**2. Problem description**

We present a new model to select an optimum project portfolio tacking into account many constraints in the multi-period planning horizon. Also, this model can be used to select the RRSs. The portfolio selection problem of the project RRSs is combined with four basic concepts (i.e., project opportunity, work breakdown structure, risk event, and risk responses) as well as three key elements (i.e., schedule, quality, and cost) are considered in these concepts. These concepts are described as project scope, work breakdown structure, risk event, risk response. There is a strategy to respond r risk events. On the other hand, N project should be evaluated with their risk responses' effects to select an optimum portfolio. The optimal portfolio will be top j projects. All parameters of the mathematical model change dynamically. In this model, an optimum portfolio is selected considering its risk response expenditure. The most enticing RRSs can be acquired by solving the mathematical model. **Figure 1** depicts the process of portfolio RRSs.

In this section, we present notations and mathematical modeling in Sections 2.1 and 2.2, respectively.

It should also be mentioned that the definition of parameters of *s w ar*,

*s w <sup>r</sup>* , *q<sup>w</sup> ar*, *<sup>ε</sup><sup>w</sup>*, *<sup>δ</sup><sup>w</sup>*, *<sup>T</sup>*´ *max*, *Qmax*,~*ear*, *<sup>q</sup><sup>w</sup> <sup>r</sup>* , *M*, *M* \$ can be found in Rahimi et al. [13]. Following is the mathematical mode.

**Figure 1.** *Process of portfolio RRSs.*

## **2.1 Mathematical programming**

$$\text{MaxZ}\_{1} = \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \times \bar{\mathbf{p}}\_{jt} + \sum\_{j=1}^{n} \sum\_{a=1}^{A} \sum\_{r=1}^{R} \mathbf{z}\_{jar} \times \bar{\mathbf{e}}\_{ar} \tag{1}$$
 
$$\begin{split} \text{MinZ}\_{2} &= \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{i=1}^{m} h\_{ij} \bar{\mathbf{C}}\_{it} + \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{k=1}^{s} m\_{kj} \bar{\mathbf{C}}\_{kt} + \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{a=1}^{z} r\_{aj} \bar{\mathbf{C}}\_{at} \\ &+ \sum\_{j=1}^{n} \sum\_{a=1}^{A} \bar{\mathbf{C}}\_{at} m\_{jr} \mathbf{x}\_{jar} \end{split} \tag{2}$$

s.t.;

$$\sum\_{\mathbf{t}=1}^{T} \mathbf{x}\_{\mathbf{j}\mathbf{t}} \le \mathbf{1}; \qquad \forall \mathbf{j} \tag{3}$$

$$\sum\_{t=1}^{T} \left( t + d\_{jt} \right) \varkappa\_{jt} \le T + \mathbf{1} + T\_{\max}; \quad \forall j \tag{4}$$

$$\sum\_{j=1}^{n} h\_{ij} \mathbf{x}\_{jt} \le H\_{it}; \quad \forall i, t \tag{5}$$

$$\sum\_{j=1}^{n} m\_{kj} \mathbf{x}\_{jt} \le \mathbf{M}\_{kt}; \quad \forall k, t \tag{6}$$

$$\sum\_{j=1}^{n} r\_{oj} \mathbf{x}\_{jt} \le R\_{ot}; \quad \forall O, t \tag{7}$$

$$\left(\sum\_{i=1}^{m} h\_{\vec{v}i} \vec{\mathbf{C}}\_{\text{it}} + \sum\_{k=1}^{s} m\_{k\vec{j}} \vec{\mathbf{C}}\_{\text{kt}} + \sum\_{o=1}^{x} r\_{o\vec{j}} \vec{\mathbf{C}}\_{\text{ot}}\right) \times \mathbf{x}\_{\text{jt}} < \bar{p}\_{jt}, j = 1, 2, \dots, n; \quad \forall t \tag{8}$$

