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1.a fuzzy

  1. 1. The current issue and full text archive of this journal is available at www.emeraldinsight.com/1463-5771.htmBIJ18,2 A fuzzy goal programming model for strategic information technology172 investment assessment Faramak Zandi Industrial Engineering Department, Faculty of Technology and Engineering, Alzahra University, Tehran, Iran, and Madjid Tavana Management Department, Lindback Distinguished Chair of Information Systems, La Salle University, Philadelphia, Pennsylvania, USA Abstract Purpose – The high expenditures in information technology (IT) and the growing usage that penetrates the core of business have resulted in a need to effectively and efficiently evaluate strategic IT investments in organizations. The purpose of this paper is to propose a novel two-dimensional approach that determines the deferrable strategy with the most value by maximizing the real option values while minimizing the risks associated with each alternative strategy. Design/methodology/approach – In the proposed approach, first, the deferrable investment strategies are prioritized according to their values using real option analysis (ROA). Then, the risks associated with each investment strategy are quantified using the group fuzzy analytic hierarchy process. Finally, the values associated with the two dimensions are integrated to determine the deferrable IT investment strategy with the most value using a fuzzy preemptive goal programming model. Findings – Managers face the difficulty that most IT investment projects are inherently risky, especially in a rapidly changing business environment. The paper proposes a framework that can be used to evaluate IT investments based on the real option concept. This simple, intuitive, generic and comprehensive approach incorporates the linkage among economic value, real option value and IT investments that could lead to a better-structured decision process. Originality/value – In contrast to the traditional ROA literature, the approach contributes to the literature by incorporating a risk dimension parameter. The paper emphasizes the importance of categorizing risk management in IT investment projects since some risk cannot be eliminated. Keywords Fuzzy control, Information technology, Value analysis, Risk analysis, Analytical hierarchy process Paper type Research paper 1. Introduction Information technology (IT) investments represent the largest capital expenditure items for many organizations and have a tremendous impact on productivity by reducing costs, improving quality and increasing value to customers. As a result, many organizationsBenchmarking: An International continue to invest large sums of money in IT in anticipation of a material return on theirJournal investment (Willcocks and Lester, 1996). The selection of appropriate IT investments hasVol. 18 No. 2, 2011pp. 172-196q Emerald Group Publishing Limited1463-5771 The authors would like to thank the anonymous reviewers and the Editor for their insightfulDOI 10.1108/14635771111121667 comments and suggestions.
  2. 2. been one of the most significant business challenges of the last decade. Powell (1992) Fuzzy goalhas studied the similarities and differences between IT investments and other capital programminginvestments in organizations. He notes that IT investments are undertaken byorganizations to gain competitive advantage, to improve productivity, to enable new ways modelof managing and organizing and to develop new businesses. Appropriate strategic ITinvestments can help companies gain and sustain a competitive advantage (Melville et al.,2004). However, many large IT investment projects often do not meet original expectations 173of cost, time or benefits. The rapid growth of IT investments has imposed tremendouspressure on management to take into consideration risks and payoffs promised by theinvestment in their decision making. A review of the current literature offers several IT investment evaluation methodsthat provide frameworks for the quantification of risks and benefits. The net presentvalue (NPV) (Hayes and Abernathy, 1980; Kaplan and Atkinson, 1998), return oninvestment (Brealey and Myers, 1998; Farbey et al., 1993; Kumar, 2002; Luehrman,1997), cost benefit analysis (Schniederjans et al., 2004), information economics (Bakosand Kemerer, 1992; Parker and Benson, 1989) and return on management (Chen et al.,2006; Stix and Reiner, 2004; Strassmann, 1997) are among most widely used methods toassess the risks and payoffs associated with IT investments. In addition to the above mentioned traditional quantitative approaches, there is astream of research studies which emphasizes real option analysis (ROA). The ROA differsfrom the traditional methods in terms of priceability of the underlying investment project(McGrath, 1997). With the traditional methods, the underlying investment project of anoption is priced as known (Black and Scholes, 1973) while in IT investment situations theprice of an underlying investment is rarely known (McGrath, 1997). The ROA uses threebasic types of data: (1) current and possible future investment options; (2) the desired capabilities sought by the organization; and (3) the relative risks and costs of other IT investment options that could be used.The method can help assess the risks associated with IT investment decisions bytaking into consideration the changing nature of business strategies andorganizational requirements. The real options are commonly valued with the Black-Scholes option pricing formula(Black and Scholes, 1973, 1974), the binomial option valuation method (Cox et al., 1979)and Monte-Carlo methods (Boyle, 1977). These methods assume that the underlyingmarkets can be imitated accurately as a process. Although this assumption may hold forsome quite efficiently traded financial securities, it may not hold for real investments thatdo not have existing markets (Collan et al., 2009). Recently, a simple novel approach toROA called the Datar-Mathews method (Datar and Mathews, 2004, 2007; Mathews andSalmon, 2007) was proposed where the real option value is calculated from a pay-offdistribution, derived from a probability distribution of the NPV for an investment projectgenerated with a Monte-Carlo simulation. This approach does suffer from the marketprocess assumptions associated with the Black-Scholes method (Black and Scholes, 1974). When valuating an investment using ROA, it is required to estimate severalparameters (i.e. expected payoffs and costs or investment deferral time). However, theestimation of uncertain parameters in this valuation process is often very challenging.Most traditional methods use probability theory in their treatment of uncertainty.
