Decision support system for portfolio components selection and prioritizing

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

Abstract

This paper shows the design and implementation of a decision support system (DSS) for portfolio selection based on optimization models. Components selection is an essential process for portfolio management and plays an important role in accomplishing organizational goals. These models 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 includes formulation and solution of the problem using 0-1 integer programming (one objective portfolio) and goal programming (multiple objectives portfolio) with one or multiple periods as planning horizon. Besides, this paper shows simple examples and the use of a DSS developed in AIMMS, an optimization software, for portfolio selection. Mathematical programming methods can improve the quality of the decision making process, reducing subjectivity and optimizing the resources allocation.

Keywords: Project selection, portfolio management, optimization models.

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 debatable.

This paper shows models for project selection, maximizing the benefits of an organization and considering its strategic goals. This paper includes a brief literature review of the project selection methods used in the industry, the portfolio selection framework, the formulation of project selection based on integer linear programming (ILP) and goal programming (GP), considering one and multiple periods in a planning horizon. The final sections are devoted to the presentation of some examples using a DSS developed in AIMMS and the conclusions with the advantages, potential improvement, and limitations of this methodology.

Portfolio and Organizational Strategy

Portfolio Management

In a broader context, a portfolio is a “collection of programs, projects or operations managed as a group to achieve strategic objectives” (Project Management Institute, 2012b, p. 3). The portfolio components are not necessarily interdependent, but their benefits should be quantifiable and must contribute to reach the strategic goals of the organization.

Portfolio management is “the coordinated management of one or more portfolio to achieve organizational strategies and objectives” (Project Management Institute, 2012b, p. 5). Portfolio management processes can be grouped into three categories: defining, aligning, and authorizing and controlling process groups. Portfolio defining includes all the activities that make possible to identify, categorize, evaluate, select, and prioritize the projects that would be undertaken by the organization. The portfolio aligning includes the balance and optimization of the portfolio and finally the portfolio authorizing and controlling includes the evaluation of portfolio performance and checks that it meets the strategic goals (Project Management Institute, 2012b).

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.

Project Selection Methods

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 (Bible & Bivins, 2011). 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.

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. Nonnumeric selection methods includes sacred cow, selection based on operating/competitive necessity and comparative approaches such as Q-sort (Mantel et al., 2011) and analytic hierarchy process (AHP) (Saaty, 2008).

Numeric Selection Methods

Numeric selection methods rate the candidate projects according quantitative and qualitative normalized criteria and includes the following:

  • Financial methods based on discounted cash flow (DCF: NPV, IRR) models and non-DCF models (pay-back). Financial methods are broadly employed. Blocher, Stout, and Cokins (2010) claimed that three of four firms use both NPV and IRR for capital-budgeting purposes. All of these financial methodologies are powerful tools to evaluate the economic benefits of a project; however, they ignore nonmonetary factors giving priority to shareholders.
  • 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). 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 methods in different kinds of projects and industry sectors. This model has some limitations, for example, weight assignment to the criteria can be subjective and this method does not guarantee the optimal resource allocation.
  • 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) represented for and objective function subjected to a series of constraints (e.g., cost, technical restrictions). In the literature, there are some examples about using optimizations models. 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.

Portfolio Selection and Optimization Framework

Decision support systems based on optimization models can be used during the portfolio selection process of the organization. A critical successful factor in the implementation of a decision support system for project selection is the adoption of a basic portfolio selection framework. The Project Management Institute (PMI, 2012b) presents a standard framework for portfolio management that makes it possible to identify, categorize, evaluate, select, prioritize, and balance the portfolio. Bible and Bivins (2011) developed a detailed project portfolio management (PPM) framework with a screening phase and a selection phase. In the screening phase candidate projects are screened according to some criteria and the results of a business case. The selection phase includes the evaluation of project benefits, followed by the selection of an initial portfolio and, finally, an optimization based on a what if analysis. Archer and Ghasemzadeh (1999) proposed a general framework for project selection considering the whole project life cycle from conception to closing phase. This paper adopts this framework with some modifications such as the inclusion of reviewing for alignment with strategy in the pre-screening stage and the reviewing for feasibility in the screening stage. Exhibit 1 shows this framework and the main stages are described below.

