FUZZY LOGIC APPLICATION FOR
EXTRUDERS REPLACEMENT PROBLEM
Edison Conde Perez dos Santos
LABFUZZY – PEP – COPPE/UFRJ, Federal University of Rio
de Janeiro, Brazil
E-mail: edison.conde@hotmail.com
Fabio Luiz Peres Krykhtine
LABFUZZY – PEP – COPPE/UFRJ, Federal University of Rio
de Janeiro, Brazil
E-mail: krykhtine@labfuzzy.coppe.ufrj.br
Carlos Alberto Nunes Cosenza
LABFUZZY – PEP – COPPE/UFRJ, Federal University of Rio
de Janeiro, Brazil
E-mail: cosenzacoppe@gmail.com
Armando Celestino Gonçalves Neto
LABFUZZY – PEP – COPPE/UFRJ, Federal University of Rio
de Janeiro, Brazil
E-mail: armando1964@gmail.com
Submission: 31/05/2016
Revision: 23/06/2016
Accept: 09/08/2016
ABSTRACT
In a scenario of uncertainty and imprecision,
before taking the replacement analysis, a manager needs to consider the
uncertain reality of a problem. In this scenario, the fuzzy logic makes an
excellent option. Therefore, it is necessary to make a decision based on the
fuzzy model. This study is based on the comparison of two methodologies used in
the problem of asset replacement. The study, thus, was based on a comparison
between two extruders for polypropylene yarn bibliopegy, comparing mainly the
costs involved in maintaining the equipment.
Keywords: Fuzzy set, Fuzzy Number,
Fuzzy Replacement Problem, Fuzzy ranking, Membership Function.
1. INTRODUCTION
The
replacement decision is of critical importance to the company, as it is usually
irreversible, i.e. not having
liquidity and undertaking a significant amount of resources. Hence, a hasty
decision to replace it could cause serious problems for the working capital in
the company.
Nowadays,
the importance of this question is increasing because obsolescence happens
incredibly fast as the technology increases becoming a “new” and old “useful”
equipment, because the market always occurs new possibilities more advantages
and productivity even less maintenance and diminishing the total cost of
operation. So the well-known financial deterministic ways to determine the
economic lives of equipment’s will fail in this context of plenty of
uncertainty and “vagueness”.
To
solve the “vagueness”, which is
embedded in this cost variable problem, you have to consider maintenance cost
(Y), interest rate (i), i.e. Minimal Attractiveness Rate of
Interest – MARR, initial Investment (C), capital recovery cost, among others.
Before describing this method, a common deterministic method review will be
done.
2. FUZZY LOGIC IN REPLACEMENT ANALYSIS
The
theory of fuzzy was designed by L.A. Zadeh in 1975, in order to provide a
mathematical tool for the treatment of “uncertain” or “vagueness” background
information. The fuzzy logic based on this theory was initially constructed
from the concepts already established in classical logic; operators are defined
similarly to the traditionally ones, and others have been introduced over time,
often eminently practical character needs.
The
fuzzy logic allowed us to deal with the “vagueness” problem in different areas.
In 1975, Zadeh created the fuzzy logic that can treat complex models with set
fuzzy logic. The fuzzy logic can be used to solve different kinds of problems.
However, even fuzzy systems, as they are posed now, can be described as shallow
models in the sense that they are primarily used in deductive reasoning.
In
this article use logic structures, fuzzy numbers () and membership function will be specified (μ). Then, it
will be used the fuzzy logic and arithmetics fuzzy will be used to calculate
the Equivalent Uniform Annual Worth Fuzzy Value (EUAW) and compare with
classical replacement analysis, the Equivalent Uniform Annual Cost (EUAC).
BISWAS
et al., in 2011, studied replacement analysis. He proposed a methodology for
the Equivalent Uniform Annual Worth Fuzzy Value (EUAW). It employed the Yager's
ranking index for maintenance cost, where Y() presents the representative value of the fuzzy number A.
Here, we consider for Equivalent Uniform in general:
EUA (x) = , equation 2.1
Now,
when we calculate the Equivalent Uniform Annual Worth Fuzzy Value (EUAW) then
the capital cost (), the scrap value (
), the maintenance cost (
) are fuzzy numbers and rate period n-years (a) is
a discrete variable.
3. DETERMINISTIC REPLACEMENT METHOD
As
stated by Thuessen & Fabrick, the equipment in operation is a denoted defensor
(d) and the alternative or alternatives are called challengers (D). Some remarkable notes are the following: for
“D” and “d” differing so much in operational working characteristics for “D”
requires an high Initial investment value. Notwithstanding the maintenance cost
is very low compared to the defensor’s one; the challenger’s cash flow is very
different from the cash flow of the old asset; the recovery cost of capital of
d is decreasing while its operational cost is high and always increasing; the
salvage value of d, in most cases, is negligible or has a low amount.
