Project portfolio selection problems: Two decades review from 1999 to 2019

Abstract

Project portfolio selection (PSP) problems have been explored for more than two decades. As a result, in the last two decades, the literature of PSP has experienced a sharp increase. Despite this increase, the literature has not been thoroughly reviewed. The main goal of this paper is to investigate and explain the features, state of the art, research directions, and real-world applications of PSP. To pursue this goal, the paper investigates PSP form aspects of PPS criteria, uncertainty modelling tools, selection and optimization methods, and application areas. More than 105 journal papers have been reviewed. The review has resulted in the categorization of applied tools in PPS in addition to the presentation of new research directions.

Keywords

Project portfolio selection problem real-world applications uncertainty project decision making

1 Project portfolio selection problems

1.1 Definitions, importance, and significance of PPS

Project portfolio selection (PPS) is an important tool that is applied to find the “right” projects and implement the stated strategy [13]. In other words, PPS is a vital process of multi-project management deals with a synergistic process for managing multi-projects to reach organizational strategic goals [104]. To better address project portfolio management, it is necessary to solve project scheduling while addressing project selection. The main goal of project portfolio selection and scheduling (PPSS) is to find a proper set of projects from a number of competing projects and plan a proper implementation scheme [112]. In the PPSS, one area of difficulty is the existence of tradeoffs. Another area of complexity is uncertainty that exists in this process.

PPS is a common problem in the areas of new product development (NPD) and research and development (R&D). Besides, PPS could be applied in technology selection problems, and it has a considerable impact on other practices [45, 76]. PPS has been studied by many scholars in the last two decades. Almost two decades ago, Archer and Ghasemzadeh [1] presented a framework for project portfolio selection that influenced the studies in the following years. To present the significance of PPS studies, the term “project portfolio selection” was searched in SCOPUS on 14 June 2019. The results imply that PPS studies have become increasingly popular since early 2010s. To depict the increase in the last two decades, Fig. 1 is presented. This figure compares the number of published studies over the previous two decades. Figure 2 compares the number of published PPS studies in different years from 1999 to 2019. The results show that the problem has kept its significance and is still an interesting topic for scholars and practitioners.

Fig.1

Comparing the number of published studies in the last two decades.

Fig.2

Number of published studies from 1999 to 2019.

The main results of the search are reported as follows:

In total, 257 documents were discovered.

The main fields of PPS studies are computer science, engineering, decision science, business, management and accounting, mathematics, social sciences, environmental sciences, and earth and planetary sciences. To better illustrate the importance of different fields of PPS studies, Fig. 3 is presented. This figure compares the number of published studies in each field.

Journals with the highest number of published PPS papers were discovered. The journals with the highest number of published papers are European Journal of Operational Research, Annals of Operations Research, Journal of the Operational Research Society and Expert Systems with Applications.

Authors with the highest number of published PPS papers are Cruz-Reyes, L., Gutjahr, W.J., Tavana, M., Fernandez, E., Liesiö, J., Mousavi, S.M., Mohagheghi, V. and Khalili-Damghani, K. Figure 4 presents the authors with the most PPS documents.

Among the PPS documents, North China Electric Power University, Universidad de Malaga, Universitat Wien, Aalto University, Islamic Azad University Science and Research Branch, Universidad Autónoma de Sinaloa, La Salle University, Philadelphia, Northwestern Polytechnical University, and Shahed University had the highest affiliations.

Fig.3

Different fields of PPS studies.

Fig.4

Authors publishing PPS documents.

1.2 Selection objectives

The main objective in project portfolio optimization is to find an optimal portfolio of projects that simultaneously achieve the company’s strategic objectives and considers the limitations that are imposed on the process [6]. In the early studies, the main focus was aimed at the financial criteria of projects. Later, scholars developed frameworks to address PPS with a focus on strategic financial criteria. In recent years, there has been scattered focus on other criteria, e.g., sustainable development [54].

The rest of this review paper is organized as follows: research methodology is highlighted in section 2. In section 3, selection and evaluation criteria applied in PPS are reviewed. Section 4 presents various uncertainty modelling tools applied in PPS studies. Modelling methods and their solution approaches are reviewed in section 5. In section 6, a review of case studies and application examples is given. Finally, section 7 presents the concluding remarks of this review paper.

2 Research methodology

This paper presents a comprehensive literature review addressing concepts, such as project portfolio optimization, project uncertainty, project portfolio case study, and project evaluation. A structured keyword search was carried out to databases and major publisher websites to find proper documents for this paper. Keywords such as “uncertainty”, “evaluation”, “selection” and “mathematical modelling” were combined with phrases like “project portfolio”, “project management”, “construction project”, “research and development” and “new product development” to find proper documents. All the appropriate documents between 1999 and 2019 were extracted from Scopus or Web of Science.

