portfolio selection of interdependent r\u0026d projects

$: Portfolio Selection of Interdependent R\u0026D Projects$
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INDIAN JOURNAL OF INDUSTRlAL AND APPLIED MATHEMATICS Copyright © 2012 Indian Society ofIndustrial and Applied Mathematics Vol. 3 No.1 January-June 2012, pp. 62-73 Print ISSN: 0973-4317

Portfolio Selection of Interdependent R&D Projects

Rupak Bhattacharyya1*, Pankaj Kumar2, Samarjit Kar3 and Seema Sarkar (Mondal)4

National Institute of Technology, Durgapur-713209, West Bengal, India

E-mails:[email protected]@[email protected]\ [email protected]

Abstract: Worldwide competition of markets has forced organizations to invest in targeted R&D projects so that resources can be focussed on successful outcomes. A number of alternatives are encountered to select the most appropriate projects in R&D project selection model. In order to maintain technicaVrisklresourcel benefit competitiveness, the decision-maker assigns limited amount of assets to a set of competing projects. Interdependencies between applicant projects have great impact on the project outcome. But very little effects are noticed to involve them while modelling the problem. In this paper, we have considered outcome, technical, resource and risk interdependencies to develop a bi-objective non-linear R&D project portfolio selection model under the constraints on resources, budget, interdependencies and outcome, and discuss how our methodology can be used to make decision support tools for optimal R&D project selection in a corporate environment. The objectives are minimization of risk and maximization of benefit. Suitable numerical results are provided to illustrate our method. The solution is done by using fuzzy additive goal programming technique.

Keywords: R&D project portfolio; Project interdependencies; Multi-objective optimization; Fuzzy additive goal programming.

2000 Mathematics Subject Classification: 97M40

1. Introduction In today's environment, with increasing competition and limitations of financial resources, the way of selection of R&D projects that maximize some measure of utility or benefit to the organization

* Author for Correspondence. E-mail: [email protected]

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RUPAK BHATTACHARYYA ET AL 63

has become a critical one. The purposes of project portfolio decision are to allocate a limited set of resources to projects in a way that balances risk, reward and alignment with corporate strategy. The R&D project portfolio decision deals with future events and opportunities, much of the information required to make portfolio decisions is at best uncertain and at worst very unreliable. Project selection is usually described in terms of constraint optimization problem. Given a set of project proposals, the goal is to select a subset of projects to maximize some objective without violating the constraints. The subset of selected projects is called a portfolio. Though some methods for R&D project portfolio selection already exist (Rabbani et al. [1], Wang and Hwang [2], Fang et al. [3], Carlsson et al. [4] Medaglia [5], Ringuest et al. [6], Bhattacharyya et al. [7]), unfortunately, R&D project managers have not been able to adopt many of these mechanisms.

Interdependent projects render an increase in benefit. When interdependencies occur, the parameters associated with a particular project depend upon which other projects have been selected so that the total cost and benefit obtained from a portfolio of the projects is not equal to the sum of the individual project costs and benefit. Not much work has been done in the field of R&D project portfolio selection with consideration of interdependencies. Schmidt [8] presents a model that accounts for the combined effect of benefit, outcome and resource interactions within a single set of projects. The model also allows for the allocation of several different resources. Santhanam and Kyparisis [9] discuss a non-linear 0 - 1 goal programming model for interdependent information system project selection formulating benefit, resource and technical interdependencies among candidate projects. Stummer and Heidenberger [10] consider three phases of R&D project selection. First, proposal candidates are identified by a score based screening process. Next, an integer linear programming model determines all efficient portfolios considering multiple objectives, project interdependencies and time. Finally, an interactive procedure matches portfolios with aspired benefit and resource. Verma and Sinha [11] develop a theoretical framework for understanding the interdependencies between projects theirre1ationship to project performance in a multiple-concurrent R&D environment. Eilat et al. [12] propose and demonstrate a methodology for the construction and analysis of efficient, effective and balanced portfolio of R&D project with interactions. Guo et al. [13] propose a 0 1 non-linear mathematic programming method based R&D project portfolio selection model in which outcome, technical, resource and risk interdependencies are considered.

