Franck Bariscanbaz

Marle

Construction of project portfolio considering efficiency, strategic effectiveness, balance and project interdependencies

Keywords: Project management, Project portfolio selection, Project interactions, Nonlinear programming, Constraints satisfaction

Construction of project portfolio considering efficiency, strategic effectiveness, balance and project interdependencies

Baris Canbaz, Franck

Introduction

While project management is a process that focuses on the extent to which a specific and temporary endeavor establishes, maintains and achieves its intended targets within cost, time, and performance parameters, project portfolio management focuses attention at a more aggregated level. Its primary objective is to build and maintain the appropriate mix of projects necessary to achieve organizational goals, with a short-term or long-term range. A portfolio may mix projects of different natures, of different sizes and of different status (from 0.1% to 99.9% of advancement). Portfolio-level decisions are crucial since they affect strategic effectiveness, resource allocation efficiency and balance at a portfoliolevel rather than at a project-level perspective [START_REF] Floricel | Using R&D portfolio management to deal with dynamic risk[END_REF]; Petit

Challenges for PPM research

Consequence of complexity is a potential distortion in assessment and prediction of project attributes and interdependencies. This is an important issue since choosing the right combination of projects may help to obtain more value than the sum of individual projects, because of positive interactions (called synergies), or less value in case of negative interactions (called cannibalizations). Organizations therefore need to pay more attention to managing the mix of various projects to avoid negative interactions among the projects and to benefit from possible positive interactions. Historically, many PPM tools and techniques treated each project as an isolated entity, despite the gap with reality and despite the existence of literature for decades [START_REF] Reiter | Choosing an investment program among interdependent projects[END_REF][START_REF] Aaker | Model for the selection of interdependent R&D projects[END_REF]T. E. Gear & Cowie 1980).

Several difficulties are associated with project portfolio selection [START_REF] Kerzner | Advanced project management -Best practices on implementation[END_REF][START_REF] Ghasemzadeh | Portfolio selection of capital investment projects in the Durban metropolitan region[END_REF]. Firstly, the inclusion of a project and its input and output values may be dependent on the other projects in the same portfolio. For instance, when projects share the same resource (human or material), the resource requirements of projects may be reduced if they are included together in the portfolio. Interactions can also be negative when there are cannibalizations between projects in the same portfolio, in terms of market share for instance. Projects may be mutually exclusive, which means that they cannot simultaneously exist in the portfolio. On the other hand, mutually inclusive projects cannot be started without the inclusion of the other projects in the portfolio. Determination of these interactions and interdependencies, in terms of existence and value, is an extremely complex process.

Secondly, random and epistemic uncertainties are inherent to the evaluation data of a project, due to the stochastic nature of the project activity and its environment, and because evaluation is undertaken before the project starts. In reality, everything is potential and some criteria may sometimes be evaluated years before they occur, like the final profit or the final cost for instance.

Thirdly, variables may be qualitative or quantitative, and it is difficult to get a global evaluation by mixing different kinds of data (for example, mixing the image of the organization with the expected long-term profit and with the improvement of customer satisfaction). This requires some work to calibrate the evaluation scales in order to make them comparable.

Lastly, the portfolio may need to be balanced. Some zones have to be defined in order to analyse the balance in terms of number of projects in each of them. There are two key points when studying balance: zones must be correctly defined, and particular attention must be paid to projects which are close to a frontier.

Literature review on project portfolio selection

From the beginning of the development of the subject, numerous project selection tools and approaches have been proposed, as this activity being recognized as critical [START_REF] Guimaraes | Organizational bias in the selection of MIS projects[END_REF][START_REF] Mckeen | Selecting MIS projects by steering committee[END_REF][START_REF] Melone | Strategies for MIS project selection[END_REF]Santhanam & Schniederjans 1993). We divide these various tools into four categories: portfolio mapping tools, multi-criteria ranking tools, mathematical programming tools and hybrid tools [START_REF] Hall | An interactive approach for selecting R&D projects[END_REF][START_REF] Cooper | Best Practices for Managing R&D Portfolios[END_REF].

Portfolio mapping tools

As introduced in previous Section, portfolio mapping tools are generally simple, two-axis diagrams that easily display the trade-off between two criteria [START_REF] Dickinson | Technology portfolio management: Optimizing interdependent projects over multiple time periods[END_REF][START_REF] Mikkola | Portfolio management of R&D projects: Implications for innovation management[END_REF]). Assessment may be qualitative or quantitative, precise or imprecise. The use of bubbles instead of points has the advantage of adding complementary information illustrated with the size, color and shape of the bubble (R. G. Cooper, E. S. Edgett, et al. 1997;R. G. Cooper, S. J. Edgett, et al. 1997). These tools use graphical and charting techniques to help to prioritize projects (depending on their position on the graph) and to estimate the balance of the portfolio (the number of projects in each zone of the graph). This supposes that the criteria used for mapping has been previously defined, as well as their evaluation scale (ordinal, cardinal, quantitative…) and the different zones for interpretation of the position of the project. Interdependencies between projects are sometimes considered [START_REF] Groenveld | Roadmapping integrates business and technology[END_REF]Killen 2012).