  3. 3. BIJ Fuzzy logic and fuzzy sets can represent ambiguous, uncertain or imprecise information18,2 in ROA by formalizing inaccuracy in human decision making (Collan et al., 2009). For example, fuzzy sets allow for graduation of belonging in future cash-flow estimation (i.e. future cash flow at year 5 is about 5,000 dollars). Fuzzy set algebra developed by Zadeh (1965) is the formal body of theory that allows the treatment of imprecise estimates in uncertain environments.174 In recent years, several researchers have combined fuzzy sets theory with ROA. ´ Carlsson and Fuller (2003) introduced a (heuristic) real option rule in a fuzzy setting, where the present values of expected cash flows and expected costs are estimated by trapezoidal fuzzy numbers. Chen et al. (2007) developed a comprehensive but simple methodology to evaluate IT investment in a nuclear power station based on fuzzy risk analysis and real option approach. Frode (2007) used the conceptual real option framework of Dixit and Pindyck (1994) to estimate the value of investment opportunities in the Norwegian hydropower industry. Villani (2008) combined two successful theories, namely real options and game theory, to value the investment opportunity and the value of flexibility as a real option while analyzing the competition with game theory. Collan et al. (2009) presented a new method for real option valuation using fuzzy numbers. Their method considered the dynamic nature of the profitability assessment, that is, the assessment changes when information changes. As cash flows taking place in the future come closer, information changes and uncertainty is reduced. Chrysafis and Papadopoulos (2009) presented an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function using statistical data. Wang and Hwang (2007) developed a fuzzy research and development portfolio selection model to hedge against the environmental uncertainties. They applied fuzzy set theory to model uncertain and flexible project information. Since traditional project valuation methods often underestimate the risky project, a fuzzy compound-options model was used to evaluate the value of each project. Their portfolio selection problem was formulated as a fuzzy zero-one integer programming model that could handle both uncertain and flexible parameters and determine the optimal project portfolio. A new transformation method based on qualitative possibility theory was developed to convert the fuzzy portfolio selection model into a crisp mathematical model from the risk-averse perspective. The transformed model was solved by an optimization technique. We propose a novel two-dimensional approach that determines the deferrable strategy with the most value by maximizing the real option values while minimizing the risks associated with each alternative strategy. First, the deferrable investment strategies are prioritized according to their values using the ROA. Then, the risks associated with each investment strategy are quantified using the group fuzzy analytic hierarchy process (GFAHP). Finally, the values associated with the two dimensions are integrated to determine the deferrable IT investment strategy with the most value using a fuzzy preemptive goal programming model. The proposed framework: . addresses the gaps in the IT investment assessment literature on the effective and efficient evaluation of IT investment strategies; . provides a comprehensive and systematic framework that combines ROA with a group fuzzy approach to assess IT investment strategies; . considers fuzzy logic and fuzzy sets to represent ambiguous, uncertain or imprecise information; and
  4. 4. . it uses a real-world case study to demonstrate the applicability of the proposed Fuzzy goal framework and exhibit the efficacy of the procedures and algorithms. programmingThis paper is organized into five sections. In Section 2, we illustrate the details of the modelproposed framework followed by a case study in Section 3. In Section 4, we presentdiscussion and practical perspectives and in Section 5, we conclude with our conclusionsand future research directions. 