(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 6, No. 3, 2009
Application of a Fuzzy Programming Technique to
Production Planning in the Textile Industry
J. F. Webb
*I. Elamvazuthi , T. Ganesan, P. Vasant
Swinburne University of Technology Sarawak Campus,
Kuching, Sarawak, Malaysia
Universiti Technologi PETRONAS
Tronoh, Malaysia
*
analysis of real-world problems [3]. The period of development
of fuzzy theory from 1965 to 1977, is often referred to as the
academic phase. The outcome was a rather small number of
publications of a predominantly theoretical nature by a few
contributors, mainly from the academic community. At this
time, not much work in the area of fuzzy decision making was
reported. The period from 1978 to 1988, has been called the
transformation phase during which significant advances in
fuzzy set theory were made and some real-life problems were
solved. In this period, some important principles in fuzzy set
theory and its applications were established. However, work on
fuzzy decision making was not very active, in the area of
engineering applications. Some earlier work on fuzzy decision
making can be found in [4] and [5]. From 1989 to the present
work on fuzzy techniques has boomed . In this period, many
problems concerning applications in industry and business have
been tackled successfully. In the early 1990s, fuzzy techniques
were used to aid the solution of some soft computing problems.
The aim of soft computing is to exploit, whenever possible, the
tolerance for imprecision and uncertainty in order to achieve
computational tractability, robustness, and low cost, by
methods that produce approximate but acceptable solutions to
complex problems which often have no precise solution.
Abstract—Many engineering optimization problems can be
considered as linear programming problems where all or some of
the parameters involved are linguistic in nature. These can only
be quantified using fuzzy sets. The aim of this paper is to solve a
fuzzy linear programming problem in which the parameters
involved are fuzzy quantities with logistic membership functions.
To explore the applicability of the method a numerical example is
considered to determine the monthly production planning quotas
and profit of a home-textile group.
Keywords: fuzzy set theory, fuzzy linear programming, logistic
membership function, decision making
I.
INTRODUCTION
Many problems in science and engineering have been
considered from the point of view optimization. As the
environment is much influenced by the disturbance of social
and economic factors, the optimization approach is not always
the best. This is because, under such turbulent conditions, many
problems are ill-defined. Therefore, a degree-of-satisfaction
approach may be better than optimization. Here, we discuss
how to deal with decision making problems that are described
by fuzzy linear programming (FLP) models and formulated
with elements of imprecision and uncertainty. More precisely,
we will study FLP models in which the parameters are known
only partially to some degree of precision.
Currently, fuzzy techniques are often applied in the field of
decision making. Fuzzy methods have been developed in
virtually all branches of decision making, including multiobjective, multi-person, and multi-stage decision making [6].
Apart from this, other research work connected to fuzzy
decision making includes applications of fuzzy theory in
management, business and operational research [7]. Some
representative publications can be found in [8], [9], [10], [11]
and [12].
Even though the information is incomplete, the model
builder is able to provide realistic intervals for the parameters
in these FLP models. We will demonstrate that the modeling
complications can be handled with the help of some results
which have been developed in fuzzy set theory. The FLP
problem which we will be considering in this work is to find
ways to handle fuzziness in the parameters. We will develop a
FLP model in which the parameters are known with only some
degree of precision. We will also show that the model can be
parameterized in such a way that a satisfactory solution
becomes a function of the membership values. The FLP model
derived in this way is flexible and easy to handle
computationally [1].
Decision making is an important and much studied
application of mathematical methods in various fields of
human activity. In real-world situations, decisions are nearly
always made on the basis of information which, at least in
part, is fuzzy in nature. In some cases fuzzy information is
used as an approximation to more precise information. This
form of approximation can be convenient and sufficient for
making good enough decisions in some situations. In other
cases, fuzzy information is the only form of information
available.