In this study, the material collection has been carried out using the literature search. The review is conducted similar to the method by Seuring [90]. Figure 5 depicts the taxonomy of the applied literature review method. Two main approaches of deductive and inductive methods are often employed to establish criteria for content analysis. The following dimensions will be discussed, explained, and justified:

The selection and evaluation criteria will be reviewed. In this review, the papers applying various approaches to PPS selection criteria are categorized. The criteria are mainly grouped into financial and non-financial (strategic) criteria. Tables are used to better depict the review.

Given the uncertain environment of projects, uncertainty is discussed in a separate section (Section 4). Various forms of fuzzy sets in addition to grey theory, uncertainty theory, and stochastic uncertainty are addressed.

The modelling approaches applied to PPS are reviewed in Section 5. The main identified categories are frameworks and DSSs, optimizing, and scoring methods. For optimizing methods, solution approaches are categorized by two main groups of exact, heuristics and meta-heuristics solutions approaches. Frameworks and DSSs form a more general approach that could include optimizing and scoring methods. However, given their importance in the application, they are reviewed separately.

Given the practicality of PPS problems, a separate section is presented to mention the applied cases of PPS. This section aims to offer the areas in which PPS studies have beencarried out.

Fig.5

Taxonomy of the literature review.

3 Selection and evaluation criteria

In this section, the criteria which have been applied in project portfolio selection problems by scholars in previous studies between 1999 and 2019 are identified and categorized. As mentioned earlier, the initial PPS studies were mainly focused on financial criteria. Then, other strategic criteria, such as social benefits or sustainability, have been addressed. Due to this reason, the selection criteria are grouped into two main groups of financial and non-financial (strategic) criteria. Financial criteria refer to the criteria that are directly linked to the financial conditions of the projects and firms. On the other hand, non-financial or strategic group refers to the criteria that are mainly related to the strategic goals of the firms and how projects support those goals. Therefore, such criteria can have financial impacts, but since they primarily have the strategic impact they are presented in the strategic group. Also, some of the criteria that were more unique are placed in the third category of the other criteria.

3.1 Financial criteria

Given the importance of financial issues in any project-oriented firm, the main factor for PPS is financial. However, addressing financial issues in PPS does not follow a unique method, and a variety of methods are available for this purpose. Some of these methods areas as follows: Net present value (NPV), financial return and return on investment (ROI). There is another group of financial factors that deals with risks associated with the financial aspects of projects. Such factors minimize risks that could lead to bankruptcy or financial failure of the portfolio. Table 1 presents a summary of some of the researches using financial assessment criteria.

Table 1
Studies using financial criteria

Researchers Criteria

Conka et al. [14], Nikkhah Nasab and Najafi [80], Li et al. [59], Li et al. [58], Sefair et al. [89], Li et al. [61] Net present value

Mohagheghi et al. [76], Mohagheghi et al. [77], Mohagheghi et al. [78], Xu et al. [100], Zhang et al. [111] Ratio of fuzzy net present value to lower semi-variance

Bas [5] Risks regarding cash

Khalili-Damghani et al. [53] Risk of investment

Yan and Ji [103] Risk of Bankruptcy

Tang et al. [92] Investment risk tolerance

Zhong et al. [113] Opportunity cost management

Mohagheghi et al. [76, 78], Zhang et al. [111] A new risk index based on lower semivariance

Sefair et al. [89], Kettunen and Salo [50], Mohagheghi et al. [77], Xu et al. [100], Li et al. [61] Downside risk measure, Mean-semivariance, Lower semivariance, skewness risk, mean-variance

Researchers	Criteria
Conka et al. [14], Nikkhah Nasab and Najafi [80], Li et al. [59], Li et al. [58], Sefair et al. [89], Li et al. [61]	Net present value
Mohagheghi et al. [76], Mohagheghi et al. [77], Mohagheghi et al. [78], Xu et al. [100], Zhang et al. [111]	Ratio of fuzzy net present value to lower semi-variance
Bas [5]	Risks regarding cash
Khalili-Damghani et al. [53]	Risk of investment
Yan and Ji [103]	Risk of Bankruptcy
Tang et al. [92]	Investment risk tolerance
Zhong et al. [113]	Opportunity cost management
Mohagheghi et al. [76, 78], Zhang et al. [111]	A new risk index based on lower semivariance
Sefair et al. [89], Kettunen and Salo [50], Mohagheghi et al. [77], Xu et al. [100], Li et al. [61]	Downside risk measure, Mean-semivariance, Lower semivariance, skewness risk, mean-variance

Table 1 presents that net present value is one of the most applicable methods in the literature. However, it is worth noting that this measure requires proper information to provide reliable results. Given the fact that at the initial phases of projects, little or no exact data is given, the accuracy of this method is weak at such phases. This can be dealt with improving the level of knowledge by applying experienced experts’ opinions or using historical data of similar projects. Therefore, dealing with uncertainty can improve financial methods. As presented in Table 1, several attempts have been made to address fuzzy net present value method. In recent years, considering downside risk has been receiving more attention. Risk of bankruptcy is another new approach applied in the literature.