1752. The proposed frameworkThe mathematical notations and definitions used in our model are presented in theAppendix. The framework shown in Figure 1 is proposed to assess alternative ITinvestment strategies. The framework consists of several steps modularized into fivephases.Phase 1: establishment of the IT investment boardWe institute a strategic IT investment board to acquire pertinent investmentinformation. Executive management is typically responsible for creating the board,specifying its responsibilities and defining its resources. Let us assume that l strategicIT investment board members are selected to participate in the evaluation process: ITIB ¼ ½ðITIBÞ1 ; ðITIBÞ2 ; . . . ; ðITIBÞk ; . . . ; ðITIBÞl ŠPhase 2: identification of the IT investment strategiesNext, the strategic IT investment board identifies a set of alternative deferrable ITinvestment strategies. Let us assume that n alternative IT investments with themaximum deferral time of Tm are under consideration: a ¼ ½a1 ; a2 ; . . . ; ai ; . . .an ŠPhase 3: prioritization of the IT investment strategies: real option considerationsIn this phase, the real options equations suggested by Dos Santos (1994) are used toprioritize IT investments strategies. This phase is divided into the following three steps. Step 3.1: construction of the individual real option matrices. The following individualreal option matrices are given by each strategic IT investment board member: ~ BðT 1 Þ ~ ~ ~ ~ BðT 2 Þ . . . BðT m Þ CðT 1 Þ CðT 2 Þ . . . CðT m Þ ~ 2 3 a1 ~k ~K ~k B1 ðT 1 Þ B1 ðT 2 Þ . . . B1 ðT m Þ ~k ~k ~k C1 ðT 1 Þ C1 ðT 2 Þ . . . C1 ðT m Þ 6 7 6 ~k ~k ~k ~k ~k ~k 7 ~ k ¼ a2 6 B2 ðT 1 Þ B2 ðT 2 Þ . . . B2 ðT m Þ C2 ðT 1 Þ C2 ðT 2 Þ . . . C2 ðT m Þ 7 ARO1 6 7 . 6 . . 6 . . . . . . 7 ð1Þ . 6 . . . . . . 7 . ... . . . ... . 7 4 k 5 k an B ðT Þ BK ðT Þ . . . BK ðT Þ C ðT Þ C k ðT Þ . . . C k ðT Þ ~ ~ 1 2 m m 2 m n n n n n n For k ¼ 1; 2; . . . ; l:Fuzzy numbers are often represented by triangular or trapezoidal fuzzy sets. In thisstudy, we use trapezoidal fuzzy sets. A major advantage of trapezoidal fuzzy numbers is
  5. 5. BIJ Phase 118,2 Establishment of the IT investment board Phase 2 Identification of the IT investment strategies176 Phase 3 Prioritization of the IT investment strategies: real option considerations Step 3.1 Construction of the individual real option matrices Step 3.2 Construction of the weighted collective real option matrix Step 3.3 Computation of the vector of the real option value for the IT investment strategies Phase 4 Prioritization of the IT investment strategies: risk considerations Step 4.1 Identification of the criteria and sub-criteria for the GFAHP model Step 4.2 Construction of the individual fuzzy pairwise comparison matrices Step 4.3 Construction of the weighted collective fuzzy pairwise comparison matrix Step 4.4 Computation of the vector of the risk value for the IT investment strategies Phase 5 Development of the strategic IT investment plan Step 5.1 Determination of the goal and priority levels Step 5.2 Computation of the goal values Step 5.3 Construction of the proposed goalFigure 1. programming modelThe proposed framework
  6. 6. that many operations based on the max-min convolution can be replaced by direct Fuzzy goalarithmetic operations (Dubois and Prade, 1988). The following trapezoidal fuzzy numbersare used for the individual fuzzy present values of the expected cash flows and the cost of programmingthe ith IT investment at time Tj by strategic IT investment board member (ITIB)k: model b o a g ~ k ðT j Þ ¼ Bk ðT j Þ ; Bk ðT j Þ ; Bk ðT j Þ ; Bk ðT j Þ Bi i i i i o a b g 177 ~k Ci ¼ C k ðT j Þ ; C k ðT j Þ ; C k ðT j Þ ; C k ðT j Þ ð2Þ i i i i For j ¼ 1; 2; . . . ; m:That is, we have the following intervals: j o k a Bk ðT j Þ ; Bk ðT j Þ i i the most possible values for the expected cash flows of the ith IT investment at time Tj evaluated by strategic IT investment board member (ITIB)k. o g k k Bi ðT j Þ þ Bi ðT j Þ the upward potential for the expected cash flows of the ith IT investment at time Tj evaluated by strategic IT b investment board member (ITIB)k. o Bk ðT j Þ 2 Bk ðT j Þ i i the downward potential for the expected cash flows of the ith IT investment at time Tj evaluated by strategic IT investment board member (ITIB)k. j o k a k k C i ðT j Þ ; C i ðT j Þ the most possible values of the expected cost of the ith IT investment at time Tj evaluated by strategic IT investment board member (ITIB)k. o g k k C i ðT j Þ þ C i ðT j Þ the upward potential for the expected cost of the ith IT investment at time Tj evaluated by strategic IT b investment board member (ITIB)k. o C k ðT j Þ 2 C k ðT j Þ i i the downward potential for the expected cash flows of the ith IT investment at time Tj evaluated by strategic IT investment board member (ITIB)k.Consequently, substituting equation (2) into matrix (1), the individual real optionmatrices can be rewritten as: ~ BðT i Þ ~ CðT i Þ 2 o a b g o a b g 3 6 Bk ðT i Þ ; Bk ðT i Þ ; Bk ðT i Þ ; Bk ðT i Þ 1 1 1 1 C k ðT i Þ ; C k ðT i Þ ; C k ðT i Þ ; C k ðT i Þ 1 1 1 1 7 a1 6 7 6 6 o a b g o a b g 7 7 k k k k 6 B2 ðT i Þ ; B2 ðT i Þ ; B2 ðT i Þ ; B2 ðT i Þ k k k k 7 6 C 2 ðT i Þ ; C 2 ðT i Þ ; C 2 ðT i Þ ; C 2 ðT i Þ 7~kARO1 ðT i Þ ¼ a2 6 7 6 7 . 