The first and most meaningful impetus towards the
mathematical formalization of fuzziness was pioneered by
Zadeh [2]. Its further development is in progress, with
numerous attempts being made to explore the ability of fuzzy
set theory to become a useful tool for adequate mathematical
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The first step in mathematically tackling a practical
decision-making problem consists of formulating a suitable
mathematical model of a system or situation. If we intend to
make reasonably adequate mathematical models of situations
that help practicing decision makers in searching for rational
decisions, we should be able to introduce fuzziness into our
models and to suggest means of processing fuzzy information.
In this paper a methodology to solve an FLP problem by
using a logistic membership function is considered. The rest of
the paper is organized as follows. In section 2, the basic fuzzy
model is defined and this is followed by a numerical example
in section 3. Section 4 provides the results and discussion, and
finally, concluding remarks are made in section 5.
II.
μa%ij
III.
THE MODEL
Maximize Cx
Ax ≤ b, x ≥ 0.
(1)
in which the components of a 1×n vector C, an m×n matrix A
and an n×1 vector b are all crisp parameters and x is an ndimensional decision variable vector.
The system (1) may be redefined in a fuzzy environment
with the following more elaborate structure:
n
Maximize
if aija ≤ aij ≤ aijb
(4)
if aij ≥ aijb
NUMERICAL EXAMPLE
TABLE I. REQUIRED PROCESS TIME FOR SHEET, PILLOW CASE
AND OF A QUILT [14]
∑cjxj
~
j =1
Departments
Subject to
n
if aij ≤ aija
In this example the profit for a unit of sheet sales is around
1.05 Euro; a unit of pillow case sales is around 0.3 Euro and a
unit of quilt sales is around 1.8 Euro. The firm concerned
would like to sell approximately 25.000 sheet units, 40.000
pillow case units and 10.000 units quilt units. The monthly
working capacity and required process time for the production
of sheets, pillow cases and quilts are given in Table 1 [14].
In view of this, let us determine monthly production
planning details and profit for a home-textile group. X1
presents the quantity of sheets that will be produced, X2
presents the quantity of pillow cases and X3 presents the
quantity of quilts. The profit figures with logistic membership
functions as given in Table I.
A conventional linear programming problem is defined by
Subject to
⎧
⎪
⎪1
⎪⎪
B
=⎨
⎛ aij − aija ⎞
α⎜ b a ⎟
⎪
⎜
⎟
⎪ 1 + Ce ⎝ aij − aij ⎠
⎪
⎪⎩ 0
∑ aij x j ≤~ bi ,
~
j =1
i = 1,2 L m
Cutting
Sewing
Pleating
Packaging
(2)
~ a b
~ a b
~
~
All fuzzy data c j ≡ S (c j , c j ) and aij ≡ S (aij , aij ) are
fuzzy variables with the following logistic membership
functions [13],
⎧
⎪
⎪1
⎪
B
⎪
μc% j = ⎨
⎛ c j −caj ⎞
⎪
α⎜ b a ⎟
⎜ c j −c j ⎟
⎪
⎝
⎠
⎪ 1+ Ce
⎪0
⎩
Required unit time(hour)
Sheet
Pillow
case
Quilt
Working
hours per
month
0.0033
0.056
0.0067
0.01
0.001
0.025
0.004
0.01
0.0033
0.1
0.017
0.01
208
4368
520
780
If we consider, around 1.05 ≡ S% (1.02,1.08) , around 0.3 ≡
S% (0.2, 0.4) , and around 1.8 ≡ S% (1.7, 2.0) , then, the
mathematical model of the above problem with fuzzy
objective coefficients can be described as follows.
if c j ≤ caj
if caj ≤ c j ≤ cbj
Maximize
~
~
~
S (1.02,1.08) x1+ S (0.2,0.4) x 2 + S (1.7, 2.0) x 3
(3)
subject to
0 . 033 x1 + 0 . 01 x 2 + 0 .0033 x 3 ≤ 208 ;
if c j ≥ cbj
0 . 056 x1 + 0 .25 x 2 + 0 .1 x 3 ≤ 4368 ;
0 . 0067 x1 + 0 .04 x 2 + 0 .17 x 3 ≤ 520 ;
0 . 1 x1 + 0 . 1 x 2 + 0 .01 x 3 ≤ 780 ;
x1 ≥ 25000 ;
x 2 ≥ 40000 ;
x 3 ≥ 10000 ;
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(5)
and we set
B = 1, C = .001, ε = 0.2 and d = 13.8 [15].
