Pharmaceutical R &D portfolio optimization with minimum borrowed capital based on fuzzy set theory

Due to long lead times, uncertain outcomes and lack of enough historical data, pharmaceutical research and development (R &D) portfolio selection is a often very complex decision issue. The aim of this paper is to investigate pharmaceutical R &D portfolio selection with unavailable and unreliable project information, where the borrowed capital is allowed. Based on fuzzy set theory, we propose a pharmaceutical R &D portfolio optimization model with minimum borrowed capital by taking into account corporate strategy in developing new products, scarcity of resources, lack of investment budget and cardinality constraint. In the proposed model, the pharmaceutical R &D company is assumed to achieve the objectives of maximizing terminal wealth and minimizing the cumulative borrowed capital over the whole investment horizon. Then, we transform the proposed bi-objective model into the corresponding single-objective model by using the weighted sum approach and employ the modified artificial bee colony (MABC) algorithm to solve the transformed model. Finally, we provide a numerical example to illustrate the application of our model.


Introduction
R&D project portfolio selection is an organizational decision making task commonly found in organizations like government-funding agencies, universities, research institutes and technology-intensive firms Tian et al. (2005). Along with the economic development and globalized marketplace, social competition becomes more and more fierce. As a result, the investment markets are changing faster and faster. Especially, global pharmaceutical R&D has witnessed an upsurge development, which is strongly motivated by highly unsatisfied medical demands, such as in cardiovascular, neo- plastic and autoimmune diseases. Once approved on the market, a new drug's profit is prosperous. However, almost all the potential drug candidates face unpredictable performance with technical and market risks. Meanwhile, pharmaceutical R&D is a tough process characterized by long lead times and huge amount of investment. Over the past few decades, the expenditure on R&D has been increasing dramatically year by year. According to Mikulic (2020), the global pharmaceutical R&D expenditure in 2019 closed to 186 billion U.S. dollars. It is estimated that worldwide pharmaceutical R&D expenditure in 2022 will exceed 200 billion U.S. dollars. Actually, R&D has become a survival tool for any pharmaceutical company striving to achieve and maintain a strong competitive position in the future due to patent expiration of previous innovator drugs. The aim of R&D project portfolio selection is to select a subset of promising projects to construct a portfolio from a set of candidate projects based on multiple decision criteria. R&D project portfolio selection is always constrained by limited resources such as budget, research staff, laboratory space, and other technical scarcities. In many cases, organizations are often forced to pick out a certain number of projects from all candidate projects by means of project portfolio techniques mitigating the corresponding risks and enhancing the overall value of portfolio. Therefore, R&D project investment is a risky venture.
There are numerous decision models and methods that address R&D portfolio selection under the framework of probability theory. Among them, the investors are assumed to be able to produce precise values for the future attributes of R&D projects. For example, Abbassi et al. (2014) proposed a 0-1 nonlinear integer programming model for balancing portfolio values and risks of R&D project portfolio. Nishimura and Okada (2014) examined how R&D portfolios of drug pipelines affected pharmaceutical licensing, controlling firm size, diversity, and competition. Arratia and ópez I F L, Schaeffer S E, Cruz-Reyes L, (2016) studied a static R&D project portfolio selection in public organizations. Arratia-Martinez et al. (2018) developed a fuzzy programming model for the research and development project portfolio selection under uncertainty environment. Beaujon et al. (2001) presented an R&D project portfolio selection model with a wide variety of constraints (e.g., capital, headcount, strategic intent, etc.). Bistline (2016) analyzed energy R&D decisions with uncertainty in research outcomes and markets. Çaǧlar and Gürel (2017) studied a public R&D project portfolio selection with project cancellations. Shafahi and Haghani (2018) gave a mathematical model for project selection and scheduling problem, where some of the available projects were phased projects. Chen and Zhu (2011) proposed an integrated approach including nonparametric efficiency analysis, bootstrapping, and mean-variance optimization model to resource allocation problems, and applied it to R&D project budgeting. Gemici-Ozkan and Wu (2010) introduced a decision-support framework for the R&D portfolio selection problem faced by a major U.S. semiconductor manufacturer. Girotra et al. (2007) conducted an event study around the failure of phase III clinical trials and their effect on the market valuation of the firm. Hassanzadeh et al. (2014a) developed a robust optimization model to assist contract research organizations in making their primary business decision, i.e., selection and scheduling of new drug development project opportunities. Hassanzadeh et al. (2014b) presented a multi-objective binary integer programming model for R&D project portfolio selection with competing objectives and uncertain coefficients in both objective functions and constraints. Huchzermeier and Loch (2001) developed a simple real option model for R&D project selection. Ringuest et al. (2000) described a methodology for the selection of R&D projects to add to or remove from an existing R&D portfolio. Rogers et al. (2002) proposed a stochastic optimization model for pharmaceutical R&D project portfolio selection by using a real options approaches. Solak et al. (2010) presented a multistage stochastic programming model for R&D project portfolios under endogenous uncertainty. Stummer and Heidenberger (2003) described a three-phase approach to assist R&D managers in obtaining the most attractive project portfolio. Tohumcu and Karasakal (2010) developed an approach based on analytic network process (ANP) and data envelopment analysis (DEA) to evaluate the performance of R&D projects. Pennings and Sereno (2011) presented a compound option valuation approach for pharmaceutical R&D project portfolio with technical and economic uncertainties. Karasakal and Aker (2017) developed multiple criteria sorting methods based on data envelopment analysis to evaluate R&D projects. Wang et al. (2018) established a two-stage stochastic programming model for portfolio optimization of R&D on low-carbon energy technology.
