Project Portfolio Selection via Harmony Search Algorithm and Modern ...

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ScienceDirect Procedia - Social and Behavioral Sciences 226 (2016) 51 – 58

29th World Congress International Project Management Association (IPMA) 2015, IPMA WC 2015, 28-30 September – 1 October 2015, Westin Playa Bonita, Panama

Project portfolio selection via harmony search algorithm and modern portfolio theory Hamed Nasr Esfahani a,*, Mohammad hossein Sobhiyah a, Vahid Reza Yousefi b a

Tarbiat Modares University, Jalal Ale Ahmad Highway, P.O.Box: 14115-111, Tehran, Iran b University of Tehran, Enghelab Ave., Tehran, Iran.

Abstract Since the success of all organizations depends on aligning their projects with strategic plan of theirs, selecting appropriate project portfolio is a key decision in the path to fruitful and thriving organization. Modern portfolio theory (MPT), proposed by Harry Markowitz, attempts to maximize portfolio expected return for a specific amount of risk or equivalently, minimize risk when the expected return is specified. In this paper we first go through the literature of the portfolio selection. Secondly, the definition and mathematical formulation of modern portfolio theory is presented; thirdly in the next section the fundamental of classic harmony search algorithm (a metaheuristic algorithm) is illustrated, and finally, the numerical example of solving a benchmark problem of project portfolio selection and its results is presented. Provided results demonstrates that this algorithm solves the hard problem to almost optimality faster and robuster than other exact algorithms. © 2016 2016Published The Authors. Published Ltd. © by Elsevier Ltd. by ThisElsevier is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of IPMA WC 2015. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of IPMA WC 2015. Keywords: project portfolio selection; modern portfolio theory; harmony search algorithm; metaheuristic.

1.

Introduction

Today, organizations should opt for appropriate portfolios of projects to advance their goals. Generally speaking, there are so many projects that an organization cannot fund or support. Ideally, corporate managers try to choose an optimal subset of projects to meet the corporate`s goals while satisfying budgetary constraints and restrictions. On the other hand, they try their best to refrain from the overall risk of a portfolio of projects and ensure

* Corresponding author. Tel.: +989128345182; fax: +982122356255. E-mail address: [email protected]

1877-0428 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of IPMA WC 2015. doi:10.1016/j.sbspro.2016.06.161

