Benefit cost ratio using ivhfs

fuzzy application

1. Benefit / cost ratio analysis using interval-valued hesitant fuzzy sets Aydin S . Mamaghani Israa Y. Ismail, Selçuk Başarıcı

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

3. 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.

4. 1 . Benefit Cost Ratio Analysis 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?

5. 1 . Benefit Cost Ratio Analysis 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.

6. 1. Benefit Cost Ratio Analysis 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.

7. 1. Benefit Cost Ratio Analysis 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

8. 1. Benefit Cost Ratio Analysis Steps of BCR Analysis Determine the benefits, costs, and the resulting B/C ratio for each countermeasure. List countermeasures with a B/C ratio greater than 1.0 in order of increasing cost. 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. Continue in order of increasing costs to calculate the incremental B/C ratio for each countermeasure compared to the last-picked countermeasure. Stop when the incremental B/C ratio (disregarding negative ratios) is less than

9. 1. Benefit Cost Ratio Analysis 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

10. 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)

11. 2.Fuzzy Benefit Cost Ratio Analysis Differences of both projects ’ benefits and costs are described and calculated

12. 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 .

13. 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

14. 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 )

15. 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 {< C ij , h E ( C ij )>} Expected Net Annual Benefit {< A ij , h E ( A ij )>} Interest rate {< r ij , h E ( r ij )>}

16. 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 (A 1 ) and the challenger (A 2 ) 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.

17. 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:

18. 3. BCR using IVHFSs Step 3.1: Calculate the term for the 1 st alternative Calculate the formula for all combinations of possible values of A 1 and r 1. 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 {< , h E ( A j )∩ h E ( r j )>} where h E ( A j )∩ h E ( r j ) means having the minimum lower and upper values for all intervals’ combinations. The number of possible outputs = K A1 *K r1 where K A and K r is the number of possible inputs to A 1 and r 1 ; respectively

19. 3. BCR using IVHFSs Step 3.2: Calculate the term for the 2 nd Alternative Calculate the formula for all combinations of possible values of A 2 and r 2. For the membership intervals, apply the extension principle similar to the procedure in step2.1. The output of this step is {< , h E ( A j )∩ h E ( r j )>} The number of possible outputs = K A2 *K r2 where K A and K r is the number of possible inputs to A 2 and r 2 ; respectively

20. 3. BCR using IVHFSs Step 3.3: Calculate the term Calculate the formula for all combinations of possible values of and . The output of this step is {< , h E ( )∩ h E ( )>} The number of possible outputs for = ( K A1 *K r1 )*( K A2 *K r2 ). Step 3.4: Calculate the term Similarly we calculate for all possible combinations of C 1 and C 2 The output of this step is {< , h E ( C 1 )∩ h E ( C 2 )>} The number of possible outputs is K C1 *K C2

21. 3. BCR using IVHFSs Step 3.5: Calculate the term Calculate the formula for all combinations of possible values of and The output of this step is {< , h E ( )∩ h E ( )>} The number of possible outputs for = ( K A1 *K r1 )*( K A2 *K r2 )*( K C1 *K C2 )

22. 3. BCR using IVHFSs Step 4 Decide which Alternative to select based on the following rules: If all possible values of = (B 2 -B 1 )/(C 2 -C 1 ) are greater than 1 ; A 2 (the challenger) is selected. If all possible values of = (B 2 -B 1 )/(C 2 -C 1 ) are less than 1 ; A 1 (the defender) is selected. If some possible values of are > 1 and some are < 1, then, go to step 4.1

23. 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: A 1 - Supportive combinations (with >1) A 2 - 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.

24. 3. BCR using IVHFSs Compare the global score for A 1 -supportive and A 2 -supportive classes using the definition: Select the Alternative with the higher score of possibility

25. 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)

26. See Excel For computational Analysis