The main drawback of the above literatures is that none of them has considered interdependencies among all combination of candidate projects. They are restricted to pair or triplet of candidate projects. The aim of this paper is to develop a bi-objective R&D project portfolio selection model based on interdependencies over all possible combination of candidate projects. The rest of the paper is organized as follows. Section 2 presents outcome, technical, resource and risk interdependencies among all candidate projects. In Section 3, we construct a new bi-objective R&D project portfolio selection model which maximize the benefit and minimize the risk of the portfolio. In Section 4, we provide a suitable case study to illustrate our methodology. Section 5 provides some conclusions.

INDIAN JOURNAL OF INDUSTRIAL AND ApPLIED MATHEMATICS Vol. 3 No.1 January-June 2012

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64 Portfolio Selection of Interdependent R&D Projects

2. Project Interdependencies In this section, we first describe the notations used in the construction of the R&D project portfolio selection model. A discussion of different interdependencies and their effects on the problem is discussed in the following subsection.

2.1 Notations

N = Number of candidate projects.

T = Number of periods.

{I if project i is selected in period t

Xii = 0 otherwise, i = 1, 2, ... , N, t = 1,2, ... , T.

a = Number of projects having outcome interdependency, a::; N.

f3 = Number ofprojects having technical interdependency, f3::; N.

T] = Number of projects having resource interdependency, T] ::; N.

Vii = Projected outcome of project i in period t.

Vij i2 •. .in t = Additional outcome due to outcome interdependency for projects

il' i2,·· .,in (n ::; a) in period t.

aij i2 ... int = Additional outcome due to technical interdependency for projects il' i2, •• • ,in (n ::; T]) in period t.

ri/ = Projected risk of implementing project i in period t.

rUt = Additional risk due to risk interdependency between the projects i and j in period t.

Cit = Expected cost required by ith project in period t.

c. . t = Cost shared by projects iI' i2, •• • ,i (n::; N) in period t. ~~ .. ~ n

Bt = Budget available for stage t.

~ = Minimum expected outcome in period t.

R~t = Amount of resource of type s required for implementation of project i individually in period t.

R4iz .. .int = Amount of resources of type s needed for implement ofprojects il' i2, •• • ,in (n ::; {J) together in period t.

R:t = Amount of available resources of type s in period t.

Rs Total amount of available resources of type s.

0; = Index set representing the projects having outcome interdependency with ith project.

r; = Index set representing the projects having technical interdependency with ith project.

Ri = Index set representing the projects having resource interdependency with ith project.

Li = Index set representing the projects having risk interdependency with ith project.


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2.2 Effect of Interdependencies

In this subsection, we will discuss the effects of different type of interdependencies.

2.2.1 Outcome Interdependency

Outcome interdependency affect the overall outcome obtained from a project portfolio. When the outcome interdependency occurs, the total value of a project portfolio is greater than the sum of the individual project values. Let VOl be the additional outcome due to the outcome interdependency between project i and

N-I N

project) in period t. Then L L vijtxitXjt represents the additional outcome due to the outcome i~1 j~i+1

interdependency between all pair of candidate projects. If Viji2 .... iat is the additional outcome for the outcome interdependency between the projects il' i2, ••• , ia in period t, then the additional outcome due to the outcome interdependencies between

N-(a-I)N-(a-2) N

every a-tuple of candidate projects is L L .... L Viji2 .... iatXijtXi2t···xiat·

ij ~I i2 ~ij + 1 ia ~ia_j + 1

Therefore, the total contribution to the outcome due to a number of outcome interdependencies in period t is

N-I N N-2 N-I N N-(a-I)N-(a-2)

L L vijtxitXjt + L L L vijktxitxjtxkt + .... + L L i~1 j~i+1 i~1 j~i+lk~j+1 ij~1 i2~ij+1

N

L Viji2 .... iatXijtXi2t ",xia t

ia=ia_t +1

As we have considered every possible n-tuples (n ::; a), some of the terms in the above expression maybe zero.