6 . . . . 7 . . 6 . . 7 6 7 6 o a b g 7 an 4 Bk ðT Þ o ; Bk ðT Þ a ; Bk ðT Þ b ; Bk ðT Þ g k k k k C n ðT i Þ ; C n ðT i Þ ; C n ðT i Þ ; C n ðT i Þ 5 n i n i n i n i ð3Þ
  7. 7. BIJ Step 3.2: construction of the weighted collective real option matrix. This framework allows for assigning different voting power weights given to each investment board18,2 member: W ðvpÞ ¼ ½wðvpÞ1 ; wðvpÞ2 ; . . . ; wðvpÞj ; . . . ; wðvpÞl Š ð4Þ Therefore, in order to form a fuzzy weighted collective real option matrix, the individual178 fuzzy real option matrices will be aggregated by the voting powers as follows: ~ BðT i Þ ~ CðT i Þ 2 3 a1 ~ B1 ðT i Þ ~ C1 ðT i Þ 6~ ~ 7 6 B2 ðT i Þ C2 ðT i Þ 7 ARO2 ðT i Þ ¼ a2 ~ 6 7 ð5Þ . 6 . . 7 . 6 . . 7 . 6 . . 7 4 5 an ~ Bn ðT i Þ ~ n ðT i Þ C where: Pl ~k k¼1 ðwðvpÞk Þ Bi ðT i Þ ~ Bi ðT i Þ ¼ Pl ð6Þ k¼1 wðvpÞk Pl ~k k¼1 ðwðvpÞk Þ Ci ðT i Þ ~ Ci ðT i Þ ¼ Pl ð7Þ k¼1 wðvpÞk Step 3.3: Computation of the vector of the real option value for the IT investment strategies. The real option values of the investment strategies at times T 1 ; T 2 ; . . . ; T m can be determined by the following fuzzy real option value matrix: T1 T2 ... Tm 2 3 a1 FROV 1 ðT 1 Þ FROV 1 ðT 2 Þ ... FROV 1 ðT m Þ 6 7 6 FROV 2 ðT 1 Þ FROV 2 ðT 2 Þ ... FROV 2 T m 7 AFROV ¼ a2 6 ~ 7 ð8Þ . 6 . 6 . . . . . . 7 7 . 6 . . ... . 7 4 5 a4 FROV n ðT 1 Þ FROV n ðT 2 Þ ... FROV n T m or: 2 3 2 3 ~ ~ a1 B1 ðT i Þ·e 2dT i ·N ðD11 ðT i ÞÞ2 C1 ðT i Þ·e 2rT i ·NðD21 ðT i ÞÞ FROV 1 ðT i Þ 6~ ~ 7 6 7 a2 6 B2 ðT i Þ·e 2dT i ·N ðD12 ðT i ÞÞ2 C2 ðT i Þ·e 2rT i ·NðD22 ðT i ÞÞ 7 6 FROV 2 ðT i Þ 7 6 7 6 7 AFROV ðT i Þ ¼ . 6 ~ . 7¼6 . 7 ð9Þ .6 .6 . . 7 6 7 6 . . 7 7 4 5 4 5 ~ ~ a4 Bn ðT i Þ·e 2dT i ·N ðD1n ðT i ÞÞ2 Cn ðT i Þ·e 2rT i ·NðD2n ðT i ÞÞ FROV n ðT i Þ
  8. 8. where the IT investment strategy ith cumulative normal probabilities for the D1and D2 Fuzzy goalare as follows: programming NðD1 ðT i ÞÞ N ðD2 ðT i ÞÞ model 2 3 a1 N ðD11 ðT i ÞÞ N ðD21 ðT i ÞÞ 6 7 6 N ðD12 ðT i ÞÞ ARO3 ðT i Þ ¼ a2 6 N ðD22 ðT i ÞÞ 7 7 ð10Þ 179 . 6 . 6 . . . . 7 7 . 6 . . 7 4 5 an N ðD1n ðT i ÞÞ N ðD2n ðT i ÞÞ D1 ðT i Þ D2 ðT i Þ 2 3 a1 D11 ðT i Þ D21 ðT i Þ 6 7 6 D ðT Þ D22 ðT i Þ 7 ARO4 ðTÞ ¼ a2 6 12 i 7 ð11Þ . 6 . 6 . . . . 7 7 . 6 . . 7 4 5 an D1n ðT i Þ D2n ðT i Þor equivalently: D1 ðT i Þ D2 ðT i Þ a1 2 3 ~ ~ LnðEðB1 ðT i ÞÞ=EðC1 ðT i ÞÞÞþð ðr 1 2d1 þs2 ðT i ÞÞ=2Þ · T i ~ LnðEðB1 ðT i ÞÞ=EðC1 ðT i ÞÞÞþð ðr1 2d1 2s2 ðT i ÞÞ=2Þ · T i ~ pffiffiffiffi 1 pffiffiffiffi 1 6 s1 ðT i Þ T i s2 ðT i Þ Ti 7 6 1 7 a2 6 LnðEðB2 ðT i ÞÞ=EðC2 ðT i ÞÞÞþð ðr2 2d2 þs2 ðT i ÞÞ=2Þ · T i 6 ~ ~ ~ LnðEðB2 ðT i ÞÞ=EðC2 ðT i ÞÞÞþð ðr2 2d2 2s2 ðT i ÞÞ=2Þ · T i ~ 7 7 6 pffiffiffiffi 2 pffiffiffi 2 7ARO4 ðT i Þ ¼ 6 s2 ðT i Þ T i s2 ðT i Þ T 7 6 7 . 6 . 6 . . . . 7 7 . 6 . . 7 6 LnðEðB ðT ÞÞ=EðC ðT ÞÞÞþ r 2d þs2 ðT Þ =2 · T ~n ~ 7 4 ~n i i ð ðffiffiffiffi n n i Þ Þ i pn LnðEðBn ðT i ÞÞ=EðCn ðT i ÞÞÞþð ðr n 2dn 2s2 ðT i ÞÞ=2Þ · T i 5 ~ pffiffiffiffi n an sn ðT i Þ T i sn ðT i Þ T i ð12Þ 2where E and s denote the possibilistic mean value and possibilistic varianceoperators as follows: ~ EðBðT i ÞÞ ~ EðCðT i ÞÞ s 2 ðT i Þ 2 3 a1 ~ EðB1 ðT i ÞÞ ~ EðC1 ðT i ÞÞ s2 ðT i Þ 1 6 7 6 EðB ðT ÞÞ ~ s2 ðT i Þ 7 ARO5 ðT i Þ ¼ a2 6 ~2 i EðC2 ðT i ÞÞ 2 7 ð13Þ 6 7 . . 6 . . . 7 . 6 . . . . . 7 . 7 6 4 5 ~ an EðBn ðT i ÞÞ ~ EðCn ðT i ÞÞ s2 ðT i Þ n
