A Real Options Approach For Software Development

J. of Mult.-Valued Logic & Soft Computing, Vol. 18, pp. 541–560 Reprints available directly from the publisher Photocopying permitted by license only

©2012 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group

A Real Options Approach For Software Development Projects Using Fuzzy Electre A. Çagri Tolga1,* 1

Industrial Engineering Department, Galatasaray University, 34357 Ortaköy, Istanbul, Turkey

Software development project selection decision is a vital process in technological organizations. These decisions are very important in two ways. First, in technovation organizations, software development project budget necessitates huge investment and this project selection decisions should be thought with the strategic objectives of the firm. Second, multidimensionality of the software development projects’ organizational returns is naturally risky in terms of expected outcome. Real options approach helps to calculate this risky side of the selection process. This paper considers the software development project selection process in multi-criteria thinking. Vagueness is another consideration in the evaluation process. The fuzzy ELECTRE takes both fuzzy real option value criteria and nonmonetary criteria into account. In this study integration of fuzzy real options valuation to fuzzy ELECTRE is offered for a selection process among software development projects. Keywords: Real Options, Software Development Projects, Multi-Criteria, Fuzzy; ELECTRE, Black-Scholes Approach.

1. INTRODUCTION Software development (also known as Software Application Development; Software Engineering, Software Design) is improving a software product in a planned and structured process. This software could be generated for the three most common purposes which are to meet specific needs of a specific client/business, to meet a perceived need of some set of potential users (the case with commercial and open source software), or for personal use (e.g. a

*Corresponding author: E-mail: [email protected]; Tel: +90212 2274480 Ext.378

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scientist may write software to automate a mundane task). In this work the case with commercial use is investigated. In general, the term software development is used to refer to the activity of computer programming that means the process of writing and maintaining the source code, whereas the broader sense of the term includes all that is involved between the conceptions of the desired software through to the final display of the software. Therefore, software development may include research, new development, modification, reuse, re-engineering, maintenance, or any other activities that result in software products. Basically, a project that searches these activities to present enhanced software for any purpose to general users would be a software development project (SDP). Software development process resembles to research and development (R&D) process according to the stages cited above. Thinking only cost, revenue or financial base takes just one dimension into consideration in software development. However any failed SDP make many contributions to know-how of the experts and to company’s database. Or expanded hardware for any development project contributes to accumulation of property. Dynamic nature of the software development projects drives us to evaluate them with more than one criterion. Technical success probability or marketing strategy criteria have to assist in evaluation process. In literature there are few studies about software development projects. Shehabuddeen et. al. [1] offered a framework for technology selection especially in the application of a software tool. Buyukozkan et al. [2] proposed a methodology to improve the quality of decision-making in the software development project under uncertain conditions. Multi dimensional side of software development drives us to use one of useful multi-criteria methods named ELECTRE in this paper. One of the most important criteria for the evaluation of software development projects is the financial attribute. Many researchers handle this attribute with a simple point of view whereas this attribute has a complex dynamic role in the evaluation process. The dynamic role of this attribute can be handled using a real options valuation (ROV) model. A financial option is defined as the right to buy (if a call) or sell (if a put) a nominated asset by paying a preset price on or before a specified date but it does not contain an obligation. Real options are based on financial options. However, the nature of real options involves permanent, fixed or immovable assets. In contrast to financial options, real options are not tradable -e.g. any retailer cannot sell the right to extend her/his branch against her/his competitor, she/he can only make this decision. The key advantage and value of real option analysis is to integrate managerial flexibility into the valuation process and thereby assist in making the best decisions [3]. Real options give a right but not an obligation to make or not to make an investment for a certain period. For instance, investing in the expansion of a firm’s software development department gives the company right to produce more brilliant output and more certain products but not the obligation. Real options were first