3.2 Strategic criteria

An important fact is that PPS is a strategic level decision-making problem. In other words, in this process, the organization is trying to reach its strategic goals through project implementation. Consequently, addressing strategic criteria is an essential and practical aspect of PPS. Some of the most important strategic criteria applied in the literature of PPS are summed up in Table 2. Aspects, e.g., social benefits, green and environmental impacts and sustainable development, are some of the recent aspects addressed in PPS literature.

Table 2
Various forms of addressing strategy

Researchers Strategic criteria

Carlsson et al. [12], Zhu and Wang [116] Strategic fit and balance

Gutjahr et al. [33], Gutjahr and Reiter [31] The strategic benefits

Stummer and Kiesling [91] Strategic weights of competencies in objective function

Mohagheghi et al. [77], Khalili-Damghani et al. [53], Rad and Rowzan [84] Strategic alignment

Jafarzadeh et al. [47], Liu and Zhang [64] Reinvestment strategy

Khalili-Damghani et al. [53], Khalili-Damghani and Sadi-Nezhad [51], Khalili-Damghani and Tavana [52], Yang et al. [105], Tavana et al. [93], Mohagheghi et al. [76], Mohagheghi et al. [77], Debnath et al. [17] Environmental and green aspects

Mavrotas and Pechak [70, 71] Considering clean development mechanism (CDM) in PPS

Fernandez et al. [23], Zaras et al. [110], Cruz-Reyes et al. [15], Khalili-Damghani et al. [53], Rivera et al. [86], Khalili-Damghani and Sadi-Nezhad [54], Khalili-Damghani and Tavana [52], Mohagheghi et al. [77], Schaeffer and Cruz-Reyes [88], Mohagheghi et al. [75], Ça $\overset{˘}{g}$ lar and Gürel [9] Social aspects

Khalili-Damghani et al. [53], Khalili-Damghani and Tavana [52], Zaras et al. [110], Khalili-Damghani and Sadi-Nezhad [51], Mohagheghi et al. [75], Mohagheghi et al. [77], Dobrovolskien $\dot{e}$ and Tamosiuniene [19], Martins et al. [69], Wu et al. [99] Sustainable aspect

Researchers	Strategic criteria
Carlsson et al. [12], Zhu and Wang [116]	Strategic fit and balance
Gutjahr et al. [33], Gutjahr and Reiter [31]	The strategic benefits
Stummer and Kiesling [91]	Strategic weights of competencies in objective function
Mohagheghi et al. [77], Khalili-Damghani et al. [53], Rad and Rowzan [84]	Strategic alignment
Jafarzadeh et al. [47], Liu and Zhang [64]	Reinvestment strategy
Khalili-Damghani et al. [53], Khalili-Damghani and Sadi-Nezhad [51], Khalili-Damghani and Tavana [52], Yang et al. [105], Tavana et al. [93], Mohagheghi et al. [76], Mohagheghi et al. [77], Debnath et al. [17]	Environmental and green aspects
Mavrotas and Pechak [70, 71]	Considering clean development mechanism (CDM) in PPS
Fernandez et al. [23], Zaras et al. [110], Cruz-Reyes et al. [15], Khalili-Damghani et al. [53], Rivera et al. [86], Khalili-Damghani and Sadi-Nezhad [54], Khalili-Damghani and Tavana [52], Mohagheghi et al. [77], Schaeffer and Cruz-Reyes [88], Mohagheghi et al. [75], Ça $\overset{˘}{g}$ lar and Gürel [9]	Social aspects
Khalili-Damghani et al. [53], Khalili-Damghani and Tavana [52], Zaras et al. [110], Khalili-Damghani and Sadi-Nezhad [51], Mohagheghi et al. [75], Mohagheghi et al. [77], Dobrovolskien $\dot{e}$ and Tamosiuniene [19], Martins et al. [69], Wu et al. [99]	Sustainable aspect

Projects have different environmental impacts that have to be considered while making project-related decisions. Today’s environmental condition has improved the necessity of considering green and environmental issues in various decision-making problems. Another aspect is the social impact of projects. There is a category of PPS studies that is mainly aimed at social PPS. These projects have different characteristics and hence, cannot be addressed by using financial approaches. Some of the studies focus on staff assignment and issue such as learning in PPS (e.g., [31, 32]). The concept of sustainability is based on the interrelationship among social, environmental and financial development [44]. Table 2 presents that the importance of the strategic factors and non-financial issues has increased over the years.

3.3 Other criteria

Numerous application areas of PSS have caused this problem to be associated with various sorts of evaluation and selection criteria. Selecting the best portfolio calls for choosing project evaluation criteria according to the application environment. This makes it almost impossible to categorize all PPS criteria. Thus, this paper uses a third category to present some of the other criteria applied in PPS studies.

