Advances in Systems Science and Applications (2014) Vol.14 No.1 22-35 Optimizing Investment Decisions for a Set of Projects Under Uncetainty of Future Returns Konstantin N. Nechval1 and Nicholas A. Nechval2 1Applied Mathematics Department, Transport and Telecommunication Institute Lomonosov Street 1, LV-1019, Riga, Latvia 2Statistics Department, EVF Research Institute, University of Latvia Raina Blvd 19, LV-1050, Riga, Latvia Abstract Project portfolio selection is a crucial decision in many organizations, which must make informed decisions on investment, where the appropriate distribution of in- vestment is complex, due to varying levels of risk, resource requirements, and interaction among the proposed projects. In this paper, we present a mathemat- ical model of investment for a set of projects under uncertainty of future returns. The model addresses the problem of an investor with access to a limited pool of capital, who makes decisions on investments. The problem is to decide how much to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. We discuss optimal investment decisions for the cases where the return from investment is a random variable. The single-period and multi period cases of investment decisions are considered. The paper presents closed form solutions for commonly adopted utility functions. Keywords Projects, future return, uncertainty, investment decisions, optimiza- tion 1 Introduction Project portfolio investment is the periodic activity involved in investing a port- folio of projects, that meets an organizations stated objectives without exceeding available resources or violating other constraints. Some of the issues that have to be addressed in this process are the organization’s objectives and priorities, financial benefits, intangible benefits, availability of resources, and risk level of the project portfolio [1]. Difficulties associated with project portfolio investment result from several factors: (i) there are multiple and often-conflicting objectives, (ii) some of the objectives may be qualitative, (iii) uncertainty and risk can affect projects, (iv) the project portfolio may need to be balanced in terms of important factors, such as risk and time to completion, (v) some projects may be interdependent, and (vi) the number of feasible portfolios is often enormous. In addition to these difficulties, due to resource limitations there are usually constraints such as finance, work force, and facilities or equipment, to be consid- ered. As some researchers have noted [2], the major reason why some projects are selected but not completed is that resource limitations are not always formally Advances in Systems Science and Applications (2014) Vol.14 No.1 23 included in the project selection process. In cases where resource limitations are at fault for a failed project, a selection model that incorporated resource limita- tions could have aided the decision maker in avoiding such mistakes [1]. Portfolio selection becomes more complex when resource availability and consumption are not uniform over time. There are many different techniques that can be used to estimate, evaluate, and choose project portfolios [3-4]. Some of these techniques are not widely used because they address only some of the above issues, they are too complex and require too much input data, they may be too difficult for decision makers to understand and use, or they may not be used in the form of an organized process [5]. Among all of the techniques that are available, optimization techniques are the most fundamental quantitative tool for project portfolio selection and address most of the important issues [6]. However, they have largely failed to gain user acceptance [7], and few modeling approaches, from a variety of optimization ap- proaches that have been developed, are being utilized as aids to decision making in this area [8]. According to Hess “management science has failed altogether to implement project selection models; we have proposed more and more sophisti- cation with less and less practical impact” [9]. One of the major reasons for the failure of traditional optimization techniques is that