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introduced by Trigeorgis [4]. Benaroch and Kauffman [5] applied real options to information technology investment that is timing of the deployment of point-of-sale debit services. Huchzermeier and Loch’s [6] model built real option for R&D managers as to when it is and when it is not worthwhile to delay commitments. In workaday life real situations are very often vague and uncertain in several ways. When there is a shortfall for information or unwillingness to lead the financial data out the company, a system might not be known completely. Zadeh [7] suggested a strict mathematical outline named fuzzy set theory that overcomes these inadequacies. The fuzzy approach to real option valuation (FROV) was first studied by Carlsson and Fuller [8]. This work was based on Black-Scholes’ real option valuation, but under fuzziness. Then, Wang and Hwang [9] offered fuzzy compound options for R&D project selection based on the Black-Scholes real option valuation. Another fuzzy real option valuation based on Cox et al.’s [10] method was investigated by Allenotor and Thulasiram [11]. They modeled pricing of grid/distributed computing resources as a problem of real option pricing. Tolga et al. [12] studied a comparison between fuzzy trinomial lattice and fuzzy compound options applied to a call center in Turkey. ELECTRE, an acronym for Elimination Et Choix TRaduisant la REalité in French, is one of the multi-criteria decision-making (MCDM) methods based on outranking of alternatives using concordance and discordance indexes. ELECTRE is an adaptable method that allows combining both qualitative and quantitative data. This method requires less detailed data, thus careful use of time and manpower resources is the important factor in the choice of the method. The sorts of ELECTRE methods distinguish from each other while the degree of complexity, or the nature of the main problem, or the richness of the information required differentiate. There are six main versions of ELECTRE: 1, 2, 3, 4, Tri, and 1S. Roy [13] introduced the first ELECTRE method. Obtaining a subset or kernel N of project options such that any alternative which is not in N is outranked by at least one alternative in N was the aim of that work. ELECTRE 1S method was enhanced by Roy and Skalka [14] to adopt the ELECTRE 1 into the fuzzy logic. The best option is selected from other options within the kernel. ELECTRE methods are applied to many areas since they are stated. For example Raoot and Rakshit [15] used the method for the multiple criteria evaluation and ranking of efficient layout alternatives. Montazer et al. [16] discussed the architecture of a fuzzy system based on ELECTRE III including both modules, utilizing fuzzy concept for dealing with the uncertainty of the problem. Brito et al. [17] proposed a multi-criteria model that integrates utility theory and the ELECTRE TRI method for assessing risk in natural gas pipelines, and for classifying sections of pipeline into risk categories. In this study; the authority of the firm does not agree to share the financial data and that was the main reason of fuzzy logic

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usage. Fuzzy real options integrated ELECTRE 1S method is offered according to the structure of the problem. Risky and dynamic side of the problem will be investigated by fuzzy real options which represent management flexibility. The rest of the study is organized as follows: The second section includes the evaluation attributes for software development projects and illustrates the criteria of the problem. In Section 3, fuzzy real options valuation model is presented. The preceding section contains an alternative fuzzy ranking approach integrated fuzzy ELECTRE method. The proposed model’s steps are given in Section 5. Application of the proposed model is the main subject of Section 6. And discussions and findings of the study are given in the last section. 2. SOFTWARE DEVELOPMENT PROJECTS and SELECTION CRITERIA Larger software systems are usually developed by a team of people, some form of process is typically followed to guide the stages of production of the software. Therefore, software development project includes stages; research, new development, modification, reuse, re-engineering, maintenance, and market introduction that result in software products. Especially the first phase containing research and new development stages in the software development process may involve many departments, including marketing, engineering, research and development and general management. Marketing department serves to build up the customer needs; engineering and R&D department construct the software data received from the previous department and general management department organizes the orders of the studies. The second phase includes modification, reuse, reengineering and maintenance stages. At that stage only engineering department performs useful studies to present high quality product. However the last phase i.e. market introduction needs whole departments. The financial criteria ranking of the valuation process will be evaluated by real options approach, however because of the vagueness in data collection fuzzy form of real options will be utilized. As stated in the introduction section fuzzy real options valuation will be made according to the phases cited above. There will be three phases and two options which are shown in Figure 1. The rest of the criteria are derived from recent works and they are determined by departments mentioned above according to these studies. There are many studies in R&D evaluation using AHP, ANP and ELECTRE. As said before because of analogy between R&D projects and SDPs the criteria are derived from these works. Then a DELPHI work with employees is made and criteria below are derived for the application of ELECTRE 1S method.

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Figure 1 Phases and options of a software development project.