  • Candidate projects definitions. During this stage, a set of candidate projects are proposed. These initiatives can come from customer requirements, legal/environmental requirements, the development of new products, etc.
  • Pre-screening stage. During this stage, candidate projects are reviewed if they are linked to at least one strategic goal of the organization. Any project that does not meet this criterion should be removed.
  • Project study. This stage has as goal to develop a business case for the potential projects with the purpose of defining some main attributes of the projects such as NPV, costs, duration, risk level, etc.
  • Screening stage. This stage considers the assessment of different criteria that are critical success factor for any project and includes the economic and technical feasibility and sustainability. The economic evaluation ensures the project is profitable for the organization. Technical feasibility ensures the organization can obtain the technology to undertake the project. The sustainability assessment includes environmental and social impact.
  • Selection stage. This stage has two parts: project selection and portfolio adjustment. This paper shows the design of a decision support system (DSS), based on mathematical programming that finds the optimal portfolio that maximizes the benefits subjected to customized constraints (technical requirements, resources constraints, and interdependence among projects). After an optimal solution is found, the decision maker can make adjustment in the final portfolio.
  • Execution stage. This stage includes the activities required to develop all deliverables of the projects according to the scope, time, and cost approved.
  • Closing stage. This stage should include the assessment of the portfolio performance and the verification that the goals were meet. This stage gives valuable information and learned lessons to the organization.

Mathematical Programming Models for Portfolio Components Selection

The basic objective of the mathematical programming problem is to maximize or minimize an objective function and meet some constraints. The formulation of the linear programming problem includes the definition of decision variables, objective function, and constraints. There are many forms of mathematical programming for optimization that can be used in portfolio selection including linear and non-linear programming, integer programming, goal programming, dynamic and stochastic programming (Heidenberger & Stummer, 1999). Nonetheless, two approaches seem to be easier to apply in project selection problems: integer linear programming (ILP) model, when the decision maker is focused on optimizing one objective; and goal programming (GP) model, when the decision maker considers satisfying multiple goals. This paper focuses on these two approaches to the problem.

Framework for project portfolio selection. Adapted from Archer & Ghasemzadeh, 1999

Exhibit 1 – Framework for project portfolio selection. Adapted from Archer & Ghasemzadeh, 1999.

Portfolio Components Selection Optimizing One Objective: Integer Linear Programming Approach

The integer programming model selects a set of projects which maximize a benefit (strategic objective). This section focuses on the formulation of project selection problems using integer programming considering two cases: in the first one, it is assumed the projects are executed at the same time, so the resources are available to be used by the selected projects in one period of time. In the second case, project selection and scheduling during a time horizon is considered, so the projects can be executed in different moments according to resources’ availability during each period and relationship between candidate projects.

Portfolio Components Selection optimizing one objective without Scheduling (01-ILP)

This model is the most simplified approach and assumes all resources are available to execute the selected candidate projects at the same time, that is, the resources are available to be used for simultaneous project execution. This problem, known as capital budgeting problem, is described in Chen, Batson and Dang (2010) and the formulation is shown in Equations (1) to (3). This model considers n candidate projects and each project i have an associated decision variable which is defined as follows:

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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:

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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. Usually Z is the overall NPV of the portfolio.

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:

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Where aiq is the use of resource q by the project i and bq is the availability of the resource q 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 equal (=) constraint.

Mathematical programming models can consider interdependent project within a portfolio such as contingent projects, mutually exclusive projects and mandatory projects (Heidenberger & Stummer, 1999). These conditions are described using constraints equations relating candidate projects. For example, consider the case of dependent projects where if project e is selected, then project i must also be selected, but the opposite is not a condition. This case is described by Equation 4 (Winston & Venkataramanan, 2003).

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Another case is when there are two projects that are mutually exclusive, that is, if for example project e is selected, then project i cannot be selected. This case is described by Equation 5 (Winston & Venkataramanan, 2003):

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

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Portfolio Components Selection optimizing one objective with Scheduling (01-ILP)

More complex models can consider the starting time and duration of the candidate projects in the decision variables (Heidenberger & Stummer, 1999). This is a more real approach according to portfolio management in corporate environments and can be used for the optimal distribution of the resources over the planning horizon when a project portfolio should be executed. Ghasemzadeh, Archer, and Iyogun (1999) present a model for project selection and scheduling using zero-one integer linear programming and the basic formulation is shown in Equations 7 through 11. This model considers n candidate projects and t periods of time. The decision variables are defined as follows:

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For i = 1, …, n, where n is the total number of projects being considered and j = 1, …, t, where t is the total number of periods considered in the planning horizon.

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

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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. Usually Z is the overall NPV of the portfolio.

The inclusion of the time for starting a project implies the use of some set of constraints to control the flow of execution, the availability of resources in each period j and the interdependence relationship of some candidate projects. The constraint represented in Equation 9 ensures that each project, if selected, will be started only once during the planning horizon.

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Another important condition is that all selected projects should be finished within the planning horizon. In this case, all projects selected should be finished by the end of period t. This is described in Equation 10:

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Where di is the duration of project i (number of periods required to be completed)

The availability of the resources (financial resources, machinery, workforce) may vary during the planning horizon. For example, the organization may have availability of financial resources according a cash flow (budget). This set of constraints is shown in Equation 11:

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Where bk is the cumulated amount of resource available in period k and ak+1-j is the cumulated amount of resources required by project i in the period k.

