# Project portfolio selection using mathematical programming and optimization methods

**Shweta Chopra, PhD Candidate Purdue University**

**Edie K. Schmidt, MBA, PhD, Professor College of Technology Purdue University**

**Abstract**

This paper explores the implementation of a project selection tool using mathematical programming. Project selection is an essential process for portfolio management and plays an important role in accomplishing organizational goals. This paper presents a literature review of the techniques used in project selection. Numerical methods include financial models, scoring models, and optimization models. This paper focuses on project selection using optimization models. This method select a set of projects which deliver the maximum benefit (e.g., net present value [NPV], profit) represented for objective functions subjected to a series of constraints (e.g., budget, manpower). This paper shows simple examples, which includes formulation and solution of the problem using 0-1 integer programming (one objective portfolio) and goal programming (multiple objectives portfolio). Mathematical programming methods can improves the quality of the decision making process reducing subjectivity and optimizing the resources allocation in the projects that add more value to the organization.

Keywords: Project selection, portfolio management, mathematical programming.

**Introduction**

Portfolio categorization, evaluation, and prioritization are essential processes for portfolio management and play important roles in efforts to accomplish organizational strategic goals. Selection processes based on qualitative and quantitative criteria have been used for decision making to justify capital investment and resources allocations. In many cases, financial criteria are the only criteria considered in project selection decisions. In others, the decision making process is still based on the experience and feeling of the top management. Usually the decision that results from these methodologies can be very debatable.

This paper shows models for project selection maximizing the benefits of an organization and considering its strategic goals. This research includes a literature review of the project selection methodologies used in the industry. Following the theoretical framework, this paper shows the use of mathematical programming for project selection. The final sections are devoted to the presentation of some examples and a discussion of the advantages, potential improvement and limitations of this methodology.

**Projects, Portfolio, and Organizational Strategy**

**Project Management**

In general terms, projects can be defined as a planned sequence of managerial and technical activities which employ resources to produce a particular desired outcome. The Project Management Institute (2008a) defines a project as “a temporary endeavor undertaken to create a unique product, service, or result” (p. 5). This definition shows two main features of projects: their temporary nature and unique outcome. These two features make any project different because the time, place, product or process specifications, or people who work in the project make the difference.

Project Management Institute (2008a) defines project management as “the application of knowledge, skills, and techniques to project activities to meet the project requirements” (p. 6). A project that meets requirements is a project which produces the expected results within the scope, budget, and on schedule. In addition, projects must add the maximum value possible to the organization.

**Portfolio Management**

Projects can be grouped into programs and portfolios. A program is defined as “a group of related projects managed in a coordinated way to obtain benefits and control not available from managing them individually” (Project Management Institute, 2008a, p.7). Programs allow companies to enhance the performance of related projects sharing resources and synchronizing efforts. In a broader context, a portfolio is a “collection of projects or programs and other work that are grouped together to facilitate effective management of that work to meet strategic business objectives” (Project Management Institute, 2008a, p.8). In the case of portfolios, the projects and programs associated are not necessarily interdependent but usually they contribute to reach the strategic goals of the organization.

Portfolio management refers to the activities to manage the components of a portfolio (projects and programs) in a coordinated manner to reach organizational objectives (Project Management Institute, 2008b). Portfolio management processes can be grouped into two categories: portfolio planning and portfolio monitoring and control. Portfolio planning includes all the activities that make possible to identify, categorize, evaluate, select, prioritize, balance, and authorize projects that would be undertaken by the organization. The portfolio monitoring and control process includes the evaluation of portfolio performance during the execution phase and checks that it meets a strategic goal (Project Management Institute, 2008b).

**Projects and Corporate Strategy**

At a high organizational level, portfolio management links the single projects with the strategic planning of the organization. The vision, mission, and strategic objectives are the result of the strategic planning cycle of the organization. The vision represents the future desired position for the organization, the mission represents the current statement to add value to customers and shareholders, and the strategic objectives represent the individual achievements that allow the organization to meet the vision. In an organizational context, projects are the driver that makes it possible to reach the vision of a company. In general, companies perform projects to maintain competitiveness and the sustainability of their operations. The motivations to execute projects include these:

• Increase production capacity (new facilities);

• Operations optimization (new technology and process);

• Business opportunities (new products development);

• Customer or market requirements; and

• Legal/environmental requirement.

The portfolio and projects must be associated to specific strategic objectives in order to guarantee that they add the maximum value to the organization (Archer & Ghasemzadeh, 2004). Exhibit 1 shows this relationship.