In
deterministic analysis (EUAC), besides obsolescence, tear and wear (physical
degradation) is the other main reason for replacement. This is a very
conflicting question, mainly when the firm is dealing with insurers/reinsurers
prationioneers in order to set up the depreciation with an absolute accuracy of
the life of certain equipment or infrastructure building.
Although
engineering specific methods with higher complicated mathematical development
for some specific areas help accountants and financial professionals to
determine the depreciation of some equipment, there is no universally accepted
depreciation methods.
Coming
back to the economic aspects, in an eventual replacement, it should be taken
into account in the Initial Investment the following costs denoted as first
costs needed for the equipment to become operational: freight; package;
insurance; foundations; special connections and pipelines; assembling (if this
cost is not included in the original budget); other unexpected costs that are
not object of insurance cover.
In
regard to the defensor, the unit to be replaced, the remotion cost should
include the disassembling cost, foundations remotion costs, closing electric connections
costs and other similar costs. These costs must be deduced from the salvage
value of the equipment, generating the Net Salvage Value (NSV). Even if the
former amount was negative, it should be respected in depreciation
calculations.
3.1.
Deterministic replacement method considering sunk cost and equal lives
for equipments
According
to Sharpe and Chan Park, one of the main mistakes in financial engineering is
to disregard that cost occurred in the past with the defensor, at the time of
replacement, a decision couldn't have been. They are sunk costs. At a first
glance, this concept is not easy to understand for non-financial professionals.
To justify this statement, the Outsider View Point (OVP) consideration is
introduced. From an outsider viewpoint, at the moment of replacement decision,
all costs already incurred with defensor equipment including the acquiring cost
of the same, do not need to be taken into account. All these costs incurred are
sunk costs. The only cost, at the moment of replacement decision, if there is
any, is its net salvage value (NSV) or even an offering proposal (OP) that the
seller of the new equipment makes for the old asset.
Another
remarkable point is to establish the Time Study Period (TSP) which means the
time that the cash flows of the alternatives are being compared. In order to
clarify all these concepts, an example that deals with an extruder’s
replacement is proposed below, whole data are the following:
Table 1: Data for the replacement
in Defensor (d).
|
Defensor (d) |
|
Data of Acquisition
01/01/2008 |
|
Initial Investment
US$ 400,000.00 |
|
Estimated life
10 years |
|
Total Operational cost
US$ 1,500.00 |
|
NSV at the end of Life
US$ 40,000.00 |
|
|
Table 2: Data for the replacement
in Challenger (D).
|
Challenger (D): |
|
Data of
Acquisition
01/01/2013 |
|
Initial
Investment
US$ 500,000.00 |
|
Estimated
life
10 years |
|
Total
Operational cost
US$ 1,000.00 |
|
NSV at the end
of Life
US$ 50,000.00 |
Note: consider MARR = 10% per year.
According
to the data, the TSP = 10 years. It will be used to solve the Equivalent
Uniform Annual Cost (EUAC) method in BISWAS et al., 2011. So EUAC value for the
defender U$ 195 M and a EUAC for
challenger equal to U$ 179 M.
4. FUZZY REPLACEMENT METHOD
The
method of analysis of fuzzy equipment replacement is one of the most important
and most common types of alternative comparisons found in practice. In a
replacement analysis of fuzzy, one of the feasible alternatives involves the
option for continued equipment operation.
There
are different reasons which can contribute to an equipment replacement choice.
Initially, the current asset (defender) may have a number of deficiencies
including high set-up cost, excessive maintenance, declining production
efficiency, heavy energy consumption, and physical impairment. And finally,
potential replacement assets (challengers) may take advantage of new technology
and be easily set up, maintained at low cost, high in output, energy efficient,
and possess increased capabilities, perhaps at a vastly reduced cost (CENGIS;
MURAT, 1998).
As
it has been said, a great contribution of fuzzy set theory was its capability
of representing “vagueness”. The theory also allowed mathematical operators and
a programming to apply to the fuzzy domain. A fuzzy set is a class of objects with
a continuum of grades of membership. Such a set is characterized by a
membership function, which assigns to each object a grade of membership ranging
between zero and one. Picture 1 demonstrates a triangular fuzzy number (TFN) : Fuzzy set
theory has also been applied to many engineering economic areas. Buckley (1987)
developed fuzzy mathematics for compound interest problems. The theory can
determine the fuzzy present value and fuzzy future value of fuzzy cash amounts,
using fuzzy interest rates. Chiu and Park (1994) developed comprehensive left
and right side representation of fuzzy finance (THOGA, MURAT, CENGIZ, 2005).