Multi-criteria decision-making tools

Multi-criteria decision-making tools include scoring and ranking/sorting models. These methods maximize the value of the portfolio with both financial and/or non-financial measures. With scoring models, projects are assessed on a number of different criteria that can be weighted regarding their importance (Moore & N. R. Baker 1969;[START_REF] Lucas | A multiple-criterion scoring approach to information system project selection[END_REF][START_REF] Henriksen | A practical R&D project-selection scoring tool[END_REF][START_REF] Jiang | Information system project selection criteria variations within strategic classes[END_REF][START_REF] Lootsma | Multi-criteria analysis and budget reallocation in longterm research planning[END_REF][START_REF] Shoval | Models for computer system evaluation and selection[END_REF].

Ranking and outranking methods propose a respectively absolute or relative ranking of alternatives [START_REF] Buss | How to rank computer projects[END_REF][START_REF] Odusote | An examination of the importance of resource considerations when contractors make project selection decisions[END_REF]. For instance, Analytic Hierarchy Process (T. [START_REF] Saaty | The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation[END_REF]T. L. Saaty 2001) and the Electre family [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF][START_REF] Figueira | ELECTRE methods with interaction between criteria: An extension of the concordance index[END_REF][START_REF] Doumpos | An evolutionary approach to construction of outranking models for multicriteria classification: the case of the ELECTRE TRI method[END_REF][START_REF] Skitmore | Bid-spread[END_REF]) are multi-criteria scoring/ranking methods that use hierarchical criteria and pair wise comparisons. AHP has been particularly used in numerous research works, including some refinements and variations around the initial version of the method [START_REF] Muralidhar | Using the analytic hierarchy process for information system project selection[END_REF][START_REF] Khalil | Selecting the appropriate project delivery method using AHP[END_REF][START_REF] Brenner | Practical R&D project prioritization[END_REF][START_REF] Mohanty | Project selection by a multiple-criteria decision-making method: an example from a developing country[END_REF][START_REF] Alidi | Use of the analytic hierarchy process to measure the initial viability of industrial projects[END_REF][START_REF] Xidonas | Portfolio engineering using the IPSSIS multiobjective optimisation decision support system[END_REF][START_REF] Gerogiannis | A case study for project and portfolio management information system selection: a group AHP-scoring model approach[END_REF].

Generally, refinements consist of including interdependencies using the Analytic Network Process (E. W. L. [START_REF] Cheng | Analytic network process applied to project selection[END_REF][START_REF] Meade | R&D project selection using the analytic network process[END_REF][START_REF] Aragonés-Beltrán | An ANP-based approach for the selection of photovoltaic solar power plant investment projects[END_REF]; J. W. [START_REF] Lee | An integrated approach for interdependent information system project selection[END_REF] or introducing uncertainty using fuzzy numbers [START_REF] Mahmoodzadeh | Project Selection by Using Fuzzy AHP and TOPSIS Technique[END_REF]C. C. Huang et al. 2008;[START_REF] Tiryaki | Fuzzy portfolio selection using fuzzy analytic hierarchy process[END_REF][START_REF] Büyüközkan | A fuzzy-logic based decision-making approach for new product development[END_REF]. The checklist method is a variation on the scoring model with multiple criteria in multiple categories assessed with a yes/no answer based on a minimum threshold acceptance value (R. G. Cooper et al. 1998).

Mathematical programming

Mathematical models developed for portfolio selection focused historically on maximizing the financial portfolio value within some budget constraints (N. R. [START_REF] Baker | R&D project selection model: An assessment[END_REF][START_REF] Danila | Strategic evaluation and selection of R&D projects[END_REF][START_REF] Liberatore | The practice of management science in R&D project management[END_REF][START_REF] Pillai | Performance measurement of R&D projects in a multiproject, concurrent engineering environment[END_REF]. They focused mainly on quantitative tools and values, like cash flow, net present value or return on investment or payback period [START_REF] Liberatore | An extension of the analytic hierarchy process for industrial R&D project selection and resource allocation[END_REF][START_REF] Okpala | Evaluation and selection of construction projects in Nigeria[END_REF]Weingartner 1966b).

In order to overcome limitation of such models and of some multi-criteria decision-making models (the compensatory bias), some mathematical programming models have been introduced, such as goal programming (S. [START_REF] Lee | Goal programming for decision-making[END_REF][START_REF] Benjamin | A linear goal programming model for public sector project selection[END_REF][START_REF] Badri | A comprehensive 0-1 goal programming model for project selection[END_REF][START_REF] Santhanam | A zero-one goal programming approach for information system project selection[END_REF][START_REF] Khorramshahgol | An integrated approach to project evaluation and selection[END_REF][START_REF] Ghasemzadeh | Portfolio selection of capital investment projects in the Durban metropolitan region[END_REF], multi-objective or multi-attribute decision-making (G. [START_REF] Hwang | Multiple attributes decision-making methods and applications[END_REF]Schniederjans & Santhanam 1993;[START_REF] Anadalingam | A multi-stage multi-attribute decision model for project selection[END_REF][START_REF] Ringuest | The linear multi-objective R&D project selection problem[END_REF][START_REF] Molenaar | Model for public sector design-build project selection[END_REF], quadratic or nonlinear programming [START_REF] Weber | Planning Models for Research and Development[END_REF][START_REF] Santhanam | A multiple criteria decision model for information system project selection[END_REF] and dynamic programming [START_REF] Ullmann | Dynamic programming and the capital allocation problem[END_REF][START_REF] Nemhauser | Discrete dynamic programming and capital allocation[END_REF].