*Fuzzy Approach Model to Portfolio Risk Response Strategies DOI: http://dx.doi.org/10.5772/intechopen.95009*

$$\sum\_{j=1}^{n} \sum\_{a=1}^{A} \tilde{\mathbf{C}}\_{a} \max\_{r} z\_{jar} + \left[ \sum\_{i=1}^{m} h\_{\tilde{\mathbf{y}}} \tilde{\mathbf{C}}\_{\mathbf{i}t} + \sum\_{k=1}^{s} m\_{k\tilde{\mathbf{y}}} \tilde{\mathbf{C}}\_{\mathbf{k}t} + \sum\_{o=1}^{x} \tilde{\mathbf{C}}\_{o} r\_{o\tilde{j}} \right] \times \mathbf{x}\_{\tilde{\mathbf{y}}} \le \tilde{B}\_{\tilde{\mathbf{y}}}; \quad \forall r, j, t \tag{9}$$

$$\sum\_{r=1}^{R} \mathbf{s}\_r^w - \sum\_{r=1}^{R} \sum\_{a=1}^{A} \left( \mathbf{s}\_{ar}^w \mathbf{z}\_{jar} \right) \le \varepsilon^w; \quad \forall j, w \tag{10}$$

$$\sum\_{r=1}^{R} q\_r^w - \sum\_{r=1}^{R} \sum\_{a=1}^{A} (q\_{ar}^w z\_{jar}) \le \delta^w; \quad \forall j, w \tag{11}$$

$$\sum\_{r=1}^{R} \mathbf{s}\_r^W - \sum\_{r=1}^{R} \sum\_{a=1}^{A} \left( \mathbf{s}\_{ar}^W \mathbf{z}\_{jar} \right) \le \acute{T}\_{\text{max}}; \quad j=n \tag{12}$$

$$\sum\_{r=1}^{R} q\_r^W - \sum\_{r=1}^{R} \sum\_{a=1}^{A} (q\_{ar}^W z\_{jar}) \le \mathcal{Q}\_{\text{max}}; \quad j = n \tag{13}$$

$$\sum\_{j=1}^{n} \mathbf{x}\_{jt} \cdot \left( \mathbf{M} A R R \mathbf{R}\_t - I\_{jt} \right) \le \mathbf{0}; \quad \forall t \tag{14}$$

$$\sum\_{j=1}^{n} \mathbf{x}\_{jt} \ge \mathbf{0}; \qquad \forall t \tag{15}$$

$$z\_{jat} + z\_{\;j\acute{a}\!\!\!} \le \mathbf{1} \left( A\_a, A\_{\acute{a}} \right) \in \overrightarrow{\mathcal{M}}; \quad \forall j, a, \acute{a}, r, \acute{r} \tag{16}$$

$$z\_{\dot{x}r} + z\_{\dot{y}\acute{a}\dot{r}} = \mathbf{1}\left(A\_a, A\_{\acute{a}}\right) \in \overset{\leftrightarrow}{\mathbf{M}}; \quad \forall \mathbf{j}, a, \acute{a}, r, \acute{r} \tag{17}$$

$$z\_{\dot{j}ar} - z\_{\dot{j}\dot{a}\dot{r}} \le 0 \ (A\_a, A\_{\dot{a}}) \in \overline{M}; \quad \forall \dot{j}, a, \acute{a}, r, \acute{r} \tag{18}$$

$$z\_{jar}, z\_{\;j\;\!r\;} \in \{0, 1\}; \quad \forall j, a, \not{a}, r, \not{r} \tag{19}$$

$$x\_{\dagger} \in \{0, 1\}; \quad \forall j, t \tag{20}$$

Objective function value (OFV) (1) maximizes the NP of the selected portfolio and effects on all RRSa for each project of the selected portfolio. Objective function value (2) is minimizing the total cost of the chosen projects consisting of four terms. These terms are the human resource expenditure, the machine resource expenditure, the raw materials resource cost, and implementing the RRSs, respectively.