Capacity of employees, resources, and success probability etc. criteria are determined and explained in Table 1. Experts are asked to make a pair-wise comparison between alternatives according to the determined criteria. Contribution to firms’ know-how is an obligatory criterion and this criteria needs to be evaluated in a fuzzy manner because there are no strict expression for that criterion. Also trends/flexibility criterion necessitates a fuzzy form evaluation because of its vagueness in future customer needs. Capacity criterion and marketing potential are also evaluated in fuzzy form. For joining the integrity fuzzy form will be applied to whole criteria although the other criteria could be assessed by crisp numbers.

Criteria

Explanation

Capacity (C1)

Staff numbers and skills dedicated to any SDP in software development process

Resources (C2)

Requirements for additional equipment and facilities

Expandability (C3)

Enhancing current functions or adding new functionality capability

Quality (C4)

Correctness and verifiability of the project

Reliability (C5)

The certain performance of a software for a specified period of time

Success Probability (C6)

Probability of technical success

Technical Contribution (C7)

Contribution to Firm’s know-how

Performance (C8)

Efficiency and usability of the software

Marketing Potential (C9)

Probability of commercial success against competitors, customer acceptance

Distribution Capacity (C10)

Likely sales volume and market share

Trends/Flexibility (C11)

Adequacy for customer future preferences

FROV (C12)

Fuzzy real option value of software development projects

Table 2-1 Explanation of Criteria

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3. REAL OPTIONS APPROACH There are mostly more than two stages in software development project as stated in Section 1 and the decision maker (DM) has to determine whether to exercise or to carry on the option (which means to stop or to defer the project). Technically, if any stage of the project is succeeded option to expand should be made real and more investment is needed. However, adversely if there is no success in any stage, no need to continue the investment more and thereby that means downside risk limit of the software development project is reached. Expenses up to that point correspond to option premium in financial options. Stages of the SDP were investigated deeply in Section 2 and they will be used hereafter. In this study, we use Wang and Hwang’s [9] fuzzy real option valuation method which is based on Geske compound options model for the evaluation of financial attribute in fuzzy ELECTRE. A SDP project containing three phases as shown in Figure 1 will be examined. Let K be the present value of investment cost for stages i = 1, 2, 3, and G i

be the present value of the project return after market introduction. K i (i = 1, 2, 3) and G should be fuzzy numbers. Time to maturities of first and second options for the project are T1 and T2, respectively. Fuzzy real option value of the SDP project is:  - ∆T2 M (u ; z ; T / T ) - K e-η T2 M (u ; z ; T / T ) - K e-ηηT1 N (u ) (1) Z = Ge 1 1 1 2 3 2 2 1 2 2 2 where

u1 =

σ T1

,

(2)

u2 = u1 − σ T1 , (3)

ln[ E (G ) / G c ] + (η − ∆ + σ 2 / 2)T1

z1 =

ln[ E (G ) / E ( K 3 )] + (η − ∆ + σσ2 / 2)T2 σ T2

, (4)

z2 = z1 − σ T2 , (5)

Here, D is the dividend yield; η is the interest rate; σ is the volatility of the project return; N is the cumulative normal distribution; and M (u, z, ρ) is the

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Figure 2 A trapezoidal fuzzy number, Ã = (a1, a2, a3, a4)

bivariate cumulative normal distribution function with u and z that are upper and lower integral limits and correlation coefficient ρ. Risk neutral expectation of the project return is given at the first term of Eq. (1), the second term is the expected investment at T2, and the last term gives the expected investment at T1. We can calculate the volatility (σ) of the rate of the change of the project return as Var(G ) / E (G ) and the dividend yield (D) is calculated as E ( K 1 ) / E (G ). Possibilistic mean (E ( A )) and variance ( Var( A )) of fuzzy number A (shown in Figure 2) which could de denoted as (a1, a2, a3, a4) are calculated from Eqs. (6) and (7) [8]: E ( A ) =

Var( A ) =

a2 + a3 a1 − a2 − a3 + a4 (6) + , 2 6

(a3 − a2 )2 (a3 − a2 )(a2 + a4 − a1 − a3 ) (a2 + a4 − a1 − a3 )2 + + , (7) 4 6 24

We obtain the critical value Gc from the equation below with interpolation.