A group of these criteria is related to risk. Risk of investment, risk of project implementation, and risk of uncertainty are some examples. Synergy among projects was addressed in the study of Rivera et al. [86]. Hu and Szmerekovsky [39] addressed budget allocation and budget slack. Dynamic allocation of resources was considered by Martínez-Vega et al. [68]. Employee competencies and staff assignment were used in the studies of Gutjahr et al. [31] and Gutjahr and Reiter [32]. Ghaeli et al. [24] considered public acceptance, political acceptance and customer risk. Risk of possibly overtime for subcontractors was addressed by Gutjahr and Reiter [31]. Gutjahr and Froeschl [30] and Zhou et al. [114] addressed risk preference of decision makers. Risk of each project in a mathematical modelling approach is addressed by Gurgur [29] and Bhattacharyya et al. [8]. Underperformance risk [34] and conditional value at risk [57] are some of the other criteria applied in PPS.

PPS criteria play a crucial role in the final results of PPS, therefore, it is necessary to properly address evaluation and selection criteria. This section reviewed selection criteria in three main categories of financial, non-financial, and other criteria. In the financial criteria, the ones closely related to the financial condition of the firm were reviewed. The review showed that several financial measures have been used and developed in PPS studies. One practical improvement is addressing uncertainty in the financial criteria. This would improve the process by attending to uncertain project environments. Another category was strategic criteria. Given the fact that PPS is a strategic decision-making process, this section separately addressed strategic criteria. Such criteria are dependent on the mission and the stated goals of the firms. In details, firms set long term goals and missions and their way of reaching such goals is through project implementation. Therefore, strategic criteria are derived from goals of the firms, and then the projects that support such goals are selected. The review of strategic criteria showed a variety of such criteria. Actually, this variety shows that it is necessary to establish designed criteria for each area of PPS application. The third group of criteria consisted of criteria that could be better represented in a separate category. Some of these criteria are related to the operational level of projects. Performance of projects, technical aspects, time considerations, interdependency, synergies, and staffing are some of such criteria. These criteria deal with successful completion of selected projects in a portfolio. A review of these criteria showed that this category can be improved by using the literature on success factors of projects to develop PPS criteria.

4 Uncertainty

In fact, in any real-world PPS process, one of the concepts that increases the complexity of the process is the uncertainty that exists in the project evaluation [70, 71]. Insufficient information and lack of expertise are almost always common in investment decision making processes. In project management, it is necessary to properly address uncertainty. This section reviews different fuzzy extensions that have been used to address uncertainty in PPS.

4.1 Stochastic uncertainty

Using stochastic tools to address uncertainty is a common and popular approach in many decision-making environments. Stochastic theory applies historical data to address uncertainty. However, using stochastic theory in a project environment is not very popular since projects are unique and do not have sufficient historical data. One solution to deal with this issue is utilizing data from similar past projects. Graves et al. [28] used conditional stochastic dominance to address PPS. Mavrotas and Pechak [70, 71] used stochastic parameters in Monte Carlo simulation in PPS. Stochastic multi-criteria acceptability analysis was used by Yang et al. [105]. Tofighian et al. [95] considered stochastic income in PPS. Stochastic optimization has been applied in the studies of Gutjahr and Froeschl [31], Panadero et al. [81], Gurgur [29] and Gutjahr and Reiter [31].

4.2 Classic fuzzy sets theory

Zadeh [108] introduced fuzzy sets to address uncertainty. In projects, imprecision of information and lack of sufficient data are some of the factors that make it necessary to apply experts’ judgments. This is done by using fuzzy sets. Fuzzy PPS has been the subject of many studies. Carlsson et al. [12] developed fuzzy mixed integer programming model by using trapezoidal fuzzy number. Credibilistic fuzzy measure was used by Zhang et al. [111], Xu et al. [100] and Li et al. [57]. Bas [5] developed a fuzzy multidimensional 0–1 knapsack model. Fuzzy rule-based approach was used in the studies of Riddell and Wallace [85] and Khalili-Damghani et al. [53]. Fernandez et al. [23] applied fuzzy outranking relations. Using fuzzy constraints in mathematical modelling can be found in the study of Perez and Gomez [82]. Lifshits and Avdoshin [62] used fuzzy multi-objective modelling approach. Liu and Liu [65] employed robust fuzzy optimization in PPS. Lukovac et al. [66] applied Neuro-fuzzy modelling in PPS. Through the years, the necessity for improving fuzzy sets theory arose as it was more applied in real-world situations. In the following sections, applications of fuzzy extensions in PPS are reviewed.