Constraint (3) ensures that each project selection will happen only one time on the planning horizon. Constraint (4) states that the completion time of each selected project is less than the planning horizon plus the upper bound for project delivery delay. Constraints (5)–(7) define the maximum limits of all three resources. Constraint (5) states that the number of human resources of all types needed for projects during selection cannot exceed the maximum available human resources for all types and all planning terms. Constraint (6) ensures that all machine-hour resources of all types needed for projects during selection do not exceed the maximum available machine-hour resources for all types and all planning terms. Constraint (7) ensures that all raw materials resources of all types needed for projects during selection do not exceed the maximum available raw materials resources of all types and for all planning terms. Constraint (8) certifies that the total cost of each selected project is less than its net profit for all planning terms. Constraint (9) certifies that the total cost of a selected project including human resource expenditure, machine resource expenditure, raw material cost, and implementing the RRSs, is less than its budget for all projects and all planning terms.

Constraint (10) certifies that, in each project, each work packages (except the last one) is completed in the due date, otherwise (if it takes more), it does not affect the schedule of its successors'start times. Constraint (11) ensures that, in each project, each work packages (except the last one) maintain a certain level of quality. Constraint (12) indicates that, in each project, the last work package must be finished in the project deadline. Constraint (13) indicates that in each project, the last work packages must conform to project quality standards. Constraint (14) ensures if a project is selected, it is attractive and that means the internal RoR of the chosen projects should be greater than or equal to the MARR. Constraint (15) indicates that in each period, projects can be chosen. Constraints (16)–(18) are about strategies. Constraint (16) ensures that strategies *Aa*, and *Aa*´prevent each other for each project. Constraint (17) ensures that for each project, only one strategy must be selected if strategies*Aa*, and *Aa*´exclude each other. Constraint (18) states that projects cooperate if one strategy is chosen another strategy must be chosen too. Also, in constraint (19) attributes a binary variable for each project. Constraint (20) refers to binary decision variables.

#### **2.2 Proposed uncertainty programming**

Uncertainty in data can be grouped into two categories: randomness and fuzziness. Randomness originates from the random nature of data and Fuzziness refers to the vague parameters Infected with epistemic uncertainty-ambiguity of these parameters stems from the lack of knowledge regarding the exact value of these parameters. The proposed model for this problem is a fuzzy multi-objective nonlinear programming (FMONLP). There are a number of adopted methods to transform this model into its equivalent crisp match, from which a two-phase approach is offered [13–20]. Firstly, using an efficient method introduced by Jimenez et al., [21], the basic model is transformed into an equivalent auxiliary crisp multiobjective model. Secondly, the fuzzy aggregation function, developed by [20], is used to solve the crisp multi-objective mode. To do this, a single-objective parametric model to find the final preferred compromise solution replaces the crisp multi-objective model.

Several methods have been proposed to convert a probabilistic model into an equivalent non-probabilistic one. Probabilistic constraints transform into nonprobabilistic ones using fuzzy measures, which was introduced, in the literature review section. The possibility (Pos) and necessity (Nec) are the general fuzzy measures respectively showing the optimistic and pessimistic attitudes of the decision maker. The Pos measure shows the possibility degree of occurrence of a probabilistic event, and the Nec measure indicates the minimum possibility degree of occurrence of a probabilistic event. Certainty degree of occurrence of an uncertain event is measured by credibility (Cr), which equals the average of the Pos and Nec measures [22]. New fuzzy measure Me, which is a developed Cr measure is presented by [23]. The main advantage of this measure is its flexibility to avoid excessive views. In the following, the three measures of a fuzzy event, including possibility, necessity and credibility, are described. Variable ξ is determined as a fuzzy variable on probabilistic space ð Þ *Θ*, *P*ð Þ Θ , *Pos* and its membership function, obtained from the probability measure Pos, is as follows:

$$\mathbb{P}(X) = \text{Pos}\{\theta \in \Theta | \xi(\theta) = \infty\}, \mathfrak{x} \in \mathbb{R} \tag{21}$$

Set A is in *P*ð Þ Θ . The necessity and credibility measures of are defined as follows:

$$\operatorname{Vec}\{A\} = \mathbf{1} - \operatorname{Pos}\{A^\circ\} \tag{22}$$

*Fuzzy Approach Model to Portfolio Risk Response Strategies DOI: http://dx.doi.org/10.5772/intechopen.95009*

$$\operatorname{Cr}\{A\} = \frac{1}{2} (\operatorname{Pos}\{A\} + \operatorname{New}\{A\})\tag{23}$$