Like in multi-criteria methods, uncertainty is often modeled through the fuzzy set theory (K. Chen & Gorla 1998; C.-T. Chen & H.-L. Cheng 2009; J. Wang & W. L. Hwang 2007;[START_REF] Machacha | A fuzzy logic-based approach to project selection[END_REF][START_REF] Coffin | Multiple criteria R&D project selection and scheduling using fuzzy logic[END_REF][START_REF] Wong | A fuzzy stochastic technique for project selection[END_REF][START_REF] Carlsson | A fuzzy approach to R&D project portfolio selection[END_REF][START_REF] Huang | Optimal project selection with random fuzzy parameters[END_REF][START_REF] Ghapanchi | A methodology for selecting portfolios of projects with interactions and under uncertainty[END_REF][START_REF] Iyigun | A decision support system for R&D project selection and resource allocation under uncertainty[END_REF][START_REF] Tiryaki | Fuzzy stock selection using a new fuzzy ranking and weighting algorithm[END_REF][START_REF] Lin | A fuzzy decision support system for strategic portfolio management[END_REF][START_REF] Zhu | Developing an R&D projects portfolio selection decision system based on fuzzy logic[END_REF]. Some models do not need computational support to make calculations, like decision trees and game theory techniques [START_REF] Hess | Swinging on a branch of a tree: project selection application[END_REF][START_REF] Ali | Selecting product development projects: pioneering versus incremental innovation strategies[END_REF].

Data Envelopment Analysis (DEA) helps select the best portfolio taking both project uncertainties and projects interdependencies into consideration simultaneously [START_REF] Eilat | Constructing and evaluating balanced portfolios of R&D projects with interactions: A DEA based methodology[END_REF]. Classically, fuzzy DEA (FDEA) is an extension considering a particular way to model uncertainty on data [START_REF] Ghapanchi | A methodology for selecting portfolios of projects with interactions and under uncertainty[END_REF].Generally, all of these models not direct attention to balancing the portfolio. Recent developments enrich the parameters considered for optimizing the portfolio [START_REF] Liesiö | Preference Programming for Robust Portfolio Modeling and Project Selection[END_REF][START_REF] Eilat | Constructing and evaluating balanced portfolios of R&D projects with interactions: A DEA based methodology[END_REF].

Hybrid decision-making tools

Hybrid decision-making tools are combinations of a number of different tools [START_REF] Conka | A combined decision model for R&D project portfolio selection[END_REF][START_REF] Sun | A packing-multiple-boxes model for R&D project selection and scheduling[END_REF][START_REF] Riddell | The use of fuzzy logic and expert judgment in the R&D project portfolio selection process[END_REF]. For instance, AHP is mixed with goal programming [START_REF] Schniederjans | Using the analytic hierarchy process and goal programming for information system project selection[END_REF] or multiattribute decision-making [START_REF] Dey | Integrated project evaluation and selection using multiple-attribute decisionmaking[END_REF]. A mix of different techniques for considering uncertain and imprecise assessments and project interdependencies has been proposed [START_REF] Ravanshadnia | Hybrid fuzzy MADM project-selection model for diversified construction companies[END_REF]. The Contingent Portfolio Programming is a combination of decision trees and mathematical programming [START_REF] Gustafsson | Contingent Portfolio Programming of Risky Projects[END_REF]. A similar model existed before (Lockett & A. E. Gear 1973), but CPP integrates new dimensions like uncertainties with certainty equivalencies, and the support of multiple additive decision criteria regarding the staged nature of R&D projects.

Problem formulation

4.1. Modelling and integrating project interdependencies The challenge of managing multiple complex projects in a dynamic and uncertain environment is amplified by the presence of project interdependencies, which are simultaneously crucial for reliable decisions and difficult to capture and pre-dict (Aritua, N. J. Smith, and Bower 2009;Blau et al. 2004;Collyer and Warren 2009;Dahlgren and Söderlund 2010;Elonen and Artto 2003;Perminova, M. Gustafsson, and Wikström 2008;Rungi 2010;Verma and Sinha 2002).Focus is made here on project interdependencies consideration, as Baker and Freeland and more recently Eilat and co-authors underlined the weaknesses of these portfolio selection models and criticized them (N. [START_REF] Baker | Recent advances in R&D benefit measurement and project selection methods[END_REF][START_REF] Eilat | Constructing and evaluating balanced portfolios of R&D projects with interactions: A DEA based methodology[END_REF]): ""one of the most important limitations of present R&D project selection models is the inadequate treatment of project interrelationships with respect to both value and resource utilization."" Several types of project interdependencies exist and have been introduced in previous works [START_REF] Dickinson | Technology portfolio management: Optimizing interdependent projects over multiple time periods[END_REF]T. E. Gear and Cowie 1980;Killen and Kjaer 2012;Newell et al. 2008;Verma and Sinha 2002). The originality of our contribution is to consider simultaneously four types of interactions between projects, in terms of outcomes, of resource consumption, of probability of success and of mutual inclusion/exclusion. Resource may include human, material and financial resources. The benefit interdependency may be considered by different ways, for instance complementary or competitive effects on market shares. Here, it is a combined probability of success if both projects are run simultaneously which is considered. A fifth type of interdependency is called learning dependency: "the need to incorporate the capabilities and knowledge gained through another project" (Killen and Kjaer 2012).