2.2.2 Technical Interdependency

Technical interdependencies result from leveraging common technology across multiple projects. Then, as above, the total contribution to the outcome due to technical interdependency in period t is

N-I N N-2 N-I N N-(Tf-I)N-(Tf-2) N

L L aijtxitxjt + L L L aijktXi/XjtXkt+····+ L L .... L aiji2 ... .i~tXijtXi2t···Xi~t· i=1 j=i+1 i~1 j~i+lk~ j+1 ij=1 i2=ij +1 i~~i~_j +1

As we have considered every possible n-tuples (n ::; 1]), some of the terms in the above expression maybe zero.

2.2.3 Resource Interdependency

Resource interdependencies result from sharing limited resources between different projects. The resource allocation for each project is inversely related to resources for each concurrent project, an increase in the resource level for one project would lead to a decreased level of another project. We assume that some resources may be shared among one or more projects in such way that the implementation of one project reduces the resource consumption of interrelated projects.

INDIAN JOURNAL OF INDUSTRIAL AND ApPLIED MATHEMATICS Vol. 3 No. I January-June 2012

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66 Porifolio Selection of Interdependent R&D Projects

For R!t,Rijt,Rfjla, ... ,R4i2 ... ipt, we clearly have R!t?Rijt?Rijkt?.. ... ?..RtA .. ipt" Now, ifprojects i andj are implemented (i.e., if f3 = 2), then the total amount of resources of type s

. d' S S S reqUIre IS Rit + R jt - Rijt"

Similarly, if f3 = 3, i.e., if projects i,j, k are selected, the total amount of resources of type s . d· RS R S RS R S RS R S RS require IS it + jt + kt - ijt - ikt - jkt + ijkr

Thus, the total amount of resources of type s required is

N N-I N N-2N-I N

T LRftxit-L L Rijtxitxjl + L L L RijklXitXjlXkt + ...

L i=1 i=1 j=i+1 i=1 j=1 k= j+1

N-(j3-I)N-(j3-2) N

1=1 + (-1)f3-1 ~ ~ .... ~ R~ . tXi/Xi 1 ."Xipl L..J L..J L..J '1'2···'p 1 2 ip=ip_l +1

As we have considered every possible n-tuples (n :s; /3), some of the terms in the above expression maybe zero.

2.2.4 Risk Interdependency

For successful implementation of R&D project portfolio, the risk attached with the projects must be as less as possible. Consideration of two or more projects in a period may increase the risk of the portfolio by a large amount. Let r ijl?" 0 is the amount the decision-maker may lose for the implementation of the ith and jth projects together at period t. Therefore, the total contribution to the risk due to risk interdependencies

N-I N

between pair of projects in period t is L L rijtXilXjt· i=1 j=i+1

As the measurement of risk is too tough and in someway imprecise, we have considered risk interdependencies between pair of projects only. Then the total risk involved in the project portfolio is

t [ ~,uxu + ~ jt;;,~,Xi'}

3. R&D Project Portfolio Selection Model In this section, we will formulate the R&D project portfolio selection model. The notations used are given in Section 2.1.

3.1 Formulation of Objective Functions

In this R&D project portfolio selection problem, we have considered two objectives: maximization of the benefit and minimization of the risk.


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3.1.1 Maximization of Benefit

The total outcome from the projects will be obtained by considering the total individual outcomes and additional outcomes due to outcome interdependency and technical interdependency. Thus, the total outcome is

N N-I N N-2 N-I N

LVitXit + L L vijtXitxjt + L L L vijktXitxjtxkt + .... i=1 i=1 j=i+1 i=1 j=i+lk=j+1

N-(a-I)N-(a-2) N

T + L L L Viri2 .... iatXirtXizt ",Xia t

Zo=L N-I N N-2 N-I N t=1

+ L L aijtxitxjt + L L L aijktxitXjtXkt + .... i=1 j=i+1 i=1 j=i+lk=j+1

N-(1)-I) N-(1)-2) N

+ L L .... L airi2· ... ~tXirtXi2t···Xi~t i~=~_r+1

The total cost will be obtained by the total individual costs for each project subtracted by the savings due to the sharing costs between projects. Then the total cost is

Then the total benefit is defined as

3.1.2 Minimization of Project Risk

Minimization of risk will decrease the chance of failure of the projects and therefore that of the decision-maker. So the objective is to minimize total risk

Thus, we are with the following bi-objective optimization problem

3.2 Formulation of the Constraints

{MaxZB(X)

MinZcCx).

