Benefit cost ratio using ivhfs

BENEFIT / COST RATIO ANALYSIS
USING INTERVAL-VALUED HESITANT
FUZZY SETS
Aydin S. Mamaghani
Israa Y. Ismail,
Selçuk Başarıcı

OUTLINE
1) Benefit Cost Ratio Analysis – Crisp Case
2) Fuzzy Benefit Cost Ratio Analysis (Using TFN)
3) A proposed Heuristic for Benefit Cost Ratio
Analysis using Interval Valued Hesitant Fuzzy
Sets (IVHFSs)

1. BENEFIT COST RATIO ANALYSIS
History
 Jules Dupuit, an engineer from France, first introduced
the concept of benefit cost ratio in 1848.
 Alfred Marshall, a British economist further enhanced
the formula that became the basis for benefit cost ratio.
However, the formalized development of it did not occur
until the Federal Navigation Act of 1936 was introduced.

 A benefit-cost analysis is a systematic evaluation of the
economic advantages (benefits) and disadvantages
(costs) of a set of investment alternatives.
 The analysis evaluates incremental differences between
the Base Case and the Alternative(s). In other words, a
benefit-cost analysis tries to answer the question: What
additional benefits will result if this Alternative is
undertaken, and what additional costs are needed to bring
it about?

 A benefit-cost analysis provides monetary measure of the
relative economic desirability of project alternatives, but
decision-makers often weigh the results against other
non-monetized effects and impacts of the project, such as
environmental effects.
 The B/C ratio for an alternative is calculated as
equivalent benefits divided by equivalent costs. When
evaluating multiple alternatives with differing lives, use
equivalent annual worth for equivalent benefits and
costs.

 One of two decision criteria should be used when
performing benefit-cost (B/C) ratio analysis, as follows:
Only one investment
alternative is under
consideration
Invest in the alternative if the B/C ratio is
greater than or equal to 1.0. Otherwise do
not invest in the alternative.
Two or more investment
alternatives are under
consideration
Perform incremental B/C analysis. At each
step, choose the higher cost alternative if the
incremental B/C ratio is greater than or equal
to 1.0. Otherwise choose the lower cost
alternative.

 The incremental B/C ratio method may be used to determine
whether extra increments of cost are justified for a particular
location or for considering improvements at two or more
locations. This method assumes that the relative merit of a
project is measured by its change in benefits and costs,
compared to the next lower-cost alternative

 Steps of BCR Analysis
1. Determine the benefits, costs, and the resulting B/C ratio for each
countermeasure.
2. List countermeasures with a B/C ratio greater than 1.0 in order of
increasing cost.
3. Calculate the incremental B/C ratio of the second lowest-cost
countermeasure compared to the lowest-cost countermeasure. Pick
the second lowest-cost countermeasure if this ratio is positive; else
pick the lowest-cost countermeasure.
4. Continue in order of increasing costs to calculate the incremental B/C
ratio for each countermeasure compared to the last-picked
countermeasure.
5. Stop when the incremental B/C ratio (disregarding negative ratios) is
less than

 Disadvantages
 The required data might be hard to quantify a priori;
 It disregards the problem of economic inequalities, i.e.,
one part of the population benefits at the expense of the
other part;
 It takes no notice to any qualitative information

2.FUZZY BENEFIT COST RATIO ANALYSIS
BCR Using TFN
 BCR=B/C
 B represents equivalent value of benefits
 C represents project’s net cost
B/C ratio which is greater than or equal to 1
indicates that the project is economically
advantageous
While calculating BCR, we can utilize either net
present value (NPV) or net equivalent uniform
annual value (NEUAV)

2.FUZZY BENEFIT COST RATIO ANALYSIS
 Differences of both projects’ benefits and costs are
described and calculated

FUZZY (TFN) BCR – NET PRESENT VALUE
Step 1 -
n: Crisp life cycle ; = (1,1,1)
r: Fuzzy interest rate
r(y) and l(y) : Right and Left side representations of fuzzy
interest rates
formula is used to deduct future value into
present value.