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that return of their investment or other such objectives are maximized (Better and Glover, 2006). Organizations are often confronted with having more projects to choose from than the resources to carry them out and thus one of the main management tasks is to select from an array of projects those better adapted to the organization's objectives (Ghasemzadeh F. et al. 1999). Wrong decision making in selecting projects of a portfolio has two negative consequences. Firstly, resources are consumed on inappropriate projects; Secondly, the company cannot have the advantages it may have gained if these resources had been spent on better projects (Martino JP., 1995). In the field of project management and engineering, the selection of projects for portfolio is a popular and wide investigating research topic (Aaker and Tyebjee 1978; Bouyssou et al. 2006; Carraway and Schmidt 1991; Dickinson et al. 2001; Ewing et al. 2006; Golabi et al. 1981; Talias 2007; Fox et al. 1984; Henriksen and Palocsay 2008; Kleinmuntz and Kleinmuntz 1999; Kleinmuntz 2007; Kuei et al. 1994; Liesio et al. 2007; Stummer et al. 2009; Mavrotas et al. 2006; Medaglia et al. 2007a, 2007b; Peng et al. 2008a; Santhanam and Kyparisis 1995; Golabi 1987; Stummer et al. 2003; Liesio 2006; Cooper et al. 1999). Generally, selecting a subset of projects from a bigger set of projects which is called project portfolio, on the basis of multiple selection criteria is a typical multi-criteria decision-making (MCDM) problem in public organization (Medaglia et al. 2007b; Golabi 1987; Kleinmuntz and Kleinmuntz 1999; Kleinmuntz 2007; Ewing et al. 2006) and in industrial firms (Mavrotas et al. 2006; Golabi et al. 1981; Talias 2007; Stummer et al. 2003). Since project portfolio selection is a very important issue in project and engineering management, a lot of studies proposed approaches (most with successful practical applications) to solve project portfolio selection problem (PPSP) (Aaker and Tyebjee 1978; Bouyssou et al. 2006; Carraway and Schmidt 1991; Dickinson et al. 2001;Fox et al. 1984; Peng et al. 2008a;Golabi et al. 1981; Talias 2007; Kleinmuntz 2007; Ewing et al. 2006; Kuei et al. 1994; Liesio 2006; Cooper et al. 1999; Liesio et al. 2007; Stummer et al. 2009; Henriksen and Palocsay 2008; Mavrotas et al. 2006; Medaglia et al. 2007a,2007b; Santhanam and Kyparisis 1995; Golabi 1987; Kleinmuntz and Kleinmuntz 1999; Stummer et al. 2003). As an example, Aaker and Tyebjee (1978) utilized a quadratic 0–1 programming method to select interdependent R&D projects. Likewise, Mavrotas et al. (2006) combined multicriteria decision analysis (MCDA) with 0–1 integer programming (IP) for project prioritization under policy restrictions. Fox et al. (1984) presented an economic model that included benefit interactions (called present value interactions) between R&D projects and produced some promising results for project portfolio selection problem. Carraway and Schmidt (1991) proposed an improved discrete dynamic programming (DDP) algorithm to allocate resources among interdependent projects. Medaglia et al. (2007a) proposed a multi-objective evolutionary method for linearly constrained projects selection under uncertainty. Particularly, Golabi et al. (1981) proposed an additive multi-criteria value model for project portfolio selection problems. U.S. Department of Energy (DOE) utilized an integer linear programming (ILP) problem subject to resource constraints to select portfolio of solar energy projects. Similarly, Kleinmuntz and Kleinmuntz (1999) and Kleinmuntz (2007) also applied the additive model that has been presented by Golabi et al. (1981) to assist capital budgeting in over 750 hospitals and healthcare organizations in the United States. Furthermore, a three-phase approach including multi-objective integer linear programming (MOILP) has been proposed by Stummer and Heidenberger (2003) for R&D project portfolio selection and presented results of implementing a decision support system (DSS) based on this model in an industrial enterprise. likewise, Stummer et al. (2009) and Henriksen and Palocsay (2008) developed some multi-criteria DSS for evaluating and selecting different projects and obtained some interesting results about project portfolio selection. Carazo et al. (2009) solved a multi-objective project portfolio selection model. Yu et al. (2012) proposed a genetic algorithm based multi-criteria algorithm for project portfolio selection problem. Tupia et al. (2013) proposed an improved genetic algorithm for selection of IT projects with a portfolio problem approach. Nowak (2013) presented a concept of a new methodology based on interactive approach for project portfolio selection. Naderi (2013) proposed a mathematical formulation in form of mixed integer linear programming model and uses three effective metaheuristics in form of the imperialist competitive algorithm, simulated annealing and genetic algorithm are developed to solve the problem of selecting and scheduling a set of projects among available projects.

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2.

Modern Portfolio Theory

Modern portfolio theory is a theory in finance which was developed in 1952 by Markowitz and attempts to maximize portfolio expected return of the portfolio for a given amount of risk. In other words, it seeks to minimize the risk of portfolio for a specified expected return by appropriately selecting the elements of the portfolio (Markowitz (1952, 1959, and 1987). Markowitz originally formulated the fundamental theorem of mean-variance portfolio framework, which explains the trade-off between mean and variance each representing expected returns and risk of a portfolio, respectively (Lin & Gen 2007). The formulation of the mean-variance method can be described as follows: N

N

min ¦¦ wi w j σ ij

,

(1)

R* ,

(2)

i 1 j 1

N

Subject to

¦wP i

i

i 1

N

¦w

i

1,

(3)

i 1

0 d wi d 1

, i = 1,…, N

(4)

Where, N is the number of projects available, P i is the expected return of project i (i = 1,…, N), V ij is the covariance between projects i and j (i = 1,…, N , j = 1,…, N), R* is the total desired expected return. And, the decision variables are: wi the proportion of the total capital invested in project i (i = 1,…, N) ( 0 d wi d 1 ) * In order to determine different objective functions produced by varying R values, standard practice suggests a risk aversion parameter λ ‫[ א‬0, 1]. With this new parameter, the model can be described as:

ªN N º ªN º min O «¦¦ wi w j σ ij »  (1  O ) «¦ wi Pi » , ¬i 1 ¼ ¬i 1 j 1 ¼

(5)

N

Subject to

¦w

i

1,

(6)

i 1

0 d wi d 1 , i = 1,…, N

(7)