In this subsection, we will formulate the constraints required to model the problem realistically.


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3.2.1 Outcome Constraints

As the minimum expected outcome for the projects in period t is ~, we have N

3.2.2 Resource Constraints

L V;tXu?' V; 'lit. ;=1

The projects are implemented by using limited amount of resources. As the available resources are always finite, the required resource with particular type should be within the resource available of that type for each period. Thus, we have

N

LRftxu:S: R;tVs,t. ;=1

The total amount of resources available is limited. So, the amount of resource required should not be more than the total resource available for each type of resources. Thus, we have

N N-I N N-ZN-I N

T Ll~Xi/ - L L Rijtxi/xjt + L L L RijktXi/XjtXkt + ... ;=1 ;=1 j=i+1 ;=1 j=1 k=j+1

L N-(f3-I) N-(f3-Z) N t=1 +(_1)/3-1 L L L

3.2.3 Budget Constraints

The project expenses during the planning horizon should not exceed the predetermined budget for each stage or period. So, we have

3.2.4 Interdependency Constraints

N

LCuxu :S:Bt 'lit. ;=1

Let the ith project be interdependent with some projects. Let 0i be the index set representing the projects having outcome interdependency with ith project. Then, we have

Xu? I1 Xjto i = 1, 2, ... ,N. JEO,

Similarly for technical, resource and risk interdependencies, we have

Xu? I1 Xjt, xu? I1 Xjt, xu? I1 Xjt, i = 1, 2, ... ,N, j~ j~, j~

r;, Ri, Li being the respective index sets representing the projects having technical, resource and risk interdependencies with ith project.


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3.3 The Model

Thus, we are with the following R&D project portfolio selection model

MaxZB(X)

MinZR(x)

Subject to N N N

L vitxit;;;:: Vr "It, LRftxit::;; R;t V s,t, LCitXit::;; Bt "It, i=1 i=1 i=1

(1)

T

L

N N-l N N-2N-l N

LRftxit - L L Rfjtxitxjt + L L L RijktXitXjtXkt+··· i=1 i=1 j=i+l i=1 j=1 k= j+l

N-(f3-1) N-(f3-2) N

t=1 + (_1)13-1 L L .... L R4iz ... iptXi\tXi2t",XifJt i\=1 i2=il +l ifJ=ifJ-I +1

Xit;;;:: II Xjt,Xit;;;:: II Xjt,Xit;;;:: II Xjt' Xit;;;:: II Xjt· jeO; jeT; jeR; jeL;

To convert the above bi-objective optimization problem into preference based single objective optimization problem, let us consider the two single-objective optimization problems optimizing separately the objectives of the model subject to the satisfaction of the constraints of the problem. The optimum values as well as the values of the remaining objective function in both cases are calculated. Considering both problems, let the minimum values of the two objectives are Bmin and Rmin respectively. Also, let the maximum values of the obj ectives be Bmax and Rmax respectively. Then by fuzzy additive goal programming method (Narasimhan [14]), the bi-objective portfolio selection model (1) is transformed into the following model:

Max {Wjal +W2a2}

Subject to

ZB(X) - Z]fn _ Z}F - ZR(X) _ < < < < ----=--'--'-----="::-, - - ab ' - a2, 0 - al _ 1, 0 - a2 _ 1, ZB - zwm zWax - Zytn N N N

(2) L VitXit ;;;:: Vr "It, LRftxit ::;; R~t "Is, t, LCitXit ::;; Bt "It, i=1 i=1 i=1