they prescribe solutions to portfolio selection problems without allowing for the judgment, experience and insight of the decision-maker [7]. A literature review we conducted in this field clearly showed that, although there are many different methods for project evaluation and portfolio selection that have their own advantages, no single technique addresses all of the issues that should be considered in project portfolio selection [10]. Among published methodologies for project portfolio selection, there has been little progress to- wards achieving an integrated framework that: (a) simultaneously considers all the different criteria in determining the most suitable project portfolio, (b) takes advantage of the best characteristics of existing methods by decomposing the process into a flexible and logical series of activities and applying the most ap- propriate technique(s) at each stage. Well-known pragmatic difficulties that make project selection challenging in- clude the following: (1) Success uncertainty (unknown rewards): Whether (or how well) a project will succeed, both technically and in the market, may be un- certain. (2) Changing opportunities (randomly arriving opportunities): The op- portunity of funding a project may be uncertain. New projects are continually be- ing proposed throughout the year, stimulated by changes in technology and mar- ket opportunities that might help the company to achieve its goal. (3) Need for quick funding decisions (on-line decision): The organizations proposing projects lack a comprehensive view of all projects activities company-wide. It is essential 24 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... for top management to give the organization prompt accept/revise/reject deci- sions, since such feedback helps to coordinate their activities. (4) Combinatorial complexity: The costs and benefits from different projects may interact. 2 Problem Statement The problem we will examine can be formulated as follows. Let w0 denote the initial wealth (measured in monetary units) of the investor and assume that there are m projects (opportunities) for investments which belong to a set of m risk categories, with corresponding random rates of return r1, r2, ..., rm, among which the investor can allocate his wealth. The investor can also invest in a project of a riskless category offering a sure rate of return s. If we denote by u1, ..., um the corresponding amounts to invest in the projects belonging to the set of m risk categories and by (w0 − u1 −Λ− um) the amount of investment in the project of the riskless category, the final wealth is given by w1 = s(w0 − u1 − · · · − um) + m∑ j=1 rjuj , (1) or equivalently w1 = sw0 + m∑ j=1 (rj − s)uj . (2) The objective is to maximize over u1, ..., um, E{g(ω1)} (3) where g is a known utility function for the investor, w1 is a random variable. We assume that the given expected value is well defined and finite for all w0, ui, and that g is concave and twice continuously differentiable. We will not impose constraints on u1, ..., um. This is necessary in order to obtain the results in convenient form. A few additional assumptions will be made later. If a given amount w0 of the initial wealth is available for investment within N time periods, we are interested in determining decision rules on how much to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. 3 Investment Decisions for the Single Case Let us consider the preceding problem for every value of initial wealth and denote by µ∗ j = µ∗ j (ω0), j = 1, ...,m, the optimal amounts to invest in projects belonging to the set of m risk categories when the initial wealth is w0. We say that the investment portfolio {µ1 j (ω0), ..., µ m j (ω0)} is partially separated if u∗j (w0) = cjh(w0), j = 1,...,m (4) Advances in Systems Science and Applications (2014) Vol.14 No.1 25 where cj , j = 1, ...,m, are fixed constants and h(w0) is a function of w0 (which is the same for all j). When partial separation holds, the ratios of amounts invested in projects be- longing to the set of m risk categories are fixed and independent of the initial wealth; that is, u∗j (w0) u∗k(w0) = cj ck , forj, k ∈ {1, ...,m}, ck ̸= 0 (5) Actually, in the cases we will examine, when partial separation holds, the invest- ment portfolio {µ1 j (ω0), ..., µ m j (ω0)} will be shown to consist of affine (linear plus constant) functions of w0 that have the form u∗j (w0) = cj [a+ bsw0], j = 1,...,m (6) where a and b are constants characterizing the utility function g. In the special case where a=0 in (6), we say that the optimal investment portfolio is completely separated in the sense that the ratios of the amounts invested in projects belong- ing to both the set of m risk categories and the riskless category are fixed and independent of initial wealth. Here the following theorem holds. Theorem 1. If the utility function satisfies − g′(w1) g′′(w1) = a+ bw1, forallw1 (7) where g′ and g′′ denote the first and second derivatives of g, respectively, and a and b are some scalars, then the optimal investment portfolio is given by (6). Furthermore, if G(w0) is the optimal value of the problem, i.e., G(w0) = max u1,...,um E{g(w1)} (8) then we have −G′(w0) G′′(w0) = a s + bw0, forallw0 (9) Proof. Let us assume that an optimal investment portfolio exists and is of the form u∗j (w0) = cj(w0)[a+ bsw0], j = 1,...,m (10) where cj(w0), j = 1, ...,m, are some differentiable functions. We will prove that dcj(w0)/dw0 = 0 for all w0 and hence the functions cj must be constant. We have for every w0, by the optimality of µ∗ j (ω0) for j = 1, ...,m, dE{g(w1)} duj = E { g′ [ sw0 + m∑ k=1 (rk − s)ck(w0)(a+ bsw0) ] (rj − s) } = 0 26 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... j = 1(1)m (11) Differentiating the m equations in (11) with respect to w0 yields E   (r1 − s)2 · · · (r1 − s)(rm − s) · · · · · · · · · · · · (rm − s)(r1 − s) · · · (rm − s)2  g′′(w1)(a+ bsw0) ×  dc1(w0)/dw0 ... dcm(w0)/dw0  = −  E { g′′(w1)(r1 − s)s [ 1 + m∑ k=1 (rk − s)ck(w0)b ]} ... E { g′′(w1)(rm − s)s [ 1 + m∑ k=1 (rk − s)ck(w0)b ]}  = −  E { g′′(w1)(r1 − s)s [ 1 + m∑ j=1 (rj − s)cj(w0)b ]} ... E { g′′(w1)(rm − s)s [ 1 + m∑ j=1 (rj − s)cj(w0)b ]}  . (12) Using relation (7), we have g′′(w1) = − g′(w1) a+ b [ sw0 + m∑ j=1 (rj − s)cj(w0)(a+ bsw0) ] = − g′(w1) (a+ bsw0) [ 1 + m∑ j=1 (rj − s)cj(w0)b ] . (13) Substituting in (12) and using (11), we have that the right side of (12) is the zero vector. The matrix on the left in (12), except for degenerate cases, can be shown to be nonsingular. Assuming that it is indeed nonsingular, we obtain dcj(w0) dw0 = 0, j = 1,...,m (14) Advances in Systems Science and Applications (2014) Vol.14 No.1 27 and cj(w0) = cj , where cj are some constants, thus proving (6). We now turn our attention to proving relation (9). We have G(w0) = E{g(w1)} = E g s 1 + m∑ j=1 (rj − s)cjb w0 + m∑ j=1 (rj − s)cja  (15) and hence G′(w0) = E g′(w1)s 1 + m∑ j=1 (rj − s)cjb  (16) G′′(w0) = E g′′(w1)s 2 1 + m∑ j=1 (rj − s)cjb 2 (17) The last relation after some calculation and using (13) yields G′′(w0) = − E { g′(w1)s [ 1 + m∑ j=1 (rj − s)cjb ]} s a+ bsw0 (18) By combining (16) and (18), we obtain the desired result: −G′(w0) G′′(w0) = a s + bw0 (19) This ends the proof. It can be shown that the following utility functions satisfy (19): exponential: −e−w/a, forb = 0 (20) Logarithmic ln(ω + a), forb = 1 (21) power [1/(b− 1)](a+ bω)1−(1/b), otherwise (22) Naturally in our problem only concave utility functions from this class are ad- missible. Furthermore, if a utility function that is not defined over the whole real line is used, the problem should be formulated in a way that ensures that all possible values of the resulting final wealth are within the domain of definition of the utility function. 28 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... 