When λ equals zero, the equations maximize the mean return of the portfolio, regardless of the risk of portfolio (variance). In contrast, when λ equals unity, the model minimizes the risk of the portfolio regardless of the mean return. Hence, it can be claimed that, the risk of the portfolio becomes more important as λ approaches unity, while when λ approaches zero, the return of portfolio becomes more valuable and important. As a result, with different λ values there will be different objective functions (which is composed of mean return and variance) for project portfolio selection problem. Tracing the mean return and variance intersections, we draw a continuous curve that is called an efficient frontier in the Markowitz theory. Because each point on an efficient frontier curve specifies an optimum portfolio, this problem is a multi-objective optimization problem (Cura T., 2009). Overall, by varying λ and solving equations (5-7) we can trace out exactly the same efficient frontier curve as we would obtain by solving * equations (1-4) for varying values of R . Figure 1 represents a curve (efficient frontier) and portfolios (dots) that are suboptimal. The standard of portfolio management 3 rd edition, considers portfolio efficient frontier as one of the tools and techniques of managing portfolio value (PMI, 2013).

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Fig. 1. Example of Efficient Frontier, source: APIfunds.com

3.

Harmony Search Algorithm

The harmony search algorithm is one of the meta-heuristic algorithms which is inspired by the art of music and has been proposed by Geem et al. to optimize the mathematical functions and engineering problems (Geem et al., 2001). Analogous to the way the musical instruments are played with certain discrete musical notes based on musician`s experiences or randomness in an improvisation process, the design variables can also be assigned with certain discrete values based on computational intelligence or randomness in the optimization process. A musician improves the music he has produced on an aesthetic standard. Likewise, design variables in a computer memory can be improved based on objective function. Steps of the harmony search algorithm are summarized as follows: Step 1. Define the objective function of the problem and the algorithm parameters (HMCR, PAR, bw). Step 2. Construct the harmony memory (HM). Step 3. Improvise a new harmony. Step 4. Update the harmony memory. Step 5. Check the stopping criterion. 3.1.

Defining the objective function and algorithm parameters

In this step, the Objective function and its constraints are described as follows:

Minimize

(8)

f(X)

Subject to: Li d xi d U i

i 1,2,..., N

(9)

The harmony memory is a matrix (memory location) where best harmonies (solution vectors) are saved. Harmony memory is analogous to the genetic pool in the genetic algorithm. HMCR, PAR and bw are parameters which are used to control the diversification and intensification of the harmony search algorithm and to improve the harmonies in the harmony memory (Geem , 2009). 3.2.

Construct the harmony memory

According to the steps that were mentioned before, in this step we initialize the harmony memory. In order to generate that, the algorithm produces solution vectors randomly (harmonies), HMS times and put them in the HM matrix:

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HM

ª x11  x1N º » «   » «  « x1HMS  xNHMS » ¼ ¬

(10)

After constructing the HM and when the algorithm is going to start, each row of the harmony memory matrix represents a random solution vector. 3.3.

Improvise a new harmony

After the harmony memory matrix is produced, the algorithm begins the operation by producing a new harmony and iterations of algorithm are executed one after another. X ( x1 , x2 ,..., xn ) is a new harmony (solution vector) that is improvised regarding the following rules: x

Memory consideration with probability HMCR,

x

Pitch adjusting with probability PAR and

x

Random selection with probability (1-HMCR).

In the new harmony (solution vector) x1 is the value of the first decision variable and this value is selected HMS 1 2 with probability randomly from the values in the first column of the harmony memory matrix x1 , x1 ,..., x1 HMCR (the value of HMCR parameter is variable between 0 and 1. HMCR is the probability of selecting one value from the historical values stored in the HM. On the contrary, (1-HMCR) is the probability of randomly selecting one value from all of the values that a decision variable is able to opt for (Geem, 2009). Pitch adjustment is an operation in which the selected value, is pitch adjusted, regarding the PAR parameter. This parameter is selected between 0 and 1 by the user of the algorithm and is the probability of pitch adjusting the selected value.

^

3.4.

`

Update harmony memory

After the algorithm produces the new harmony ( X ( x1 , x2 ,..., xn ) , new solution vector) then the value of this new harmony is calculated via objective function, and it is compared to the worst harmony in the HM. If the new harmony is better than the worst harmony in the HM, based on the objective function value, the new harmony is included in the harmony memory and the existing worst harmony is excluded from the harmony memory. 3.5.