It is possible to consider interdependence among candidate projects, such as complementary, mutually exclusive, and mandatory projects. The modeling of this constraints are shown in Equations 12, 13 and 14 (Ghasemzadeh, Archer, & Iyogun, 1999). In the case of complementary projects, if project A depends on project B and C, then if project A is selected, projects B and C must be included in the portfolio. However, projects B and C could be selected even if project A is not included. This condition is considered in the following set of constraints

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Where Sl is the set of complementary projects for a particular project l. If the precursor projects must be finished before the dependent project l, the following set of constraints is necessary:

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Regarding mutually exclusive projects, here only one project of a mutually exclusive set of project can be selected. If P sets of mutually exclusive projects are considered, the corresponding relationship is described by Equation 20.

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Where Sp is a set of mutually exclusive projects.

It is important to consider the set of mandatory projects because these projects consume part of the available resources of the organization during the planning horizon. The following set of constraints allows the inclusion of mandatory projects in the final portfolio:

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Where Sm is the set of mandatory projects

Ongoing projects should be also included in the final portfolio because organizations may decide they should be continued in the following planning horizon and these projects also consume some resources of the organization. The following constraints guarantee the inclusion of ongoing projects in the final portfolio:

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Where So is the set of ongoing projects. It is assumed here that mandatory projects are not interrupted and they continue in the period 1 of the planning horizon.

Portfolio Components Selection Optimizing Multiple Goals: Goal Programming approach

Goal programming models select a set of projects which exactly or approximately meet some target goals while satisfying some constraints. Goal programming models can be linear or non-linear, and integer or non-integer in their objective function or constraints (Heidenberger & Stummer, 1999). There are two approaches of goal programming that can be applied to the project selection problem, depending how the decision maker values the importance of the target goals and the way the objective function is defined: the weighted and the lexicographic goal programming. This paper focuses on weighted goal programming.

Portfolio Components Selection satisfying multiple goals without Scheduling (Weighted GP)

The general goal programming formulation is shown by Jones and Tamiz (2010). A specific formulation for the project selection problem is shown in Equations 17 through 19. This model considers n candidate projects, m goals and some constraints. Each project i have an associated decision variable which was defined by Equation 1.

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

Sep : amount by which the project set numerically exceeds the pth goal

Sup : amount by which the project set is numerically under the pth goal

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

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Qp is a normalization constant associated with the pth goal. This constant ensures that the objective function is consistent with the units when the problem in consideration has goals with different units.

The goals are a set of m equations in the model, one equation for each goal, as shown in Equation 19:

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Where cpi is the contribution to the pth goal by the project i and gp is the target of goal p

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 Equation 3.

The interdependence between projects can be represented with the Equations 4, 5 and 6 as was described before. The solution of the problem is the set of projects that minimize the objective function Z (i.e. the set of projects that minimizes the deviation from the goals).

Portfolio Components Selection satisfying multiple goals with Scheduling (Weighted GP)

The formulation for the project selection with multiple objectives can be modified in order to consider the availability of resources during a planning horizon. This model considers n candidate projects, m goals, t periods, and some constraints. The decision variables are defined by Equation 7.

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

Sep : amount by which the project set numerically exceeds the pth goal

Sup : amount by which the project set is numerically under the pth goal

The objective function Z is the total deviation of the any project set from the goals. The solution seeks minimize Z and is described by Equation 17. The goals are defines as a set of m equations in the model, one equation for each goal, as shown in Equation 19:

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Where cpi is the contribution to the pth goal by the project i and gp is the target of goal p

The constraints describing flow execution (Equations 9 and 10), resources availability in the planning horizon (Equation 11), projects interdependence (Equations 12 to 14), mandatory projects (Equation 15), and ongoing projects (Equation 16) are also applicable in goal programming with scheduling. The solution of the problem is the set of projects that minimize the objective function Z (i.e. the set that minimizes the deviation from the goals).

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

A decision support system (DSS) is a computer-based system that integrates data and some algorithms to produce some information that helps in a decision making process. Mathematical programming can be used to develop a DSS 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 DSS 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 consider:

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

A DSS includes the components shown below:

Components of a decision system support for project selection

Exhibit 2 – Components of a decision system support for project selection

  • Mathematical programming (MP) solver which applies the algorithms to solve the optimization problem.
  • Project portfolio database that keeps all the information of the candidate and selected projects.
  • Graphical user interface (GUI) that allows the decision maker to interact with the system.
  • Management module that addresses the flow of information between the different components of the system.

A DSS can provide the following information to the decision makers:

  • The set of projects that maximize the benefit (objective function), meeting all the constraints (one objective).
  • The set of projects that satisfy the target goals, meeting all the constraints (multiple-goals).
  • Sequence of project execution (scheduling) in case of defining a planning horizon.
  • Sensitivity analysis for the optimal portfolio for different scenarios (change in projects or constraints).