G c e- ∆ (T2 −T1 ) N (c1 ) - E ( K 3 )e-η (T2 −T1 ) N (c2 ) - E ( K 2 ) = 0 (8)

where

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c1 =

ln[G c / E ( K 3 )] + (η − ∆ + σ 2 / 2)(T2 − T1 )

σ T2 − T1

, (9)

c2 = c1 − σ T2 − T1 . (10)

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For simplifying the calculation of the project value, investment costs  defined in Eqs. ( K 2 and K 3 ) of stages 2 and 3 and future project return (G) (2), (3), (8), and (9) are replaced by their possibilistic mean and variance values [8].

4. FUZZY ELECTRE This work is based on selection procedure and though only vague data can be taken out from the company, fuzzy logic will be used. As stated in Section 1, ELECTRE 1S is selected for modification by using alternative ranking method because of its appropriateness to that selection procedure. The basic steps of proposed fuzzy ELECTRE method can be viewed as follows: Suppose a MCDM problem has m decision criteria (C1, C2, …, Cm) and n alternatives (A1, A2,…, An). Alternatives are evaluated by m criteria separately. The decision matrix is formed by the fuzzy ratings assigned to the alternatives w.r.t. each criterion denoted by T = (tij )n×m . The relative fuzzy weight vector about criteria is displayed by W = (w 1 , w 2 ,..., w m ). Step 1: Constitute the decision matrix from the decision maker’s thoughts as below:

 t11  t T =  21    tn1

t12 t1m   t22 t2 m  (11)   tn 2 tnm 

Step 2: If the objective is a minimization like cost criteria, apply the normalization procedure as seen in Eq. (12), otherwise like benefit criteria apply the normalization in Eq. (13):

rij = 1 / tij

rij = tij

∑

1/ tij i =1 n

∑ tij n

i =1

2

2

(12)

(13)

where i = 1, 2 ,…,n and j = 1, 2 ,…,m. At that point division of fuzzy numbers (shown in Section 3) has to be explained that one can easily calculate with Eq. (14)

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1 / A = (1 / a4 ,1 / a3 ,1 / a2 ,1 / a1 ) (14)

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Step 3: Calculate the weighted fuzzy normalized decision matrix V = (vij )n×m for i = 1,2 ,…, n and j = 1,2 ,…, m, where vij = w j × rij . Step 4: Designate the fuzzy concordance and discordance sets. The fuzzy concordance set, which is formed by all criteria for which alternative Ap is preferred to alternative Aq, can be written as

{

}

O( p, q ) = j v pj ≥ vqj (15)

The discordance set, the complement of O( p, q ) which includes all criteria for which Ap, is worse than Aq, can be written as

{

}

D( p, q ) = j v pj < vqj (16)

Here, the alternative fuzzy ranking method offered by Kahraman and Tolga [18] is going to be used. For detailed information see the said study. All the fuzzy comparisons in Eqs. (15) and (16) are made with the conclusion of Eq. (18). An index that measures the possibility of one fuzzy number being greater than another will be determined in said method. That preference index will be illustrated by I(ω) ∈ [0,1] and it is determined by Eq. (17):

0   (a4 − b1 )2   (b2 − b1 − a3 + a4 )   (a4 + a3 − a2 − a1 ) + (b4 + b3 − b2 − b1 )  a4 + a3 − b2 − b1  I (ω) =  ( a + a − a 3 2 − a1 ) + (b4 + b3 − b2 − b1 )  4  (a2 − b3 )2  (a4 + a3 − b2 − b1 ) − (b4 − b3 + a2 − a1 )   ( a + a − a − a ) + (b + b − b − b ) 4 3 2 1 3 2 1  4  1

,

b1 ≥ a4

, b2 ≥ a3 , b1 < a4 , b3 ≥ a2 , b2 < a3 (17)

, b3 < a2 , b4 > a1 ,

b4 ≤ a1

The fuzzy preference relation (PKT ) of the fuzzy numbers will be determined as following:

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 A B if I (w) ∈ (0.5,1]  PKT ( A, B) =  A = B if I (w) = 0.5 (18)  B A if I (w) ∈ [0, 0.5) 