More details and descriptions of the fuzzy theory are explained in [22]. In this research, the Me-based probabilistic programming method is selected to deal with the uncertain parameters of the presented model. The fuzzy measure Me is defined, according to [22], as follows:

$$\text{Me } \{A\} = \text{Vec}\{A\} + \mathcal{E} \left(\text{Pos}\{A\} - \text{Vec}\{A\}\right) \tag{24}$$

Where ε as a parameter shows the optimistic-pessimistic attitude of a decision maker. Mathematical programming problem (25) with fuzzy parameters is as follows:

$$\operatorname{Min} f(x, \bar{c})$$

Subjected to

$$\begin{aligned} \tilde{A}\mathfrak{x} &\geq \tilde{b} \\ \tilde{N}\mathfrak{x} &\leq \tilde{d} \\ \mathfrak{x} &\geq 0 \end{aligned} \tag{25}$$

In this notation~*<sup>c</sup>* <sup>¼</sup> ð Þ <sup>~</sup>*c*1,~*c*2, … ,~*cn* ,*A*<sup>~</sup> <sup>¼</sup> *<sup>a</sup>*~*ij* � � *<sup>m</sup>*�*<sup>n</sup>*,*N*<sup>~</sup> <sup>¼</sup> *<sup>n</sup>*~*ij* � � *<sup>m</sup>*�*<sup>n</sup>*, <sup>~</sup> *<sup>b</sup>* <sup>¼</sup> <sup>~</sup> *b*1, ~ *b*1, … , ~ *bn* � �*<sup>t</sup>*

and ~ *<sup>d</sup>* <sup>¼</sup> <sup>~</sup> *d*1, ~ *d*1, … , ~ *dn* � �*<sup>t</sup>* represent the triangular fuzzy numbers which are used in the objective function and constraints, respectively. Furthermore, the fuzzy number *x* ¼ ð Þ *x*1, *x*1, … , *xn* is the crisp decision vector, which shows the possibility distribution for fuzzy parameters.

To deal with the probabilistic objective functions and constraints, the expected value and chance-constrained operators based on the Me measure in this method are used. Accordingly, we can rewrite this model (26) as below:

$$Min\ E[f(\mathbf{x}, \tilde{c})]$$

Subjected to

$$\text{Me}\left\{\bar{A}\mathbb{x}\geq\bar{b}\right\}\geq a\tag{26}$$

$$\text{Me}\left\{\bar{N}\mathbb{x}\leq\bar{b}\right\}\geq\beta$$

$$\mathbb{x}\geq\mathbf{0}$$

In this notation, E is the expected value operator, α and β are respectively the decision maker's minimum confidence level for satisfaction of probabilistic constraints. Jiménez et al. [21] defined the expected value operator based on Me measure as follows:

$$E[\xi] = \frac{1-\varepsilon}{2}\xi\_1 + \frac{1}{2}\xi\_2 + \frac{\varepsilon}{2}\xi\_3\tag{27}$$

According to [22] we can transform the aforementioned model (26) into two approximation models including the upper approximation model (UAM) and the lower approximation model (LAM). These models are presented as follows:

$$(UAM)\left\{ \begin{aligned} &\min\left(\frac{1-\varepsilon}{2}c\_1 + \frac{1}{2}c\_2 + \frac{\varepsilon}{2}c\_3\right) \mathbf{x} \\ &A\_{(2)}\mathbf{x} + (1-\alpha)(A\_{(3)} - A\_{(2)})\mathbf{x} \ge b\_2 - (1-\alpha)(b\_{(2)} - b\_{(1)}) \\ &N\_{(2)}\mathbf{x} - (1-\beta)\left(N\_{(2)} - N\_{(1)}\right) \mathbf{x} \le d\_2 + (1-\alpha)(d\_{(3)} - d\_{(2)}) \end{aligned} \right. \\ &(LAM) \begin{cases} \min\left(\frac{1-\varepsilon}{2}c\_1 + \frac{1}{2}c\_2 + \frac{\varepsilon}{2}c\_3\right) \mathbf{x} \\ &A\_{(2)}\mathbf{x} + \alpha(A\_{(3)} - A\_{(2)}) \mathbf{x} \ge b\_2 + (1-\alpha)(b\_{(3)} - b\_{(2)}) \\ &N\_{(2)}\mathbf{x} + (1-\beta)\left(N\_{(3)} - N\_{(2)}\right) \mathbf{x} \le d\_2 + \beta(d\_{(2)} - d\_{(1)}) \end{cases} \end{aligned} \tag{29}$$