Check the stopping criterion

The algorithm iterates over and over again, until the stopping criterion (maximum number of iterations, obtaining the optimum value for the problem and etc.) is satisfied and computation is terminated. Otherwise, steps 3 and 4 are repeated (Geem, 2009). Harmony search algorithm has been used by a lot of researchers to solve optimization and reengineering problems. Furthermore, to make a better and robuster algorithm, it has been modified in some studies. Mahdavi et al. (2007) defined variable PAR (VarPAR) and variable bw (Varbw), Jaberipour and Khorram (2010) presented a new way of adjusting bw. Kaveh and Nasr combined the harmony search algorithm with the greedy heuristic (Kaveh and Nasr, 2011, 2012). Kaveh and Ahangaran proposed a social harmony search model to optimize the cost of composite floor system (Kaveh and Ahangaran, 2012). 4.

Numerical Example

In order to undrestand the effectiveness and roubustness of the harmony search algorithm for solving project portfolio selection problem, we utilized this algorithm for solving a benchmark problem which can be found at: http://people.brunel.ac.uk/%7Emastjjb/jeb/orlib/portinfo.html. These data files are the test problems used in the paper of chan et al. (2000). We solved the first benchmark problem (N=31) by the harmony search algorithm.

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The algorithm is coded in MATLAB run on AMD Athlon™ II X2 250 processor 2.99 GHz and 4 GB of RAM. 1.1. Data of this problem is as follows: Number of projects are: N=31, Expected return for each project (asset) is:

Pi (0.0013 0.0042 0.0015 0.0045 0.0109 0.0018 0.0026 0.0050 0.0071 0.0032 0.0021 0.0052 0.0045 0.0036 0.0040 0.0001 0.0003 0.0004 0.0053 0.0048 0.0027 0.0019 0.0047 0.0038 0.0027 0.0048 0.0033 0.0023 0.0058 0.0020 0.0024) The variance of each project and the covariance between projects i and j ( V ij ) are easily calculated according to the data in the OR-Library website. We assign λ, 50 values between 0 and 1. To start the algorithm, λ is assigned 0.02, then according to equation (5) algorithm starts with the following objective function:

º ªN N ªN º min 0.02«¦¦ wi w j σ ij »  (1  0.02) «¦ wi Pi » ¬i 1 ¼ ¼ ¬i 1 j 1

(11)

As it is obvious, at the beginning, the concentration of the algorithm is on maximizing the return of the portfolio. As the algorithm gets through, λ increases and finally equals unity. Because of this and at the end, the major attempt of the algorithm is on minimizing the risk of the portfolio. After finishing the algorithm, 50 ordered pairs (first number * * is risk of the portfolio and the second number is return of the portfolio, ( x , y ) are obtained. If these pairs are illustrated, they represent the Efficient Frontier with a very good accuracy. After setting HMCR, PAR and bw an appropriate amount between 0 and 1, and iteration number to 3000, the algorithm is ready to start. For each value of “λ” the algorithm improves the harmony memory 3000 times. Eventually, 50 ordered pairs (risk, return) are obtained, and are depicted in the same place where the real efficient frontier is illustrated. Figure 2 illustrates the real efficient frontier (green curve) and the blue dots are the ones we obtained from the algorithm. It is evident that the blue dots almost exactly fit the real efficient frontier curve.

Fig. 2. Standard efficient frontier and benchmark problem with N=31

In order to measure the accuracy of the algorithm to find the portfolios and figure out how well they fit the real efficient frontier, we calculate the percentage deviation (distance) of all of the portfolios found by the algorithm, from the real efficient frontier. Figure 3 illustrates the continuous efficient frontier and a portfolio near it. The distance of the portfolio from efficient frontier can be calculated from two directions, horizontally and vertically.

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Here we calculate the distance both horizontally and vertically, then considering the minimum of these two deviations.

Fig. 3. Deviation of the portfolio from efficient frontier *

*

Error measure for each ( x , y ) is regarded as the minimum of the x-direction, y-direction percentage deviations * * (Chang et al., 2000). The mean percentage error of the obtained portfolios ( x , y ) was calculated as %0.022486, and the median percentage error of the obtained portfolios was calculated as %0.000103. CPU time for implementing whole algorithm was 25.641 seconds. 5.

Conclusion

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