Notice that R&D project investment decision is often full of changes due to long lead times, uncertain outcomes and lack of enough historical data. In many cases, the collected data information for R&D project management are often at best uncertain and at worst very unreliable. Fuzzy set theory in Zadeh (1965) has been proved to be a powerful tool to deal with imprecise and uncertain data in many applications, which can also been used to characterize the uncertain information on project parameters. Some researchers have investigated R&D project selection problems by using fuzzy set theory to capture and model the uncertain project information. For example, Coffin and Taylor (1996) proposed a multiple criteria model for R&D project selection and scheduling using fuzzy logic. Mohanty et al. (2005) formulated a fuzzy ANP (analytic network process) model for R&D project selection. Wang et al. (2005) proposed a fuzzy AHP model for evaluating R&D projects from multiple disciplines. Carlsson et al. (2007) developed a fuzzy mixed integer programming model for R&D portfolio selection, where future cash flows were characterized by trapezoidal fuzzy number. Wang and Hwang (2007) used fuzzy set theory to model uncertain and flexible project information, and presented a fuzzy zero-one integer programming model for R&D project portfolio selection. Huang et al. (2008) proposed a fuzzy AHP method for governmentsponsored R&D project selection. Sun et al. (2008) presented a group decision support approach to evaluate experts for R&D project selection. Chen and Hung (2010) developed an integrated fuzzy evaluation method for selecting a suitable outsourcing manufacturing partners in pharmaceutical R&D. Bhattacharyya et al. (2011) presented a fuzzy triobjective programming model for R&D project portfolio selection. Collan and Luukka (2014) presented four new fuzzy similarity measure-based TOPSIS variants for R&D project ranking and evaluation. Hassanzadeh et al. (2012) adopted fuzzy pay-off method for R&D project evaluation and developed a fuzzy R&D project portfolio model. Biancardi and Villani (2017) presented a fuzzy approach to value R&D investments. Hesarsorkh et al. (2021) formulated a robust possibilistic optimization approach for pharmaceutical R&D project portfolio selection and scheduling under uncertainty. It is important to remark that all the studies above do not account for R&D project portfolio with borrowed capital.
In the real world, due to the incomplete and imprecise information, investors may be forced to borrow a certain capital to make up budget shortfalls for the selected R&D projects. So, it is necessary to investigate fuzzy R&D project portfolio selection problem, where the borrowed capital is allowed in any intermediate or the final time period. To the best of our knowledge, no existing literature has addressed R&D project portfolio selection with borrowed capital in fuzzy environment. An integration of the borrowed capital over bankruptcy in R&D project portfolio is evidently needed. For this, we propose a fuzzy pharmaceutical R&D portfolio selection model with minimum borrowed capital, in which the optimal investment strategies can be generated to help investors not only maximize terminal wealth, but also minimize the cumulative borrowed capital over bankruptcy during the whole investment horizon. In the proposed model, we take into account corporate strategy in developing new products, scarcity of resources, lack of investment budget and cardinality constraint. By using the weighted sum approach, we transform the proposed model into the corresponding single-objective model. Then, we utilize a fuzzy simulationbased MABC algorithm for solution. Our paper contributes to the existing literature on R&D in two ways. (i) This paper is the first time to investigate fuzzy R&D portfolio selection with borrowed capital. Our study gives an approach to handle R&D portfolio selection with borrowing constraint. (ii) The proposed model can provide decision maker's different aspiration levels with different investment strategies by varying the credibility levels for the practical investment requirements on R&D portfolio selection. Table 1 displays a feature comparison of the proposed R&D project portfolio models with the existing closely related researches.
The remainder of this paper is organized as follows. In Sect. 2, we introduce some basic conceptions about fuzzy variables for measuring the uncertainty associated with pharmaceutical R&D projects in a fuzzy environment. In Sect. 3, we formulate a bi-objective fuzzy programming model for pharmaceutical R&D project portfolio selection in fuzzy environment. In Sect. 4, we first use the weighted sum approach to transform the proposed model into the corresponding single-objective fuzzy programming problems. Then, we use the self-dual credibility measure to deal with the fuzzy objectives and the fuzzy constraints in the transformed model, and express investor's satisfaction degrees for the aforementioned fuzzy constraints. Thereupon, the corresponding deterministic pharmaceutical R&D project portfolio selection model can be obtained. A fuzzy simulated simulation-based artificial bee colony (MABC) algorithm is given for solution. In Sect. 5, we provide a numerical exam-ple to demonstrate the application of our model. Finally, we conclude the paper in Sect. 6.