Like project attributes, interdependencies are by nature difficult to assess, since even their existence is sometimes potential, meaning that we have a potential chance to have interdependency between two projects with an uncertain value. This makes all the more difficult the introduction of project interdependencies into exact calculation models. The results should be analysed considering the error or uncertainty on data, meaning that the gap between two proposed solutions should be superior to the error margin to be considered reliable enough to make a decision.

Several methods have included project interdependencies in the portfolio selection problem formulation [START_REF] Czajkowski | Selecting interrelated R&D projects in space technology planning[END_REF]Weingartner 1966a; J. W. [START_REF] Lee | An integrated approach for interdependent information system project selection[END_REF][START_REF] Reiter | Choosing an investment program among interdependent projects[END_REF][START_REF] Aaker | Model for the selection of interdependent R&D projects[END_REF][START_REF] Santhanam | A decision model for interdependent system project selection[END_REF]. Schmidt presents three different types of interactions in his portfolio construction model [START_REF] Schmidt | A model for R&D project selection with combined benefit, outcome and resource interactions[END_REF]. It combines the effects of resource interactions, benefit interactions and outcome interactions among projects. These interactions are represented with matrices. The combined effect of these different interactions among projects was initially considered by Aaker and Tyebjee, but they did not define an objective function and they did not construct a methodology for solving the problem [START_REF] Aaker | Model for the selection of interdependent R&D projects[END_REF].

Dickinson and co-workers presented a real world application of portfolio management process at Boeing Company [START_REF] Dickinson | Technology portfolio management: Optimizing interdependent projects over multiple time periods[END_REF]). The proposed method accounts for revenue interactions among projects with a dependency matrix which is constructed with qualitative assessments. An objective function maximizes the net present value of the entire portfolio. The model contains constraints about resource budget, about the maximum number of projects in a portfolio and about the minimum number of projects that must support each of the strategic objectives of the program. One project can support only one strategic objective.Liesiö and co-workers extended their Robust Portfolio Modelling technique [START_REF] Liesiö | Preference Programming for Robust Portfolio Modeling and Project Selection[END_REF]) to account for project interactions [START_REF] Liesiö | Robust Portfolio Modeling with Incomplete Cost Information and Project Interdependencies[END_REF]. Interactions among projects are represented with dummy projects that are triggered by synergy and cannibalization constraints. However, these constraints do not contain the combined effect of different type of interactions that Schmidt considered.

Schmidt"s approach is the most complete one in terms of its description and formulation of project interactions. Considering the three portfolio objectives (Efficiency, Balance, and Strategic Effectiveness), all approaches maximize the portfolio efficiency. Dickinson and co-workers support portfolio balance with graphical tools. Liesiö and co-workers provide portfolio balance more effectively with logical constraints. Regarding the strategic effectiveness of a portfolio, Schmidt"s approach does not support this portfolio objective except with budget constraints. Since there is no complete method which satisfies the entire project portfolio construction objectives while integrating the combined effect of different interactions among the projects, we combined the strongest aspects of these models with additional features.

This Section introduces our formulation proposal, including objective function and numerous constraints. The strength of this proposal is the introduction of useful and feasible constraints, meaning that they have an influence on the performance of the final proposed portfolio (influencing parameter), but simultaneously they make sense for practitioners and are feasible in terms of assessment (precision and energy to get the data).  Efficiency: the goal is to maximize a value which is often defined by profitability or some other financial measure, relating outcomes to resources used to obtain them.  Balance: the portfolio can be balanced in dimensions such as long-term vs.

Nomenclature

short-term, low risk vs. high risk and breakdown by project types or market areas.  Strategic Effectiveness: investments should contribute to corporate objectives.

For example, even a highly profitable investment can be turned down if it does not relate to core competencies of the firm.

It is possible to run a multi-objective optimization or to focus on a specific objective and consider the two other parameters as constraints. We decide to use the latter form and generate the portfolio which has the maximal efficiency value while not violating the constraints of balance and strategic effectiveness.