**Literature Review of Project Selection Process**

Portfolio selection is a process that involves the assessment of a set of available project proposals in order to undertake a group of them that make possible to achieve some strategic goals (Mantel, Meredith, Shafer, & Sutton, 2011). Portfolio selection is a periodic process that must guarantee that projects selected are inside the resource constraints of the organization (Ghasemzadeh & Archer, 2000). Portfolio selection looks for the best balance in terms of return, capital investment, risk, timing, sustainability, and other factors according to the industry sector.

A successful project implies not only doing the project right but also doing the right project. For this reason project selection methodologies play an important role in portfolio management. However, there are a plethora of project selection methodologies, and there is no agreement on which is the most effective (Archer & Ghasemzadeh, 2004). Consequently, organizations choose the methodology that best reflects their project management maturity level, organizational culture, and kind of projects developed. Mantel et al. (2011) classify the project selection methods in two categories: nonnumeric and numeric. The following sections describe the main methodologies according to these two categories.

**Nonnumeric Selection Methods**

Nonnumeric selection methods are used in the industry because these methods are simple and take in consideration the experience and know-how of the decision makers. Some of these methods are described in the following.

*Sacred cow*

*Sacred cow*

In this method, a high level executive based on her or his experience, knowledge, and authority level decides that the organization must develop a specific project (Mantel et al., 2011). This method is common in many kinds of business; however, decisions resulting might be questionable due to subjective assessment of the decision maker or poor technical and economical justification.

*Operating/competitive necessity*

*Operating/competitive necessity*

This method selects the projects that are needed to keep the business running (Mantel et al., 2011). Under certain circumstances, an organization must undertake some projects to assure its sustainability in the long term.

*Comparative approaches*

*Comparative approaches*

Q-sort uses a pool of experts that rank a set of alternatives in a sequence taking into consideration quantitative and qualitative criteria. At the end, this methodology produces a list of ranked projects according to the judgment of the members in the decision pool (Mantel et al., 2011).

The analytic hierarchy process (AHP) is a multicriteria decision method that can use qualitative and quantitative factors and is based on pair-wise comparison by which the judgment of experts produces a recommendation (Saaty, 2008). As project selection is a decision making process, AHP can be used as a project selection methodology. For example, Dey (2006) applied AHP for a project selection case study of a cross-country petroleum pipelines project in India. This case includes identification of alternatives, identification of factors to be considered (technical, environmental, and socio-economic criteria), creation of the AHP framework for deployment of the main and secondary decision factors according to each criteria, and comparison of pair-wise alternatives for each factor and finally aggregating the results.

Vaidya and Kumar (2006) claimed that “the specialty of AHP is its flexibility to be integrated with different techniques like Linear Programming, Quality Function Deployment, Fuzzy Logic, etc” (p.2). This makes it possible to combine AHP with other project selection methodologies and takes advantage of their strengths.

**Numeric Selection Methods**

Numeric selection methods rate the candidate projects according quantitative and qualitative normalized criteria. Criteria usually include financial benefits, productivity, reliability, environmental impact, and risks associated with each project alternative. The main numeric methods are shown in the following section.

*Financial assessment methods*

*Financial assessment methods*

Blocher, Stout, and Cokins (2010) described the financial methods for capital investments evaluation according to two categories: discounted cash flow (DCF) models and non-DCF models.

DCF models consider the value of money in time and include performance indicators such as the net present value (NPV), the internal rate of return (IRR), and the profitability index (PI). The NPV is the difference between the present value of cash inflow and outflow for an investment as calculated in equation 1. A positive NPV means the project earns more than the required rate of return and that the project may be accepted.

Where: *Io* is the initial investment

*Ft* is the net cash flow in the period t

*k* is the required rate of return

*n* is the number of periods in life of the project

The internal rate of return (IRR) is an estimate of rate of return of the investment that produces NPV zero. The project is accepted if the IRR exceeds the discount rate set by the organization.

The profitability index (PI) is the ratio between net present values per invested amount. Equation 2 shows this relationship:

Where: *Io* is the investment

NPV is the net present value

Non-DCF models include the payback period (PBP). The PBP is the time required for the cumulative cash inflow (after-tax) to recover the initial investment. The PBP is considered a measure of the risk of the investment, longer PBP means higher risk to the organization (Mantel et al., 2011). Equation 3 shows how to determine the PBP with an uniform annual cash inflow.

Where: *Io* is the investment

*F* is the annual after tax cash inflow

Financial methods are broadly employed. Blocher et al. (2010) claimed that three of four firms use both NPV and IRR for capital-budgeting purposes. All these financial methodologies are powerful tools to evaluate the economic benefits of a project; however, they ignore nonmonetary factors giving priority to shareholders.