4.3 Type-2 fuzzy sets

Zadeh [109] developed type-2 and higher-types fuzzy sets. Type-2 fuzzy sets possess fuzzy membership functions (FMFs), which are also referred to as “membership of membership”. In type-2 fuzzy sets, each element has membership value expressed by fuzzy set in [0, 1]. Furthermore, type-2 FMFs are three-dimensional which consider possibilities by using the weights in the membership domain. The novel third dimension gives new design degrees of freedom for coping with secondary uncertainties [73, 67]. It can be concluded that type-2 fuzzy sets have a higher capability in coping with situations where judgments are more subjective and more imprecise. Interval type-2 fuzzy sets were developed to apply the advantages of type-2 fuzzy sets with less complexity [72]. The computational intensiveness of general type-2 fuzzy sets has caused the popularity of interval type-2 fuzzy sets [55]. Mohagheghi et al. [78] applied interval type-2 fuzzy sets to present a PPS mathematical model under uncertainty. Their model employed the ratio of lower semivariance to return index to evaluate projects and form a portfolio of projects. Another recent application of type-2 fuzzy sets is the study of Wu et al. [99]. They applied type-2 fuzzy sets in PPS in energy section. Their model addresses sustainability in renewableenergy projects.

4.4 Interval-valued fuzzy sets

Using intervals instead of a crisp value to address membership degrees has led to the development of interval-valued fuzzy sets [27]. Thus, interval-valued fuzzy sets express unknown and vague membership degrees by intervals in [0, 1] instead of traditional [0, 1] - valued membership degrees [106]. Mohagheghi et al. [76] applied interval-valued fuzzy sets to present a mathematical model to address PPS. Another example of these sets is the study of Mohagheghi et al. [78]. They presented a MADM approach based on interval-valued fuzzy sets to develop aPPS model.

4.5 Intuitionistic fuzzy sets

Intuitionistic fuzzy sets (IFSs) [2] improve classic fuzzy sets by providing the ability to express the degree of belonging, not belonging, and hesitation. However, these sets are not able to fully express the ideas of the DMs since the values of membership, non-membership, and hesitancy cannot form a value bigger than 1. One of the applications of intuitionistic fuzzy sets in PPS is the study of Mohagheghi et al. [75]. Another application is the study of Wu et al. [98]. They presented a framework for energy project portfolio selection.

4.6 Pythagorean fuzzy sets

There are times when the degree of membership, non-membership, and hesitancy make a number bigger than 1. Yager [101, 102] developed the concept of Pythagorean fuzzy sets (PFSs) to address these conditions. These sets are an extension of intuitionistic sets. This extension has resulted in sets that are more powerful and flexible in expressing uncertainty. By using these sets, the DM can express opinions that could not be expressed in intuitionistic sets. Mohagheghi and Mousavi [74] introduced a mathematical model for high-technology project portfolio selection by using PFSs. Their study was the first study to apply these sets in PPS.

4.7 Hesitant fuzzy sets

Hesitant fuzzy sets (HFSs), initially developed by Torra [96], are the extensions of normal fuzzy sets which handle the situations where a set of values are possible for the membership of a single element. Zhou and Xu [115] presented portfolio models in hesitant fuzzy environments. Their study, however, is not focused on a portfolio of projects and deals with an investment portfolio. Thus, these sets have not been applied in PPS studies.

4.8 Grey theory

Grey theory is another uncertainty modelling tool that has been used in project environments. These sets employ intervals instead of crisp values to address uncertain elements. Bhattacharyya [7] presented a grey theory-based method for R&D PPS. Balderas et al. [3] introduced a TOPSIS-Grey method to address PPS. Balderas et al. [4] developed a grey mathematical approach to address PPS.

4.9 Uncertainty theory

Using uncertainty theory introduced by Liu [63] is a new approach to address project uncertainty. Liu [63] developed a new uncertain measure and further extended the presented theory based on normality, duality, sub-additivity, and product axioms. His presented approach has been used by Huang and Zhao [40], Huang et al. [42], Huang and Zhao [41], and Yan and Ji [103] in project selectionstudies.

5 Modelling approaches

Project evaluation and PPS studies have applied a wide variety of approaches. Two of the most known classifications of project evaluation methods are as follows: (1) Archer and Ghasemzade [1] classified the approaches into five main groups of ad hoc methods, comparative methods, scoring approaches, portfolio matrices, and optimization approaches. (2) Iamratanakul et al. [45] reviewed and categorized the project selection models into groups of scoring methods, economic methods, mathematical programming, real options analysis, simulation modelling, and heuristics methods. In this paper, to illustrate different approaches used to address this problem, first optimization methods and their solution approaches are reviewed. Then, scoring methods are presented and reviewed. Finally, studies introducing frameworks and DSSs arereviewed.

5.1 Optimization methods and solution approaches

Optimization is a popular approach in PPS. In fact, due to the features of PPS and the requirements of organizations, different forms of mathematical modelling approaches have been developed. Models based on their objectives can be categorized as single, bi and multi-objective. In case of uncertainty, optimizing methods based on stochastic, fuzzy, and robust techniques have been developed. Different forms of mathematical programming, such as integer programming (IP), mixed integer programming (MIP), linear programming (LP), non-linear programming (NLP), and quadratic programming (QP), have been proposed. To review mathematical programming methods applied in PPS, Table 3 is presented.

