1.a fuzzy

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BIJ
18,2 A fuzzy goal programming
model for strategic
information technology
172
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 organizations
Benchmarking: An International continue to invest large sums of money in IT in anticipation of a material return on their
Journal investment (Willcocks and Lester, 1996). The selection of appropriate IT investments has
Vol. 18 No. 2, 2011
pp. 172-196
q Emerald Group Publishing Limited
1463-5771
The authors would like to thank the anonymous reviewers and the Editor for their insightful
DOI 10.1108/14635771111121667 comments and suggestions.

been one of the most significant business challenges of the last decade. Powell (1992) Fuzzy goal
has studied the similarities and differences between IT investments and other capital programming
investments in organizations. He notes that IT investments are undertaken by
organizations to gain competitive advantage, to improve productivity, to enable new ways model
of managing and organizing and to develop new businesses. Appropriate strategic IT
investments 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 173
of cost, time or benefits. The rapid growth of IT investments has imposed tremendous
pressure on management to take into consideration risks and payoffs promised by the
investment in their decision making.
A review of the current literature offers several IT investment evaluation methods
that provide frameworks for the quantification of risks and benefits. The net present
value (NPV) (Hayes and Abernathy, 1980; Kaplan and Atkinson, 1998), return on
investment (Brealey and Myers, 1998; Farbey et al., 1993; Kumar, 2002; Luehrman,
1997), cost benefit analysis (Schniederjans et al., 2004), information economics (Bakos
and 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 to
assess the risks and payoffs associated with IT investments.
In addition to the above mentioned traditional quantitative approaches, there is a
stream of research studies which emphasizes real option analysis (ROA). The ROA differs
from the traditional methods in terms of priceability of the underlying investment project
(McGrath, 1997). With the traditional methods, the underlying investment project of an
option is priced as known (Black and Scholes, 1973) while in IT investment situations the
price of an underlying investment is rarely known (McGrath, 1997). The ROA uses three
basic 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 by
taking into consideration the changing nature of business strategies and
organizational 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 underlying
markets can be imitated accurately as a process. Although this assumption may hold for
some quite efficiently traded financial securities, it may not hold for real investments that
do not have existing markets (Collan et al., 2009). Recently, a simple novel approach to
ROA called the Datar-Mathews method (Datar and Mathews, 2004, 2007; Mathews and
Salmon, 2007) was proposed where the real option value is calculated from a pay-off
distribution, derived from a probability distribution of the NPV for an investment project
generated with a Monte-Carlo simulation. This approach does suffer from the market
process assumptions associated with the Black-Scholes method (Black and Scholes, 1974).
When valuating an investment using ROA, it is required to estimate several
parameters (i.e. expected payoffs and costs or investment deferral time). However, the
estimation of uncertain parameters in this valuation process is often very challenging.
Most traditional methods use probability theory in their treatment of uncertainty.

BIJ Fuzzy logic and fuzzy sets can represent ambiguous, uncertain or imprecise information
18,2 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

.
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.
programming
This paper is organized into five sections. In Section 2, we illustrate the details of the model
proposed framework followed by a case study in Section 3. In Section 4, we present
discussion and practical perspectives and in Section 5, we conclude with our conclusions
and future research directions. 175
2. The proposed framework
The mathematical notations and definitions used in our model are presented in the
Appendix. The framework shown in Figure 1 is proposed to assess alternative IT
investment strategies. The framework consists of several steps modularized into five
phases.

Phase 1: establishment of the IT investment board
We institute a strategic IT investment board to acquire pertinent investment
information. Executive management is typically responsible for creating the board,
specifying its responsibilities and defining its resources. Let us assume that l strategic
IT investment board members are selected to participate in the evaluation process:
ITIB ¼ ½ðITIBÞ1 ; ðITIBÞ2 ; . . . ; ðITIBÞk ; . . . ; ðITIBÞl Š

Phase 2: identification of the IT investment strategies
Next, the strategic IT investment board identifies a set of alternative deferrable IT
investment strategies. Let us assume that n alternative IT investments with the
maximum deferral time of Tm are under consideration:
a ¼ ½a1 ; a2 ; . . . ; ai ; . . .an Š

Phase 3: prioritization of the IT investment strategies: real option considerations
In this phase, the real options equations suggested by Dos Santos (1994) are used to
prioritize IT investments strategies. This phase is divided into the following three steps.
Step 3.1: construction of the individual real option matrices. The following individual
real 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 this
study, we use trapezoidal fuzzy sets. A major advantage of trapezoidal fuzzy numbers is

BIJ
Phase 1
18,2 Establishment of the IT investment board

Phase 2
Identification of the IT investment strategies
176
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 goal
Figure 1. programming model
The proposed framework

that many operations based on the max-min convolution can be replaced by direct Fuzzy goal
arithmetic operations (Dubois and Prade, 1988). The following trapezoidal fuzzy numbers
are used for the individual fuzzy present values of the expected cash flows and the cost of
programming
the 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
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
Consequently, substituting equation (2) into matrix (1), the individual real option
matrices 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
~k
ARO1 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