Basic conceptions
This section briefly reviews some concepts about fuzzy numbers, which have appeared in the literature and are necessary for our discussion in this paper.
Let ξ be a fuzzy variable defined on the possibility space ( , P( ), Pos) with membership function μ(x), and let u be a real number. Then, the credibility of fuzzy event {ξ ≤ u} is defined by Liu and Liu (2002b) represent the possibility and necessity measures of {ξ ≤ u}, respectively. Notice that Cr is self-dual measure, i.e., Cr{ξ ≤ u}+Cr{ξ ≥ u} = 1. When the credibility value of a fuzzy event achieves 1, it indicates that the fuzzy event will surely happen. In Liu (2002a), we can obtain the axioms about credibility measure Cr as follows.
Theorem 1 Liu (2002a). Assume that is a nonempty set, P is the power set of , and Cr is the credibility measure. Then, for any A, B ∈ P, we have Definition 1 Liu and Liu (2002b). Let ξ be a fuzzy variable. The expected value of ξ is defined as provided that at least one of the two integrals is finite. Then, the following results can be found in Liu and Liu (2002b).
Theorem 2 Let ξ be fuzzy variable with finite expected values, and let λ and μ be two any given two real numbers. Then E(λξ + μ) = λE(ξ ) + μ. (4) Example 1 Let ξ be a trapezoidal fuzzy number determined by quadruplet (a, b, c, d) of crisp numbers such that a < b < c < d. Then, its membership function has the following form By Definition 1, the expected value of the trapezoidal fuzzy variable ξ = (a, b, c, d) is computed by

Mathematical formulation of pharmaceutical R&D portfolio selection
In this section, we discuss a pharmaceutical R&D portfolio selection problem with minimum borrowed capital in a fuzzy environment. Based on fuzzy set theory, we propose a pharmaceutical R&D portfolio selection model with the objectives of maximizing the terminal wealth and minimizing the cumulative borrowed capital over the whole investment periods.

Problem description and notations
Assume that a pharmaceutical company with inadequate initial capital W 0 intends to invest n candidate pharmaceutical drug R&D projects. Each candidate pharmaceutical drug R&D project contributes to a particular corporate strategy and has T 2 -continuous investment periods. The whole investment horizon of each candidate pharmaceutical drug R&D project is divided into three stages including drug discovery, testing and market introduction. The investment periods of drug discovery and testing of each project are set as T 1 and (T 2 − T 1 ), respectively. The time axis of each pharmaceutical drug R&D project is displayed in Fig. 1. The investment returns of these selected pharmaceutical drug projects are obtained at the end of period T 2 after market introduction. The execution of each candidate project requires the exclusive use of a number of resources (e.g., budget, human resources, etc.), while the availability of each resource type is usually limited. To avoid a possibility of bankruptcy before reaching the end of an investment horizon, we assume that the company can borrow a certain capital with borrowing rate r b (t) at the beginning of period t when the available capital at current period is less than the corresponding actual implementation costs. In addition, we assume that the company can use the surplus capital, if possible, to invest a risk-free asset with return rate r f (t) at period t (t = 1, 2, . . . , T 2 ). Consider the fact that pharmaceutical R&D investment decision is very difficult since it is investment intensive, the required resources may be scarce and markets are ever-changing. As a result, the collected data information for portfolio management is highly uncertain and often very inaccurate.
Sometimes even much of the information for project investment decision is unavailable and unreliable. Fuzzy set theory is a powerful tool to describe imprecise information in such environments. For this, we use fuzzy numbers to capture and model the uncertain pharmaceutical R&D project information. Notice that trapezoidal fuzzy variables are often used to describe the fuzzy uncertainty associated with financial markets due to their simple to estimate and easy to generalize to the LR-type forms (Carlsson et al. 2002). With the same consideration as Carlsson et al. (2002), we assume that the implement costs and the required labor to implement project l j minimum budget to be spent on projects contributing to strategy j; the return rate of the risk-free asset at period To state the model formulation, the objective functions and constraints of the proposed model are separately stated in the following two subsections.

