
in two ways (1)by rounding the continuous variables []
to the nearest integers and (2) by truncating the values to
integers [].
In the present study, rounding o method is used because
it has an equal probability to choose between the nearest
lower and the nearest upper integer values. For example, if the
continuous variable has a value of . in one case and . in
the second case, then rounding o the digits takes the nearest
higher integer of  in the rst case and the nearest lower
integer of  in the second case. Truncation, on the other hand,
takes the value of  in both of the cases since it always takes
the nearest lower integer value. us, it can be seen that the
former method is unbiased and is therefore more reasonable.
Binary variables are also handled in the same fashion
as that for integers except that in this case the bounds are
restricted between  and .
4.2. Handling of Constraints. For constraint handling, the
following methodology is used [].
(i) Between two feasible solutions, the one with the best
objective function value is preferred.
(ii)Ifonesolutionisinfeasibleandtheothersolution
is feasible, the feasible solution is preferred without
considering the cost of the objective function.
(iii) Between two infeasible solutions, the solution corre-
sponding to the lowest sum of constraint violation is
preferred regardless of the objective function value.
Besides, following the previous three rules, equality con-
straints were transformed into inequations as explained. Sup-
pose that 𝑘()=0,=1,2,...,, are equality constraints
then these are transformed to inequalities using a tolerance
value =10−04 as |𝑘()| ≤ 0for all =1,...,.
4.3. Control Parameter Settings. Fine tuning of SDE param-
eters was done to obtain the appropriate value of control
parameters for solving the TLP. A series of experiments were
conducted, and it was observed that a smaller crossover rate
(<.) gave good results for TLP. In this study, the value of Cr
is therefore taken as .. Scaling factor =0.5and population
size NP = 100. Considering the complexity of the problem,
thenumberoffunctionevaluations(NFE)waskeptquite
high as  ×5. Finally, a reasonable accuracy of −04 was
taken to analyze the performance of SDE. e algorithm is
executed  times. In order to demonstrate the eciency of
SDE, the results are also compared with GMIN-BB [],
ILXPSO [].
Mathematical Problems in Engineering
T : Problems taken in this study.
Problem  Problem  Problem  Problem 
Product Wid. Qty. Product Wid. Qty. Product Wid. Qty. Product Wid. Qty.
        
        
       
        
 