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From fuzzy preference relation one can easily deduce that if the outcome of Eq. (17) is larger than 0.5, this means that the fuzzy number Ã is preferred  to B. Step 4: Calculate the concordance and discordance indexes. The degree of confidence in the pair-wise judgments of (Ap→Aq) is represented by Opq and it is calculated by Eq. (19) O pq =

∑

j*

w j* (19)

where j* are criteria contained in the concordance set O (p, q). The discordance index of D (p, q) can be defined as D pq = (

∑

j0

v pj 0 − vqj 0 ) (

∑

j

v pj − vqj ) (20)

where j0 are criteria contained in the discordance set D (p, q). Though the absolute value clause in Eq. (20), all the compared values in Eqs. (15) and (16) should be defuzzified by equation below and these values should be used in Eqs. (19) and (20). a2

d ( A i ) =

∫

a4

a1

∫

x ⋅ µ A ( x )dx

a4

a1

∫

=

a1

 x − a1   x ⋅ a − a  dx + 2 1

µ A ( x )dx

a2

∫

a1

a3

∫

a4

 a −x  dx x.1dx +  x ⋅ 4  a4 − a3 

∫

a3

a2 a3

a4

a2

a3

 a3 − x   x − a1   a − a  dx + 1.dx +  a − a  dx 2 1 3 2

∫

∫

(21)

(a + 2 a2 + 2 a3 + a4 ) = 1 6

Step 5: Outrank the relationships. A higher concordance index Opq and a lower discordance index Dpq means the dominance relationship of alternative Ap becomes stronger over alternative Aq. When the Opq ≥ O and Dpq < D, that represents Ap outranks Aq. Here, O and D are the averages of Opq and Dpq, respectively.

5. STEPS of THE FUZZY REAL OPTIONS VALUE INTEGRATED ELECTRE 1S MODEL In selection procedure, this study argues risky side of the software development projects by fuzzy real options, and this evaluation method for economical side is integrated to ELECTRE 1S method which is called fuzzy

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ELECTRE. This proposed method finds the solution with more realistic way however it does not disregard the dynamic environment of SDPs and it dissolves the problem at a quick pace. The steps of the offered method are as below: Step 1. Determine the weights of criteria. Experts have to determine the importance levels of criteria. Step 2. Construct the decision matrix. For criteria from 1 to 11 shown in Table 2-1 utilize the experts’ opinions derived from fuzzy conversion scale and initialize them to Eq. (11) like below.

 t11  t T =  21    tn1

t12 t22 tn 2

t111   t211    tn11 

Step 3. Compute the economic criteria. Economic side -fuzzy real options value- of the selection process is calculated in this step with sub steps below. Step 3.1. For fuzzy real options value, calculate E (G ), E ( K 1 ), E ( K 2 ) , E ( K 3 ), and Var(G ) values at first for each alternative separately via Eqs. (6) and (7). Then find Gc with interpolation. Step 3.2. Calculate u1, u2, u3, and, z1 values respectively by Eqs. (2), (3), (4), and (5). Step 3.3. Find Z values for each alternative by Eq. (1) which might become input values for the next step. T Step 4. Integration. Integrate the t112 tn12 n   values found in Step 3.3 to the decision matrix as the twelfth criteria. Step 5. Normalization. Normalize tij values for criteria 1 to 12 by applying Eq. (13). Step 6: Calculate the weighted fuzzy normalized decision matrix V = (vij )n × m for i = 1,2 ,…, n and j = 1,2 ,…, m, where vij = w j × rij . Step 7. Designate the fuzzy concordance and discordance sets. Calculate the concordance sets by using Eq. (15) and applying the fuzzy number ranking procedure explained in Section 4. Then compute the discordance sets by using Eq. (16) and also apply the ranking procedure given in Eqs. (17) and (18). Step 8: Calculate the concordance and discordance indexes. Concordance index is calculated by Eq. (19) and discordance index is calculated by Eq. (20). Defuzzify (by using Eq. (21)) all discordance values because of the absolute value clause in Eq. (20). Step 9. Outrank the relationships. When the Opq ≥ O and Dpq < D, that represents Ap outranks Aq. Here, O and D are the averages of Opq and Dpq, respectively.