Objectives
The total implementation costs of the n candidate projects at period t (t = 1, 2, . . . , T 2 ) can be described by During the process of investment, a capital shortage occurs when the available wealth is less than or equal to the required investment costs on the selected projects at current period. To avoid the occurrence of bankrupt, we assume that the borrowing capital is allowed when the company suffers capital shortage in any intermediate or the final time period. Let B E t be the borrowing capital event at period t. Then, the borrowing capital event at period t can be described by where θ k is the bankruptcy level at period k. It can be seen from Eq. (8) that the company does not need to borrow capital during the first t −1 investment periods. The available wealth obtained at the end of period k (k ∈ {1, 2, . . . , t − 1}) can be described as the following formula Then, we have It follows from Eq. (10) that the wealth obtained at the end of period k (k ∈ {1, 2, . . . , t − 1}) can be rewritten as the following specific form According to the analysis above, due to the shortage of capital, the company needs to borrow a certain amount of capital at each period of the following T 2 − t consecutive investment periods. Then, the wealth obtained at the end of period t is calculated by It follows from Eq. (12) that the formulas of the wealth obtained at the end of the following T 2 −t consecutive investment periods can be represented by . .
Thus, the general form of the wealth obtained at the end of period r (r ∈ {t, t + 1, . . . , T 2 }) is computed by Notice that, at stage 3, the total implement costs for market introduction on the n projects is After finishing market introduction, the investment returns (IR) generated by the n projects is Thus, by Eqs. (11)-(15), the terminal wealth W T obtained at the end of the whole investment horizon can be given by Assume that the first objective of the pharmaceutical company is to maximize the terminal wealth. Then, we have Meanwhile, we assume that the pharmaceutical company is required to minimize the total borrowed capital (TBC) during the whole investment horizon. Then, we can obtain the second objective as follows:

Practical investment constraints
• Borrowing constraint. To avoid the shortage of capital, the company requires that the borrowed capital at period t, B t , should be no less than max{0, E(C t − W t−1 )}. Then, the borrowing constraint can be expressed by • Budget constraint. To keep the selected projects proceed normally, the following budget constraint about the portfolio at period t holds • Personnel constraint. Assume that the required personnel at stage h, n i=1 x i l i,h , should be less than or equal to the available man-power capacity L h . Then, the personnel constraint at stage h can be given by • Strategy selection constraint. The company requires that the expenditure on projects contributing to strategy j should be restricted in the range of [ l j , u j ]. Then, the strategy selection constraint can be expressed by • Cardinality constraint In the process of investment decision, cardinality constraint is a very important factor affected portfolio management, which controls the total number of assets in portfolio selection for the purpose of monitoring and control. In this paper, we assume that the pharmaceutical company requires the maximum holding number of R&D projects in the portfolio must be not more than K . Then • Borrowed capital constraint. Assume that the total borrowed capital should be not more than M times of the initial wealth W 0 of the pharmaceutical company. Then, we have

The proposed pharmaceutical R&D portfolio optimization model
Notice that, in real life, pharmaceutical R&D is often very expensive and requires substantial capital expenditure. Pharmaceutical R&D project portfolio is a very complex decision-making process since it is affected by many critical factors such as funds and resources. During the process of pharmaceutical R&D project portfolio, due to the limitation of funds and resources, it is very important for pharmaceutical company to determine a suitable number of R&D projects in a portfolio. By using fuzzy set theory, we propose a pharmaceutical R&D project portfolio selection model with minimum borrowed capital. In the proposed model, we assume that the pharmaceutical company intends to seek optimal investment strategies with the objectives of maximizing terminal wealth and minimizing the cumulative borrowing capital over the whole investment horizon. Assume that the pharmaceutical company considers decision criteria including budget constraint (21), personnel constraint (22) and strategy selection constraint (23) and borrowed capital constraint (25). Moreover, it requires that the maximum holding number of projects in the portfolio must not exceed K . Following the idea, we formulate a fuzzy programming model (P 1 ) for pharmaceutical R&D portfolio selection as follows

Solution approach
Notice that the proposed model is bi-objective programming problem with fuzzy coefficients in both the objective functions and constraints. In this section, we give a fuzzy simulation-based MABC algorithm for solution. The model (P 1 ) can be equivalently rewritten into a vector optimization problem with minimization objectives as follows: Notice that the model (P 1 ) is a bi-objective programming problem. To handle this kind of problem, the most widely used methods are the weighted sum approach in Marler and Arora (2010) and the -constraint method in Chankong and Haimes (1983). The main defect of the -constraint method is that it is very sensitive to the number of objectives (Copado-Méndez et al. 2016). As mentioned by Marler and Arora (2010), the weighted sum approach can not only generate multiple solutions by varying the weights, but also provide a single solution that describes preferences presumably associated with the selection of a group of weights. For this, we use the weighted sum approach to transform the model (P 1 ) into the following single objective programming problem where ω ∈ [0, 1] is the relative preference weight of the pharmaceutical company with respect to f TBC . ω = 0 indicates that the company only focuses on the objective W T . ω = 0.5 indicates that the company shows indifferent attitudes to both the objectives W T and f TBC . ω = 1 indicates that the company only considers the objective f TBC .