4 Investment Decisions for the Multiperiod Case It is now easy to extend the one-period result of the preceding analysis to the multiperiod case. We will assume that the current wealth can be used to invest amounts in projects at the beginning of each of N consecutive time periods. If we denote: wυ the wealth of the investor at the beginning of the υth period, uj(υ) the amount to invest in project belonging to the jth risk category at the beginning of the υth period, rj(υ) the rate of return of project which belongs to the jth risk category at the beginning of the υth period, sυ the rate of return of project which belongs to the riskless category at the beginning of the υth period then we have (in accordance with the single-period model) the system equation wν+1 = sνwν + m∑ j=1 (rj(ν) − sν)uj(ν),v = 0, 1, ..., N − 1 (23) We assume that the vectors rυ = (r1(υ), ..., rm(υ)), υ = 0, ..., N − 1, are inde- pendent with given probability distributions that result in finite expected values throughout the following analysis. The objective is to maximize E{g(wN )}, the expected utility of the terminal wealth wN , where we assume that g satisfies for all w − g′(w) g′′(w) = a+ bw (24) Applying the dynamic programming algorithm to this problem [11-12] , we have GN (ωN ) = g(ωN ) (25) Gν(wν) = max u1(ν),...,um(ν) E Gν+1 sνwν + m∑ j=1 (rj(ν) − sν)uj(ν)  , ν = 0, 1, ..., N−1 (26) From the solution of the one-period problem we have that the optimal policy at the beginning of period N − 1 is of the form u∗N−1(wN−1) = cN−1[a+ bsN−1wN−1], (27) where cN−1 is an appropriate m-dimensional vector, µ∗ N−1 = [µ∗ 1(N−1), ..., µ ∗ m(N−1)] ′. Furthermore, we have −G′ N−1(w) G′′ N−1(w) = a sN−1 + bw. (28) Advances in Systems Science and Applications (2014) Vol.14 No.1 29 Hence, applying the result of this section in (26) for the next to the last period, we obtain the optimal policy u∗N−2(wN−2) = cN−2 ( a sN−1 + bsN−2wN−2 ) (29) where cN−2 is again an appropriate m-dimensional vector. Proceedings similarly, we have for the υth period u∗ν(wν) = cν ( a sN−1 · · · sν+1 + bsνwν ) (30) where cυ, υ = 0, 1, ..., N − 1, are m-dimensional vectors that depend on the prob- ability distributions of the rates of return rj(υ) of projects belonging to the risk categories and are determined by optimization of the expected value of the opti- mal cost-to-go functions Gυ. These functions satisfy −G′ ν(w) G′′ ν(w) = a sN−1 · · · sν + bw, ν = 0, 1, ..., N − 1 (31) Thus one can see that the investor, when faced with the opportunity to reuse sequentially his wealth, uses a policy similar to that of the single-period case. 5 Multiperiod Project Portfolio Investment with Markowitz Mean-Variance Optimization To our knowledge, no analytical or efficient numerical method for finding the optimal multiperiod portfolio policy for the constrained mean-variance formula- tion of Markowitz has been reported in the literature. This section presents an optimal solution to the constrained mean-variance formulation of the multiperi- od project portfolio investment problem. We consider a portfolio with (m + 1) risky projects, with random rates of returns. Let w0 be an initial wealth of an investor at time 0. The investor can allocate his wealth among the (m + 1) projects. The wealth can be reallocated among the (m + 1) projects at the be- ginning of each of the following (T − 1) consecutive time periods. The rates of return of the risky projects at time period τ within the planning horizon are denoted by a vector rτ = [rτ(0), rτ(1), ..., rτ(m)], where rτ(j) is the random return for project j at time period τ . It is assumed in this paper that vec- tors rτ , = 0, 1, ..., T − 1, are statistically independent and return rτ has a mean E{rτ} = [E{rτ(0)}, E{rτ(1)}, ..., E{rτ(m)}]′ and a covariance Cov{rτ} =  στ(00) · · · στ(0m) ... . . . ... στ(0m) · · · στ(mm)  (32) 30 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... which can be found from the data of observations. Let wτ be the wealth of the investor at the beginning of the τth period, and let uτ(j), j ∈ {1, ...,m}, be the amount invested in the jth risky project at the beginning of the τth time period. The amount investigated in the 0th risky project at the beginning of the τth time period is equal to uτ(0) = wτ − m∑ j=1 uτ(j). (33) An investor is seeking a best multiperiod investment strategy, (uτ(0), uτ(1), ..., uτ(m)) for τ = 0, 1, 2, ..., T − 1, such that either (i) the expected value of the termi- nal wealth wT , E{wT }, is maximized if the variance of the terminal wealth, V ar{wT }, is not greater than a preassigned risk level v∗, or (ii) the variance of the terminal wealth, V ar{wT }, is minimized