A portfolio is defined as a binary vector z, which indicates for each project whether it belongs or not to the portfolio. The efficiency value of the portfolio refers to its expected revenue which is the difference between the total expected output y of the portfolio and the total input x invested into the portfolio, where inputs are both variable and fixed costs. The aim of the objective function is to maximize the expected revenue of the portfolio:

Max (Revenue) = y r R r=1 - x i I i=1
The interactions handled in this methodology are considered for pairs of projects. They consider the marginal effect of the selection or rejection of a project on the other projects in the portfolio. External interactions may appear over time due to social, political and economic changes, but are not considered in this study. Interactions among projects will change the amount of inputs required during the execution of the portfolio and the amount of expected outputs generated with this execution. These interactions are resource interactions, benefit interactions, and outcome interactions, following previous works [START_REF] Schmidt | A model for R&D project selection with combined benefit, outcome and resource interactions[END_REF][START_REF] Fox | Economic models for R&D project selection in the presence of project interactions[END_REF][START_REF] Eilat | Constructing and evaluating balanced portfolios of R&D projects with interactions: A DEA based methodology[END_REF]. Input values, output values, resource interactions and benefit interactions should be scaled with the same unit of measurement; this can be a common financial unit or a qualitative score.

Benefit interactions occur if the total amount of the benefits (direct outputs) of interacted projects which are executed simultaneously is different from the situation in which the projects are executed individually. If the projects are complementary, this difference is positive (synergy effect); if they are competitive, then the difference is negative (antagonism). 𝑽 𝒓 is the benefit (value of direct output) interaction matrix of output r which is a lower triangular matrix (v kj r = 0, ifj > 𝑘). The diagonal element of this matrix v jj r represents the amount of direct output r from the individual project j and the off-diagonal element v ij r represents the benefit interaction of output r between projects i and j.

Outcome interactions occur if the probability of success of a project changes by undertaking another project in the same portfolio. This interaction reflects the relationship among the project successes. 𝑷is the probability of success interaction matrix. The diagonal element p jj is the probability of success of project j and the off-diagonal element p jk represents the probability of success interaction among two projects, which is the marginal change in the probability of success of the project j when project k is included in the portfolio. In general, 𝑷 matrix is neither symmetrical nor triangular, because the reciprocal impacts can be different; project j may have greater impact on project k than vice versa. The detailed formulation of the amount of output expected from the portfolio is thus given by:

y r = z j z l p jl n p l=1 v jj r + v ji r z i p il z l n p l=1 j-1 i=1 n p j=1 for∀r j, l, i, h ∈ ℤ z j ∈ D j , ∀j = 1 ⋯ n p
Resource interactions occur if the total resource requirements of a project portfolio cannot be represented as the sum of its individual project requirements. It occurs when the projects share the same resources, where portfolio resource requirements are less than the sum of individual project requirements. U i is defined as the resource interaction matrix of input i. This is a lower triangular matrix(u kj i = 0, if j > 𝑘). The diagonal element u jj i represents the amount of input i that is required for the individual project j and the off-diagonal element u kj i represents the resource interaction of input i between projects j and k.The detailed formulation of the amount of input i required for the portfolio is thus given by:

x i = z j n p j=1 u jj i + z j n p h=j+1 z h u hj i n p j=1 for∀i j, l, i, h ∈ ℤ z j ∈ D j , ∀j = 1 ⋯ n p
The CSP requires a solution that maximizes the objective function. Variables of the CSP are :z j ∀j = 1 ⋯ n p . All of the variables are binary so the domains of the variables include only 0 and 1, D j = 0,1 , ∀j = 1 ⋯ n p . Constraints of the CSP are described hereunder.

Portfolio Balance Constraints

We define balance constraints in order to control which projects must be executed together and which must not be executed together in the same portfolio. For instance, two very risky projects should not be launched simultaneously, since they may both fail. If they share the same market or the same scarce resource, we would prefer not to undertake them together. On the other hand, it is possible to decide that a project cannot be started without starting another one, or that they can be complementary. It might be more efficient to fund these two projects simultaneously.

With these balance constraints, we can control the diversification of the projects in the portfolio in terms of various trade-offs. This helps to build balanced portfolios and to express constraints of mutual exclusion or mutual inclusion. For example, if project 3 must always be included when project 1 is already in the portfolio, then the related constraint is conditional:

𝐼𝐹𝑧 1 = 1 𝑇𝐻𝐸𝑁𝑧 1 + 𝑧 3 = 2
If project 5 and project 2 are mutually exclusive projects and project 5 and project 3 likewise, only one of these projects can be selected. The related constraint is: IF𝑧 5 = 1 𝑇𝐻𝐸𝑁𝑧 5 + 𝑧 2 + 𝑧 3 = 1 Portfolio Strategic Effectiveness Constraints described hereunder will guarantee economic fit, strategic fit and feasibility of the portfolio.

Portfolio Size Constraints

Organizations could have preferences on maximum and/or minimum number of projects in a single portfolio, this defines portfolio size limits. For example, they can prefer the portfolios which have equal or less than a given number of projects, since the portfolios which have more projects than this limit would be too complex to be executed by the organization and outside the feasibility limits. Thus, organizations would like to define the maximum number of projects in a portfolio to maintain the portfolio feasible and to avoid funding very complex portfolios.