The aspiration of the objective function is calculated by
solving the following:
Maximize
1.08 x 1 +0.4 x2 + 2.0 x3
subject to
.0033x1 + .001x2 + .0033 x3 ≤ 208;
.056 x1 + .025 x2 + .1x3 ≤ 4368;
.0067 x1 + .004 x2 + .017 x3 ≤ 520;
(6)
.01x1 + .01x2 + .01x3 ≤ 780;
x1 ≥ 25000;
Figure 1. 3D plot for iterations M=748.
x2 ≥ 40000;
x3 ≥ 10000;
Fig. 2 shows the 3D outcome for M = 749 iterations and
various alpha values with respect to G. The optimum values
for the objective function as per this figure are 86,691.8
(maximum) and 86,639.5 (minimum).
which gives the optimal value of the objective function as
67203.88 for x1 = 29126.21, x2 =35000.00 and x3 =10873.79
[15].
With the help of the program LINGO version 10.0 we
obtain the following results [15]:
λ = 0.5323011, x1 = 27766.99, x2 = 40000.00,
x3 = 10233.01, η = 0.4911863
Therefore, to achieve maximum profit the home-textile
group should plan for a monthly production of 27766.99 sheet
units, 40000 pillow case units and 102333.01 quilt units. This
plan gives an overall satisfaction of 0.5323011. The decision
making method may be improved further by adopting a
recursive iteration methodology.
IV.
Figure 2. 3D plot for iterations M = 749.
RESULTS AND DISCUSSION
Fig. 3 shows the 3D outcome for M = 750 iterations and
various alpha values with respect to G. The optimum values
for the objective function as per this figure are 86,576.2
(maximum) and 86,524.0 (minimum).
The numerical example is solved by using a recursive
method for various iterations. This was carried out using the
C++ programming language on a personal computer with a
dual core processor running at 2 GHz [16]–[17]. Fig. 1 shows
the 3D outcome of the iterations with M = 748 for various
alpha values with respect to the objective function G. The
values of α 1 and α 2 vary from 0 to 1. The optimum values
for the objective function as per Fig. 1 are 86,807.7
(maximum) and 86,755.4 (minimum).
Figure 3. 3D plot for iterations M=750.
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Fig. 4 shows the 3D outcome for M = 751 iterations and
various alpha values with respect to G. The optimum value for
the objective function as per this figure are 86,440.7
(maximum) and 86,408.0 (minimum).
Figure 6. Decision variable, X1 versus M iterations.
Figure 4. 3D plot for iterations M=751.
Fig. 5 shows the linear approximation for G with respect to
iterations 748 to 751. It can be seen that as the iterations are
increased, the values of the objective function decrease. The
percentage error is minimum at iteration, M = 748; however,
after that it increases until it peaks at M = 750; thereafter, the
percentage error decreases again to a level lower than that at
M = 748. This shows that the maximum number of iterations
that can be used for similar cases in the future can be limited
to M = 750.
Figure 7. Decision variable, X2 versus M iterations.
Figure 8. Decision variable, X2 versus M iterations.
Figure 5. Objective Function (G) versus iterations
Table II presents results that involve
Figs. 6, 7 and 8 show the linear approximation for the decision
variables x1, x2 and x3 with respect to the number of iterations.
It can be observed that x1, x2 and x3 decrease as the iterations
are increased from M = 748 to M = 751.