if the expected terminal wealth, V ar{wT }, is not smaller than a preassigned level e∗. Mathematically, a mean- variance formulation for multiperiod project portfolio investment can be posed as one of the following two forms: (i) Maximize V ar{wT } (34) subject to V ar{wT } ≤ ν∗ (35) m∑ j=1 uτ(j) ≤ E{wτ} (36) µτ(j) ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., T − 1 (37) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (38) and (ii) Minimize V ar{wT } (39) subject to E{ωT } ≥ e∗ (40) m∑ j=1 uτ(j) ≤ E{wτ} (41) µτ(i) ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., T − 1 (42) Advances in Systems Science and Applications (2014) Vol.14 No.1 31 with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ , (43) where ∆τ = [∆τ(1),∆τ(2), ...,∆τ(m)] ′ = [rτ(1) − rτ(0), rτ(2) − rτ(0), ...,rτ(m) − rτ(0)] ′ (44) Formulation (i) or (ii) enables an investor to specify a risk level he can afford when he is seeking to maximize his expected terminal wealth or specify an ex- pected terminal wealth he would like to achieve when he is seeking to minimize the corresponding risk. A strategy of multiperiod project portfolio investment is an investment se- quence, UT = [u0, u1, u2, ..., uN−1] (45) where uτ = [uτ1 , uτ2 , ..., uτm ] ′, ∀τ = 0(1)T − 1 (46) More specifically, UT is a feedback strategy and uτ maps the wealth at the be- ginning of the τth period, wτ , into a project portfolio decision in the τth pe- riod, i.e., uτ = uτ (wτ ). A multiperiod project portfolio investment strategy, U∗ T , is said to be efficient if there exists no other portfolio one, UT , such that E{wT ;UT } ≥ E{wT ;U ∗ T } and V ar{wT ;UT } ≤ V ar{wT ;U ∗ T } with at least one equality strictly. By varying the value of v∗ in (i) or the value of e∗ in (ii), the set of efficient multiperiod project portfolio investment strategies can be generated. An equivalent formulation to either (i) or (ii) in generating efficient multiperi- od portfolio strategies is (iii) Maximize E{wT } − ϑV ar{wT } (47) subject to m∑ j=1 uτ(j) ≤ E{wτ} (48) uτj ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., T − 1 (49) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (50) 32 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... Table 1 The observations of returns for three risky projects. Year(t) Project 0 1 2 1 1.3000 1.2250 1.1490 2 1.1030 1.2900 1.2600 3 1.2160 1.2160 1.4190 4 0.9540 0.7280 0.9220 5 0.9290 1.1440 1.1690 6 1.0560 1.1070 0.9650 7 1.0380 1.3210 1.1330 8 1.0890 1.3050 1.7320 9 1.0900 1.1950 1.0210 10 1.0830 1.3900 1.1310 11 1.0350 0.9280 1.0060 12 1.1760 1.7150 1.9080 where ϑ ∈ [0,∞). It will be noted that if U∗ T } solves (iii), then U∗ T } solves (i) with v∗ = V arwT ;U∗ T , and solves (ii) with e∗ = EwT ;U∗ T . Note that ϑ is equal to ∂E{wT }/∂V ar{wT } at the optimal solution of (iii). Problem formulation (iii) is preferable to be adopted in investment situations where an investor is able to specify his desirable trade-off between the expected terminal wealth and the associated risk. 6 Numerical Example Consider the case of a stationary multiperiod process with T = 2. An investor has one unit of wealth at the very beginning of the planning horizon, i.e., w0 = 1. The investor is trying to find the best allocation of his wealth among three risky projects, 0, 1, and 2 in order to maximize E{w2} while keeping his risk not exceeding 0.07; that is, v∗ = 0.07. The observations of returns for risky projects, 0, 1, and 2 are given in Table 1. At first consider the case of a stationary multiperiod process with T = 1. The problem is to maximize E{ω1} (51) subject to V ar{ω1} ≤ ν∗ = 0.07 (52) 2∑ j=1 u0(j) ≤ E{w0} = w0 = 1 (53) Advances in Systems Science and Applications (2014) Vol.14 No.1 33 u0(j) ≥ 0, j = 1, 2, ... (54) with w1 = 2∑ j=1 r0(j)u0(j) + w0 − 2∑ j=1 u0(j)  r0(0) = r0(0)w0 +∆′ 0u0. (55) Using (55) and Table 1, it can be shown that E{w1} = E{r0(0)}w0 + E{∆′ 0}u0, (56) where E{R0(0)} = 1.0891, E{∆′ 0} = [0.1246, 0.1455] (57) V ar{ω1} = [ω0, u ′ 0]Cov{[r0(0),△′ 0] ′}[ω0, u ′ 0] ′ (58) with Cov{[r0(0),∆′ 0] ′} =  0.0099 0.0015 0.0021 0.0015 0.0407 0.0374 0.0021 0.0374 0.0723  (59) Using Solver software (MS Excel), we obtain