As well as defining maximum number of projects in a portfolio, organizations can also define the minimum number of projects in a single portfolio. They can stipulate that each portfolio must have equal or more than a given number of projects, since the portfolios which do not contain enough projects generally do not provide enough profit and do not benefit from synergies between projects. The definition of the maximum and minimum number of projects in a portfolio will support effectiveness, and this will help organizations to align the mix of projects in the portfolio with organizational needs and limits.

𝑧 𝑗 𝑛 𝑝 𝑗 =1 ≤ 𝑀𝑎𝑥 𝑧 𝑗 𝑛 𝑝 𝑗 =1 ≥ 𝑀𝑖𝑛
Portfolio size limits can also be dependent on portfolio elements, namely the projects which are selected in the portfolio. For example, if a project is very complex, an organization can consider having a maximum limited number of other projects in a portfolio with this project. An organization wants to maintain feasibility, otherwise the portfolio will be so complex to handle that it will no longer be feasible. An organization can also consider a project too simple to be executed with less than a minimum number of other projects.

𝐼𝐹𝑧 𝑗 = 1 𝑇𝐻𝐸𝑁 𝑧 𝑗 𝑛 𝑝 𝑗 =1 ≤ 𝑀𝑎𝑥 𝑗 𝑓𝑜𝑟∀𝑗 𝐼𝐹𝑧 𝑗 = 1 𝑇𝐻𝐸𝑁 𝑧 𝑗 𝑛 𝑝 𝑗 =1
≥ 𝑀𝑖𝑛 𝑗 𝑓𝑜𝑟∀𝑗 While Max and Min define the limits of maximum and minimum number of projects in the portfolio for any combination of the projects regardless of the projects in the portfolio and their characteristics, 𝑀𝑎𝑥 𝑗 and 𝑀𝑖𝑛 𝑗 determine the limits of the size of the portfolio for a particular project j included in this portfolio.

Input and Output Threshold Constraints

Projects use different type of input to create different type of output. In general, these inputs are limited and the organization has minimum output expectancies. Therefore, a selection model that incorporates resource limitations may aid the decision maker. Typically, an organization is not able to simultaneously engage more than a certain quantity of inputs, called R i . It would like to obtain at least a certain amount of outputs from their project portfolios, called Q r , otherwise they find it worthless to undertake. 𝑥 𝑖 ≤ 𝑅 𝑖 𝑦 𝑟 ≥ 𝑄 𝑟

Strategic Objective Constraints

Every organization aims to link the project portfolio to its business strategy and to its general strategic objectives. The main focus is to ensure that the final portfolio truly reflects the strategy of the company. In order to maintain this alignment of the portfolios with organizational strategy, we use a projectobjective contribution score matrix, called S. In this matrix, we give a score to every project P j for each objective O o , this score is called s jo . This matrix shows the contribution level of projects to strategic objectives of the company. The scoring scale can be adjusted according to organizational preferences.

The management team evaluates the s jo with qualitative and/or normalized quantitative assessments. They also define threshold values for each objective, threshold of the objective o is called 𝑀𝑖𝑛𝑆𝑐𝑜𝑟𝑒 𝑜 .These are the minimum objective levels that must be provided for each objective. If the minimum score of an objective is exceeded, then the portfolio is considered as aligned with this objective.

𝒛 𝒋 𝒏 𝒑

𝒋=𝟏

𝒔 𝒋𝒐 ≥ 𝑴𝒊𝒏𝑺𝒄𝒐𝒓𝒆 𝒐 𝒇𝒐𝒓 ∀𝒐

Mandatory Projects

Finally, mandatory projects which absolutely must be in the portfolio are defined, as suggested by [START_REF] Stewart | A multi criteria decision support system for R&D project selection[END_REF]. This would include projects with a very high importance for strategic objectives, or that may use a specific resource. In these cases, the organization cannot ignore them and the project is automatically included in the portfolio. For example, if project 1 is a mandatory project, then its corresponding binary value is 1 in the vector: 𝒛 𝟏 = 𝟏

Solving strategy

Three previous reviews of project selection methods and models have been used and completed [START_REF] Badri | A comprehensive 0-1 goal programming model for project selection[END_REF]E. W. L. Cheng and H. Li 2005;Jin Wang et al. 2009). Methods have evolved from static quantitative programming techniques to more flexible techniques such as fuzzy multi-attribute decisionmaking or fuzzy mathematical programming. The criteria are not only economical and technical, but also environmental and societal. They are evaluated not only with quantitative and measurable scales.

Our interest is the portfolio selection tools that are capable of handling interactions among project investments and resources. Cooper and co-authors highlight the fact that no existing portfolio selection method is superior in all aspects, dominant in the sense of Pareto (R. Cooper et al. 1998). They all have strengths and weaknesses in different aspects. In this study, we focus only on methods which can take into account interactions between project investments and project resources. While mathematical programming provides variability and flexibility in the problem definition, we can also model very complex interactions between projects. This is why we have chosen mathematical modelling to study interactions in this paper.