*Scoring methods*

*Scoring methods*

Scoring methods consider more than one criterion and can combine qualitative and quantitative factors. Scoring methods include the unweighted 0-1 factor method and the weighted factor scoring method.

The unweighted 0–1 factor method lists some factors which are desirable for the projects under review and a decision committee check off which criteria are satisfied (Mantel et al., 2011).

The weighted factor scoring method considers a set of factors that have their relative importance weight associated which can be estimated according expert judgment or consensus in a decision committee. How well a project alternative meets a criterion is evaluated, and the final score for each alternative is the product of criterion score and weight (Mantel et al., 2011). Equation 4 shows how to determine the final score for each alternative.

Where: *Si*, is the total score of the *ith* project

*sij* is the score of the ith project on the jth criterion

*wj* is the weight of importance of the *jth* criterion

According to Archer and Ghasemzadeh (2004) “scoring models are probably the easiest to use of all the models” (p. 244). There are numerous examples of application of weighted factor scoring method in different kind of projects and industry sectors.

Sarkis, Presley, and Liles (1997) illustrated a framework for strategic multi-attribute evaluation for business process reengineering (BPR) projects. In this work, a link between the projects and the strategic goals of the organization is established. Three types of strategic metrics categories are used in the analysis: financial, quantitative, and qualitative criteria. The scores for each criterion are normalized using linear utility functions. Finally, weights of criteria are assigned for a decision team.

Project Management Institute (2008b) uses this kind of model as the standard process for evaluation, selection, and prioritization of portfolio components (project and programs). The standard for evaluation presents a scoring model comprising weighted key criteria using a simple 1-5-10 scale for each criterion and then evaluating components according to groups of criteria. This model has some limitations. The problems of weight assignment to the criteria and the introduction of reliability in the values for each alternative are not considered.

Strang (2011) showed an action research case study using a weighted multi-criteria scoring model in a proposals selection process for a nuclear project. In this case study, the mandate was to select and evaluate the best alternative to commissioning a new tritium extraction facility. This model applies AHP to estimate the weights of criteria and the transformation of original-scaled values into dimensionless values to get the total score of each alternative. In general, this case study considers some important elements in the decision process that are omitted in the examples reviewed before such as the estimation of the factor weights using AHP, which takes into account the opinion of experts, and reliability factors for the values of the main variables for each project under consideration.

*Optimization models*

*Optimization models*

Optimization models are based on operation research tools for optimization and use some form of mathematical programming to select a set of projects which deliver the maximum benefit (e.g., NPV, IRR, PBP) represented for and objective function subjected to a series of constraints (e.g., cost, people, technical restrictions). In the literature, there are some examples about using optimizations models combined with some of the other models mentioned earlier. However, according to Archer and Ghasemzadeh (2004), the use of mathematical programming models is not generalized because they can be highly complex and require a significant amount of data.

Ghasemzadeh, Archer, and Iyogun (1999) showed a methodology for project selection and scheduling using a zero-one linear programming model taking into account organizational objectives and constraints such as resources limitations. This particular model looks for maximizing the overall NPV. Lee and Kim (2000) showed a methodology for project selection that uses a zero-one linear programming model for information system (IS) projects, and the objective is minimizing the costs associated with several projects that have some interdependency. In this model, AHP is used in conjunction with linear programming.

**Mathematical Programming Models for Project Selection**

The project selection process can be undertaken using mathematical programming models. The basic problem of linear programming is to maximize or minimize an objective function and meet some constraints at the same time. The formulation of the linear programming problem includes the definition of decision variables, objective function, and constraints. This paper shows two approaches to the problem: the 0–1 Integer Linear Programming model when the decision maker is focus in optimize one objective, and the Goal Programming model when the decision maker considers multiple objectives.

**0-1 Integer Programming Model**

The 0–1 Integer programming model selects a set of projects which deliver the maximum benefit. The formulation of several mathematical programming models can be found in many texts of operations research and mathematical programming. This section focuses on the formulation of the project selection model.

The 0–1 Integer programming model considers *n* candidate projects and each project *i* has an associated decision variable which is defined as follows:

for i=1,…,n, where n is the total number of projects being considered.

The objective function *Z* is the total benefit of the any project set. The solution seeks maximize Z as follows:

Where *Z* is the criterion to be maximized and corresponds to the total benefit of the portfolio and *ci* is the benefit provided by the project *i.* This criterion should be related with the portfolio goals and the organizational strategy.