  9. 9. ˜ ˜BIJ Since Bi and Ci are trapezoidal fuzzy numbers, we use the formulas proposed by ´ Carlsson and Fuller (2003) to find their expected value and the variance:18,2 ~ ðBðT j ÞÞo þ ðBðT j ÞÞa ðBðT j ÞÞg 2 ðBðT j ÞÞb EðBi ðT j ÞÞ ¼ þ 2 6 o a ~ ðCðT j ÞÞ þ ðCðT j ÞÞ ðCðT j ÞÞ 2 ðCðT j ÞÞb g180 EðCi ðT j ÞÞ ¼ þ 2 6 ððBðT j ÞÞa 2 ðBðT j ÞÞo Þ2 ððBðT j ÞÞa 2 ðBðT j ÞÞo ÞððBðT j ÞÞb þ ðBðT j ÞÞg Þ s2 ðT j Þ ¼ i þ 4 6 ððBðT j ÞÞb þ ðBðT j ÞÞg Þ2 þ 24 ð14Þ Phase 4: prioritization of the IT investment strategies: risk considerations In this phase, the strategic IT investment board identifies the evaluation criteria and sub-criteria and uses GFAHP to measure the risk for each criterion and sub-criterion associated with the investment projects. This phase is divided into the following four steps. Step 4.1: identification of the criteria and sub-criteria for the GFAHP model. In this step, the strategic IT investment board will determine a list of the criteria and sub-criteria for the GFAHP model. Let c1 ; c2 ; . . . ; cp and sc1 ; sc2 ; . . . ; scq be the criteria and sub-criteria, respectively. Step 4.2: construction of the individual fuzzy pairwise comparison matrices. The hierarchal structure for ranking the IT Investments strategies in the risk dimension consists of four levels. The top level consists of a single element and each element of a given level dominates or covers some or all of the elements in the level immediately below. At the second level, the individual fuzzy pairwise comparison matrix of the p criteria of IT investment risk evaluated by strategic IT investment board member (ITIB)k will be as follows: c1 c2 . . . cp 2 k 3 ~ ~k ~k c1 6 b11 b12 . . . b1p 7 2 k 6 7 c 6 ~k ~k ~k 7 AR ¼ 2 6 b21 b22 . . . b2p 7 ~ ð15Þ . 6 . . 6 . . 7 . 7 . 6 . . ... . 7 . . 7 6 4 k cp b k k 5 ~ ~ b ... b~ p1 p2 pp Let the individual fuzzy comparison qualification between criteria i and j evaluated by strategic IT investment board member (ITIB)k be the following trapezoidal fuzzy numbers: o a b g ~k ¼ bij bk ; bk ; bk ; bk ð16Þ ij ij ij ij
  10. 10. Consequently, substituting equation (18) into matrix (17), the individual fuzzy Fuzzy goalcomparison qualification between criteria i and j evaluated by strategic IT investmentboard member (ITIB)k can be rewritten as: programming model C1 c2 ... Cp c1 2 3 ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ 11 11 11 11 12 12 12 12 ... ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ 1p 1p 1p 1p 2 k 6 c2 6 ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ 7 ... ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ 7 181ðAR Þ ¼ 6 21 ~ 6 21 21 21 22 22 22 22 2p 2p 2p 2p 7 7 .6 . .6 . . . . . . 7 7 6 . . ... . 7 4 5 cp ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ ððbk Þo ;ðbk Þa ;ðbk Þb ;ðbk Þg Þ k o k a k b k g ... ððbpp Þ ;ðbpp Þ ;ðbpp Þ ;ðbpp Þ Þ p1 p1 p1 p1 p2 p2 p2 p2 ð17ÞAt the third level, the individual fuzzy pairwise comparison matrix of IT investmentrisk sub-criteria with respect to p IT investment risk criteria evaluated by strategic ITinvestment board member (ITIB)k will be as follows: sc1 sc2 ... scq 2 k k k 3 d~ ~ d12 ... ~ d1q sc1 6 11 P k P P 7 3 k 6 k 7 ~ sc 6 d ~ ~ d22 ... ~k d2q 7 AR ¼ 2 6 21 P 7 ð18Þ . 6 P P7 . . 6 . . . 7 6 . . . 7 6 . k. ... k. 7 scq 4 ~k ~ ~ 5 dq1 dq2 ... dqq P P PThe individual fuzzy comparison qualification between sub-criterions i withsub-criterion j with respect to criterion p evaluated by strategic IT investment boardmember (ITIB)k are the following trapezoidal fuzzy numbers: k o a b g dij ¼ dk ; dk ; d k ; d k ~ ij ij ij ij ð19Þ p pTherefore, we have: sc1 sc2 ... scq sc1 2 3 ððdk Þo ;ðdk Þa ;ðdk Þb ;ðdk Þg Þp ððd k Þo ;ðdk Þa ;ðd k Þb ;ðd k Þg Þp 11 11 11 11 12 12 12 12 ... ððdk Þo ;ðd k Þa ;ðdk Þb ;ðdk Þg Þp 1q 1q 1q 1q 6 7 6 ððdk Þo ;ðdk Þa ;ðdk Þb ;ðdk Þg Þ ððd k Þo ;ðdk Þa ;ðd k Þb ;ðd k Þg Þ ... ððd k Þo ;ðd k Þa ;ðdk Þb ;ðdk Þg Þ 7 6 21 21 21 21 p 22 22 22 22 p 2q 2q 2q 2q 7 ~3 sc 6 7ðAR Þk ¼ 2 6 . . . 7 . 66 . . . 7 . . 4 . . ... . 7 5 ððdq1 Þ ;ðdq1 Þ ;ðdq1 Þ ;ðdq1 Þ Þp ððdq2 Þ ;ðdq2 Þ ;ðd k Þb ;ðd k Þg Þp k o k a k b k g k o k a q2 q2 ... ððd k Þo ;ðd k Þa ;ðdk Þb ;ðdk Þg Þp qq qq qq qq scq ð20ÞAt the fourth level, the individual fuzzy pairwise comparison matrix of n IT investmentstrategies with respect to q IT investment risk sub-criteria evaluated by strategicIT investment board member (ITIB)k will be as follows:
  11. 11. BIJ a1 a2 ... an18,2 2À k Á À Á À Á 3 r ~ ~k r12 ... ~k r1n a1 6 11 q q 7 q 4 k 6À k Á À k Á À Á 7 ~ a2 6 r21 q 6 ~ r22 ~ q ... ~k 7 r2n q 7 AR ¼ 6 7 ð21Þ182 . 6 . . 6 . . . 7 . 6 . . . ... . 7 . 7 6 7 4À k Á À Á À kÁ 5 an rn1 q ~ ~k rn2 q ... rnn q ~ The individual fuzzy comparison qualification between IT investment strategies i with IT investment strategy j with respect to sub-criterion q evaluated by strategic IT investment board member (ITIB)k are the following trapezoidal fuzzy numbers: o a b g ~k k k k k rij ¼ r ij ; r ij ; r ij ; r ij ð22Þ q q or equivalently: a1 a2 ... an a1 2 3 ððr 11 Þo ;ðr11 Þa ;ðr11 Þb ;ðr 11 Þg Þq ððr 12 Þo ;ðr12 Þa ;ðr 12 Þb ;ðr 12 Þg Þq ... ððr 1n Þo ;ðr 1n Þa ;ðr1n Þb ;ðr 1n Þg Þq k k k k k k k k k k k k 6 7 6 ððr k Þo ;ðr k Þa ;ðr k Þb ;ðr k Þg Þ ððr k Þo ;ðr k Þa ;ðr k Þb ;ðr k Þg Þ ... ððr k Þo ;ðr k Þa ;ðr k Þb ;ðr k Þg Þ 7 ðAR Þ k ¼ a2 6 21 ~4 6 21 21 21 q 22 22 22 22 q 2n 2n 2n 2n q7 7 6 7 . 