Table 3
Optimization approaches in PPS

Authors Models

Ghasemzadeh et al. [25], Urli and Terrien [97], Zhu and Wang [116], Yu et al. [107], Nikkhahnasab and Najafi [80], Tavana et al. [94], Hassanzadeh et al. [37] Integer programming

Doerner et al. [20], Jafarzadeh et al. [47] Integer linear programming

Carlsson et al. [12], Bas [5], Zhu and Wang [116], Li et al. [57], Wu et al. [98] Fuzzy programming

Gutjahr et al. [33], Gutjahr and Reiter [32], Naderi [79], Li et al. [58], Li et al. [59], Ghodoosi et al. [26], Sefair et al. [89], Schaeffer and Cruz-Reyes [88], Li et al. [60], Zhong et al. [113] Mixed integer programming

Hu et al. [38], Gutjahr et al. [32], Urli and Terrien [97], Carazo et al. [11], Fernandez et al. [23], Hassanzadeh et al. [36], Perez and Gomez [82], Roland et al. [87], Wu et al. [98], Mohagheghi et al. [77], Rabbani et al. [83] Multi-objective programming

Gutjahr et al. [33], Gutjahr and Reiter [31], Urli and Terrien [97], Yu et al. [107], Hall et al. [34], Carazo [10], Ghodoosi et al. [26], Li et al. [60] Non-linear programming

Gurgur [29], Gutjahr and Reiter [31], Gutjahr and Froeschl [30], Mavrotas and Pechak [70, 71], Yang et al. [105] Stochastic programming

Zhang et al. [111], Mohagheghi et al., [76], Mohagheghi et al. [77], Li et al. [60] Semi-variance, mean variance models

Sefair et al. [89] Quadratic programming

Authors	Models
Ghasemzadeh et al. [25], Urli and Terrien [97], Zhu and Wang [116], Yu et al. [107], Nikkhahnasab and Najafi [80], Tavana et al. [94], Hassanzadeh et al. [37]	Integer programming
Doerner et al. [20], Jafarzadeh et al. [47]	Integer linear programming
Carlsson et al. [12], Bas [5], Zhu and Wang [116], Li et al. [57], Wu et al. [98]	Fuzzy programming
Gutjahr et al. [33], Gutjahr and Reiter [32], Naderi [79], Li et al. [58], Li et al. [59], Ghodoosi et al. [26], Sefair et al. [89], Schaeffer and Cruz-Reyes [88], Li et al. [60], Zhong et al. [113]	Mixed integer programming
Hu et al. [38], Gutjahr et al. [32], Urli and Terrien [97], Carazo et al. [11], Fernandez et al. [23], Hassanzadeh et al. [36], Perez and Gomez [82], Roland et al. [87], Wu et al. [98], Mohagheghi et al. [77], Rabbani et al. [83]	Multi-objective programming
Gutjahr et al. [33], Gutjahr and Reiter [31], Urli and Terrien [97], Yu et al. [107], Hall et al. [34], Carazo [10], Ghodoosi et al. [26], Li et al. [60]	Non-linear programming
Gurgur [29], Gutjahr and Reiter [31], Gutjahr and Froeschl [30], Mavrotas and Pechak [70, 71], Yang et al. [105]	Stochastic programming
Zhang et al. [111], Mohagheghi et al., [76], Mohagheghi et al. [77], Li et al. [60]	Semi-variance, mean variance models
Sefair et al. [89]	Quadratic programming

To further review the optimization techniques and mathematical programming methods of PPS, two main solution methods of exact, heuristics, and meta-heuristics are reviewed. Several exact solution algorithms have been proposed to address PPS mathematical models. For example, various extensions of Bender’s decomposition have been used to discuss solution approaches of PPS. Hassanzadeh et al. [36] proposed a robust optimization method for uncertain linear programming. Hassanzadeh et al. [37] also applied robust augmented weighted Tchebycheff program to address PPS. Benders decomposition was applied by Hall et al. [34] and Sefair et al. [89]. Another example is the study of Roland et al. [87] that applied cutting-plane approach.