Constrains are functions that have in consideration the availability of resources (money, people, facilities, etc.) for project execution or can describe some requirements (technical, environmental, etc.) that projects must meet. In general, resources constraints can be defined by:

Where *aij* is the use of resource *j* by the project *i* and *b*j is the availability of the resource *j* to be used for the execution of the project portfolio. In the case of constraints related with requirements, these constraints can be represented by an inequality (> or <) or strictly equal (=) constraint.

Sometimes some constraints that describe any dependent relationship between candidate projects are needed. That is the case of complementary projects where if, for example, project *j* is selected, then project *i* must also be selected, but the opposite is not a condition. This case is described by the following constraint:

Another case is when there are two projects that are disjointed, that is, if for example project *j* is selected, then project *i* cannot be selected. This case is described by the following constraint:

Finally, if for example, project *i* is mandatory, but their execution affect the resources available for the other projects, it must be include in the project selection model with the constraint:

The solution of the set of equations is the set that produce the maximum value for the objective function Z, this corresponds to the set of projects that maximizes the benefit. Linear programming problems are solved applying some algorithms including simplex or Karmakar's method and integer linear programming problems are solved using branch-and-bound method or cutting plane algorithm (Winston & Venkataramanan, 2003). Computational software such as GAMS, Lindo, Lingo, AIMMS and Excel has solvers that employ these algorithms to solve linear and integer programming problems.

*Example*

*Example*

The following basic example shows the use of integer programming in project selection. The company ABC has a pool of six candidate projects and wants to know what projects increase its overall NPV. The budget and NPV associated for each project are shown in Exhibit 2. The company has $18,000 available for investment.

The decision variable indicates if the project is selected or not and it applies the condition described in equation 5. The objective function for this problem is the overall NPV and is given by:

The constraint related with the money that ABC has available for investment can be described by:

Exhibit 3 shows the solution using the Excel solver after making the code of the problem and setting up the constraints. According Exhibit 3, to maximize the NPV, ABC must develop projects 1, 2, 4, and 5 and the maximum NPV possible is $57,000.

**Goal Programming Model**

The goal programming model selects a set of projects which meets (exactly or approximately) some goals while satisfy some constraints. This section describes the formulation of this model.

The goal programming model considers *n* candidate projects, *k* goals and some constraints. In the same way as the 0–1 Integer programming model, each project *i* has an associated decision variable which is defined as follows:

For i = 1, 2,…, n, where n is the total number of projects being considered.

Each goal *j* has associated a target value *g*j and a goal weight *Wj* according its relative importance. Any possible solution (set of projects) has two deviational variables defined as follows:

*Sej* = amount by which the project set numerically exceeds the jth goal*Su*j = amount by which the project set is numerically under the jth goal

The objective function *Z* is the total deviation of the any project set from the goals. The solution seeks minimize Z as follows:

Where

The goals are defines a set of *k* equations in the model, one equation for each goal as follows:

Where: *cji* is the contribution to goal *j* by the project *i* and *gj* is the target of goal *j*

As in the 0–1 Integer programming model, constrains are functions that limit resources for project execution or enforce some requirements (technical, environmental, etc.) that projects must satisfy. In general, resources constraints can be defined by:

Where *aij* is the use of resource *j* by the project *i* and *b*j is the availability of the resource *j* to be used for the execution of the project portfolio. In the case of constraints related with requirements, these constraints can be represented by an inequality (> or <) or a strictly equality (=) constraint.

The solution of problem is the project set that minimize the objective function *Z* (i.e. the set of projects that minimizes the deviation from the goals).

*Example*

*Example*

The following basic example shows the use of goal programming in project selection. This example shows a solution for a proposed problem in Winston and Venkataramanan (2003).

An small aerospace company is considering eight projects. Each project has been rated on five attributes: ROI, cost, productivity improvement, worker requirements, and degree of technological risk. The company have the following goals:

Goal 1 : Achieve a return on investment of at least $3,250

Goal 2: Limit cost to $1,300

Goal 3: Achieve a productivity improvement of at least 6

Goal 4: Limit manpower use to 108

Goal 5: Limit technological risk to a total of 4

The Exhibit 4 shows the information for the candidate projects and goals for this problem.

The decision variable indicates if the project is selected or not and it applies the condition described in equation 13. The objective function for this problem is the total deviation of the goals *Z* and is given by:

Where: *Z* is the total deviation of portfolio of meeting goals and *Wj* is the weight of goal j.

The next set of equations represents the five goals:

Exhibit 5 shows an end user interface developed using AIMMS (Paragon Decision Technology) for this problem. For this input data, the optimal portfolio is the set with projects 1 and 7. Exhibit 6 shows the 5 best portfolios ranked and the corresponding deviation from the goals.