6 . 6 . . . . . . 7 . 6 . . ... . 7 7 4 5 k o k a k b k g k o k a k b k g k o k a k b k g an ððr n1 Þ ;ðr n1 Þ ;ðrn1 Þ ;ðrn1 Þ Þq ððrn2 Þ ;ðrn2 Þ ;ðr n2 Þ ;ðrn2 Þ Þq ... ððrnn Þ ;ðrnn Þ ;ðrnn Þ ;ðrnn Þ Þq ð23Þ Step 4.3: construction of the weighted collective fuzzy pairwise comparison matrix. At the second level, the fuzzy weighted collective pairwise comparison matrix of p IT investment risk criteria will be as follows: c1 c2 ... cp c1 2 3 ððb11 Þo ;ðb11 Þa ;ðb11 Þb ;ðb11 Þg Þ ððb12 Þo ;ðb12 Þa ;ðb12 Þb ;ðb12 Þg Þ ... ððb1p Þo ;ðb1p Þa ;ðb1p Þb ;ðb1p Þg Þ 6 7 6 ððb Þo ;ðb Þa ;ðb Þb ;ðb Þg Þ ððb Þo ;ðb Þa ;ðb Þb ;ðb Þg Þ ... ððb Þo ;ðb Þa ;ðb Þb ;ðb Þg Þ 7 6 21 21 21 21 22 22 22 22 2p 2p 2p 2p 7 6 7 ~2 c AR ¼ 2 6 7 6 . . . . . . 7 .6 .6 . . ... . 7 7 .4 5 ððbp1 Þo ;ðbp1 Þa ;ðbp1 Þb ;ðbp1 Þg Þ ððbp2 Þo ;ðbp2 Þa ;ðbp2 Þb ;ðbp2 Þg Þ ... ððbpp Þo ;ðbpp Þa ;ðbpp Þb ;ðbpp Þg Þ cp ð24Þ
  12. 12. or: Fuzzy goal c1 c2 . . . cp programming 2~ ~ ~ 3 model c1 b11 b12 ... b1p 6~ ~ ~ 7 ~2 c 6 b21 b22 ... b2p 7 AR ¼ 2 6 7 ð25Þ 6 . . 7 . . 6 . 6 . . . . 7 183 . . ... . 7 4 5 cp ~ bp1 ~ bp2 ... ~ bppwhere: Pl k ! ~ k¼1 ðwðvpÞk Þ bij j ~ ðbij Þj ¼ Pl ð26Þ k¼1 wðvpÞkAt the third level, the fuzzy weighted collective pairwise comparison matrix of the ITinvestment risk sub-criteria with respect to the p IT investment risk criteria will be asfollows: sc1 sc2 ... scq 2 o a b g o a b g 3 sc1 ððd 11 Þ ; ðd 11 Þ ; ðd 11 Þ ; ðd 11 Þ Þp ððd 12 Þ ; ðd 12 Þ ; ðd12 Þ ; ðd 12 Þ Þp ... ððd 1q Þ ; ðd 1q Þa ; ðd 1q Þb ; ðd 1q Þg Þp o 6 7~3 sc 6 ððd 21 Þo ; ðd 21 Þa ; ðd 21 Þb ; ðd 21 Þg Þp ððd 22 Þo ; ðd 22 Þa ; ðd22 Þb ; ðd 22 Þg Þp ... ððd 2q Þo ; ðd 2q Þa ; ðd 2q Þb ; ðd 2q Þg Þ 7AR ¼ 2 6 7 . 6 . . . 7 . 6 . . . 7 . 6 . . ... . 7 4 5 scq ððd q1 Þ ; ðd q1 Þ ; ðd q1 Þb ; ðd q1 Þg Þp o a ððd q2 Þo ; ðd q2 Þa ; ðdq2 Þb ; ðd q2 Þg Þp ... o a b ððd qq Þ ; ðd qq Þ ; ðd qq Þ ; ðd qq Þ Þpg ð27Þor: sc1 sc2 ... scq 2 ~ ~ ~ 3 sc1 ðd11 ÞP ðd12 ÞP ... ðd1q ÞP 6 ~ ~ ~ 7 ~3 sc 6 ðd21 ÞP ðd22 ÞP ... ðd2q ÞP 7 AR ¼ 2 6 7 ð28Þ . 6 . . . 7 . 6 . . . 7 . 6 . . ... . 7 4 5 scq ~ ðdq1 ÞP ~ ðdq2 ÞP ... ~ ðdqq ÞPwhere: Pl k ! ~ ðwðvpÞk Þ dij k¼1 p ~ ðdij Þj ¼ Pl ð29Þ k¼1 wðvpÞkAt the fourth level, the fuzzy weighted collective pairwise comparison matrix of the nIT investment strategies with respect to the q IT investment risk sub-criteria will be asfollows:
  13. 13. BIJ a1 a1 a2 ... an 2 318,2 ððr 11 Þ ;ðr 11 Þ ;ðr 11 Þ ;ðr 11 Þ Þq ððr 12 Þ ;ðr 12 Þ ;ðr12 Þ ;ðr 12 Þ Þq ... ððr 1n Þ ;ðr 1n Þ ;ðr 1n Þb ;ðr1n Þg Þq o a b g o a b g o a 6 7 6 ððr 21 Þo ;ðr 21 Þa ;ðr 21 Þb ;ðr 21 Þg Þq ððr 22 Þo ;ðr 22 Þa ;ðr22 Þb ;ðr 22 Þg Þq ... ððr 2n Þo ;ðr 2n Þa ;ðr 2n Þb ;ðr2n Þg Þq 7 AR ¼ a2 6 ~4 7 6 . 6 . . . 7 . 6 . . . 7 . . . ... . 7 4 5184 o a b g o a b g o a an ððr n1 Þ ;ðr n1 Þ ;ðr n1 Þ ;ðrn1 Þ Þq ððr n2 Þ ;ðr n2 Þ ;ðrn2 Þ ;ðrn2 Þ Þq ... ððr nn Þ ;ðrnn Þ ;ðr nn Þ ;ðr nn Þ Þq b g ð30Þ or: a1 a2 ... an 2 3 a1 ð~11 Þq r ð~12 Þq r ... ð~1n Þq r 6 7 6 ð~21 Þq r ð~22 Þq r ... ð~2n Þq 7 r A 4 ¼ a2 6 ~ 7 ð31Þ 6 . . 6 . . . 7 . 6 . . . . ... . 7 . 7 4 5 an ð~n1 Þq r ð~n2 Þq r ... ð~nn Þq r where: Pl k¼1 ðwðvpÞk Þ rk ~ij rij ¼ ~ Pl ð32Þ k¼1 wðvpÞk Step 4.4: computation of the vector of the risk value for the IT investment strategies. The fuzzy composite vector of the deferrable IT investment strategies at the fourth level will be calculated based on the corresponding eigenvectors: ~ ~ ~2 FRV ¼ A 4 · A 3 · W R ¼ ½ FRV 1 FRV 2 ... FRV n ŠT ð33Þ or: FRV ¼ ½ððFRV Þo ; ðFRV Þa ; ðFRV Þb ; ðFRV Þg ÞR1 ððFRV Þo ; ðFRV Þa ; ðFRV Þb ; ðFRV Þg ÞR2 . . . ððFRV Þo ; ðFRV Þa ; ðFRV Þb ; ðFRV Þg ÞRn ފT ð34Þ where: ~ ~4 A4 ¼ b W R 1 ~4 W R2 ... ~4 W Rq c ð35Þ ~ ~3 A 3 ¼ b W R1 ~3 W R2 ... ~3 W Rp c ð36Þ h ~2 AR · e ~2 W R ¼ Lim 2 h h!1 ð37Þ ~ e T · AR · e