T : Parameters of the problems.
Problem max 
𝑗𝑗max 𝑗 = 1,...,
  . [0,30]4∩4
  . [0,15]×[0,12]×[0,9]×[0,6]4
  . [0,15]×[0,12]×[0,9]×[0,6]×[0,6]5
  . [0,15]×[0,12]×[0,8]×[0,7]×[0,4]×[0,2]6
T : Results for the trim loss problem .
Algorithm Fitness  
GMIN-BB .
1
1
1
0
1
3
2
0
1010
2000
0530
2010
ILXPSO .
1
1
1
0
14
3
2
0
1010
2000
0530
2010
SDE .
1
1
1
0
9
7
3
0
1020
2110
0300
2120
4.4. Numerical Results. To evalu a t e t he per f o r mance of S D E ,
four hypothetical cases of the problem described in Section 
have been taken. e problems specication and problem
parameters are given in Tables and . In order to minimize
the eect of the stochastic nature of the algorithm, each
problem is executed  times taking dierent random seeds,
and the average of tness values at the best solutions through-
out the optimization run is recorded. An Intel Dual Core
personal computer with GB RAM is used for experiment.
e experimental results in terms of best tness value as
well as best solution are given in Tables ,and the cutting
patterns are given in Tables ,,and .FromTables
and , we observed that all the algorithms provided the same
objective value but dierent solutions. However, from Tables
and it is clear that our algorithm gives the same objective
value as GMIN-BB algorithm which is also the optimal
value, but the ILXPSO, a PSO variant used for solving this
problem, does not achieve the global optimum. In order to
T : Results for the trim loss problem .
Algorithm Fitness  
GMIN-BB .
1
0
0
0
11
0
0
0
1000
1000
2000
1000
ILXPSO .
1
1
1
0
5
2
1
0
1210
1020
2100
1220
SDE .
1
1
1
0
4
3
1
0
1130
0210
3000
1210
T : Results for the trim loss problem .
Algorithm Fitness  
GMIN-BB .
1
0
0
0
0
15
0
0
0
0
10000
10000
10000
10000
10000
ILXPSO .
1
1
1
1
1
3
2
2
2
1
23000
10101
00234
00120
22000
SDE .
1
1
0
0
0
6
4
0
0
0
20000
10000
04000
10000
11000
Mathematical Problems in Engineering
T : Results for the trim loss problem .
Algorithm Fitness  
GMIN-BB .
1
0
0
0
0
0
8
7
0
0
0
0
100000
200000
020000
010000
020000
100000
ILXPSO .
1
1
0
0
0
0
9
7
0
0
0
0
100000
110000
020000
100000
020000
200000
SDE .
1
1
0
0
0
0
8
7
0
0
0
0
100000
200000
020000
010000
020000
200000
T : Solution results for problem .
Cutting
pattern no. Cutting pattern generated Trim loss
(290×1)+(315×2)+(455×2) 
(315×1)+(350×3)+(455×1) 
(290×2)+(315×1)+(455×2) 
Total trim loss 
T : Solution results for problem .
Cutting
pattern no. Cutting pattern generated Trim loss
(330×1)+(385×3)+(415×1)
(330×1)+(360×2)+(415×2) 
(330×3)+(360×1)+(415×1) 
Total trim loss 
T : Solution results for problem .
Cutting
pattern no. Cutting pattern generated Trim
loss
(330×2)+(360×1)+(415×1)+(435×1) 
(370×4)+(435×1) 
Total trim loss 
satisfy the order number , the total trim loss in terms of
widthcomputedbyILXPSOis,whilebySDE,itis.
So our algorithmobtained the optimal solution for all the
four cases while ILXPSO obtained it for only two problems.
Tables ,,,and give the optimal cutting pattern
T : Solution results for problem .
Cutting pattern
no. Cutting pattern generated Trim loss
(330×1)+(360×2)+(530×2) 
(380×2)+(430×1)+(490×2) 
Total trim loss 
corresponding to the optimal solution obtained by SDE, and
the last column of these tables gives the trim loss in terms of
width. From a close observation of these tables it can be said
that the trim loss is much less. ere may exist more than one
optimal solution of an optimization problem which are called
alternate optimal solutions.
AlternatesolutionsobtainedbytheSDEarelistedin
Tables  and  for the problems  and , respectively. For the
alternatesolutions,objectivevalueisthesamewhilecutting
pattern is dierent which is obvious from Tables  and .
Other results which consist of the best and the worst results,
standard deviation (Std.), and average values of the obtained
results for all problems are recorded in Tabl e  . Additionally,
the computational times, the number of function evaluation
(NFE), and success rate (SR) are also included in Table .e
success rate in the last column of Ta b l e shows the reliability
of the algorithm for that particular problem. Also variance is
much less which shows the robustness of the scheme.
e convergence graphs of SDE, illustrating the best
tness versus number of function evaluations are given in
Figures and . Furthermore, the convergence graphs of
constraint violation are also illustrated in the same gures on
the secondary axis.
Comparison of SDE with another algorithm ILXPSO on
the basis of NFE and SR is given in Table .esuccessrate
of SDE for problems – is , , , and , respectively
while in case of ILXPSO it is , , ,  respectively. It
means that SDE is more reliable than ILXPSO. From Tab l e  ,
it is clear that the performance of SDE is better than that of
other algorithms in both criteria. CPU absorbance time of
thealgorithmsisnotcomparedhere,sincetheywereimple-
mented in completely dierent computational environments,
while it is recorded for SDE in Ta b l e  .
5. Conclusions
In this paper, the performance of SDE is analyzed on a real-
lifeproblemoftrimlossorTLP,arisingfrequentlyinpaper
industries. TLP is especially suited to investigate the eciency
of an optimization algorithm like that of SDE because of
its complex mathematical model which is nonlinear and
nonconvex and contains integer as well as binary variables.
Also, it has several constraints associated with it. Conclusions
that can be drawn at the end of this study can be summarized
as follows.
(i) SDE can be easily modied for solving the problems
having integer or/and binary restrictions imposed on
it.
Mathematical Problems in Engineering
T : Alternate optimal solutions of problem .
S. no. 12311 12 13 21 22 23 31 32 33 41 42 43

   ———    
 
—   ——   
  — —— — 
T : Alternate optimal solutions of problem .
S. no. 12311 12 13 21 22 23 31 32 33 41 42 43
 

T : Best, worst, mean tness, standard deviation, average NFE, time, and success rate for all problems.
Pro. Best Worst Mean Std. Average NFE Time % SR
.   . .3.5707101  . 
. . . 1.7763615  . 
. . . 4.3301301  . 
. . . 4.9749401  . 
0
10
20
30
40
50
60
70
80
90
100
0
500
1000
1500
2000
2500
3000
3500
4000
0 5000 10000 15000 20000 25000
NFE
Fitness
Const. violation
Const. violation
Fitness
F : Plot of tness and constraint violation versus NFE for
problem .
0
5
10
15
20
25
30
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 10000 20000 30000 40000
Fitness
Const. violation
NFE
Const. violation
Fitness
F : Plot of tness and constraint violation versus NFE for
problem .
T : Comparison of SDE and ILXPSO on average NFE and suc-
cess rate for all problems.
Pro. NFE % SR
SDE ILXPSO SDE ILXPSO
 
   
   
   
(ii) SDE can deal eciently with nonlinear/nonconvex
optimization problems subject to several constraints.
is is important in real-life scenarios, where usually
the problems are complex in nature.
(iii) SDE outperformed some of the contemporary opti-
mization algorithms in terms of solution quality as
well as convergence rate.
Acknowledgments
is research was supported by MSIP, Korea, under ITRC
NIPA--(H--) and also was supported by the
National Research Foundation of Korea (NRF) Grant funded
by the Korea Government (no. --).
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