NET PRESENT VALUE
 Step 2 - Assign lower cost as the defender and the
next lowest as the challenger
 Step 3 - Determine incremental benefits and costs
between the challenger and the defender
 Step 4 - Calculate Fuzzy BCR as follows;
If fuzzy BCR is equal or greater than (1,1,1), alternative 2 is
prefered

NET EQUIVALENT UNIFORM ANNUAL
VALUE (NEUAV)
In case of regular annuity ;
A : Net annual benefit ; C : First cost ;
γ(n,r)=((1+r)ⁿ-1)/(1+r)ⁿr)

3. BCR USING IVHFSS
The proposed heuristic assumes a uniform Annual cash
flow for BCR analysis and compares alternatives through
the following steps.
 Step1:
For each alternative, estimate the possible values for the
problem parameters; with the associated membership degrees
in the form of IVHFS.
For j = 1 to m; m= number of alternatives, i = 1to k; k is the number of
possible values for alternative j ; we have:
 Investment Cost {<Cij, hE(Cij)>}
 Expected Net Annual Benefit {<Aij, hE(Aij)>}
 Interest rate {<rij, hE(rij)>}

3. BCR USING IVHFSS
 Step 2:
 Calculate B/C for each alternative and exclude alternatives with
B/C <1 (as will be described in the following steps)
 Assign the defender (A1) and the challenger (A2) as described
earlier.
 In Steps 3 and 4 , we describe the process of finding for
the purpose of comparing alternatives. Note that the same
procedure is followed to find B/C for each alternative (in step 2) to
check its feasibility before being considered in the comparison.

3. BCR USING IVHFSS
 Step 3
In this step, we aim to find the equation using the
previously described input.
where, , n = project life.
In the following we detail how to calculate each term in
the ratio formula using IVHFSs:

3. BCR USING IVHFSS
 Step 3.1: Calculate the term for the 1st alternative
 Calculate the formula for all combinations of possible values of A1
and r1.
 For the membership intervals, apply the extension principle; (i.e.,
take the intersection, minimum, of the membership intervals for A
and r, and if the same result is found by different combinations,
select the maximum.
 The output of this step is {< , hE(Aj)∩ hE(rj)>}
where hE(Aj)∩ hE(rj) means having the minimum lower and upper
values for all intervals’ combinations.
 The number of possible outputs = KA1*Kr1 where KA and Kr is the
number of possible inputs to A1 and r1; respectively

3. BCR USING IVHFSS
 Step 3.2: Calculate the term for the 2nd
Alternative
 Calculate the formula for all combinations of possible values of
A2 and r2.
 For the membership intervals, apply the extension principle
similar to the procedure in step2.1.
 The output of this step is {< , hE(Aj)∩ hE(rj)>}
 The number of possible outputs = KA2*Kr2 where KA and Kr is the
number of possible inputs to A2 and r2; respectively

3. BCR USING IVHFSS
 Step 3.3: Calculate the term
and .
 The output of this step is {< , hE( )∩ hE( )>}
 The number of possible outputs for = (KA1*Kr1 )*(KA2*Kr2 ).
 Similarly we calculate for all possible combinations of C1 and C2
 The output of this step is {< , hE( C1)∩ hE( C2)>}
 The number of possible outputs is KC1*KC2

3. BCR USING IVHFSS
and
 The output of this step is {< , hE( )∩ hE( )>}
 The number of possible outputs for
= (KA1*Kr1 )*(KA2*Kr2 )*(KC1*KC2)

3. BCR USING IVHFSS
 Step 4
Decide which Alternative to select based on the
following rules:
 If all possible values of = (B2-B1)/(C2-C1) are
greater than 1 ; A2 (the challenger) is selected.
 If all possible values of = (B2-B1)/(C2-C1) are less
than 1 ; A1 (the defender) is selected.
 If some possible values of are > 1 and some
are < 1, then, go to step 4.1

3. BCR USING IVHFSS
 Step 4.1:
 Calculate the score for the membership intervals for all
possible outputs from the relation ; where #h
is the number of elements (intervals) in the set
 Classify the output combinations into two classes based on
their ratio value:
 A1 - Supportive combinations (with >1)
 A2 - Supportive combinations (with <1)
 Find the global score for each class by taking the maximum
lower and maximum upper values of all elements under this
class.

3. BCR USING IVHFSS
 Compare the global score for A1-supportive and A2-
supportive classes using the definition:
 Select the Alternative with the higher score of possibility

EXAMPLE
Alternative 2
Alternative 1
Brazil
France
Factory Location
30 (Crisp value)
45 [0.3,0.5] [0.55,0.7]
40 [0.8,0.9]
Initial Investment
(C) ($m)
2.8[0.3, 0.4] [0.6,0.7]
2.2 [0.5,0.6] [0.7,0.8]
3.2 [0.2,0.4] [0.5,0.7]
3.6 [0.4,0.5] [0.7,0.8]
Expected Annual Net
Benefits (A) ($m)
2.8% [0.3, 0.6]
3.4% [0.4, 0.7] [0.7, 0.8]
3% [0.3,0.5]
3.5% [0.5,0.6] [0.7,0.8]
Interest Rate (r)
30 years
30 years
Project Life (n)

 See Excel For computational Analysis

Benefit cost ratio using ivhfs

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