Where ε is the optimistic-pessimistic parameter. Solving the LAM and UAM models provides the decision maker with the lower and upper bound of the optimal decision respectively. In this research, we use UAM models to solve problem. Accordingly, the auxiliary crisp equivalent of the presented model with triangular fuzzy parameters is presented as follows:

UAM:

$$\text{Max } Z\_1 = \sum\_{t=1}^{T} \sum\_{j=1}^{n} x\_{jt} \times \left(\frac{1-\varepsilon}{2} p\_{jt(1)} + \frac{1}{2} p\_{jt(2)} + \frac{\varepsilon}{2} p\_{jt(3)}\right) + \sum\_{j=1}^{n} \sum\_{a=1}^{A} \sum\_{r=1}^{R} z\_{jar}$$

$$\times \left(\frac{1-\varepsilon}{2} \mathfrak{e}\_{ar(1)} + \frac{1}{2} \mathfrak{e}\_{ar(2)} + \frac{\varepsilon}{2} \mathfrak{e}\_{dr(3)}\right) \tag{30}$$

$$\begin{split} \text{Min } Z\_{2} &= \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{i=1}^{m} h\_{ij} \cdot \left( \frac{1-\varepsilon}{2} \mathbf{C}\_{it(1)} + \frac{1}{2} \mathbf{C}\_{it(2)} + \frac{\varepsilon}{2} \mathbf{C}\_{it(3)} \right) \\ &+ \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{k=1}^{\varepsilon} m\_{kj} \cdot \left( \frac{1-\varepsilon}{2} \mathbf{C}\_{kt(1)} + \frac{1}{2} \mathbf{C}\_{kt(2)} + \frac{\varepsilon}{2} \mathbf{C}\_{kt(3)} \right) \\ &+ \sum\_{t=1}^{T} \sum\_{j=1}^{n} \mathbf{x}\_{jt} \sum\_{o=1}^{x} r\_{oj} \cdot \left( \frac{1-\varepsilon}{2} \mathbf{C}\_{ot(1)} + \frac{1}{2} \mathbf{C}\_{ot(2)} + \frac{\varepsilon}{2} \mathbf{C}\_{ot(3)} \right) \\ &+ \sum\_{j=1}^{n} \sum\_{a=1}^{A} \left( \frac{1-\varepsilon}{2} \mathbf{C}\_{a(1)} + \frac{1}{2} \mathbf{C}\_{a(2)} + \frac{\varepsilon}{2} \mathbf{C}\_{a(3)} \right) \max\_{r} z\_{jar} \end{split} \tag{31}$$

Subjected to

$$\begin{aligned} & \left(\sum\_{i=1}^{m} h\_{\vec{y}\cdot \cdot} \left(\mathbf{C}\_{\text{it}(2)} \mathbf{x} - (1-\beta) \left(\mathbf{C}\_{\text{it}(2)} - \mathbf{C}\_{\text{it}(1)} \right) \right) \\ & + \sum\_{k=1}^{s} m\_{\vec{k}\cdot\cdot} \left(\mathbf{C}\_{\text{kt}(2)} \mathbf{x} - (1-\beta) \left(\mathbf{C}\_{\text{kt}(2)} - \mathbf{C}\_{\text{kt}(1)} \right) \right) \\ & + \sum\_{o=1}^{x} r\_{o\cdot\cdot} \left(\mathbf{C}\_{\text{ot}(2)} \mathbf{x} - (1-\beta) \left(\mathbf{C}\_{\text{ot}(2)} - \mathbf{C}\_{\text{ot}(1)} \right) \right) \right) \times x\_{\vec{y}} \\ & \leq p\_{j\mathbf{c}(2)} + (1-\beta) \left(p\_{j\mathbf{j}(3)} - p\_{\vec{y}\mathbf{1}(2)} \right) \end{aligned} \tag{32}$$