Heuristics and meta-heuristics approaches have also been used to address PPS mathematical models. Ant colony optimization was used in the studies of Doerner et al. [20], Gutjahr et al. [33], Gutjahr et al. [32] and Rivera et al. [86]. Gutjahr et al. [38] introduced a greedy heuristic to address PPS. NSGAII as a popular tool was applied in the studies of Gutjahr et al. [32], Gutjahr and Reiter [31], Khalili-Damghani et al. [53], Fernandez et al. [23], Wu et al. [98] and Balderas et al. [3]. Scatter Search was used by Carazo et al. [11]. Application examples of variable neighborhood search (VNS) are the studies of Gutjahr and Froeschl [30] and Panadero et al. [81]. Another popular tool in PPS studies is the genetic algorithm. It was employed in the studies of Zhang et al. [111], Gutjahr et al. [38], Yu et al. [107], Nikkhahnasab and Najafi [80], Naderi [79], Li et al. [57] and Liu and Zhang [64]. Nikkhahnasab and Najafi [80] and Naderi [79] used simulated annealing. The imperialist competitive algorithm was applied by Naderi [79] to address PPS. Also, harmony search algorithm is another tool applied in PPS [22].

5.2 Scoring methods

Another main group of PPS studies focuses on scoring methods to find a portfolio of projects. These methods are mainly based on Multi-attribute decision making (MADM) methods. Table 4 reviews these methods.

Table 4
Scoring methods of PPS

Authors Models

Ghaeli et al. [24], Conka et al. [14], Hashemizadeh and Ju [35] AHP

Conka et al. [14], Khalili-Damghani et al. [53], Tavana et al. [94], Jafarzadeh et al. [46] Data envelopment analysis (DEA)

Riddell and Wallace [85] Fuzzy rule-based aggregation procedure

Khalili-Damghani et al. [53] Evolutionary algorithm (EA)

Khalili-Damghani et al. [51] Balance score card (BSC)

Khalili-Damghani and Tavana [52] Structural equation modelling

Jeng and Huang [48] Modified Delphi method (MDM), a decision-making trial and evaluation laboratory (DEMATEL) method, and an analytic network process (ANP)

Tavana et al. [94], Balderas et al. [3] TOPSIS

Hummel et al. [43], Debnath et al. [17] Measuring attractiveness by a categorical based evaluation technique (MACBETH)

Debnath et al. [17] Decision-making trial and evaluation laboratory (DEMATEL)

Authors	Models
Ghaeli et al. [24], Conka et al. [14], Hashemizadeh and Ju [35]	AHP
Conka et al. [14], Khalili-Damghani et al. [53], Tavana et al. [94], Jafarzadeh et al. [46]	Data envelopment analysis (DEA)
Riddell and Wallace [85]	Fuzzy rule-based aggregation procedure
Khalili-Damghani et al. [53]	Evolutionary algorithm (EA)
Khalili-Damghani et al. [51]	Balance score card (BSC)
Khalili-Damghani and Tavana [52]	Structural equation modelling
Jeng and Huang [48]	Modified Delphi method (MDM), a decision-making trial and evaluation laboratory (DEMATEL) method, and an analytic network process (ANP)
Tavana et al. [94], Balderas et al. [3]	TOPSIS
Hummel et al. [43], Debnath et al. [17]	Measuring attractiveness by a categorical based evaluation technique (MACBETH)
Debnath et al. [17]	Decision-making trial and evaluation laboratory (DEMATEL)

5.3 Frameworks and decision support systems

Frameworks are used to simplify the PPS process. The framework is also employed as a basis for decision support. In a DSS, the software helps in integrating user activities in each of the decision making process stages smoothly while providing a high level of usability that is required for effective managerial decision making [1]. In Table 10, a review of frameworks and DSSs introduced and used in PPS studies is presented. Archer and Ghasemzadeh [1] presented one of the first decision-making frameworks for PPS. Dong et al. [21], Martins et al. [69] introduced web-based DSSs for PPS. Another example of DSS is the study of Hu et al. [38]. Stummer and Kiesling [91] developed a multi-criteria DSS to address PPS. Later, Khalili-Damghani et al. [53] presented a hybrid multi-objective framework. This framework integrated data mining model with the results from a data envelopment analysis (DEA) model and an evolutionary algorithm (EA). Cruz-Reyes et al. [15] developed a method by DSS and SMART method. Cruz-Reyes et al. [16] also presented a DSS by using argumentation theory. M-MACBETH DSS was presented by Hummel et al. [43].

In this section, different modelling approaches were reviewed. Such models limit the capability of top managers in the decision-making process. Different scoring methods have also been developed in PPS (e.g., [3, 17]). Such methods are mainly based on judgments, and unlike mathematical models are easily affected by opinions of decision makers. Given the mentioned facts and due to the fact that PPS is concerned with qualitative and quantitative data, it is often more practical to develop frameworks that apply both the scoring methods and mathematical models (e.g., [77, 94]). While comparing PPS with other managerial problems, it is obvious that PPS studies suffer from a lack of comprehensive expert systems.