**Implementation of a Decision Support System (DSS) for Project Selection**

Mathematical programming can be used to develop a decision support system that assists the decision maker in the project selection process. This tool does not replace the knowledge and experience of experts, but it gives insights for the decision making team. A decision support system can be customized according to the needs of the organization, policies regarding resources allocation and portfolio management. The objective function can be chosen according to most important criteria for the organization. The constraints can considerer these elements:

• The resources the company needs to develop the project including investment, people, and equipment.

• Technical requirements such as productivity, quality, degree of risk, and environmental requirements.

• Relationship between projects including complementary, disjoint, and mandatory projects.

A decision support system includes the components shown in Exhibit 7

• The user interface that allows the decision maker to interact with the system.

• The management module that address the flow of information between the different components of the system.

• The project portfolio database that keeps the information of the candidate and selected projects.

• The mathematical programming solver applies the algorithms to solve the optimization problem.

A decision support system can provide this information for the users:

• Set of projects that maximize the benefit (objective function) meeting all the constraints established; and

• Sensitivity analysis to check the ranges of project features and constraints; the selected set remains optimal.

**Conclusion**

This paper describes the different methodologies used in project selection emphasizing mathematical programming and optimization techniques. The complexity of mathematical programming can be reduced for the end user with the development of a decision support system, which assists the decision maker in choosing the set of projects that adds more value to the organization.

Some of the advantages of optimization models are: they lead to optimal project selection without bias and subjectivity. Optimization techniques consider relationships between projects and other factors that other methods do not consider. Optimization techniques allow the user to explore scenarios through sensitivity analysis for each factor in the objective function and the constraints. Mathematical programming and optimization techniques rely on the availability and quality of the information about the candidate projects. The more knowledge of the candidate projects, the more accurate the evaluation and selection process.

The high potential of mathematical programming and optimization techniques is based on their capacity to customization according to the needs of the decision-making team. The objectivity and robustness of the project selection process is improved setting the objective function and constraints that best reflect a particular situation.

**References**

Archer, N., & Ghasemzadeh, F. (2004). Project portfolio selection and management. In P. W. G. Morris & J. K. Pinto (Eds.), *The Wiley guide to managing projects* (pp. 237–255). Hoboken, NJ: John Wiley & Sons.

Blocher, E., Stout, D., & Cokins, G. (2010). *Cost management: A strategic emphasis.* (5th ed.). New York, NY: McGraw-Hill Irwin

Dey, P. K. (2006). Integrated project evaluation and selection using multiple-attribute decision-making technique. *International Journal of Production Economics*, 103(1), 90–103.

Ghasemzadeh, F., Archer, N., & Iyogun, P. (1999). A Zero-one model for project portfolio selection and schedulling. *The Journal of the Operation Research Society, 50* (7), 745–755.

Ghasemzadeh, F., & Archer, N. P. (2000). Project portfolio selection through decision support. *Decision Support Systems*, 29(1), 73–88.

Lee, J. W., & Kim, S. H. (2000). Using analytic network process and goal programming for interdependent information system project selection. *Computer and Operation research*, 27(4), 367–382.

Mantel, S. J., Meredith, J. R., Shafer, S. M., & Sutton, M. M. (2011). *Project management in practice* (4th ed.). Hoboken, NJ: John Wiley & Sons.

Project Management Institute. (2011). *What is PMI?.* Retrieved from http://www.pmi.org/en/About-Us/About-Us-What-is-PMI.aspx

Project Management Institute. (2008a). *A guide to the project management body of knowledge (PMBOK ^{®} guide)* (4th ed.). Newtown Square, PA: Project Management Institute.

Project Management Institute. (2008b). *The standard for portfolio management* (2nd ed.). Newtown Square, PA: Project Management Institute.

Saaty, T. L. (2008). Decision making with the analytic hierarchy process. *International Journal of Services Sciences*, 1(1), 83–98.

Sarkis, J., Presley, A., & Liles, D. (1997). The strategic evaluation of candidate business process reengineering projects. *International Journal of Production Economics*, 50(2–3), 261–274.

Strang, K. D. (2011). Portfolio selection methodology for a nuclear project. *Project Management Journal*, 42(2), 81–93.

Vaidya, O. S., & Kumar, S. (2006). Analytic hierarchy process: An overview of applications. *European Journal of Operation Research*, 169(1), 1–29.

Winston, W. L., & Venkataramanan, M. (2003). *Introduction to mathematical programming. Operations research: Volume one* (4th ed.). Belmont, CA: Cengage.

© 2012, Hugo Caballero

Originally published as a part of 2012 PMI Global Congress Proceeding – Vancouver, Canada

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