  14. 14. h ~3 AR · e Fuzzy goal ~3 W Rp ¼ Lim 3 h h!1 ð38Þ programming eT · A~ ·e R model 4 h ~ AR · e ~4 W Rq ¼ Lim 4 h h!1 ð39Þ ~ 185 e T · AR · e e ¼ ð1 1 . . . 1 ÞT ð40ÞPhase 5: development of the strategic IT investment planDecision makers also must consider the interaction between the real option and theinvestment risks. Therefore, in this phase, the IT investment strategy with the mostvalue is determined in terms of real option and risk values in Phases 2 and 3. For thispurpose, they are considered as the coefficients of the objective functions in thefollowing fuzzy preemptive goal programming model with a series of applicableconstraints. This phase is divided into the following three steps. Step 5.1: determination of the goal and priority levels. The goals in the fuzzypreemptive goal programming model can be written as follows: For the first priority level, there are two goals. These goals are equally important sothey can have the same weight:Max Z 1 ¼ E½FROV 1 ðT 1 ފ · x11 þ E½FROV 1 ðT 2 ފ · x12 þ · · · þ E½FROV 1 ðT m ފ · x1m þ E½FROV 2 ðT 1 ފ · x21 þ E½FROV 2 ðT 2 ފ · x22 þ · · · þ E½FROV 2 ðT m ފ · x2m þ . . . E½FROV n ðT 1 ފ · xn1 þ E½FROV n ðT 2 ފ · xn2 þ · · · þ E½FROV n ðT m ފ · xnm Min Z 2 ¼ EðFRV 1 Þ · ðx11 þ x12 þ · · · þ x1m Þ þ EðFRV 2 Þ · ðx21 þ x22 þ · · · þ x2m Þþ · · · þ EðFRV n Þ · ðxn1 þ xn2 þ · · · þ xnm ÞFor the second priority level, we have: f 1 ðx11 ; x12 ; . . . ; xnm Þ # 0 f 2 ðx11 ; x12 ; . . . ; xnm Þ # 0 . . . f r ðx11 ; x12 ; . . . ; xnm Þ # 0 xi ¼ 0; 1 ði ¼ 1; 2; . . . ; nÞ
  15. 15. BIJ Max Z 1 ¼ E½FROV 1 ðT 1 ފ · x11 þ E½FROV 1 ðT 2 ފ · x12 þ · · · þ E½FROV 1 ðT m ފ · x1m þ18,2 E½FROV 2 ðT 1 ފ · x21 þ E½FROV 2 ðT 2 ފ · x22 þ · · · þ E½FROV 2 ðT m ފ · x2m þ . . . E½FROV n ðT 1 ފ · xn1 þ E½FROV n ðT 2 ފ · xn2 þ · · · þ E½FROV n ðT m ފ · xnm186 Min Z 2 ¼ EðFRV 1 Þ · ðx11 þ x12 þ · · · þ x1m Þ þ EðFRV 2 Þ · ðx21 þ x22 þ · · · þ x2m Þ þ · · · þ EðFRV n Þ · ðxn1 þ xn2 þ · · · þ xnm Þ Subject to: (Model P) x11 þ x12 þ · · · þ x1m # 1 x21 þ x22 þ · · · þ x2m # 1 . . . xn1 þ xn2 þ · · · þ xnm # 1 f 1 ðx11 ; x12 ; . . . ; xnm Þ # 0 f 2 ðx11 ; x12 ; . . . ; xnm Þ # 0 . . . f r ðx11 ; x12 ; . . . ; xnm Þ # 0 xij ¼ 0; 1 ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ where f i ðx1 ; x2 ; . . . ; xn Þ are given functions of the n investments. Step 5.2: computation of the goal values. In this step, instead of trying to optimize each objective function, the strategic IT investment board will specify a realistic goal or target value that is the most desirable value for that function. Step 5.3: construction of the proposed goal programming model. The first objective function is to be maximized and the second objective function is to be minimized. Therefore, the proposed fuzzy goal programming model for the above two-objective strategic IT investment decision will be the following single-objective model: À Á Min D ¼ P 1 sþ þ s2 þ P 2 s2 þ · · · þ P rþ2 s2 1 2 3 r Subject to: (Model F) E½FROV 1 ðT 1 ފ · x11 þ E½FROV 1 ðT 2 ފ · x12 þ · · · þ E½FROV 1 ðT m ފ · x1m þ E½FROV 2 ðT 1 ފ · x21 þ E½FROV 2 ðT 2 ފ · x22 þ · · · þ E½FROV 2 ðT m ފ · x2m þ . . . E½FROV n ðT 1 ފ · xn1 þ E½FROV n ðT 2 ފ · xn2 þ · · · þ E½FROV n ðT m ފ · xnm S2 2 Sþ ¼ l1 1 1
  16. 16. EðFRV 1 Þ · ðx11 þ x12 þ · · · þ x1m Þ þ EðFRV 2 Þ · ðx21 þ x22 þ Fuzzy goal · · · þ x2m Þ þ · · · þ EðFRV n Þ · ðxn1 þ xn2 þ · · · þ xnm Þ þ s2 2 sþ ¼ u1 2 2 programming f 1 ðx11 ; x12 ; . . . ; xnm Þ þ sþ þ sþ ¼ 0 model 3 3 f 2 ðx11 ; x12 ; . . . ; xnm Þ þ sþ þ s2 ¼ 0 4 4 . . . 187 f r ðx11 ; x12 ; . . . ; xnm Þ þ sþ þ s2 ¼ 0 rþ2 rþ2 x11 þ x12 þ · · · þ x1m # 1 x21 þ x22 þ · · · þ x2m # 1 . . . xn1 þ xn2 þ · · · þ xnm # 1 xij ¼ 0; 1 ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ sþ ; s2 h h $0 ðh ¼ 1; 2; . . . ; r þ 2Þ sþ · s2 ¼ 0 h hThe optimal solution for model (F) is the deferrable IT investment strategy with themost values at the time Ti. Next, we present a numerical example to demonstrate theimplementation process of this framework.3. Case studyWe implemented the proposed model at Mornet[1], a large mortgage company in thecity of Philadelphia with an urgent need to select an optimal IT investment strategy fortheir deferrable investment opportunities. In Phase 1, the chief executive officer instituted a committee of four strategic ITinvestment board members, including: (ITIB)1. The chief operating officer. (ITIB)2. The chief information officer. (ITIB)3. The heads of the business unit. (ITIB)4. The chief financial officer.In Phase 2, the investment board identifies five different types of deferrable investmentopportunities with the following characteristics (Table I) as suggested by Carlsson et al.(2007): a1. Project 1 has a large negative estimated NPV (due to huge uncertainties) and can be deferred up to two years (v(FNPV) , 0, T ¼ 2). a2. Project 2 includes positive NPV with low risks and has no deferral flexibility (v(FNPV) . 0, T ¼ 0).