Deterministic pharmaceutical R&D project portfolio optimization model
Since the model (P 1 ) is a fuzzy programming problem with fuzzy coefficients in both objective function and investment constraints, a commonly used technique to transform it into the corresponding deterministic programming problem. In this paper, we use the self-dual credibility measure as the basic technique tool to transform these fuzzy coefficients in the objective function and the investment constraints of the model (P 1 ) into the deterministic forms and express investor's aspiration levels for the satisfaction of the fuzzy coefficient constraints.

Treatment of the fuzzy objective function
The implementation costs on the candidate pharmaceutical R&D projects at each stage and the obtained revenues are all fuzzy numbers. Derived from the extension principle in Zadeh (1965), the scalar multiplication and additive operations on several fuzzy numbers is also a fuzzy number. It follows from Eq. (13) that the terminal wealth obtained at the end of stage 3 is also a fuzzy number. By Definition 1, Theorems 2 and 3, the expected value of the terminal wealth is Thus, the objective function of the model (P 1 ) can be expressed as the following deterministic form

Treatment of the fuzzy constraints
Due to the exist of fuzzy coefficients in investment constraints, it leads these constraints to be not clear at all. So, the above-mentioned investment constraints with fuzzy coefficients may be not strictly held. In this case, it is necessary to take into account a certain deviation and flexibility on the holding of these fuzzy constraints. Similar to Liu and Iwamura (1998), we use the self-dual credibility measure as a tool to handle these fuzzy constraints and describe the aspiration levels of the pharmaceutical R&D company for the satisfaction of these fuzzy constraints in this paper. Assume that the investment constraints (21), (22) and (23) hold with at least credibility levels λ, θ and η, respectively. Then, the three fuzzy investment constraints can be represented by the chances of the credibility as follows For the sake of description, we set Therefore, by (iii), we get According to Eqs. (31) and (32), we have By Theorem 1, Eq. (28) can be equivalently rewritten into the following form Similarly, Eq. (30) can be equivalently represented by Then, the model (P 1 ) can be rewritten into the following credibility-based chance-constrained programming problem Notice that the model (P 1 ) has some fuzzy credibility-based chance constraints. As mentioned by Liu and Iwamura (1998) and Liu (2002a), in fuzzy environment, the basic technique for handling this kind of problem is to transform it into the corresponding crisp equivalent under the predetermined confidence level. However, this approach is often very hard and only successful for several special cases. For this, we apply the fuzzy simulation technique in Liu and Iwamura (1998) to calculate the objective value and the credibility values of fuzzy variables in the model (P 1 ) .

Fuzzy simulation technique
Before solving the model (P 1 ) , we need to calculate the uncertain values as follows Expected Values :E(W T ) and E(C t − W t−1 ),

In this paper, fuzzy simulation technique introduced in Liu (2002a) is used to compute the credibility measures for the aforementioned uncertain values. • Fuzzy simulation for expected value
By Definition 1, the expected values of W T and C t − W t−1 can be, respectively, calculated by is a decision variable vector and ξ = (ξ 1 , ξ 2 , . . . , ξ n ) is a fuzzy variable vector. To calculate the value of U (x), we randomly produce ξ m and set ν m = μ(θ m ) for m = 1, 2, . . . , N , where μ is the membership function of ξ . Then, the credibility of the uncertain function U (x) is computed by The fuzzy simulation process for computing U (x) is summarized as follows: Step 1 Generate a random number θ m from the credibility space ( , P, Cr), set ν m = (2Cr{θ m }) ∧ 1 and produce ξ m = ξ(θ m ), m = 1, 2, . . . , N , respectively.
Step 2 Return U (x) via the estimation formula as shown in Eq. (37). For the uncertain function To compute U 2 (x), we set f (x, ξ) = n i=1 x i l i,k − L k . Then, its credibility can be estimated by Eq. (38). For