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6. AN APPLICATION to MID-SIZE SDP FIRM in TURKEY A mid-size software development company in Turkey wants to make outranking among three of their future software development project. This company has determined its targets strategically by courtesy of its boss with great vision –being a leader company in five 5 years- and it has also very talented employees. Though the company has great goals and it situates in development phase hereafter it will called ABC Software Company. The ABC Co. desires to choose most suitable one among three different commercial software development projects those take place in the strategic goals of the firm: SDP 1 is commercial software that directs human resources planning, recruitment and placement processes of any firm. SDP 2 is commercial software that undertakes production planning control. Then the SDP 3 is software that applies activity based costing/managing in a commercial worried firm. The application is performed in two stages. At the first stage, the financial data is taken out from the company. Though some part of the data is insufficient and the authority of the company does not want to take out the other part of the data crisply, financial info is utilized in a trapezoidal fuzzy numbers manner. At the second stage, a survey work is evaluated. Pair-wise comparisons of criteria of the projects are asked to an expert from the company. After then, non monetary and monetary criteria are consolidated in a sole table to select the most suitable software development project for the company. At the first stage, obtained financial data with trapezoidal fuzzy numbers is given in Table 6-1. The values are in thousand TL and interest rate is taken from the Central Bank of the Republic of Turkey’s (CBRT) rate (η = 5.25 %). Option value reducing dividend rate which represents downturn in profitability in real options and shown by δ is calculated by the formula E ( K 1 ) / E (G ). At the first column; mutually exclusive projects’ codes, at the second column; estimated returns of each projects after market introduction separately, at the third column; expected value of estimated returns calculated by Eq. (6), at the fourth column; present value of costs of discovery phase means costs of generation of software with all functions, at the fifth column; expected value of this cost, at the sixth column; present value of costs of testing phase for determination of the deficiencies and errors, at the seventh stage; expected value of this cost, at the eighth stage; present value of marketing costs while market introduction of the projects, at the ninth stage, expected value of this cost, at the tenth stage; first option’s maturity date, at the eleventh stage; second option’s maturity date with year rate, are given. Steps of the offered approach were given in Section 5. According to this methodology; at first we asked the experts the weights of the criteria. They evaluated the criteria by fuzzy scale shown in Figure 3. The scale is in trian-

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(60, 65, 70, 75)

(51, 52, 55, 57)

(85, 87, 90, 92)

SDP 1

SDP 2

SDP 3 88.50

53.67

67.50

 E(G)

(23, 25, 30, 32)

(9, 10, 12, 13)

(13, 15, 18, 21)

K1

PV of investment cost for Stage 1

Table 6-1 Financial data of software development project alternatives (Thousand TL).

G

Projects

PV of project return after market introduction

27.50

11.00

16.67

E(K1)

(3, 4, 5, 6)

(1.5, 2, 3, 3.5)

(1, 3, 5, 7)

K 2


4.50

2.50

4.00

E( K 2 )

(8, 10, 15, 17)

(3, 6, 8, 11)

(3, 4, 5, 6)

K 3


12.50

7.00

4.50

E(K 3)

1.5

1

1

T1 (yr.)

2

1.5

2

T2 (yr.)

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Figure 3 Fuzzy weight scale.

gular shape but it is written in trapezoidal form and it covers all details to express the exact weights in mind of experts. Then experts are questioned to compare the criteria pair-wisely according to alternatives by scale given in Table 6-2 and visualized in Figure 4. The weights of the criteria determined by expert are shown in Table 6-3. Decision matrix except economic criteria is constructed by these evaluations as determined in Step 2 and the decision matrix can be seen in Table A-1. Then computing the economic criteria in Step 3 is applied as below. In light of foregoing data in Table 6-1 values in request are calculated by Eqs. (2)-(10) among and as denoted in Steps 3.1 and 3.2. All the alternative projects’ fuzzy real option values are calculated by Eq. (1) separately and they are given in Table 6-4. These found values are integrated to the decision matrix as benefit criteria because of the nature of real options. And the last integrated decision matrix is shown in Appendix. As denoted in Step 5 normalization is applied to this

Linguistic statement

Trapezoidal fuzzy statement

Least important

(0, 10, 10, 20)

Much less important

(10, 20, 20, 30)

Less important

(20, 30, 30, 40)

Not so important

(30, 40, 40, 50)

Equal

(40, 50, 50, 60)

Important

(50, 60, 60, 70)

More Important

(60, 70, 70, 80)

Much more important

(70, 80, 80, 90)

Most important

(80, 90, 90, 100)

Table 6-2 Fuzzy Linguistic Scale

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Figure 4 Graphical expression of fuzzy scale.