from maximization of (51) a percent investment (on w0 = 1) in three risky projects, 0, 1, and 2, at period τ = 0 as follows: µ∗ 0(0) = 0%, µ∗ 0(1) = 26.92%, and µ∗ 0(2) = 73.08%. The corresponding expected terminal wealth and the risk level are given by E{w1} = 1.229 and V ar{w1} = 0.07, respectively. Now consider the case of a stationary multiperiod process with T = 2. The problem is to maximize E{w2} (60) subject to V ar{ω2} ≤ ν∗ = 0.07 (61) 2∑ j=1 u0(j) ≤ E{w0} = w0 = 1 (62) 2∑ j=1 u1(j) ≤ E{w1} (63) uτ(j) ≥ 0, j = 1, 2, τ = 0, 1 (64) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (65) 34 Konstantin N. Nechval: Optimizing Investment Decisions for a Set of Projects... Using (65) and Table 1, it can be shown that E{w2} = E{r1(0)r0(0)}w0 + E{r1(0)∆′ 0}u0 + E{∆′ 1}u1 (66) where E{r1(0)r0(0)} = 1.1960 (67) E{r1(0)∆′ 0} = [0.1371, 0.1605] (68) E{∆′ 1} = [0.1246, 0.1455] (69) V ar{w2} = [w0, u ′ 0, u ′ 1]× Cov{[r1(0)r0(0), r1(0)∆′ 0,∆ ′ 1] ′}[w0, u ′ 0, u ′ 1] (70) where Cov{[r1(0)r0(0), r1(0)∆′ 0,∆ ′ 1] ′} =  0.0491 0.0028 0.0053 0.0022 0.0037 0.0028 0.0496 0.0490 0.0447 0.0427 0.0053 0.0490 0.0942 0.0426 0.0823 0.0022 0.0447 0.0426 0.0407 0.0374 0.0037 0.0427 0.0823 0.0374 0.0723  (71) Using Solver software (MS Excel), we obtain from maximization of (60) a percent investment (on w0 = 1) in three risky projects, 0, 1, and 2, at period τ = 0 as follows: µ∗ 0(0) = 100%, µ∗ 0(1) = 0%, and µ∗ 0(2) = 0%; and a percent investment (on E{w1} = 1.0891) in three risky projects, 0, 1, and 2, at period τ = 1 as follows: µ∗ 1(0) = 39.80%, µ∗ 1(1) = 47.24% and µ∗ 1(2) = 12.96%. The corresponding expected terminal wealth and the risk level are given by E{w2} = 1.2807 and V ar{w2} = 0.07, respectively. 7 Conclusion The problem considered in this paper is to decide how much of our available resources to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. We formulate the problem in terms of dynamic programming, which allows one to obtain optimal investment decisions for a set of projects under uncertainty of future returns in a simple form. The derived optimal multiperiod project portfolio investment strategy provides investors with the best strategy to follow in a dynamic investment environment. Acknowledgments This research was supported, in part, by the Latvian Council of Science and the National Institute of Mathematics and Informatics of Latvia under Grant No. 06.1936 and Grant No. 01.0031. This support is gratefully acknowledged. Advances in Systems Science and Applications (2014) Vol.14 No.1 35 References [1] Schniederjans, M. and Santhanam, R. (1993), “A multi-objective constrained resource information system project selection. method”, European Journal of Operational Research, Vol.70, pp.244-253. [2] Lucas, H.C. (1973), “Computer-Based information systems in organization- s”, Science Research Associates, Chicago. [3] Cooper, R.G., Edgett, S.J. and Kleinschmidt, E.J. (1997), “Portfolio man- agement in new products: lessons from the leaders-I”, Research Technology Management, Vol.40, pp.16-28. [4] Dos Santos, B.L. (1989), “Selecting information system projects: problems, solutions and challenges”, Proceedings of the Hawaii Conference on System Sciences, pp.1131-1140. [5] Cooper, R.G. (1993), Winning at New Products, MA: Addison-Wesley, Read- ing. [6] Jackson, B. (1983), “Decision methods for selecting a portfolio of R&D projects”, Research Management, Vol.10, pp.21-26. [7] Mathieu, R.G.and Gibson, J.E. (1993), “A methodology for large scale R&D planning based on cluster analysis”, IEEE Transactions on Engineering Management, Vol.30, pp.283-291. [8] Liberatore, M.L. and Titus, G.J. (1983), “The practice of management sci- ence in R&D project selection”, Management Science,, Vol.29, pp.962-974. [9] Hess, S.W. (1993), “Swinging on the branch of a tree: project selection applications”, Interfaces, Vol.23, pp.5212 [10] Archer, N.P. and Ghasemzadeh, F. (1996), “Portfolio selection techniques: A review and a suggested integrated approach”, Innovation Research Center, Working Paper 46, School of Business, McMaster University, Hamilton. Corresponding author Konstantin N. Nechval can be contacted at: konstan@tsi.lv