α1 , α 2
and
α3
with M
= 748 for G, x1, x2 and x3. Other results for M = 749 to 751 are
given in the appendix.
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From Table IV, the optimum value for the objective function
using the proposed method outweighs the results obtained in
[14] and [15]. It can be deduced that the recursive iteration
method proposed here is an efficient and effective way to
solve our example fuzzy problem of production planning in
the textile industry.
TABLE II ALPHA, OBJECTIVE FUNCTION
AND DECISION VARIABLES FOR M=748
α1
α2
α3
G
x1
x2
x3
1
1
*all
86755.4
33422.5
53475.9
1
0.5
all
86780.3
33422.5
53475.9
13369
13369
0.5
1
all
86767.5
33422.5
53475.9
13369
0.5
0.5
all
86792.4
33422.5
53475.9
13369
0.3333
1
all
86770.9
33422.5
53475.9
13369
0.3333
0.5
all
86795.9
33422.5
53475.9
13369
0.25
1
all
86772.5
33422.5
53475.9
13369
0.25
0.5
all
86797.5
33422.5
53475.9
13369
0.2
1
all
86773.5
33422.5
53475.9
13369
0.2
0.5
all
86798.4
33422.5
53475.9
13369
0.1667
1
all
86774.1
33422.5
53475.9
13369
0.1667
0.5
all
86799.0
33422.5
53475.9
13369
0.1429
1
all
86774.5
33422.5
53475.9
13369
0.1429
0.5
all
86799.4
33422.5
53475.9
13369
0.125
1
all
86774.8
33422.5
53475.9
13369
0.125
0.5
all
86799.8
33422.5
53475.9
13369
0.1111
1
all
86775.1
33422.5
53475.9
13369
0.1111
0.5
all
86800.0
33422.5
53475.9
13369
V.
This paper has discussed the use of fuzzy linear
programming for solving a production planning problem in the
textile industry. It can be concluded that the recursive method
introduced is a promising method for solving such problems.
The modified s-curve membership function provides various
uncertainty levels which are very useful in the decision
making process. In this paper, only a single s-curve
membership function was considered. In the future, various
other membership functions will be considered. Apart from
providing an optimum solution for the objective functions, the
proposed method ensures high productivity. In this regard,
there is a good opportunity for developing an interactive selforganized decision making method by using hybrid soft
computing techniques.
ACKNOWLEDGMENT
The authors would like to thank Universiti Teknologi
PETRONAS and Swinburne University of Technology
Sarawak Campus for supporting this work.
Note: *all∈ (0, 1)
Table III summarizes the result for M = 748 to 751 for x1, x2
and x3 with maximum and minimum values of G. The overall
maximum value for G is 86807.7 at M = 748 and the overall
minimum value is 86408.0 at M = 751.
REFERENCES
[1]
[2]
[3]
TABLE III SUMMARY OF ITERATIONS, DECISION VARIABLES
AND OBJECTIVE FUNCTION
M
x1
x2
x3
G (max)
G(min)
748
33422.5
53475.9
13369
86807.7
86755.4
749
33377.8
53404.5
13351.1
86691.8
86639.5
750
33333.3
53333.3
13333.3
86576.2
86524.0
[6]
751
33288.9
53262.3
13315.6
86440.7
86408.0
[7]
[4]
[5]
Table IV compares the best objective function and decision
variables x1, x2 and x3 of the proposed method with previous
work by other researchers.
[8]
[9]
[10]
TABLE IV COMPARATIVE ANALYSIS
[11]
The Best
Objective
Function
x1
x2
x3
Irfan [14]
64390.999
33825.16
40000.00
9374.760
Atanu [15]
66454.369
27766.99
40000.00
10233.01
Proposed Method
86807.700
33422.50
53475.90
13369.00
Method
CONCLUSION
Decision Variables
[12]
[13]
242
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Orlovsky, S. A. 1980. On Formalization Of A General Fuzzy
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Kickert, W.J. 1978. Fuzzy Theories on Decision-Making: Frontiers in
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Zimmermann, H.J. 1987. Fuzzy Sets, Decision Making, and Experts
Systems. Boston: Kluwer.