**230**

$$\begin{aligned} &\sum\_{j=1}^{A}\sum\_{a=1}^{A}\left(\mathbf{C}\_{a(2)}\mathbf{x} - (\mathbf{1} - \boldsymbol{\beta})\left(\mathbf{C}\_{a(2)} - \mathbf{C}\_{a(1)}\right)\right) \max\_{r} \mathbf{z}\_{jdr} \\ &\quad + \left[\sum\_{i=1}^{m} h\_{\dot{\eta}}\left(\mathbf{C}\_{\dot{\mathrm{at}}(2)}\mathbf{x} - (\mathbf{1} - \boldsymbol{\beta})\left(\mathbf{C}\_{\dot{\mathrm{at}}(2)} - \mathbf{C}\_{\dot{\mathrm{at}}(1)}\right)\right) \right. \\ &\quad + \sum\_{k=1}^{s} m\_{\dot{k}\dot{\eta}}\left(\mathbf{C}\_{\dot{\mathrm{kt}}(2)}\mathbf{x} - (\mathbf{1} - \boldsymbol{\beta})\left(\mathbf{C}\_{\dot{\mathrm{kt}}(2)} - \mathbf{C}\_{\dot{\mathrm{kt}}(1)}\right)\right) \\ &\quad + \sum\_{o=1}^{x} \left(\mathbf{C}\_{\dot{\mathrm{at}}(2)}\mathbf{x} - (\mathbf{1} - \boldsymbol{\beta})\left(\mathbf{C}\_{\dot{\mathrm{ot}}(2)} - \mathbf{C}\_{\dot{\mathrm{ot}}(1)}\right)\right) r\_{\dot{g}}\right) \times \mathbf{x}\_{\dot{\mathrm{it}}} \\ &\leq \mathcal{B}\_{\dot{\mathrm{it}}(2)} + (1 - \boldsymbol{\beta})\left(\mathcal{B}\_{\dot{\mathrm{ig}}(3)} - \mathcal{B}\_{\dot{\mathrm{ig}}(2)}\right) \end{aligned} \tag{33}$$

$$\text{Max } \text{E}[\mathbf{Z}\_1] \tag{34}$$

$$\text{Min } E[Z\_2] \tag{35}$$

An efficient multi-objective method can be done as an efficient method for obtaining the satisfaction level for each OFVs according to the decision maker's preferences. For further explanations, the interested reader can refer to TH [20]. Two parameters in this method are very critical: relative importance of OFVs (i.e., weight factor) and coefficient of compensation. Details of the distribution functions of the parameters and the size of test problems are listed in **Table 1**. After that, the results on test problems for diverse values of *ϑ* and*φ* are shown in **Table 2**.

According to **Table 2**, the values of objective functions change based on the value of *ϑ*. The results indicate that satisfaction degrees displaying each objective function change based on the value of *ϑ*. In this table, the values of satisfaction degree of objective functions (1) and (2) for test problem 2 fluctuate between 0.841 and 0.965, and 0.848 and 0.961, respectively. The results show that by manipulating the value of *ϑ*, the decision-maker can make trade-offs between two objective functions and select an optimal pair. Generally, increasing the value of *ϑ* leads to higher allocated weights to acquire a higher lower bound for the satisfaction degree of objectives (*λ*0).

Based on the acquired results and considering the budget and time limitations, the most appropriate strategy for responding to the risk work packages is provided in **Table 3**. In this test problem project 8 and 3 are selected. Appendix A. shows the amount of maximum allowed time reduction (day) and the quality of each activity (in percentage). The obtained quality of each activity under acceptable and ideal condition is assumed 90% and 99% respectively (δ<sup>w</sup> <sup>∈</sup> ½ � 1%, 10% ). Appendix B. illustrates the effect of implementing risk response strategies on risks cost reduction (if occurs).


**Table 1.**

*Amount of the parameters by random generation.*


*Fuzzy Approach Model to Portfolio Risk Response Strategies DOI: http://dx.doi.org/10.5772/intechopen.95009*

#### **Table 2.** *Results of test problem 1 (*β ¼ 0*:*5*).*