  17. 17. BIJ a3. Project 3 has revenues with large upward potentials and managerial flexibility,18,2 but its “reserve costs” (c) are very high. a4. Project 4 requires a large capital expenditure once it has been undertaken and has a deferral flexibility of a maximum of one year. a5. Project 5 represents a small flexible project with low revenues, but it opens the188 possibility of further projects that are much more profitable. In Phase 3, the fuzzy real option values of the five different deferrable investment opportunities shown in Figure 2 were determined for years 1 and 2. In Phase 4, the strategic IT investment board determined the GFAHP three criteria of firm-specific risks, development risks and external environment risks as suggested by Benaroch (2002). The firm-specific risks were further divided into four sub-criteria: organizational risks, user risks, requirement risks and structural risks. Deferral time Project 1 Project 2 Project 3 Project 4 Project 5 0 FNPV ¼ ((75%), FNPV ¼ (12%, FNPV ¼ (5%, FNPV ¼ ((12%), FNPV ¼ ((5%),Table I. 17%, 15%, 126%) 20%, 45%, 56%) 24%, 17%, 218%) 85%, 71%, 6%) 12%, 4%, 358%)The five deferrable IT 1 U U U Uinvestment opportunities 2 U U U Deferral Project Project Project Project Project time 1 2 3 4 5 0 FNPV = FNPV = FNPV = FNPV = FNPV = ((75%),17%,15%,126%) (12%,20%,45%,56%) (5%,24%,17%,218%) ((12%),85%,71%,6%) ((5%),12%,4%,358%) M = (10.5%) M = 17.8% M = 48.0% M = 25.7% M = 62.5% s = 71.5% s = 24% s = 56.0% s = 62.0% s = 81.0% 1 FROV1 = FROV1 = FROV1 = FROV1 = ((90%),20%,18%,151%) (6%,26%,19%,240%) ((15%),106%,89%,8%) ((6%),13%,4%,394%) M = (12.6%) M = 52.8% M = 32.1% M = 68.8% s = 85.8% s = 61.6% s = 77.5% s = 89.1%Figure 2.The fuzzy real option 2values of the five FROV2 = FROV2 = FROV2 =deferrable IT investment ((104%),23%,21%,174%) (7%,31%,23%,288%) ((7%),14%,5%,433%) M = (14.5%) M = 63.4% M = 75.7%opportunities s = 98.7% s = 73.9% s = 98.0%
  18. 18. The development risks were further divided into two sub-criteria: team risks and Fuzzy goalcomplexity risks. External environment risks were further divided into two sub-criteria:competition risks and market risks. programming Next, the possibilistic mean risk values of the investment opportunities presented in modelTable II were calculated. In Phase 5, assuming a per annum investment, the deferrable IT investment strategywith the most value was determined using the following two-objective decision-making 189model: Min Z 2 ¼ 0:45ðx10 þ x11 þ x12 Þ þ 0:1x20 þ 0:35ðx30 þ x31 þ x32 Þ þ 0:15ðx40 þ x41 Þ þ 0:05ðx50 þ x51 þ x52 ÞSubject to: (Model P) x10 þ x11 þ x12 # 1 x21 # 1 x30 þ x31 þ x32 # 1 x40 þ x41 # 1 x50 þ x51 þ x52 # 1 x10 þ x20 þ x30 þ x40 þ x50 # 1 x11 þ x31 þ x41 þ x51 # 1 x12 þ x32 þ x52 # 1 x10 ; x11 ; x12 ; x20 ; x30 ; x31 ; x32 ; x40 ; x41 ; x50 ; x51 ; x52 ¼ 0; 1Therefore, the goal programming model for the above two-objective strategic ITinvestment decision will be the following single objective model: À Á Min D ¼ P 1 · s2 þ sþ 1 2Subject to: (Model F) ð20:105Þx10 þ ð20:126Þ · x11 þ ð20:145Þ · x12 þ 0:178x20 þ 0:48x30 þ 0:528x31 þ 0:634x32 þ 0:257x40 þ 0:321x41 þ 0:625x50 þ 0:688x51 þ 0:757x52 À Á þ s2 2 sþ ¼ 1:5 1 1 0:45ðx10 þ x11 þ x12 Þ þ 0:1x20 þ 0:35ðx30 þ x31 þ x32 Þ þ 0:15ðx40 þ x41 Þ À Á þ 0:05ðx50 þ x51 þ x52 Þ þ s2 2 sþ ¼ 0:6 2 2 x10 þ x11 þ x12 # 1 x20 # 1 Table II.Project 1 Project 2 Project 3 Project 4 Project 5 The possibilistic mean risk value of the ITE(FRV1) ¼ 0.45 E(FRV2) ¼ 0.10 E(FRV3) ¼ 0.35 E(FRV4) ¼ 0.15 E(FRV5) ¼ 0.05 investment opportunities

×