Tamiz, M. 1996. Multi-objective programming and goal programming:
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Zimmermann, H.J. 1991. Fuzzy Set Theory-and Its Applications, (2nd
rev. ed.). Boston: Kluwer.
Ross, T. J. 1995. Fuzzy Logic with Engineering Applications, New
York: McGraw- Hill.
Klir, G. J., and Yuan, B. 1995. Fuzzy Sets and Fuzzy Logic: Theory and
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Yager, R. R., Ovchinikov,S., Tong, R. M., Nguyen,H. T (eds.) 1987.
Fuzzy Sets and Applications-Selected Papers by L. A. Zadeh. New York
:John Wiley.
Zimmermman, H. J. 1985. Application of Fuzzy Set Theory To
Mathematical Programming. Information Sciences 36:25-58.
Dubois, D. and Prade, H.1980. Fuzzy Sets and Systems, Theory and
Applications, California: Academic Press Inc.
P. Vasant, A. Bhattacharya, B. Sarkar and S. K. Mukherjee, “Detection
of level of satisfaction and fuzziness patterns for MCDM model with
modified flexiable S-curve MF”, Applied Soft Computing, 7, 2007, pp.
1044-1054.
http://sites.google.com/site/ijcsis/
ISSN 1947-5500
[14] Irfan Ertugrul and Aysegül Tus, “Interactive fuzzy linear programming
and an application sample at a textile firm”, Fuzzy Optimization and
Decision Making, 6, 2007, pp. 29–49.
[15] Atanu, S., P. Vasant and Cvetko, J.A., Fuzzy Optimization with Robust
Logistic Membership Function: A Case Study In For Home Textile
Industry, Proceedings of the 17th World Congress, The International
Federation of Automatic Control, Seoul, Korea, July6-11, 2008, pp52625266.
[16] http://www.intel.com/products/processor/core2duo/index.htm
[17] http://www2.research.att.com/~bs/C++.html
0.1667
1
all
86542.7
33333.3
53333.3
13333.3
0.1667
0.5
all
86567.5
33333.3
53333.3
13333.3
0.1429
1
all
86543.1
33333.3
53333.3
13333.3
0.1429
0.5
all
86568
33333.3
53333.3
13333.3
0.125
1
all
86543.4
33333.3
53333.3
13333.3
0.125
0.5
all
86568.3
33333.3
53333.3
13333.3
0.1111
1
all
86543.7
33333.3
53333.3
13333.3
0.1111
0.5
all
86568.5
33333.3
53333.3
13333.3
Note: *all∈ (0, 1)
APPENDIX
TABLE V II ALPHA, OBJECTIVE FUNCTION
AND DECISION VARIABLES FOR M=751
TABLE V ALPHA, OBJECTIVE FUNCTION
AND DECISION VARIABLES FOR M=749
α1
α2
α3
1
1
1
0.5
0.5
1
G
x1
x2
α1
x3
α2
α3
1
1
G
x1
x2
x3
*all
86408.8
33288.9
53262.3
13315.6
*all
86639.5
33377.8
53404.5
13351.1
all
86664.4
33377.8
53404.5
13351.1
1
0.5
all
86433.6
33288.9
53262.3
13315.6
13351.1
0.5
1
all
86420.9
33288.9
53262.3
13315.6
0.5
all
86445.7
33288.9
53262.3
13315.6
13315.6
all
86651.7
33377.8
53404.5
0.5
0.5
all
86676.6
33377.8
53404.5
13351.1
0.5
0.3333
1
all
86655.1
33377.8
53404.5
13351.1
0.3333
1
all
86424.3
33288.9
53262.3
13351.1
0.3333
0.5
all
86449.1
33288.9
53262.3