Criteria

Fuzzy Weight

Criteria

Fuzzy Weight

C1

(0.45, 0.50, 0.50, 0.55)

C7

(0.50, 0.55, 0.55, 0.60)

C2

(0.25, 0.30, 0.30, 0.35)

C8

(0.35, 0.40, 0.40, 0.45)

C3

(0.20, 0.25, 0.25, 0.30)

C9

(0.60, 0.65, 0.65, 0.70)

C4

(0.60, 0.65, 0.65, 0.70)

C10

(0.40, 0.45, 0.45, 0.50)

C5

(0.25, 0.30, 0.30, 0.35)

C11

(0.50, 0.55, 0.55, 0.60)

C6

(0.30, 0.35, 0.35, 0.40)

C12

(0.80, 0.85, 0.85, 0.90)

Table 6-3 The weights of criteria

SDP Project Human Resources Management

Fuzzy Real Options Values V = (24.8, 28.6, 35.6, 39.4)

Production Planning and Control

VSDP = (24.0, 28.0, 33.0, 37.7)

Activity Based Cost Managing

VSDP = (24.6, 30.4, 36.3, 42.1)

SDP1

2

3

Table 6-4 Computed Fuzzy Real Options Values

decision matrix. Then the weighted fuzzy normalized decision matrix is computed as depicted in Step 6 of the offered method and this matrix is also shown in Table A-2. Fuzzy concordance and discordance sets are computed by Eqs. (15) and (16) as remarked in Step 7 and shown in Table 6-5. And also concordance and discordance indexes calculated with Eqs. (19) and (20) in accordance with Step 8 are denoted in Table 6-5. Let us investigate an example from Table 6-5 that how a preference index is computed for instance O(1,2) for criteria 12. From the weighted fuzzy normalized

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A. Çagri Tolga

Concordance Sets

Concordance Indexes

Discordance Sets

Discordance Indexes

O(1,2)= 2,3,4,6,7,9,10,11,12

O12= 4.600

D(1,2)= 1,5,8

D12= 0.234

O(1,3)= 2,6,7,11

O13= 1.750

D(1,3)= 1,3,4,5,8,9,10,12

D13= 0.775

O(2,1)= 1,4,5,8

O21= 1.850

D(2,1)= 2,3,6,7,9,10,11,12

D21= 0.720

O(2,3)= 2,6,7,11

O23= 1.750

D(2,3)= 1,3,4,5,8,9,10,12

D23= 0.906

O(3,1)= 1,3,4,5,8,9,10,12

O31= 4.050

D(3,1)= 2,6,7,11

D31= 0.225

O(3,2)= 1,3,4,5,7,8,9,10,12

O32= 4.600

D(3,2)= 2,6,11

D32= 0.075

Table 6-5 Concordance and Discordance Sets and Indexes

matrix denoted in Table A-2, one can easily found v112 equals (0.273, 0.476, 0.498, 0.866). With the same logic v212 = (0.260, 0.461, 0.462, 0.830) value could be found. The preference index is found by Eq. (17) and it is equal to I(ω ) = 0.542 which means from Eq. (18) PKT (v112 , v212 ) = 0.542 → v112 v212 . The other ranking of fuzzy numbers are made with the same method. Finally an outranking represented in Step 9 is made and the results of interrogations are given in Table 6-6. The relationship SDP3 → SDP1 → SDP2 can be easily inferred from the last Table. At the first order activity based cost managing software is selected to be developed by the reason of fuzzy real options integrated fuzzy ELECTRE method. SDP3’s fuzzy real option value is slightly bigger than SDP1’s fuzzy real option value; the decision maker can have conflict about to choose which one. However with help of a decision making method called fuzzy ELECTRE selection of SDP3 is more believed and more valuable in rationalist mind.

Opq

Is (Opq ≥O)?

Dpq

Is (Dpq