PROJECT SELECTION UNDER RISK AND DECISION SUPPORT Constanţa Zoie Rădulescu National Institute for Research and Development in Informatics, 8-10 Averescu Avenue, 71316 Bucharest 1, Romania Email:
[email protected] Marius Rădulescu Institute of Mathematical Statistics and Applied Mathematics “Gheorghe Mihoc - Caius Iacob”, Casa Academiei Române, Calea 13 Septembrie nr.13, Bucharest 5, RO-76100, Romania Email:
[email protected] Abstract The aim of the paper is to present the design and realization of a software package, named PROSEL (PROject analysis and SELection system), for project and product selection under risk and limited resources. An original zero-one mathematical programming model for project selection problem under risk and limited resources, that is implemented in PROSEL, is presented. Introduction A major theme in the management literature is the theme of selecting and ranking (prioritizing) projects, or project portfolio selection. It includes strategic R&D planning (selection of directions, topics or projects), the development of new commercial products, the management and the implementation of organizational change, the development and the implementation of information technology etc. In order to develop products or to perform competitive services many firms need to select the best project portfolio of products or services that meet all the requirements (such as rate of return, risk and inter-dependencies with other projects) and the resource constraints (people, financing, time, facilities etc.) In the process of project portfolio selection, decision makers must cope with significant uncertainties in the investment required, time necessary to complete the project, the availability of resources when required, and the likelihood of successful project completion. These may depend on project size, complexity, and project team experience. In addition, there may be multiple criteria to be satisfied, and the choice of projects typically is made by a committee that represents different organizations or companies that may be involved in the project. Tools for portfolio project selection and management are a recognized need in research, development, production and marketing activities for manufacturing firms
and in other sectors such as engineering, construction and software development. They are also used in the public sector, in government, health care and military. As a result of such a diversity of applications were developed several methods and decision support systems for the portfolio project selection. PROSEL a DSS for project selection under risk In 2000 at the Institute for Research in Informatics from Bucharest had started a research on the design and realization of a software package for project and product selection under risk and limited resources. As a result was designed an interactive software package, named PROSEL (PROject analysis and SELection system) intended to help managers in making high quality project portfolio selections. The potential users of such a software product may be represented by managers from various domains: decision makers from research-development institutions, marketing and human resources firm departments, army, financial institutions such as banks, mutual funds or pension funds etc. More generally the software product is intended for users who act in domains in which occur the problem of selecting from a set of actions (projects, proposals, products) a subset taking into account the risk and the resource constraints. Architecture of the software package PROSEL Problem building (goal formulation, attributes (criteria) identification, alternatives identification, resources identification, weighting preferences, alternatives assessment with respect to criteria and resources, etc)
Database - tables containing general information - tables on the specific problem
Problem choice Model choice
User parameters
Model solver
Solution of the model
Solution description
MODELS LIBRARY MULTIATTRIBUTE DECISION METHODS MULTIOBJECTIVE DECISION MODEL
The system requirements for PROSEL are as follows: MS-Windows 98, 2000, Personal computer using a PENTIUM II processor or higher, 128 MB of RAM, 40 MB of available disk space on the hard drive, one 3.5" disk drive, mouse, printer. The use of PROSEL brings power to the decision process by providing the following benefits: − Speedier construction and elucidation of the decision. − Deeper insight into factors which determine the decision. − Confidence that all important factors within the decision process have been considered. − Identification of key criteria in breaking an impasse (assuming no clear preference in alternatives occurs). − The improvement of the decision making process efficiency. The software package aids the decision maker in solving the decision problems according to his requirements. PROSEL provides support for the decision process but it does not substitute the decision maker. The package can help the decision maker in all stages of the decision process. The software product has a high degree of generality and allows the decision makers to define a wide range of project selection problems. The first step in the decision making process is the problem statement. PROSEL helps the decision maker to get a better understanding of the decision process and to obtain higher performance. It implements advanced multicriteria decision making models (several multiattribute decision methods and an original multiobjective decision model). The decision maker can use for solving the same problem one or more models from the Models Library. The multiobjective decision model The multiobjective decision model (which is an original zero-one mathematical programming model for project selection problem under risk and limited resources) is implemented in a module named RISKSEL. The fundamental problem when one allocates a budget to competing projects is to find a compromise between several conflicting criteria. It is desired to find a portfolio of projects that meets all the requirements concerning the resources constraints, maximize the performance (benefits) and minimize the risk. The multiobjective decision model represents a generalization of the classical capital budgeting model and includes several resources and experts’ opinions which generate the risk. The project risk is greater if experts’ opinions have a greater degree of dispersion. In the following we shall recall our model. More details can be found in [1] and [2].