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 board
18,2 member:
W ðvpÞ ¼ ½wðvpÞ1 ; wðvpÞ2 ; . . . ; wðvpÞj ; . . . ; wðvpÞl Š ð4Þ
Therefore, in order to form a fuzzy weighted collective real option matrix, the individual
178 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

where the IT investment strategy ith cumulative normal probabilities for the D1and D2 Fuzzy goal
are 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 Þ

or 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
7
ARO4 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
2
where E and s denote the possibilistic mean value and possibilistic variance
operators 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

˜ ˜
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
g
180 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

Consequently, substituting equation (18) into matrix (17), the individual fuzzy Fuzzy goal
comparison qualiﬁcation between criteria i and j evaluated by strategic IT investment
board 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
181
ðAR 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

ð17Þ
At the third level, the individual fuzzy pairwise comparison matrix of IT investment
risk sub-criteria with respect to p IT investment risk criteria evaluated by strategic IT
investment 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 P

The individual fuzzy comparison qualiﬁcation between sub-criterions i with
sub-criterion j with respect to criterion p evaluated by strategic IT investment board
member (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 p

Therefore, 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 7
ðAR 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

ð20Þ
At the fourth level, the individual fuzzy pairwise comparison matrix of n IT investment
strategies with respect to q IT investment risk sub-criteria evaluated by strategic
IT investment board member (ITIB)k will be as follows:

BIJ a1 a2 ... an
18,2 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 ð21Þ
182 . 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

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 ... ~
bpp

where:
Pl k !
~
k¼1 ðwðvpÞk Þ bij
j
~
ðbij Þj ¼ Pl ð26Þ
k¼1 wðvpÞk

At the third level, the fuzzy weighted collective pairwise comparison matrix of the IT
investment risk sub-criteria with respect to the p IT investment risk criteria will be as
follows:
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 7
AR 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

ð27Þ
or:
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 ÞP

where:
Pl k !
~
ðwðvpÞk Þ dij
k¼1
p
~
ðdij Þj ¼ Pl ð29Þ
k¼1 wðvpÞk

At the fourth level, the fuzzy weighted collective pairwise comparison matrix of the n
IT investment strategies with respect to the q IT investment risk sub-criteria will be as
follows:

BIJ a1
a1 a2 ... an
2 3
18,2 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 5
184 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 ÞŠT
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

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 ð40Þ

Phase 5: development of the strategic IT investment plan
Decision makers also must consider the interaction between the real option and the
investment risks. Therefore, in this phase, the IT investment strategy with the most
value is determined in terms of real option and risk values in Phases 2 and 3. For this
purpose, they are considered as the coefﬁcients of the objective functions in the
following fuzzy preemptive goal programming model with a series of applicable
constraints. This phase is divided into the following three steps.
Step 5.1: determination of the goal and priority levels. The goals in the fuzzy
preemptive goal programming model can be written as follows:
For the ﬁrst priority level, there are two goals. These goals are equally important so
they can have the same weight:

Max Z 1 ¼ E½FROV 1 ðT 1 ÞŠ · x11 þ E½FROV 1 ðT 2 ÞŠ · x12 þ · · · þ E½FROV 1 ðT m ÞŠ · x1m þ
E½FROV 2 ðT 1 ÞŠ · x21 þ E½FROV 2 ðT 2 ÞŠ · x22 þ · · · þ E½FROV 2 ðT m ÞŠ · x2m þ
.
.
.

E½FROV n ðT 1 ÞŠ · xn1 þ E½FROV n ðT 2 ÞŠ · xn2 þ · · · þ E½FROV n ðT m ÞŠ · 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 Þ

For 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

BIJ Max Z 1 ¼ E½FROV 1 ðT 1 ÞŠ · x11 þ E½FROV 1 ðT 2 ÞŠ · x12 þ · · · þ E½FROV 1 ðT m ÞŠ · x1m þ
18,2 E½FROV 2 ðT 1 ÞŠ · x21 þ E½FROV 2 ðT 2 ÞŠ · x22 þ · · · þ E½FROV 2 ðT m ÞŠ · x2m þ
.
.
.
186
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)

.
.
.

S2 2 Sþ ¼ l1
1 1

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 h

The optimal solution for model (F) is the deferrable IT investment strategy with the
most values at the time Ti. Next, we present a numerical example to demonstrate the
implementation process of this framework.

3. Case study
We implemented the proposed model at Mornet[1], a large mortgage company in the
city of Philadelphia with an urgent need to select an optimal IT investment strategy for
their deferrable investment opportunities.
In Phase 1, the chief executive officer instituted a committee of four strategic IT
investment 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 investment
opportunities 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).

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 the
188 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 U
investment 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 2
values 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%

The development risks were further divided into two sub-criteria: team risks and Fuzzy goal
complexity 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 model
Table II were calculated.
In Phase 5, assuming a per annum investment, the deferrable IT investment strategy
with the most value was determined using the following two-objective decision-making 189
model:
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 Þ
Subject 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; 1
Therefore, the 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 · s2 þ sþ
1 2
Subject 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 IT
E(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

1.a fuzzy

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