• Artificial bee colony algorithm
Notice that the proposed model with fuzzy variables in both the objective functions and constraints has some complex constraints. It is an NP-hard problem. Due to its mathematical complexity, traditional optimization methods often cannot solve it. For this, we must look for metaheuristic algorithms such as genetic algorithm (GA), particle swarm optimization (PSO) algorithm, ant colony optimization (ACO) algorithm, and artificial bee colony (ABC) algorithm. Compared with other methods, the ABC algorithm is a relatively new meta-heuristic algorithm with fewer control parameters, a simpler structure, and more convincing performance (Karaboga and Akay 2009). For this, we employ the MABC algorithm in Karaboga and Akay (2011) to solve our model. ABC algorithm is a meta-heuristic technique proposed by Karaboga (2005), which is inspired by simulating the foraging behavior of honey bees. In ABC, the food source of bees is viewed as a potential solution of the optimization problem and the nectar amount of the food source denotes the quality (fitness value) of the corresponding solution. ABC consists of three kinds of bees (i.e., employed bees, onlooker bees and scout bees). The population sizes of both employed bees and onlooker bees are the same, which is equal to the number of solutions in the population. Among them, different kinds of bees often have different tasks. The main task of employed bees is to look for food source around their hive and share the information about food source with onlooker bees by waggle dance. The main task of onlooker bees is to explore some better food sources for further search. The main task of scout bees is to perform a random search to find new food sources in the search space. When the nectar amount of a food source is low or exhausted, the food source will be abandoned by employed bees and these employed bees will become scout bees to search new food sources. ABC algorithm has been proved to be a promising performance on various complex optimization problems and received more and more attention during the past few years. For a detailed discussion, one can refer to Bajer and Zorić (2019); Li and Yang (2016); Xiang et al. (2018); Karaboga and Akay (2011) and Bansal et al. (2018).
As mentioned by Karaboga and Akay (2011), the MABC algorithm is effective for constrained optimization problems and shows better performance than that of state-of-the-art algorithms. For this, we use the modified ABC algorithm in Karaboga and Akay (2011) to solve the model (P 1 ) in this paper. In the following, let us introduce its initialization, fitness function, constraint-handling, employed bee phase, onlooker bee phase and scout bee phase.
Initialization Randomly generate a swarm of food sources (potential solutions) with population size pop − si ze. For a numerical optimization problem, each food source consists of a D-dimensional parameter vector. Here, D is equal to the number of variables in the optimization problem. Similar to the other evolutionary algorithms, ABC generates an initial population of food sources randomly. Let χ = (x, B) be a solution of the model (P 1 ) , where χ has two vectors including a binary vector x = (x 1 , x 2 , . . . , x n ) and a real-valued vector B = (B 1 , B 2 , . . . , B T 2 +1 ). The binary vector x controls the holding number of projects in the portfolio. The real-valued vector B controls the borrowed capital over the whole investment horizon. Then, a solution χ of the model (P 1 ) can be encoded as a food source with the following form where rand 1 (0, 1) and rand 2 (0, 1) are two uniformly distributed random numbers in Fitness function Notice that the model (P 1 ) is a minimization problem. After initialization, the fitness value of each solution is calculated as follows where f it i and f i represent the fitness value and objective function value of the solution χ i , respectively. Constraint-handling To satisfy Eq. (24), we sort the values of x 1 , x 2 , . . . , x n in descending order. After that, we keep the first K values of x i and set all other x i to zero. The violation value of the ith solution, V i , is computed by where g p (χ i ) ≤ 0 denote the pth inequality constraint of the model (P 1 ) . Employed bees phase In this phase, each employed bee searches around a distinct food source position and generates a new candidate food source position V = (V 1 , V 2 , . . . , V D ) with the following form After that, the fitness value of the new position V i is computed. If the fitness value of new position V i is better than the one of χ i , we select the solution vector V i , otherwise keep χ i .
After producing a new food source (solution), ABC algorithm makes a selection. In this algorithm, the feasibilitybased rule of constraint-handling mechanism in Deb (2000) is used to solve constrained optimization problems. Based on Deb's rules, for any given two solution χ i and χ j , we replace χ j by χ i with the following criteria: (i) When χ i is feasible and χ j is infeasible; (ii) When both χ i and χ j are feasible with f it i < f it j ; (iii) When both χ i and χ j are infeasible with V i < V j .
Onlooker bees phase After all employed bees complete the search process, the onlooker bees will collect the information from all of the employed bees and choose good food sources to gather honey depending on the probability value. In this algorithm, the probability of the food source i chosen by an onlooker bee is denoted by p i with the following form Notice that, in this algorithm, the infeasible solutions as well as the feasible ones are allowed to populate in the colony. The feasible solutions are selected probabilistically proportional to their fitness values, and the infeasible solutions are selected inversely proportional to their violation values. The probabilities of the infeasible solutions are between 0 and 0.5, while the probabilities of the feasible ones are between 0.5 and 1. At this phase, onlooker bee generates a modified position of the food source as shown in Eq. (43) based on the probability value and checks its nectar amount. If a parameter value produced by this operation exceeds its predetermined boundaries, the parameter can be set as an acceptable value.
Scout bees phase Define limit and fail i as the predetermined number of trials and the iteration number of failing to explore better solutions for individual i, respectively. In this phase, once a food source cannot be improved after a predetermined number of iterations, i.e., fail i >limit. Then, this food source will be abandoned from the population. By using Eqs. (39,40), to keep the population diversity, the abandoned food source will be updated by a new random food source discovered by the scouts. In this procedure, another control parameter called scout production period (SPP) is also used to enhance the convergence capability of the algorithm for constrained optimization problems. At each SPP, when an abandoned food source exceeds limit, a scout production process will be performed to provide a diversity mechanism by allowing new and probably infeasible food sources in the population.
The procedures of the algorithm are described as follows.
Step 1 Input the algorithm parameters including colony size C S, modification rate M R, maximum cycle number MC N , predetermined number of cycles limit and scout production period S P P; Step 2 Initialize food sources by Eqs. (39) and (40); Step 3 Calculate the fitness values of all food sources by Eq. (41) and memorize the best food source achieved so far; Step 4 Perform employed bees phase by Eq. (43); Step 5 Calculate the fitness value of the new food source and select the better food source position based on Deb's rules; Step 6 Calculate probabilities for onlookers by Eq. (44) and perform onlooker bees phase; Step 7 Perform scout bees phase and replace the abandoned food sources by Eqs. (39) and (40); Step 8 Return to Step 3 until the termination criteria is met.