Consider a set of n project proposals P1 , P2 ,..., Pn which can be implemented. Suppose that the projects are evaluated by m experts E1 , E 2 ,..., E m which assign scores to each project. Of course instead of experts one can consider m criteria. Denote by a ij the score given by expert E i to project Pj . Suppose that for the project implementation are available k resources R1 , R 2 ,..., R k . Denote by bij the quantity from the resource Ri
necessary to carry out the project Pj . For every
resource Ri , denote by ci the upper limit that it is available. Let x1 , x 2 ,..., x n be the decision variables of the model. x j = 1 if the project Pj is selected for funding, x j = 0 if project Pj is not selected for funding. The benefit or performance of the
portfolio x = ( x1 , x 2 ,..., x n ) of projects is defined by: see that y i =
n
∑a j =1
ij
m
n
∑ ∑ aij x j i =1 j =1
One can easily
x j is the score of the project portfolio x given by the expert
E i . The risk of the project portfolio x will be defined as the variance of the scores given by the m experts E1 , E 2 ,..., E m . Consequently the risk of the project portfolio x is: 2
2 m ⎛ n m n ⎞ ⎛ ⎞ 1 m ⎜ a ij x j − 1 a rj x j ⎟ ⎜ ⎟ − y y ∑ ⎜ i m∑ r⎟ ⎜ ⎟ m r =1 j =1 i =1 ⎝ j =1 i =1 ⎝ r =1 ⎠ ⎠ = R(x ) = m m The selection problem of a portfolio of projects for funding is a multicriteria optimization problem: ⎛ m n ⎞ max⎜⎜ ∑ ∑ a ij x j ⎟⎟ ⎝ i =1 j =1 ⎠
∑ ∑
m
⎡m ⎛ n 1 min ⎢∑ ⎜⎜ ∑ a ij x j − m ⎢ i =1 ⎝ j =1 ⎣ subject to: n
∑b j =1
ij
x j ≤ ci ,
x j ∈ {0,1},
⎞ a rj x j ⎟⎟ ∑ ∑ r =1 j =1 ⎠ m
n
i = 1,2 ,..., k
j = 1,2,..., n
2
⎤ ⎥ ⎥ ⎦
∑∑
One can easily see that in the above multicriteria problem one looks for the maximization of the performance (the benefit) and the risk minimization of the portfolio of projects that meet the constraints on the existing resources. The optimization problem is of type zero-one with two objective functions: one is linear and the other is quadratic. The constraints are linear. One can easily see that a project portfolio is considered more risky if the experts’ opinions about it have a greater degree of dispersion. Denote by θ the risk aversion coefficient of the decision maker. Parameter θ takes values in interval [0,1]. The decision makers characterized by small values of θ (near zero) are risk averse. For them the first thing is the safety degree of their return and after that the amount of the return. The decision makers characterized by large values of θ (near one) are interested first by the amount of return and after that by the safety degree of their return. By using the risk aversion coefficient θ the above bicriterial problem can be transformed in a zero-one quadratic programming problem with a single objective function. m n ⎤ ⎡ min ⎢(1 − θ ) R (x ) − θ ∑ ∑ a ij x j ⎥ i =1 j =1 ⎦⎥ ⎣⎢ n
∑b j =1
ij
x j ≤ ci ,
x j ∈ {0,1},
i = 1,2,..., k j = 1,2,..., n
To identify optimal solutions RISKSEL employs techniques based on genetic algorithms. Our experience has been that the techniques based on genetic algorithms are computationally more efficient (by an order of magnitude) and computationally more accurate in comparison to dynamic programming. In special cases, when the number of feasible solution is small RISKSEL provides optimal solutions by using smart enumeration procedures, which avoid unnecessary computations. Acknowledgements The work described in this paper was supported by the RELANSIN - National Research and Development Program of the Ministry of Education and Research, project nr. 283/2000. References [1] M.Rădulescu, C.Z. Rădulescu, Decision analysis for the project selection problem under risk, Proc. 9th IFAC / IFORS / IMACS / IFIP/ Symposium on large scale systems: theory and applications, 2001 Bucharest, Romania, (2001), 243-248. [2] M.Rădulescu, C.Z. Rădulescu, Project portfolio selection models and decision support, Studies in Informatics and Control, Vol.10, no.4, (2001), 275-286.