Figure 1: Left and Right representation of a TFN,
The equivalent uniform annual worth
(EUAW) understands that all income and expenses (irregular and uniform) must be
converted into an equivalent uniform annual value, which is the same in each
period.
The
methodology proposed by BISWAS et al., in 2011, uses the concepts seen before.
However, the fuzzy replacement method will be calculated with fuzzy theory
sets. For this, the Equivalent Uniform Annual Worth Fuzzy Value (EUAW) is obtained
from the result of the fuzzy operations in equation 2.1.
Besides that, it is important to
point out that all costs incurred between 01/01/2006 to 31/12/2016 are sunk
cost and do not matter in a replacement decision. As stated by Yager the
calculus can be performed by the formula below:
Y() =
, equation 3.1
In this case, it must be considered
TSP, optic which constrains the method when the lives of two equipment (or
project) are different and one (life) is shorter than the other. The
replacement problem will be one time in the future where cash flows are
equivalent dates.
Regarding the time horizon, the
decision maker can consider all the future or an interval time where the cash
flows are equivalents. As it is almost impossible to predict all the facts that
will affect the variables that constitute the cash flow in the future, the Net
Present Value can be considered the result of the selection between within the
time frame.
In order to make it clear, an
example of two extruders according will be show, according table 1:
Table 3: Data for the replacement
example with different lives.
|
Extruder
Defensor (d) Challenger (D) |
|
NPV (US$) NA NA |
|
Initial Investment US$ 400,000.00 US$ 500,000.00 |
|
Annual Cost of Operational US$1,500,000 US$1,000,000 |
|
Useful Life (years) 10 10 |
|
NSV at the end of Life US$ 40,000.00 US$ 50,000.00 |
Notes:
1.
NA – not appliacable;
2.
Considering MARR = 10% per year
Choosing a TSP = 10 years, it is
possible to calculate NPV (equipament) by two ways, which will be shown below.
Before introducing the calculus,
picture 2 embracing the two replacement possibilities will be presented:
(a)
Recognizing the reminiscent value
“d” (i.e the time unused between years 08 to 18)
In this way of calculus, the EUAW of
“d” will be distributed for its 10 years of useful life. Next, the Yager's
ranking index for maintenance cost, Y() for defensor (d):
Figure 2: the maintenance cost value of “d” for its 10 years and Y() = 0,4 (“low”).
Therefore, the Equivalent Uniform
Annual Worth Fuzzy Value (EUAW) for defensor (d):
Figure 3: the cost value (EUAW) of “d” for its 10 years equal to 1,0.
In this way of calculus, it will be
distributed the EUAW of “D” for its 10 years of useful life. Next, the Yager's
ranking index for maintenance cost, Y() for challenger (D):
Figure 4: the maintenance cost value of “D” for its 10 years and Y() = 0,2 (“very low”).
Therefore, the Equivalent Uniform
Annual Worth Fuzzy Value (EUAW) for challenger (D):
Figure 5: the cost value (EUAW) of “D” for its 10 years equal to 0,8.
(a) Recognizing
the reminiscent value “d” (i.e. the time unused between years 13 to
23).
With this, it can be seen that the
value of the Equivalent Uniform Annual Worth Fuzzy Value (EUAW) for challenger
(D) is equal to 0,8 (“high”), less than the value EUAW for defensor (d), which
is “very high”, equal to 1,0; suggesting replacement of the extruder.
5. CONCLUSIONS
In this work, two methodologies were
considered to deal with the problem of asset replacement. The first did not
consider the vagueness of the capital costs, residual value, maintenance or
running costs. In the second method we consider the imprecise nature of this
information, so we started using and enjoying the fuzzy logic.
It is more realistic and closer to
our daily life situation. Here, we use the Yager ranking method to handle this
type of EUAW. This method moves the EUAC for Crisp Replacement Problem with its
corresponding classification rates. This method is simple and easy to apply for
the practical resolution Fuzzy Replacement Problem.
A numerical example was used to show
the simplicity and effectiveness of the proposed method. It also shows that the
appropriate decision for replacement of Fuzzy Replacement Problem can be made
easily. We hope that the proposed method can be used for future studies in the
Fuzzy Replacement Problem when the value of money depends on time.
6. REFERENCES
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