N N-l N N-2N-l N

LRftXit- L L RijtXitXjt+ L L L RijktXitXjtXkt+· .. i=1 i=1 j=i+l i=1 j=1 k=j+l

N-(f3-1)N-(f3-2) N

T

L t=1 +(_1)13-1 ~ ~ .... ~ R~, 'tXitxit",XifJt £..J £..J £..J 11'2,·,113 1 2

i l =1 i2=il +l ip=ip_I+l


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where w) and w2 are the weights (preferences) to the objectives ZB and ZR respectively. The values ofw) and w2 will be assigned by the users.

4. Case Study In this section, we provide a numerical example where the data is obtained from an Indian medium scale organization. There are five project proposals to the company which are renamed as I, II, III, IV and V by us due to privacy. The organization investigates outcome, resource (fund and staffs) and risk interdependencies between different projects. Due to different fields of application the technical interdependencies are negligible and hence assumed to be zero.

The estimated data for outcome, funds, staffs risks and costs are given in Table 1. Table 2 provides information about interdependencies and sharing cost. Constraints on outcome, fund, staffs and budget for each period is given in Table 3.

Table 1. Projects' data

Outcome Fund Risk Cost Staff

(Million Rs.) (Million Rs.) (Million Rs.) (Million Rs.) Project

1" 2nd 1" 1" 2nd 2nd 1" 2nd 1" 2nd

Period Period Period Period period Period Period period Period period

I 7 10 0.3 0.25 30 10 0.8 0.4 3.5 4

II 2 3 0.1 0.14 12 12 0.2 0.7 5 5

III 1 3 0.1 0.19 20 28 0.7 0.5 5 10

IV 0 0 0.06 0.07 18 18 0.5 0.5 8 12

V 6 7 0.19 0.1 16 5 0.4 0.6 3 7

Table 2. Interdependencies and sharing costs

Resource Sharing Outcome Risk

Fund Staff cost

Projects Effects Projects Effects Projects Effects Projects Effects Projects Effects Projects Effects

II, III 5 I, II 0.01 II, III 5 I, II O.oI I, II 0.20 II, V 0.05

III, V 2 II, III 0.004 IV, V 3 II, IV 0.06 I, III 0.05 III, IV 0.30

IV, V 1 IV, V 0.010 III, IV,

1 III, V 0.08 II, III 0.10 III, V 0.02 V

I, III, V 4 II, III,

0.0015 IV, V 0.03 II, IV 0.02 IV,V 0.09 V


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------------------------------------------


Table 3. Outcome, resources and budget constraints

Category 1" Period 2nd Period Total

Outcome (million rupees) ? 5.7 ? 7.9 -

Fund (million rupees) ::; 0.75 ::;0.85 ::; 1.10

Workers ::; 70 ::; 50 ::; 113

Budget (million rupees) ::;21 ::; 30 -

For the above-mentioned data in Tables 1,2 and 3, model (1) and then model (2) in Section 3.3 is formulated. The solution is done by the optimization tool LINGO. The solution is obtained for different values of WI and w2• The set of solutions is given in Table 4.

Table 4. Solution set

Cases WI W2 Zs ZR Xli X21 XlI X41 X SI X l2 X 22 Xl2 X 42 X 52

a 0.5 0.5 12.233 3.5 1 1 1 0 0 1 1 1 0 0 b 0.2 0.8 10.951 1.6 1 0 0 0 1 1 0 0 0 0 c 0.8 0.2 14.666 5.6 1 0 0 0 1 1 0 0 0 0

In case a, where the decision-maker gives equal importance to both objectives, the company will select projects I, II, III in first period and projects I, II and III in second period to get the desired solution. In this scenario, the benefit and risk will be 12.233 and 3.5 million rupees. In case b, where decision-maker gives most importance to portfolio risk, the firm will select projects I, and V in first period and project I in second period. In this scenario, the benefit and risk will be 10.955 and 1.6 million rupees. Similar explanations can be given for case c.