Step 9 Report the best solution found so far as the optimum solution for the problem.
We apply the proposed model to this example and use the MABC algorithm in Karaboga and Akay (2011) for solution. In our model, the credibility levels λ, θ and η are set as 0.8, 0.8 and 0.8, respectively. We set the relative preference weight ω to be 0, 0.5 and 1. The value of M is set as 100.
The parameter values of the solution algorithm are set as follows. The value of modification rate (M R) is set as 0.85. The value of colony size is set as 100, i.e., C S = 100. The maximum cycle number is set as 2000, i.e., MC N = 2000. The value of the predetermined number of trials limit is equal to 0.5 × C S × D, where D is the dimension of the model and C S is the number of solutions in the population. The value of scout production period S P P is also equal to 0.5 × C S × D. We run the aforementioned algorithm with 2000 generations on each model. The corresponding computational results of our model about the selected projects under different strategic types, project size (PS), the amount of the borrowed capital at period t (i.e., B t ) and terminal wealth are displayed in Table  4.
It can be seen from Table 4 that the optimal investment strategies obtained by our model with different ω are distinctly different. By varying the value of ω, we can freely express the pharmaceutical company's different investment intention. If the pharmaceutical company makes a portfolio decision with the purpose of maximizing terminal wealth (i.e., ω = 0), it should follow the investment strategy listed in Column 2 of Table 4 to adjust its wealth during the whole investment horizon. In this case, the pharmaceutical company should select 9 projects (including projects P1, P2, P3, P5, P8, P9, P15, P17 and P19) to construct a portfolio. Among them, P17 and P19 should be selected for strategy 1. P5, P8, P9 and P15 should be selected for strategy 2. P1, P2 and P3 should be selected for strategy 3. At the end of investment horizon, the terminal wealth obtained by the pharmaceutical company is 16158.6654. If the pharmaceutical company shows indifferent attitudes to both the total borrowed capital and the terminal wealth (i.e., ω = 0.5), it should select 6 R&D projects including P3, P5, P10, P11, P15 and P19 to make a portfolio decision. In this case, at the end of investment horizon, the company will obtain terminal wealth 8932.9211. If the company intends to seek an optimal investment strategy with the objective of minimizing the total borrowed capital (i.e., ω = 1), it should select 5 R&D projects including P2, P5, P7, P13 and P15 to make a portfolio decision. In this case, the corresponding terminal wealth is 4300.5651. Now, we perform further numerical experiment to investigate the effects of the pharmaceutical company's different aspiration levels for the practical investment requirements including budget constraint, personnel constraint and personnel constraint on project portfolio. We assign different values on the parameters λ, θ and η to simulate the decision maker's different aspiration levels for the aforementioned practical investment requirements. For example, if the pharmaceutical company pays more attention to budget constraint, we will assign a higher credibility level λ. Then, we apply the solution approach to solve the proposed model under different parameter cases. Sensitivity comparison about our model with preference weight ω = 0 are conducted to demonstrate the effects of different credibility levels on portfolio decision. Table 5 displays the corresponding comparative computational results about the selected strategic goals, project size (PS), total borrowed capital and terminal wealth.
From Table 5, we can find that the pharmaceutical company's aspiration levels for the practical investment requirements have a significant impact on R&D portfolio selection. The larger the values of λ, θ and η are, the more prudent the pharmaceutical company is. As shown in Table 5, the investment strategies obtained by the our model under different aspiration levels (λ, θ, η) are distinctly different. It means that the pharmaceutical company's aspiration levels do affect the optimal portfolio compositions and their performance. It can be seen from the last column of Table 5 that, as the values of λ, θ and η increase from 0.6 to 0.9, the corresponding terminal wealth obtained by the proposed model decreases from 10462.7318 to 9089.5329. In other words, the more cautious the pharmaceutical company is, the less terminal wealth it gets, which is consistent with intuition. From the above results, we find that we can obtain the different investment strategies by solving the proposed model in which the different values of the parameters λ, θ and η are given. Through determining the values of the parameters λ, θ and η according to the investor's frame of mind, the investor may achieve a favorite R&D investment strategy. In a word, the obtained investment strategies under different credibility levels can freely reflect a decision maker's different investment intention, which is the advantage of our model. In the following, to analyze the effect of cardinality constraint on investment decision, we perform further experiment on our model with ω = 0 and (λ, θ, η) = (0.8, 0.8, 0.8) by varying the maximum holding number of projects K from 6 to 20. The corresponding computational results are summarized in Table 6. In addition, to examine the efficiency of the solution algorithm, we also use the ABC/best algorithm in Gao et al. (2012) to solve our model with ω = 0 and (λ, θ, η) = (0.8, 0.8, 0.8). In ABC/best algorithm, the chaotic iteration is set as 300 and the other parameters are set the same as the MABC algorithm. The comparative computational results about the total borrowed capital f TBC , objective function value (OFV), terminal wealth E(W T ) and CPU times (in seconds) obtained by the aforementioned two algorithms are listed in Table 7.
It can be seen from Table 6 that, as K increases from 6 to 20, the obtained terminal wealth first increases and then remain unchanged. In the sense of terminal wealth, the optimal investment strategy is to construct a portfolio with the 9 R&D projects including P1, P2, P3, P5, P8, P9, P15, P17 and P19. In this case, the obtained terminal wealth is 9662.3336. It is a strong evidence that it is wise for the decision maker to form the best strategy by selecting 9 R&D projects from the project pool. Thus, we can conclude that cardinality constraint plays an important role in portfolio decision. From Table 7, we can find that the MABC algorithm sacrifices some runtime for a higher search power. In terms of the objective function values and terminal wealth, the MABC algorithm performs better than the ABC/best algorithm. It is a strong evidence that the MABC algorithm is suitable for solving our model.
To test the robustness of the solution algorithm, we also take the model with ω = 0 as an example to perform the algorithm under different parameter values for thirty independent runs. The corresponding computational results are displayed in Table 8. To compare these results, we calculate the relative error (RE) as follows RE(%) = the best found average objective value − the actual found average objective value the best found average objective value × 10%.
Here, the best found average objective value is the maximum average value of all the average values obtained under different parameter cases, and the actual found average objective value is the corresponding average objective value for a certain experiment. From Table 8, we can find that the maximum RE is 1.4549%, which indicates that the solution algorithm is robust with respect to the change of parameters.