6 Case studies and application areas

One of the most vital aspects of PPS studies is their application area. Hence, this section presents a review of case studies and applications of PPS studies. The Electronic Commerce Competence Center (EC3) Austria was the main area of case study application for the studies of Gutjahr et al. [33], Stummer and Kiesling [91], Gutjahr et al. [32], Gutjahr and Reiter [31], Gutjahr and Froeschl [30]. Bas [5] and Mohagheghi et al. [76] addressed PPS in construction projects. Oil and gas industry case studies were discussed at the studies of Zhu and Wang [116], Mohagheghi et al. [78], Sefair et al. [89], Yan and Ji [103] and Stummer and Kieling [91]. An example of a financial company application of PPS can be found in the study of Khalili-Damghani and Tavana [52]. Mavrotas and Pechak [70, 71] addressed energy and power generation in PPS. An example of the food industry application can be found in the study of Dixit and Tiwari [18]. Application in the area of robotic innovations for minimal invasive surgical interventions was done by Hummel et al. [43]. Martins et al. [69] applied PPS in sustainable strategic decision making in an electricity company. Jeng and Huang [48] presented an application of PPS in research institutes. An interesting application is the study of Korotkov and Wu [56]. They carried out a risk assessment for projects participating in the Belt and Road Initiative.

7 Conclusions and directions for further research

In this paper, two decades of PPS literature was reviewed. In recent years, PPS studies have increased, and this is caused by recognition of the importance of PPS applications. This paper reviewed the evaluation and selection criteria of PPS, uncertainty modelling tools, selection methods, and application areas. This review has resulted in identifying the recent trends of PPS. The following presents the recent PPS trends:

Developing frameworks based on a combination of MADM and MODM techniques in PPS studies in increasing. One example is the study of Tavana et al. [94], in which DEA, TOPSIS and LP were used to form a PPS framework. This approach has the advantages of MADM and MODM methods and therefore provides more flexibility.

To make an efficient project portfolio, in addition to PPS, project implementation and management should be added to PPS frameworks. One of the recent trends to address this issue is simultaneous consideration of PPS and project portfolio scheduling.

Application areas of PPS studies are no longer limited to investment and construction projects. In recent years, oil and gas projects, high-technology projects, energy projects, research projects, etc. have been added to the application areas of PPS studies. This shows that PPS is becoming more important in new fields, and organizations are utilizing the advantages of project portfolio techniques in their processes.

Mathematical modelling tools have been used to model real conditions of project and firms in addition to the imposed constraints and limitations. As a result, the complexity of models has increased, and this has led to the utilization of a variety of heuristic and meta-heuristic methods.

The importance of uncertainty in project environments has led to the application of several new tools to address uncertainty. For example, uncertainty theory presented by Liu [63] and several fuzzy extensions, e.g., intuitionistic, type-2, interval-valued and Pythagorean fuzzy sets, are among new tools applied to deal with vague project conditions.

In addition to the recent trends, the review has resulted in identifying several gaps in the literature of PPS. The following presents the identified literature gaps:

Since this decision-making process requires the participation of firm managers, mathematical models, and frameworks often do not provide the final results, and the selection requires modification from managers. This calls for developing DSSs for PPS. Several DSSs have been developed in recent years but literature is rather weak when it comes to various forms of expert systems.

Using fuzzy methods to address PPS is a common approach to address uncertainty. However, using fuzzy extensions in PPS is new, and the literature is weak on this subject. Furthermore, using hybrid tools, such as fuzzy stochastic, intuitionistic type 2 fuzzy sets, etc., is a gap that has not been addressed in the literature.

In mathematical modelling of PPS scheduling issues, like time-cost-quality trade-off, are not yet addressed. In other words, PPS and scheduling studies are new to the literature and time crushing and activity scheduling form some of the gaps of PPS and scheduling.

The literature is weak in developing tailored exact solution approaches. Developing solution methods, such as lagrangian relaxation, benders decomposition, and branching techniques, for PPS models is not well-established in the literature. Furthermore, developing heuristic solution approaches for PPS is a literature gap.

In scoring methods of PPS, there are some gaps when it comes to weights of decision makers, subjective and objective weights of criteria, aggregation steps, and decision indexes.

Finally, some resulting future research directions are presented to assist researchers in developing PPS studies:

Developing mathematical models based on fuzzy-stochastic uncertainty is an interesting future research direction that can provide the model with the benefits of fuzzy and stochastic uncertainty.

Developing hybrid fuzzy mathematical models is another exciting research direction that can improve uncertain PPS studies beyond fuzzy extensions. In other words, using hybrid fuzzy uncertainty provides the model with the benefits of several fuzzy extensions instead of just one.

Developing models that incorporate operational and strategic goals in PPS is another interesting research directions. For instance, developing models that address PPS based on sustainable development and resilient scheduling could enhance PPS studies.

Developing models based on project implementation evaluation techniques like earned value analysis assists portfolio managers in reaching better PPS and scheduling results.

Developing accepted and standard criteria for each application area of PPS could be an exciting future research direction. In other words, given the various application fields of PPS, each area requires its criteria which could assist practitioners in using PPS methods.

Footnotes

Acknowledgments

The authors would like to thank four anonymous referees for their valuable comments on this paper.

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