The problem is formulated as a Constraint Satisfaction Problem (CSP). It features adequate formulations of various constraints in order to satisfy the portfolio objectives. The CSP concept is often used in artificial intelligence and is suitable for many problems in different research fields such as: operational research and logic programming [START_REF] Brailsford | Constraint satisfaction problems: algorithms and applications[END_REF]. With this concept, consistent assignments of the values to the variables are determined within a short period of time. A CSP is defined by a set of variables with non-empty domain sets that include possible values of each variable, and a set of constraints that involve subsets of the variables and define the allowable value combinations of the variables in the related subset [START_REF] Montanari | Networks of Constraints: Fundamental Properties and Applications to Picture Processing[END_REF].

Usually CSP solvers (e.g. IBM ILOG CP) utilize a branch and propagate and bound algorithm [START_REF] Khichane | Strong Combination of Ant Colony Optimization with Constraint Programming Optimization[END_REF] or branch and prune algorithm [START_REF] Hentenryck | Solving Polynomial Systems Using a Branch and Prune Approach[END_REF]) in order to determine consistent assignments of the variables. Assignments are generated on nodes through a search tree and if the constraint consistency is violated nodes are pruned so the related domains are filtered. Thus redundant assignments are avoided in the combinatorial problem and computational time is decreased. The CSP is described below with the formulation of the objective function and the description of constraints.

Application to a numerical example

In this section, a fictitious example is proposed in order to demonstrate the methodology, test its capabilities and compare the results with and without integrating project interactions. The case is introduced with its numerical data and the formulation of its constraints. The data are randomly generated, which enables to avoid confidentiality issues. Then, a computer algorithm is developed in C++ computer language and through IBM ILOG CP Solver it is used to solve the problem.

6.1. Numerical data A company wants to build a portfolio from 10 possible projects. This company cannot afford to simultaneously support more than 7 projects, because of the managerial complexity and of the difficulty in assigning project managers. They want a portfolio which has at least 3 projects, because portfolios which are too small bring too little added value to the company.

𝑧 𝑗 𝑘 10 𝑗 =1 ≤ 7 𝑓𝑜𝑟 ∀𝑘 𝑧 𝑗 𝑘 10 𝑗 =1 ≥ 3 𝑓𝑜𝑟 ∀𝑘
The company has already made some investment in the machines that will be used during the execution of Project 9, so this project is considered as a mandatory project which must be in the portfolio. 𝑧 9 𝑘 = 1 𝑓𝑜𝑟 ∀𝑘 Project 2 and Project 5 are considered as very complex projects by the top management. If at least one of them is in the portfolio, then the maximum total number of projects is 4. As Project 2 and Project 5 are very complex projects, the company doesn"t want to execute these projects together at the same time.

𝐼𝐹 𝑧

2 𝑘 = 1, 𝑇𝐻𝐸𝑁 𝑧 𝑗 𝑘 10 𝑗 =1 ≤ 4 𝑓𝑜𝑟 ∀𝑘 𝐼𝐹 𝑧 5 𝑘 = 1, 𝑇𝐻𝐸𝑁 𝑧 𝑗 𝑘 10 𝑗 =1 ≤ 4 𝑓𝑜𝑟 ∀𝑘 𝐼𝐹 𝑧 2 𝑘 = 1, 𝑇𝐻𝐸𝑁 𝑧 2 𝑘 + 𝑧 5 𝑘 = 1
𝑓𝑜𝑟 ∀𝑘 Project 8 is considered as moderately complex. If it is included in the portfolio, then the maximum number of the projects in the portfolio must be 5. They do not want to execute Project 3 with Project 4 and with Project 8, because with this combination, the risk of portfolio failure increases.

𝐼𝐹 𝑧 8 𝑘 = 1, 𝑇𝐻𝐸𝑁 𝑧 𝑗 𝑘 10 𝑗 =1
≤ 5 𝑓𝑜𝑟 ∀𝑘 𝐼𝐹 𝑧 3 𝑘 = 1, 𝑇𝐻𝐸𝑁 𝑧 3 𝑘 + 𝑧 4 𝑘 + 𝑧 8 𝑘 = 1 𝑓𝑜𝑟 ∀𝑘 Project 1 is defined as a simple project, so they want to have this project in a portfolio which has at least 4 projects. and 𝑃 should be gathered by expertise of the management of the company before formulating the model. Data gathering is out of focus of this example. Each project uses 2 different inputs to produce 2 different outputs. U i is the resource interaction matrix of input i, for i=1 or 2.V r is the output interaction matrix of output r, r=1 or 2. Resource and output matrix values are in k €. Projects are interdependent, which may change the probability of success if launched together. This is described in the probability of success interaction matrix P. ≥ 1000 𝑓𝑜𝑟∀𝑘

𝐼𝐹 𝑧

This company has 3 main strategic objectives and they want the portfolio to be totally aligned with these objectives. They give scores scaled qualitatively from 0 to 15 to every project for each objective considering the contribution of the projects to the strategic objectives (Table 1).

For each objective, they define a minimum score threshold which must be provided by the portfolio. 𝑠 𝑗𝑜 is the score of the project j (𝑃 𝑗 )for objective o (𝑂 𝑜 ).If the sum of the scores of the projects included in the portfolio exceeds the minimum threshold, then the portfolio is aligned with the objective.