13315.6
1
all
86425.9
33288.9
53262.3
13315.6
0.3333
0.5
all
86680.0
33377.8
53404.5
0.25
1
all
86656.7
33377.8
53404.5
13351.1
0.25
0.25
0.5
all
86681.6
33377.8
53404.5
13351.1
0.25
0.5
all
86450.7
33288.9
53262.3
13315.6
1
all
86426.9
33288.9
53262.3
13315.6
0.5
all
86451.7
33288.9
53262.3
13315.6
13315.6
0.2
1
all
86657.6
33377.8
53404.5
13351.1
0.2
0.2
0.5
all
86682.5
33377.8
53404.5
13351.1
0.2
0.1667
1
all
86658.2
33377.8
53404.5
13351.1
0.1667
1
all
86427.5
33288.9
53262.3
13351.1
0.1667
0.5
all
86452.3
33288.9
53262.3
13315.6
1
all
86427.9
33288.9
53262.3
13315.6
0.1667
0.5
all
86683.1
33377.8
53404.5
0.1429
1
all
86658.7
33377.8
53404.5
13351.1
0.1429
0.1429
0.5
all
86683.5
33377.8
53404.5
13351.1
0.1429
0.5
all
86452.7
33288.9
53262.3
13315.6
0.125
1
all
86659.0
33377.8
53404.5
13351.1
0.125
1
all
86428.2
33288.9
53262.3
13315.6
0.125
0.5
all
86683.9
33377.8
53404.5
13351.1
0.125
0.5
all
86453
33288.9
53262.3
13315.6
0.1111
1
all
86428.4
33288.9
53262.3
13315.6
0.1111
0.5
all
86453.3
33288.9
53262.3
13315.6
0.1111
1
all
86659.2
33377.8
53404.5
13351.1
0.1111
0.5
all
86684.4
33377.8
53404.5
13351.1
Note: *all∈ (0, 1)
Note: *all∈ (0, 1) , M= no. of iterations
AUTHOR PROFILES
TABLE VI ALPHA, OBJECTIVE FUNCTION
AND DECISION VARIABLES FOR M=750
α1
α2
α3
1
1
1
0.5
0.5
0.5
0.3333
0.3333
*all
G
x1
x2
I. Elamvazuthi is a lecturer in the Department of Electrical and Electronic
Engineering, Universiti Teknologi PETRONAS (UTP), Malaysia. His
research interests include Control Systems, Mechatronics and Robotics.
x3
86524
33333.3
53333.3
13333.3
all
86548.9
33333.3
53333.3
13333.3
1
all
86536.1
33333.3
53333.3
13333.3
0.5
all
86561
33333.3
53333.3
13333.3
1
all
86539.5
33333.3
53333.3
13333.3
0.5
all
86564.4
33333.3
53333.3
13333.3
0.25
1
all
86541.1
33333.3
53333.3
13333.3
0.25
0.5
all
86566
33333.3
53333.3
13333.3
0.2
1
all
86542.1
33333.3
53333.3
13333.3
0.2
0.5
all
86566.9
33333.3
53333.3
13333.3
T. Ganesan is currently a Graduate Assistant with the Department of
Mechanical Engineering, Universiti Teknologi PETRONAS (UTP), Malaysia,
pursuing a Masters Degree. He has a Bachelor’s Degree in Mechanical
Engineering from the same university. He specializes in Computational Fluid
Mechanics.
P. Vasant is a lecturer in the Department of Fundamental and Applied
Sciences, Universiti Teknologi PETRONAS (UTP), Malaysia. His research
interests are Soft Computing and Computational Intelligence.
J. F. Webb is a lecturer at Swinburne University of Technology, Sarawak
Campus, Kuching, Sarawak, Malaysia. He specializes in Computational
Methods, Nano-Physics and Ferroelectric Materials.
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ISSN 1947-5500