5. Conclusion The problem of selecting a suitable set of R&D projects with optimized benefit and risk is still very complex to solve, because of the difficulty of explicitly expressing the preference trade-offs, the high number offeasible candidate projects, multiple periods, different types of project interdependencies and other constraints. To deal with all of these problems, we have modelled the problem in a new way in this paper.

Previous efforts deal with project interdependencies using a limited number and type of project interdependencies. In this paper, the interdependencies are searched among each and every possible set of projects. Thus, the method discussed here is far more realistic than others. To illustrate our method, a case study is done and solved.

[NOlAN JOURNAL OF INDUSTRIAL AND ApPLIED MATHEMATICS Vol. 3 No.1 January-June 2012

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72 Portfolio Selection 0/ Interdependent R&D Projects

Our future aim is to implement our method on larger data sets for the R&D projects of multinational companies. Also, one can develop the method in fuzzy/fuzzy stochastic environments by considering the parameters as fuzzy or fuzzy random. In addition to fuzzy goal programming techniques, some other multi-objective decision-making techniques such as goal programming, compromise programming, fuzzy interactive decision-making, etc. can be used to solve the problem. Some multi-objective evolutionary algorithms such as MOGA (multi-objective genetic algorithm), VEGA (vector evaluation genetic algorithm), NEGA (Nondominated sorting genetic algorithm), NPGA (Niched Pareto-genetic algorithm) and PAES (Pareto-archived evolution strategy) may be employed to solve the problem, especially when the data set is significantly large.

References

1. Rabbani, M., Bajestani, M. A. and Khoshkhou G. B., 2010, "A Multi-objective Particle Swarm Optimization for Project Selection Problem", Expert Systems with Applications, 37(1), pp. 315-321.

2. Wang, J. and Hwang, W. L., 2007, "A Fuzzy Set Approach for R&D Portfolio Selection Using a Real Options Valuation Model", Omega, 35(3), pp. 247-257.

3. Fang, Y., Chen, L. and Fukushima, M., 2008, "A Mixed R&D Projects and Securities Portfolio Selection Model", European Journal o/Operational Research, 185(2), pp. 700-715.

4. Carlsson, C., Fuller, R., Heikkila, M. and Majlender, P., 2007, "A Fuzzy Approach to R&D Project Portfolio Selection", International Journal 0/ Approximate Reasoning, 44(2), pp. 93-105.

5. Medaglia, A. L., Graves, S. B. and Ringuest J. L., 2007, "A Multiobjective Evolutionary Approach for Linearly Constrained Project Selection under Uncertainty", European Journal 0/ Operational Research, 179, pp. 869-894.

6. Ringuest, J. L., Graves, S. B. and Case, R. H., 2000, "Conditional Stochastic Dominance in R&D Portfolio Selection", IEEE Transactions on Engineering Management, 47(4), pp. 478-484.

7. Bhattacharyya, R., Chatterjee, A., Kar S., 2010, "Uncertainty Theory Based Novel Multiobjective Optimization Technique using Embedding Theorem with Application to R&D Project Portfolio Selection", Applied Mathematics, 1, pp. 87-93.

8. Schmidt, R. L., 1993, "A Model for R&D Project Selection with Combined Benefit, Outcome, and Resource Interactions", IEEE Transactions on Engineering Management, 40(4), pp. 403-410.

9. Santhanam, R. and Kyparisis, G. J., 1996, "A Decision Model for Interdependent Information System Project Selection", European Journal o/Operational Research, 89, pp. 380-399.

10. Stummer, C. and Heidenberger, K., 2001, "Interactive R&D Portfolio Selection considers Multiple Objectives, Project Interdependencies, and Time: A Three-phase Approach", PICMET '01 Proceedings, pp. 423- 428.

11. Verma, D. and Sinha, K. K., 2000, "Toward a Theory of Project Interdependencies in High Tech R&D Environments", Journal o/Operations Management, 20, pp. 451-468.


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portfolio selection of interdependent r\u0026d projects

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