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Mathematical Problems in Engineering

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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 dierential 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

dierent 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

eciently satisfy the customers while minimizing the wastage

due to trim loss. e industries have to maintain an ecient

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 eciently. 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 classication for cutting and packing

problems is introduced in Dyckho []. Dyckho developed

a special classication 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 dened 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 dened 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 dierent 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 ecient 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 oen 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, eorts 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 dierent formulations of CSP

and the dierent 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. Dierent 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 dierential evolution

(SDE), an improved version of dierential evolution (DE)

[,]wasappliedtosolvenumericalbenchmarktest

functions, where the results clearly indicated the competence

of the algorithm. For the present study, SDE has been suitably

modied, 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 briey 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 dierential 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 rand𝑗≤Cr ∨=,

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

dierent from DE. Aer 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 specied 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 specications. In this paper, the

mathematical formulation of TLP suggested by Adjiman et

al. []andYenetal.[] is taken into consideration. e

problem is dened as follows.

It is assumed that a paper roll of width max is to be cut in

dierent sizes to satisfy the customer’s demands.

e order specications are taken as follows.

(i) 𝑖rolls of order with a width 𝑖are to be produced,

where = 1,...,indicates dierent 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,...,; dierent 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 identied 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 coecient 𝑗.etrim

loss problem may now be dened 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=1𝑗≥max 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:

𝑗≤𝑁

𝑖=1𝑖𝑗 ≤max 𝑗, =1,...,. ()

ere must be at least one pattern aer 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 modications. 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 eciency 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

𝑗𝑗max 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 specication and problem

parameters are given in Tables and . In order to minimize

the eect of the stochastic nature of the algorithm, each

problem is executed times taking dierent 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 dierent 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 dierent 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 dierent 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 eciency

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 modied 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 eciently 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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