Conclusions
In this paper, we present a fuzzy project portfolio optimization model for pharmaceutical R&D investment decision with the objectives of maximizing terminal wealth and minimizing the total borrowed capital. To the best of our knowledge, this paper is the first time to investigate R&D project portfolio selection with minimum borrowed capital in fuzzy environment. To make the proposed model more realistic, we consider several critical practical factors including borrowing constraint, budget constraint, personnel constraint, strategy selection constraint and cardinality constraint. Considering our model with fuzzy coefficients in both the objective functions and constraints, we employ credibility theory to transform it into credibility-based chanceconstrained programming problem. Then, we adopt fuzzy simulation technique to approximate the values of the aforementioned fuzzy quantities, integrate fuzzy simulation and MABC algorithm to solve our models. A numerical example is given to demonstrate the application of our model and highlight the stability of the solution algorithm. Computational results show that the proposed model is suitable for complex R&D project portfolio optimization and investors' subjective preferences can be freely incorporated into our model.
For future study, we can investigate fuzzy pharmaceutical R&D portfolio selection with several additional practical investment constraints for specific requirements such as contingency constraints, dependency constraints, flexible investment horizon, gross profit constraints, added R&D projects, budget of projects, limited budget allocation and the R&D projects from multinational companies. In addition, we will apply our model to other specific application areas and design some other effective solution algorithms to solve the complex R&D portfolio selection models.