Please insert Table 1 here

The minimum score threshold for objective 1 is 15, the minimum score threshold for objective 2 is 18, and the minimum score threshold for objective 3 is 21. 𝑠 𝑗 3 ≥ 21 𝑓𝑜𝑟∀𝑘

Results

All the feasible different project combinations are identified. Among 2 10 =1024 different portfolio alternatives (for 10 projects), 14 portfolio candidates remain after the elimination of the unfeasible portfolio alternatives. The Top 5 portfolios in terms of revenue are presented in Table 2, sorted by decreasing revenue. Lastly, the portfolio which gives the biggest expected revenue is selected between the other alternatives. This portfolio #1 gives a revenue of 1919.3k€ and consists of: Project 1, Project 3, Project 6, Project 7, Project 9, Project 10.

Please insert Table 2 here.

Table 3 below shows the results of portfolio selection considering that the projects do not have investment interactions (Resource, Benefit & Outcome interactions). In this case, all the off-diagonal elements of interaction matrices are equal to zero.

Please insert Table 3 here

In this case, 7 portfolio candidates remain after the elimination of the unfeasible portfolio alternatives. Portfolio #5 (Project 2, Project 3, Project 9, Project 10) shows the biggest revenue (1586 k€). However, this combination gives a revenue of 1 605.05k€ in the case where the investment interactions are accounted (table 3). This should be compared with the maximum revenue of 1919.3 k€ obtained with Portfolio #1 (table 2).

The presence of interactions changes three aspects of the decision. First, the remaining combinations are different. For instance, some savings and synergies due to simultaneous presence of some projects may permit keeping the combination under the resource constraint or over the minimal output threshold constraint, which would not have been the case considering the projects as independent. In this example; without considering project investment interactions, Portfolios #4, 7, 9, 10, 11, 13 and 14 are screened out as unfeasible even if they are feasible in reality where interactions occur. Even a favourable portfolio (Portfolio #1) could be screened out in different conditions.

Second, the efficiency of remaining combinations is different. This is inherent to the introduction of interactions in the objective function formulation.

Third, the decision is more reliable. Indeed, interactions are a reality. If they are not considered, as it is sometimes the case, the decision of optimal combination may be different. This involves not only a possible loss of efficiency, but more importantly a potential difference between actual and expected behaviour of the portfolio.

Conclusion

This paper addresses the challenge of building the most efficient and optimum project combination (considering their interdependencies) which meets organization's stated strategies and objectives in a desirable manner, doing so without exceeding available resources, project balance conditions or violating other constraints.

Data are basically based on expertise, and sometimes on experience (feedback on an actual past interaction between two projects). Moreover, they are basically captured bottom-up, by knowing locally for each project what its direct environment is. No systematic analysis of all the potential interdependencies with all projects for all reasons is considered for the moment. The thresholds are similarly defined by a mix of expertise and experience, but of course with unavoidable subjectivity.The presented methodology lacks also in time dimension, as it shows an instant snapshot of the portfolio and does not consider scheduling objective or constraints. Determination of the interactions is already difficult, and the expansion of the interactions definition on a time scale would be even more difficult and with a lack of reliability in the assessment. Generally, the issues of the amount of data and uncertainty associated with their assessments are important, both for data gathering, data processing and results exploitation. An important lack of reliability of the input data involves a lack of trust in the output results. These points will be studied in further research.

Finally, interactions are analysed between pairs of projects, but do not take into account the groups of projects. Interactions may change when there are multiple interaction sources at the same time. However, defining interactions between two projects is already a very difficult and complex process. If we try to define multiple-sided interactions, we increase the effort required for data gathering and the risk of miscalculation in data processing. Moreover, it is not necessary to measure and keep transitivity under control, like in other problems. AHP method for instance measures the experts" opinion inconsistency and compares it with the inconsistency of a randomly filled matrix. But in this case, it may be possible to have three projects A, B and C with positive interactions between A and B and B and C, without having positive interactions between A and C. It may even be a negative interaction. It depends on the nature of the interdependence between each pair of projects.

Our methodology proposes some solutions to this complex multidimensional problem, mainly by integrating new constraints about project interdependencies in the optimization problem formulation. It looks for the optimum project portfolio alternative which has the maximum value while respecting several constraints about strategic effectiveness, efficiency and balance. It provides decision-makers with useful insights about the positive or negative effects of project combinations. It enables decisions to be closer from the reality of portfolio behaviour, since interdependencies exist. If they are not taken into account, the decision is likely to be unrealistic and different from the real evolution and performance of the portfolio.

Table 1

 1 Scores of the projects for each objective

	𝑃 𝑗

  Table 3Results when interactions are not considered

	Portfolios	Nb of Projects	Input 1	Input 2	Obj1 Obj2 Obj3	Output 1	Output 2	Revenue
	#5	4	228 312	21 19 35	848 1299	1586
	#1	6	227 342	38 35 28	888 1301	1582,4
	#2	5	217 284	33 23 27	812 1193	1470,9
	#8	4	232 273	20 22 36	858 1114	1447
	#3	6	231 303	37 38 29	898 1116	1443,4