Software for Solution of Mathematical Programming Problems

The increasing application of mathematical programming in many areas in business (e.g., production scheduling, inventory) and the formulation of complex problems (e.g., large number of variables) have made indispensable the use of specialized software. In the last decades, both the development of efficient algorithms and the increasing capacity of processors have made possible the solution of large-scale mathematical programming in reasonable time.

In order to solve a problem of optimization, the first step is the formulation that translates the real world problem in algebraic language defining the decision variables, objective function, and constraints. After that, a computer package is used to solve the problem. During this step, the programmer must translate the formulation into a code that the software can recognize. According to Chen, Batson and Dang (2010) the main components of a software for mathematical programming include modeling language, presolver, solver, and the data and application interface.

  1. Modeling languages introduce the use of sets, symbolic parameters, indexed variables, and constraints, operators and control flow commands. The modeling languages make possible define a symbolic algebraic model of the problem, keeping separated the model and the data. This feature allows running the model with a different set of data creating instances of the same problem and compares results. Among the most popular algebraic modeling languages are AMPL, GAMS, MPL, LINGO, and AIMMS.
  2. The presolver applies preprocessing techniques in order to get a better formulation easier to solve. The presolver adjusts the variables and constraint in order to increase the computational efficiency (Chen, Batson & Dang, 2010).
  3. The solver receives the model from the algebraic modeling language and tries to find an optimal solution for a particular set of data applying the algorithm more convenient according the kind of problem considered. For example, linear programming (LP) problems are solved using the simplex algorithm, while integer programming (IP) problems can be solved using branch and bound algorithm (Winston & Venkataramanan, 2003). Among the most used solvers are CPLEX, Gurobi, Mosek, Conopt, and KNITRO.
  4. Data and application interfaces allow modeling language to read data from external data sources such as databases, spreadsheets or simple text files to generate a matrix that the solver can use to run the solution algorithm. Application interfaces (APIs) developed in commercial programming languages as Java or C++ allow to call modeling languages and solvers from customized applications (Chen, Batson & Dang, 2010).

Some modeling languages incorporate a presolver, data interface, and solvers from different solver providers in order to provide an integrated environment of application software development. A list of the main commercial modeling language and solvers is available, published by INFORMS (Fourer, 2013).

Example of application of a DSS in Portfolio Selection with multiple goals

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

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

Exhibit 3 shows the information for the candidate projects and goals for this problem.

Project information

Exhibit 3 – Project information

The decision variable indicates if the project is selected or not and it applies the condition described in equation 1. The objective function for this problem is the total deviation of the goals Z and is given by:

Min Z = 0.40Su1 + 0.30Se2 + 0.15Su3 + 0.10Se4 + 0.05Se5

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:

ROI: 2070X1 + 456X2 + 670X3 + 350X4 + 495X5 + 380X6 + 1500X7 + 480X8 + Su1Se1 =3250

Cost: 900X1 + 240X2 + 335X3 + 700X4 + 410X5 + 190X6 + 500X7 + 160X8 + Su2Se2 =1300

Productivity: 3X1 + 2X2 + 2X3 + 0X4 + 1X5 + 0X6 + 3X7 + 2X8 + Su3 – Se3 =6

People: 18X1 + 18X2 + 27X3 + 36X4 + 42X5 + 6X6 + 48X7 + 24X8 + Su4Se4 =108

Risk: 3X1 + 2X2 + 4X3 + 1X4 + 1X5 + 0X6 + 2X7 + 3X8 + Su5Se5 =4

Exhibit 4 shows an end user interface for developed using AIMMS (Paragon Decision Technology) for this problem. For this input data, the optimal portfolio is the set with projects 1, 6, and 7, and the total weighed deviation of the goals is 1.25%.

Optimal solution using AIIMS

Exhibit 4 – Optimal solution using AIIMS

Conclusion

This paper describes the use of optimization models based on mathematical programming for portfolio selection. The complexity of mathematical programming can be reduced for the end user with the development of a decision support system (DSS), which assists the decision maker in choosing the set of projects that adds more value to the organization.

Optimization models lead to optimal project selection without bias and subjectivity and consider relationships between projects and other factors that other methods do not consider. Optimization models allow the user to explore scenarios through sensitivity analysis for each factor in the objective function and the constraints. Optimization models 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 maker. The objectivity and robustness of the project selection process is improved setting the objective function and constraints that best reflect a particular situation.

References

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

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

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This material has been reproduced with the permission of the copyright owner. Unauthorized reproduction of this material is strictly prohibited. For permission to reproduce this material, please contact PMI or any listed author.

© 2014, Hugo Caballero
Originally published as a part of the 2014 PMI Global Congress Proceedings – Phoenix, Arizona, USA

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